Einstein's Cosmological Constant from The World Hologram
R(t)^2 = [1 - Lp^3|Vacuum Coherence|^2]^-1
Note when Vacuum Coherence = 0 PRE-INFLATION
R(Initial Singularity) = 1
i.e. 1 Bit thinking of universe as cosmic computer with David Finkelstein's chronons 10^-44 sec of erasures - resetting the register so to speak.
As the large-scale Vacuum Coherence limits to
|Vacuum Coherence|^2 -> 1/Lp^3
R(t) -> infinity
Vacuum Coherence can exceed this limit on smaller non-cosmological scales.
On Dec 31, 2004, at 12:43 PM, Jack Sarfatti wrote:
Inflation demands total stuff is at critical density for flat 3D space, i.e. k = 0 in large-scale FRW metric
Omega(random ZPF) + Omega(matter) + Omega(radiation) + Omega(vacuum coherence) = 1
Omega(on-mass-shell matter) ~ 1/R(t)^3, i.e. w = 0 in R(t)^-3(1 + w) for v/c << 1
Omega(on-mass-shell radiation) ~ 1/R(t)^4, i.e. w = +1/3
with Omega(random zpf) + Omega(vacuum coherence) ~ 1/R(t)^0 independent of R(t) since w = -1 for both random "normal fluid" zpf and condensate (vacuum coherence)
/\ is from random zpf causing both dark energy (negative pressure) and dark matter (positive pressure) exotic vacuum phases.
Clumps of w = -1 positive pressure, possibly like the Galactic Halo mimic w = 0 dark matter in their gravity lensing. Positive and negative pressure exotic vacua can each either universally attract or repel depending on their detailed distribution relative to the test particle. However localized negative pressure exotic vacuum clumps anti-gravitate for test particles outside their domain of support. Similarly, localized positive pressure clumps gravitate for test particles outside their support. The effective gravity strength G* of clumps of exotic vacuum regions can be larger than Newton's G.
Guv + /\zpfguv = 0
is the zero torsion field GR Poisson equation for exotic vacua.
In the Newtonian weak-field slow-speed limit, neglecting gravimagnetism, this is
Grad^2(Potential Energy per unit test mass of exotic vacuum field) ~ c^2/\zpf
Susskind's World Hologram Conjecture requires
Einstein's cosmological constant /\ ~ [LpR(t)]^-2
With critical energy density hc/Lp^4R(t)^2 = (mpc^2/Lp^3)R(t)^-2
Therefore Omega(/\) is independent of R(t) consistent with w = -1 Lorentz covariance and EEP of GR. Similarly for Omega(Vacuum Coherence), which is my new term.
OK trash what I wrote late last night on this when I was very tired. I woke up fresh with a start with the correct way to think about the problem - after taking a 600 mg ibuprofen for shoulder ache from over exercise at the health club.
Here is how Lenny Susskind's world hologram really does work and how it explains the dark energy!
1. Trash H. We don't need it.
2. The FRW space expansion factor R(t) is dimensionless. Everything is in units of Lp.
3. The Susskind-Hawking-Bekenstein-t'Hooft hologram entropy of the Universe is simply
S/kB = (1/4)R(t)^2
I have written this before of course.
R(now) = 10^61
This is a nice formula because it also explains the Arrow of Time since the entropy of the universe is 0 at the initial singularity here, i.e. R(initial singularity) = 0 or, if you prefer Finkelstein's chronons it is 1-bit at t = 0.
4. OK roughly model everything as photons.
The photon thermal distribution is
hf/(e^hf/kBT - 1) + hf/2
But the dark energy is, to first approximation virtual photons hf/2 whose mean value is hc/LpR(t).
5. Therefore we need virtual energy hf/2 to "erase" each bit every time the cosmic quantum computer clears its register to step forward another chronon to compute the history of the universe
Therefore, the dark energy density the vacuum fabric of curved spacetime needs to compute itself is
(R(t)^2/4)(hc/LpR(t))(1/Lp^3R(t))^3 = (hc/Lp^4)1/4R(t)^2 = (hc/4Lp^2)/
Einstein's cosmological constant /\ = 1/Lp^2R(t)^2 = 10^-56 cm^-2 NOW
hc/Lp^2 = c^4/G
6. So this says that Einstein's cosmological constant /\ is getting smaller as the Universe expands. On the other hand, w = pressure/(energy density) = -1 for random micro-quantum zero point energy.
The energy density of cosmic stuff scales as R(t)^-3(1 + w)
For example w = 0 for ordinary matter so that Omega(matter) ~ R(t)^-3
w = +1/3 for cosmic black body real photons so that Omega(CMB) ~ R(t)^-4
w = -1 for RANDOM micro-quantum vacuum zero point fluctuations (from covariance) so that
Omega(ZPF) ~ R(t)^-0 = constant.
7. However in my macro-quantum theory of the world hologram
/\ = (1/Lp^2)[1 - Lp^3|Vacuum Coherence|^2]
Equating this with the world hologram formula in the large-scale FRW metric limit
1/Lp^2R(t)^2 = (1/Lp^2)[1 - Lp^3|Vacuum Coherence|^2]
1/R(t)^2 = [1 - Lp^3|Vacuum Coherence|^2]
R(t)^2 = [1 - Lp^3|Vacuum Coherence|^2]^-1
Where from inflation
Omega(random ZPF) + Omega(matter) + Omega(radiation) + Omega(vacuum coherence) = 1
Thursday, December 30, 2004
Nonlocal curvature without curvature
First think of superfluid helium. The "circulation" vector field v is a "connection" like the EM vector potential A. The vorticity curl v is like magnetic field curlA. Curlv is zero outside of the vortex core in the irrotational superflow. The quantized circulation vortices are multiply connected, i.e. winding numbers or Betti numbers for 1-forms & 2-forms. They are multiply connected because they have "micro-quantum normal fluid cores" where the macro-quantum coherence drops if not to zero to a much smaller value to make a Josephson weak link of sorts. In my macro-quantum theory of gravity for Super Cosmos the "normal fluid" is the exotic vacuum dark energy of negative micro-quantum zero point pressure or the dark matter of positive zero point pressure.
Therefore, we have a Bohm-Aharonov nonlocality in superfluid helium but we do not think it is amazing because we can picture a local circulation vector field of the fluid flow.
Now we have an isomorphism with elasticity theory as shown by Hagen Kleinert. We have a kind of tetrad world crystal elastic distortion field that is essentially related to the non-tensor Levi-Civita connection field (LC), which is like A in EM and like v in superfluid helium. We can, therefore, have non-vanishing "circulation" integrals (line integrals of local g-force) even though there is no interior (covariant) curl of (LC) because the unstable dark energy thin wall is an exotic vacuum "normal fluid" core. In terms of crystal distortion theory, there is a smooth distortion (LC) field outside the source region of a thin unstable dark energy slab inside of which there is Ricci tensor which is a bunch of disclination topological defects that influence at a distance the local (LC) field of vanishing curl.
Now in EM, we say vector potential is not a local observable because it is a inhomogeneous non-tensor under the U(1) local gauge transformations. In GR these are the GCT transition functions between overlapping local coordinate charts in the atlas covering the manifold like analytical continuation of complex functions of complex variable bypassing branch points. (A power series circle of convergence stops at a branch point but you can finagle around the obstruction with a different circle.) Note in my theory the (LC) connection comes from the 4^3 3rd order partial derivatives of the more is different emergent macro-quantum world hologram Goldstone rigid phase of the post-inflationary vacuum coherence field that is the fabric of smooth OLDLRO curved spacetime. So there is no reason at all to say that the (LC) connection is not a local observable even though it is not a GCT tensor.
It's nice, and of course necessary, that the metric as second order partial derivatives of the Goldstone phase and the connection as third order partial derivatives and curvature as 4th order partial derivatives have exactly the correct number of GCT tensor indices required by Einstein's GR!
First think of superfluid helium. The "circulation" vector field v is a "connection" like the EM vector potential A. The vorticity curl v is like magnetic field curlA. Curlv is zero outside of the vortex core in the irrotational superflow. The quantized circulation vortices are multiply connected, i.e. winding numbers or Betti numbers for 1-forms & 2-forms. They are multiply connected because they have "micro-quantum normal fluid cores" where the macro-quantum coherence drops if not to zero to a much smaller value to make a Josephson weak link of sorts. In my macro-quantum theory of gravity for Super Cosmos the "normal fluid" is the exotic vacuum dark energy of negative micro-quantum zero point pressure or the dark matter of positive zero point pressure.
Therefore, we have a Bohm-Aharonov nonlocality in superfluid helium but we do not think it is amazing because we can picture a local circulation vector field of the fluid flow.
Now we have an isomorphism with elasticity theory as shown by Hagen Kleinert. We have a kind of tetrad world crystal elastic distortion field that is essentially related to the non-tensor Levi-Civita connection field (LC), which is like A in EM and like v in superfluid helium. We can, therefore, have non-vanishing "circulation" integrals (line integrals of local g-force) even though there is no interior (covariant) curl of (LC) because the unstable dark energy thin wall is an exotic vacuum "normal fluid" core. In terms of crystal distortion theory, there is a smooth distortion (LC) field outside the source region of a thin unstable dark energy slab inside of which there is Ricci tensor which is a bunch of disclination topological defects that influence at a distance the local (LC) field of vanishing curl.
Now in EM, we say vector potential is not a local observable because it is a inhomogeneous non-tensor under the U(1) local gauge transformations. In GR these are the GCT transition functions between overlapping local coordinate charts in the atlas covering the manifold like analytical continuation of complex functions of complex variable bypassing branch points. (A power series circle of convergence stops at a branch point but you can finagle around the obstruction with a different circle.) Note in my theory the (LC) connection comes from the 4^3 3rd order partial derivatives of the more is different emergent macro-quantum world hologram Goldstone rigid phase of the post-inflationary vacuum coherence field that is the fabric of smooth OLDLRO curved spacetime. So there is no reason at all to say that the (LC) connection is not a local observable even though it is not a GCT tensor.
It's nice, and of course necessary, that the metric as second order partial derivatives of the Goldstone phase and the connection as third order partial derivatives and curvature as 4th order partial derivatives have exactly the correct number of GCT tensor indices required by Einstein's GR!
The Birth of Einstein's General Relativity of the Gravitational Field
Metric Engineering Investigations 1.8
Landau & Lifshitz "Classical Theory of Fields" p. 228 explicitly say that the guv curved metric form is only for a non-inertial frame not an inertial frame. Since in 82 they are still operating in global special relativity they do not distinguish global from local noninertial frames at this stage in their pedagogical exposition:
"Thus, in a noninertial system of reference the square of the interval appears as a quadratic form of general type in the coordinate differentials, that is, it has the form
ds^2 = gikdx^idx^k (82.1)
i,k = 0,1,2,3
... Thus, when we use a noninertial system, the four-dimensional coordinate system x^0,x^1,x^2,x^3 is curvilinear. The quantities gik, determining all the geometric properties in each curvilinear system of coordinates, represent, we say, the space-time metric."
Many mathematicians have careers in relativity without paying attention to the measurement theory behind the formalism. Each choice of non-inertial coordinates is a possible physical configuration of detectors requiring non-gravity forces to sustain. For example, in the vacuum black hole Schwarzschild solution where, outside the event horizon where gtt = 0:
gtt = - grr^-1 = (1 - 2GM/c^2r)
2GM/c^2r < 1
Where A = 4pir^2 is area of sphere concentric with origin behind the event horizon
These are HOVERING REST LNIF observers that must fire their rockets toward the event horizon to maintain constant r. That is, they must fire rockets in order to stay still relative to the curvature field. This would not happen in flat Minkowski spacetime where the rocket would move away from say a flat Black Monolith from Kubrick's 2001 floating in space. The one exception seems to be the "curvature without curvature" effect from a highly unstable antigravitating thin wall of dark energy given by Vilenken 1983 with Taub's comment that the local conformal vacuum curvature is zero even though HOVERING observers need to fire their propellant away from the thin wall in order to keep a fixed distance from it. R. Kiehn points out that the LC connection field as a Cartan 1-form like any p-form splits into 3 pieces
Connection 1-form = Exact 1-form + Closed Non-Exact 1-form + Non-Closed 1-Form.
Curvature 2-form = d(Connection 1-form) = d(Non-Closed 1-Form) =/= 0 LOCALLY
However, Kiehn also mentions that the GLOBAL Closed Non-Exact 1-form describes topological defects like the Wilson loops for the dynamical nonlocal Bohm-Aharonov phase and presumably the anholonomic Berry phase that seem to give Taub's "curvature without curvature"? Since the smooth fabric of curved spacetime emerges from the single-valued local macro-quantum vacuum coherent world hologram Goldstone phase field with topological defects from the vanishing of its Higgs amplitude, local effects with nonlocal causes on the large-scale should be prevalent like the vortices in superfluid helium. Or so it appears with the dark matter Galactic Halo and the NASA Pioneer gravity anomaly of homogeneous a_g = -cH = 10^-7 cm/sec^2 starting 20 AU out from Sun - exactly like an S^2 hedgehog point topological defect in the vacuum coherence at the center of the Sun (David Thouless "Nonrelativistic Topological Quantum Mechanics Ch 1).
It is obvious, that in the full context of GR's LOCAL EEP L&L use "Galilean" the way MTW use "LIF" i.e. "In an inertial reference system, when we use Cartesian space coordinates, x^1,^2,^3 = x,y,z, and the time x^0 = ct,
goo = 1, g11 = g22 = g33 = -1, gik = 0 when i =/= k
we call a four-dimensional system with these values of gik 'Galilean'" p. 228
In Galilean relativity: "A noninertial system of reference is equivalent to a certain field of force. We now see that in (special) relativistic mechanics, these (inertial) fields are determined by the quantities gik."
Zielinski has been trying to keep a "Victorian Station Master's" ledger on just what is the "phony" gravity from the arbitrary contingent acceleration of the non-inertial frame and what is the "true" gravity from the non-contingent intrinsic curvature of the fabric of spacetime. Is that possible?
* Note that our alleged alien super-technologists in their magnificent unconventional flying machines having their way with us genetically for thousands of years, we call them Jehovah etc. The Many Masks of The Wizard of Oz with the Big Black Eyes and the spindly legs and arms
http://stardrive.org/cartoon/bovines.html
http://stardrive.org/cartoon/spectra.html
http://stardrive.org/cartoon/dan.html
http://stardrive.org/cartoon/coffee.html
are able to metric engineer the intrinsic geometry in their geodesic gliders (AKA "G-Engines" George Trimble, "acceleration fields" Paul Hill) to their will so that what we think of as "intrinsic" they think of as "contingent". A similar problem was posed by Eugene Wigner in the distinction between contingent initial conditions for the early universe and the intrinsic laws of nature. Wheeler even questions that split in his "Law without law" that we now see in Lenny Susskind's "Anthropic Landscape" making a virtue of the vice of M-Theory's lack of falsifiabilty. To such depths have The Pundits descended. :-)
Now L&L start to shift to the full GR paradigm almost imperceptibly at first.
"The same applies to 'actual' gravitational fields. Any gravitational field is just a change in the metric of spacetime as determined by the quantities gik." In ADM's canonical method gik splits into a spacelike 3-geometry with 3 shift and 1 lapse function (from local gauging of the translations) to push the time evolution of the geometrodynamic field 3-geometry forward. GCT Diff(4) requires that the physics be independent of the way of slicing. Also one needs to use an infinite dimensional superspace where each 3-geometry 3g is a "point". The same is done with the EM field Fuv in the Bohmian hidden variable version of quantum field theory. The latter is no good for practical computations however. We must still fall back on Feynman diagrams for number crunching the sorts of things Victorian Station Masters want to know. However, as Mussolini's legacy we must have the trains run on time. ;-)
Enter Einstein's Geometrodynamics (incompatible irreconcilable differences with Hal Puthoff's PV metric engineering in Eric Davis's USAF "Teleportation" paper and the NIDS "High Strangeness" UFO paper and the recent Journal Interplanetary Physics paper etc): "This important fact means that the geometrical properties of spacetime (its metric) are determined by physical phenomena, and are not fixed properties of spacetime."
Note that Alex Poltorak's method of restoring locality to gravity energy seems to demand a fundamental violation of Einstein's general relativity principle because Alex invokes a "rigid nondynamical affine connection" in contrast to the non-rigid dynamical Levi-Civita connection of Einstein's 1916 GR. I say "seems" because I do not, as yet, understand what Alex is really proposing.
Zielinski's claim that the Golden Age of Soviet Physics under the aegis of Landau eschewed Einstein at least in V.A. Fock's "meta-theoretics" is certainly not justified in Landau & Lifshitz's "The theory of gravitational fields, constructed on the basis of the theory of relativity, is called the general theory of relativity. It was established by Einstein (and finally formulated by him in 1916), and it represents probably the most beautiful of all existing physical theories. It is remarkable that it was developed by Einstein in a purely deductive manner and only later was substantiated by astronomical observations."
Note that Landau does not say that David Hilbert thought of it first nor does he say that Einstein's first wife Mileva thought of it first.
Metric Engineering Investigations 1.8
Landau & Lifshitz "Classical Theory of Fields" p. 228 explicitly say that the guv curved metric form is only for a non-inertial frame not an inertial frame. Since in 82 they are still operating in global special relativity they do not distinguish global from local noninertial frames at this stage in their pedagogical exposition:
"Thus, in a noninertial system of reference the square of the interval appears as a quadratic form of general type in the coordinate differentials, that is, it has the form
ds^2 = gikdx^idx^k (82.1)
i,k = 0,1,2,3
... Thus, when we use a noninertial system, the four-dimensional coordinate system x^0,x^1,x^2,x^3 is curvilinear. The quantities gik, determining all the geometric properties in each curvilinear system of coordinates, represent, we say, the space-time metric."
Many mathematicians have careers in relativity without paying attention to the measurement theory behind the formalism. Each choice of non-inertial coordinates is a possible physical configuration of detectors requiring non-gravity forces to sustain. For example, in the vacuum black hole Schwarzschild solution where, outside the event horizon where gtt = 0:
gtt = - grr^-1 = (1 - 2GM/c^2r)
2GM/c^2r < 1
Where A = 4pir^2 is area of sphere concentric with origin behind the event horizon
These are HOVERING REST LNIF observers that must fire their rockets toward the event horizon to maintain constant r. That is, they must fire rockets in order to stay still relative to the curvature field. This would not happen in flat Minkowski spacetime where the rocket would move away from say a flat Black Monolith from Kubrick's 2001 floating in space. The one exception seems to be the "curvature without curvature" effect from a highly unstable antigravitating thin wall of dark energy given by Vilenken 1983 with Taub's comment that the local conformal vacuum curvature is zero even though HOVERING observers need to fire their propellant away from the thin wall in order to keep a fixed distance from it. R. Kiehn points out that the LC connection field as a Cartan 1-form like any p-form splits into 3 pieces
Connection 1-form = Exact 1-form + Closed Non-Exact 1-form + Non-Closed 1-Form.
Curvature 2-form = d(Connection 1-form) = d(Non-Closed 1-Form) =/= 0 LOCALLY
However, Kiehn also mentions that the GLOBAL Closed Non-Exact 1-form describes topological defects like the Wilson loops for the dynamical nonlocal Bohm-Aharonov phase and presumably the anholonomic Berry phase that seem to give Taub's "curvature without curvature"? Since the smooth fabric of curved spacetime emerges from the single-valued local macro-quantum vacuum coherent world hologram Goldstone phase field with topological defects from the vanishing of its Higgs amplitude, local effects with nonlocal causes on the large-scale should be prevalent like the vortices in superfluid helium. Or so it appears with the dark matter Galactic Halo and the NASA Pioneer gravity anomaly of homogeneous a_g = -cH = 10^-7 cm/sec^2 starting 20 AU out from Sun - exactly like an S^2 hedgehog point topological defect in the vacuum coherence at the center of the Sun (David Thouless "Nonrelativistic Topological Quantum Mechanics Ch 1).
It is obvious, that in the full context of GR's LOCAL EEP L&L use "Galilean" the way MTW use "LIF" i.e. "In an inertial reference system, when we use Cartesian space coordinates, x^1,^2,^3 = x,y,z, and the time x^0 = ct,
goo = 1, g11 = g22 = g33 = -1, gik = 0 when i =/= k
we call a four-dimensional system with these values of gik 'Galilean'" p. 228
In Galilean relativity: "A noninertial system of reference is equivalent to a certain field of force. We now see that in (special) relativistic mechanics, these (inertial) fields are determined by the quantities gik."
Zielinski has been trying to keep a "Victorian Station Master's" ledger on just what is the "phony" gravity from the arbitrary contingent acceleration of the non-inertial frame and what is the "true" gravity from the non-contingent intrinsic curvature of the fabric of spacetime. Is that possible?
* Note that our alleged alien super-technologists in their magnificent unconventional flying machines having their way with us genetically for thousands of years, we call them Jehovah etc. The Many Masks of The Wizard of Oz with the Big Black Eyes and the spindly legs and arms
http://stardrive.org/cartoon/bovines.html
http://stardrive.org/cartoon/spectra.html
http://stardrive.org/cartoon/dan.html
http://stardrive.org/cartoon/coffee.html
are able to metric engineer the intrinsic geometry in their geodesic gliders (AKA "G-Engines" George Trimble, "acceleration fields" Paul Hill) to their will so that what we think of as "intrinsic" they think of as "contingent". A similar problem was posed by Eugene Wigner in the distinction between contingent initial conditions for the early universe and the intrinsic laws of nature. Wheeler even questions that split in his "Law without law" that we now see in Lenny Susskind's "Anthropic Landscape" making a virtue of the vice of M-Theory's lack of falsifiabilty. To such depths have The Pundits descended. :-)
Now L&L start to shift to the full GR paradigm almost imperceptibly at first.
"The same applies to 'actual' gravitational fields. Any gravitational field is just a change in the metric of spacetime as determined by the quantities gik." In ADM's canonical method gik splits into a spacelike 3-geometry with 3 shift and 1 lapse function (from local gauging of the translations) to push the time evolution of the geometrodynamic field 3-geometry forward. GCT Diff(4) requires that the physics be independent of the way of slicing. Also one needs to use an infinite dimensional superspace where each 3-geometry 3g is a "point". The same is done with the EM field Fuv in the Bohmian hidden variable version of quantum field theory. The latter is no good for practical computations however. We must still fall back on Feynman diagrams for number crunching the sorts of things Victorian Station Masters want to know. However, as Mussolini's legacy we must have the trains run on time. ;-)
Enter Einstein's Geometrodynamics (incompatible irreconcilable differences with Hal Puthoff's PV metric engineering in Eric Davis's USAF "Teleportation" paper and the NIDS "High Strangeness" UFO paper and the recent Journal Interplanetary Physics paper etc): "This important fact means that the geometrical properties of spacetime (its metric) are determined by physical phenomena, and are not fixed properties of spacetime."
Note that Alex Poltorak's method of restoring locality to gravity energy seems to demand a fundamental violation of Einstein's general relativity principle because Alex invokes a "rigid nondynamical affine connection" in contrast to the non-rigid dynamical Levi-Civita connection of Einstein's 1916 GR. I say "seems" because I do not, as yet, understand what Alex is really proposing.
Zielinski's claim that the Golden Age of Soviet Physics under the aegis of Landau eschewed Einstein at least in V.A. Fock's "meta-theoretics" is certainly not justified in Landau & Lifshitz's "The theory of gravitational fields, constructed on the basis of the theory of relativity, is called the general theory of relativity. It was established by Einstein (and finally formulated by him in 1916), and it represents probably the most beautiful of all existing physical theories. It is remarkable that it was developed by Einstein in a purely deductive manner and only later was substantiated by astronomical observations."
Note that Landau does not say that David Hilbert thought of it first nor does he say that Einstein's first wife Mileva thought of it first.
Metric Engineering Investigations 1.8
L&L p. 228 explicitly say that the guv curved metric form is only for a non-inertial frame not an inertial frame. Since in 82 they are still operating in global special relativity they do not distinguish global from local noninertial frames at this stage in their pedagogical exposition:
"Thus, in a noninertial system of reference the square of the interval appears as a quadratic form of general type in the coordinate differentials, that is, it has the form
ds^2 = gikdx^idx^k (82.1)
i,k = 0,1,2,3
... Thus, when we use a noninertial system, the four-dimensional coordinate system x^0,x^1,x^2,x^3 is curvilinear. The quantities gik, determining all the geometric properties in each curvilinear system of coordinates, represent, we say, the space-time metric."
Many mathematicians have careers in relativity without paying attention to the measurement theory behind the formalism. Each choice of non-inertial coordinates is a possible physical configuration of detectors requiring non-gravity forces to sustain. For example, in the vacuum black hole Schwarzschild solution where, outside the event horizon where gtt = 0:
gtt = - grr^-1 = (1 - 2GM/c^2r)
2GM/c^2r < 1
Where A = 4pir^2 is area of sphere concentric with origin behind the event horizon
These are HOVERING REST LNIF observers that must fire their rockets toward the event horizon to maintain constant r. That is, they must fire rockets in order to stay still relative to the curvature field.
L&L p. 228 explicitly say that the guv curved metric form is only for a non-inertial frame not an inertial frame. Since in 82 they are still operating in global special relativity they do not distinguish global from local noninertial frames at this stage in their pedagogical exposition:
"Thus, in a noninertial system of reference the square of the interval appears as a quadratic form of general type in the coordinate differentials, that is, it has the form
ds^2 = gikdx^idx^k (82.1)
i,k = 0,1,2,3
... Thus, when we use a noninertial system, the four-dimensional coordinate system x^0,x^1,x^2,x^3 is curvilinear. The quantities gik, determining all the geometric properties in each curvilinear system of coordinates, represent, we say, the space-time metric."
Many mathematicians have careers in relativity without paying attention to the measurement theory behind the formalism. Each choice of non-inertial coordinates is a possible physical configuration of detectors requiring non-gravity forces to sustain. For example, in the vacuum black hole Schwarzschild solution where, outside the event horizon where gtt = 0:
gtt = - grr^-1 = (1 - 2GM/c^2r)
2GM/c^2r < 1
Where A = 4pir^2 is area of sphere concentric with origin behind the event horizon
These are HOVERING REST LNIF observers that must fire their rockets toward the event horizon to maintain constant r. That is, they must fire rockets in order to stay still relative to the curvature field.
Boundary of a boundary vanishes & Gravimagnetic C^3?
"The boundary of a boundary vanishes." John A. Wheeler
Metric Engineering Investigations 1.7
Ray Chiao's Gravimagnetic Superconducting Radio Submarine Warfare?
On Dec 30, 2004, at 9:33 AM, RKiehn2352@aol.com wrote:
"Jack
Check out V. Fock on "harmonic coordinates" in his book Space Time and Gravity, re' the problem of fields vanishing at infinity. Also see Chapter 12 from vol4 Plasmas and non-Equilibrium Electrodynamics.
http://www22.pair.com/csdc/download/plasmas69d.pdf
Also note that a p-form decomposes into 3 parts:
(an exact part) + (a closed part, but not exact) + (non exact non closed part)."
Yes, thanks. I meant the third term in the 1-form connection as the one that gives curvature.
Exact p-form is B = dA
A is a p-1 form
d^2 = 0
Hence dB(exact) = 0
Closed, but not exact p-form is simply
C a p-form where dC(closed but non-exact) = 0
Every exact p-form is a closed p-form but not vice versa.
dD(non exact non closed part) =/= 0
Consider set of cosets of closed p-forms in set of all p-forms as well as set of cosets of exact closed p-forms in set of all closed p-forms.
That is the quotient sets whose elements are cosets
{p-forms}/{closed p-forms}
and
{closed p-forms}/{exact p-forms}
Similarly for the DUAL c-forms or "chains" where d is replaced by a boundary operator. The co-forms are integration p-dimensional manifolds with multiple connectivity I think (from memory of Wheeler's book "Geometrodynamics") given by the dimension of
Hp = dim{closed p-coforms}/{exact p-coforms} = p^th Betti number of "wormholes" on p-hypersurface.
Hodge Integral d(p-form)(p+1 manifold) = Integral(p-form)boundary(p+1 manifold)
This includes fundamental theorem of integral calculus Stokes theorem and Gauss's theorem as special cases.
Also the co-forms (chains) have a natural group composition and can be Reggeized to a linear superposition of topological graph simplices with p + 1 vertices for a p co-form.
For example for
{p-closed coforms}/{exact p-coforms} take any one element of {p-closed coforms} and "multiply" it by all elements of {exact p-coforms}, this is a coset. If there is a group structure and if the left cosets = right cosets and if the cosets do not overlap i.e. all cosets have no elements in common, then {exact p-coforms} is a normal subgroup H of the group G of {closed p-coforms}.
Here a closed p co-form has no boundary like the surface of a p = 2 sphere or the surface of a sphere with any number of wormhole handles given by the p = 2 Betti number. The exact p-coforms are boundaries of p+1 coforms.
All of the closed local GCT "classical" tensor field (GR & Maxwell for sure, also Yang-Mills) equations are expressed by John A. Wheeler as
"The boundary of a boundary vanishes."
"The last part gives the fields from the potentials. F=dA The middle part give topological defects (BA effect,etc.)"
Also non-dynamical Berry phase in addition to dynamical Bohm-Aharonov-Josephson phase? Since the classical curved fabric of spacetime emerges from the local single-valued macro-quantum coherent Goldstone phase rigid Higgs Vacuum Coherence, all of these phases will play a role in space physics on a large-scale, e.g. Pioneer Anomaly, Galactic Halo all dark energy/matter topological defects.
Ah! Thanks. That would give "curvature without curvature" like in the Vilenken-Taub solution? Also the hedgehog dark energy anomaly explaining the NASA Pioneer a_g = - cH constant acceleration between 2 spherical boundaries concentric with Sun, first boundary at 20AU. If this model is correct ALL stars should have this property that would be related to the birth of stars in the first place. Similar idea for Galactic Halo birth of galaxies. These defect seeds from pre -> post inflation vacuum breaking of translational symmetry.
"and non-trivial gauges. The exact part yields trivial gauges."
Which would be perhaps what Z is looking for in his "coordinate" part. The GCT of GR are derivative from underlying local gauge transformations of the tetrads.
"The boundary of a boundary vanishes." John A. Wheeler
Metric Engineering Investigations 1.7
Ray Chiao's Gravimagnetic Superconducting Radio Submarine Warfare?
On Dec 30, 2004, at 9:33 AM, RKiehn2352@aol.com wrote:
"Jack
Check out V. Fock on "harmonic coordinates" in his book Space Time and Gravity, re' the problem of fields vanishing at infinity. Also see Chapter 12 from vol4 Plasmas and non-Equilibrium Electrodynamics.
http://www22.pair.com/csdc/download/plasmas69d.pdf
Also note that a p-form decomposes into 3 parts:
(an exact part) + (a closed part, but not exact) + (non exact non closed part)."
Yes, thanks. I meant the third term in the 1-form connection as the one that gives curvature.
Exact p-form is B = dA
A is a p-1 form
d^2 = 0
Hence dB(exact) = 0
Closed, but not exact p-form is simply
C a p-form where dC(closed but non-exact) = 0
Every exact p-form is a closed p-form but not vice versa.
dD(non exact non closed part) =/= 0
Consider set of cosets of closed p-forms in set of all p-forms as well as set of cosets of exact closed p-forms in set of all closed p-forms.
That is the quotient sets whose elements are cosets
{p-forms}/{closed p-forms}
and
{closed p-forms}/{exact p-forms}
Similarly for the DUAL c-forms or "chains" where d is replaced by a boundary operator. The co-forms are integration p-dimensional manifolds with multiple connectivity I think (from memory of Wheeler's book "Geometrodynamics") given by the dimension of
Hp = dim{closed p-coforms}/{exact p-coforms} = p^th Betti number of "wormholes" on p-hypersurface.
Hodge Integral d(p-form)(p+1 manifold) = Integral(p-form)boundary(p+1 manifold)
This includes fundamental theorem of integral calculus Stokes theorem and Gauss's theorem as special cases.
Also the co-forms (chains) have a natural group composition and can be Reggeized to a linear superposition of topological graph simplices with p + 1 vertices for a p co-form.
For example for
{p-closed coforms}/{exact p-coforms} take any one element of {p-closed coforms} and "multiply" it by all elements of {exact p-coforms}, this is a coset. If there is a group structure and if the left cosets = right cosets and if the cosets do not overlap i.e. all cosets have no elements in common, then {exact p-coforms} is a normal subgroup H of the group G of {closed p-coforms}.
Here a closed p co-form has no boundary like the surface of a p = 2 sphere or the surface of a sphere with any number of wormhole handles given by the p = 2 Betti number. The exact p-coforms are boundaries of p+1 coforms.
All of the closed local GCT "classical" tensor field (GR & Maxwell for sure, also Yang-Mills) equations are expressed by John A. Wheeler as
"The boundary of a boundary vanishes."
"The last part gives the fields from the potentials. F=dA The middle part give topological defects (BA effect,etc.)"
Also non-dynamical Berry phase in addition to dynamical Bohm-Aharonov-Josephson phase? Since the classical curved fabric of spacetime emerges from the local single-valued macro-quantum coherent Goldstone phase rigid Higgs Vacuum Coherence, all of these phases will play a role in space physics on a large-scale, e.g. Pioneer Anomaly, Galactic Halo all dark energy/matter topological defects.
Ah! Thanks. That would give "curvature without curvature" like in the Vilenken-Taub solution? Also the hedgehog dark energy anomaly explaining the NASA Pioneer a_g = - cH constant acceleration between 2 spherical boundaries concentric with Sun, first boundary at 20AU. If this model is correct ALL stars should have this property that would be related to the birth of stars in the first place. Similar idea for Galactic Halo birth of galaxies. These defect seeds from pre -> post inflation vacuum breaking of translational symmetry.
"and non-trivial gauges. The exact part yields trivial gauges."
Which would be perhaps what Z is looking for in his "coordinate" part. The GCT of GR are derivative from underlying local gauge transformations of the tetrads.
R. Kiehn on Cartan Forms
On Dec 30, 2004, at 9:33 AM, RKiehn2352@aol.com wrote:
"Jack
Check out V. Fock on "harmonic coordinates" in his book Space Time and Gravity, re' the problem of fields vanishing at infinity.
Also see Chapter 12 from vol4 Plasmas and non-Equilibrium Electrodynamics.
http://www22.pair.com/csdc/download/plasmas69d.pdf
**
Also note that a p-form decomposes into 3 parts:
(an exact part) + (a closed part, but not exact) + (non exact non closed part)."
Yes, thanks. I meant the third term in the 1-form connection as the one that gives curvature.
"The last part gives the fields from the potentials. F=dA
The middle part give topological defects (BA effect,etc.)"
Ah! Thanks. That would give "curvature without curvature" like in the Vilenken-Taub solution? Also the hedgehog dark energy anomaly explaining the NASA Pioneer a_g = - cH constant acceleration between 2 spherical boundaries concentric with Sun, first boundary at 20AU. If this model is correct ALL stars should have this property that would be related to the birth of stars in the first place. Similar idea for Galactic Halo birth of galaxies. These defect seeds from pre -> post inflation vacuum breaking of translational symmetry.
"and non-trivial gauges.
The exact part yields trivial gauges."
Which would be perhaps what Zielinski is looking for in his "coordinate" part. The GCT of GR are derivative from underlying local gauge transformations of the tetrads.
On Dec 30, 2004, at 9:33 AM, RKiehn2352@aol.com wrote:
"Jack
Check out V. Fock on "harmonic coordinates" in his book Space Time and Gravity, re' the problem of fields vanishing at infinity.
Also see Chapter 12 from vol4 Plasmas and non-Equilibrium Electrodynamics.
http://www22.pair.com/csdc/download/plasmas69d.pdf
**
Also note that a p-form decomposes into 3 parts:
(an exact part) + (a closed part, but not exact) + (non exact non closed part)."
Yes, thanks. I meant the third term in the 1-form connection as the one that gives curvature.
"The last part gives the fields from the potentials. F=dA
The middle part give topological defects (BA effect,etc.)"
Ah! Thanks. That would give "curvature without curvature" like in the Vilenken-Taub solution? Also the hedgehog dark energy anomaly explaining the NASA Pioneer a_g = - cH constant acceleration between 2 spherical boundaries concentric with Sun, first boundary at 20AU. If this model is correct ALL stars should have this property that would be related to the birth of stars in the first place. Similar idea for Galactic Halo birth of galaxies. These defect seeds from pre -> post inflation vacuum breaking of translational symmetry.
"and non-trivial gauges.
The exact part yields trivial gauges."
Which would be perhaps what Zielinski is looking for in his "coordinate" part. The GCT of GR are derivative from underlying local gauge transformations of the tetrads.
Wednesday, December 29, 2004
Gravimagnetic Submarine Warfare?
"The Question is: What is The Question?" John Archibald Wheeler
Metric Engineering Investigations 1.6
Special Relativity considerations: In a global inertial frame in Cartesian coordinates the frame-invariant small differential space-time interval ds obeys
ds^2 = (cdt)^2 - dx^2 - dy^2 - dz^2
Any Lorentz transformation to another inertial frame in uniform relative motion to the first preserves this Cartesian form. That is under the action of O(1,3) x^u -> x^u'
ds^2 = (cdt')^2 - dx'^2 - dy'^2 - dz'^2
A transformation to an accelerated noninertial frame in 1905 special relativity sense will formally look lie a transformation to curvilinear coordinates with the possibility of off-diagonal terms. Not so however for the trivial 3D change to spherical polar coordinates where
ds^2 = (cdt)^2 - dr^2 - r^2(dtheta^2 + sin^2thetadphi^2)
Example 1 uniformly accelerating noninertial frame in the Galilean limit gt/c << 1
t' ~ t
z' ~ z - (1/2)gt^2
dz = dz' + gtdt
dz^2 = dz'^2 + (gt/c)^2(cdt)^2 + 2(gt/c)dz(cdt)
So the important part of the metric in the z'-t plane is
(cdt)^2[1 - (gt/c)^2] - dz'^2 - 2(gt/c)dz(cdt)
The mixed space-time off-diagonal cross-term is a longitudinal "gravimagnetism" effect. In this case translational acceleration of the noninertial frame is a source of gravimagnetism.
Bz = 2gt/c points along the z-axis direction of translational acceleration.
Special relativity where gt/c -> 1 changes this to the hyperbolic motion problem using hyperbolic functions.
Example 2 Galilean relativity Wx'/c << 1 etc. limit of rotating noninertial frame about z-axis
x = x'cosWt - y'sinWt
y = x'sinWt + y'cosWt
dx = dx'cosWt - x'sinWtdt - dy'sinWt - y'cosWtdt
dy = dx'sinWt + x'cosWt + dy'cosWt - y'sinWtdt
ds^ = [1 - W^2(x'^2 + y'^2)/c^2](cdt)^2 + (2Wy'/c)dx'(cdt) -(2Wx'/c)dy'(cdt) - dx'^2 - dy'^2 - dz'^2
Note the inhomogeneous transverse gravimagnetism here from the physical rotation of the noninertial frame. That is
Bx'(y') = 2Wy'/c
By'(x') = 2Wx'/c
The gravimagnetic 3-vector B = goi points in the plane perpendicular to the axis of rotation. See Ray Chiao's "Gravity Radio" A(em).B(gravity) interaction Hamiltonian papers online for the application of gravimagnetism in rotating superconductors for the high efficiency transduction between gravity waves and electromagnetic waves with application to submarine warfare C^3 and a host of other applications to the cosmology of the Big Bang if it can be achieved. Note the "Stern-Gerlach" inhomogeneous gravimagnetic "potential well" for gyroscopes in this solution. What happens in a medium c/n when n >> 1. Will that enhance the strength of gravimagnetism?
"The Question is: What is The Question?" John Archibald Wheeler
Metric Engineering Investigations 1.6
Special Relativity considerations: In a global inertial frame in Cartesian coordinates the frame-invariant small differential space-time interval ds obeys
ds^2 = (cdt)^2 - dx^2 - dy^2 - dz^2
Any Lorentz transformation to another inertial frame in uniform relative motion to the first preserves this Cartesian form. That is under the action of O(1,3) x^u -> x^u'
ds^2 = (cdt')^2 - dx'^2 - dy'^2 - dz'^2
A transformation to an accelerated noninertial frame in 1905 special relativity sense will formally look lie a transformation to curvilinear coordinates with the possibility of off-diagonal terms. Not so however for the trivial 3D change to spherical polar coordinates where
ds^2 = (cdt)^2 - dr^2 - r^2(dtheta^2 + sin^2thetadphi^2)
Example 1 uniformly accelerating noninertial frame in the Galilean limit gt/c << 1
t' ~ t
z' ~ z - (1/2)gt^2
dz = dz' + gtdt
dz^2 = dz'^2 + (gt/c)^2(cdt)^2 + 2(gt/c)dz(cdt)
So the important part of the metric in the z'-t plane is
(cdt)^2[1 - (gt/c)^2] - dz'^2 - 2(gt/c)dz(cdt)
The mixed space-time off-diagonal cross-term is a longitudinal "gravimagnetism" effect. In this case translational acceleration of the noninertial frame is a source of gravimagnetism.
Bz = 2gt/c points along the z-axis direction of translational acceleration.
Special relativity where gt/c -> 1 changes this to the hyperbolic motion problem using hyperbolic functions.
Example 2 Galilean relativity Wx'/c << 1 etc. limit of rotating noninertial frame about z-axis
x = x'cosWt - y'sinWt
y = x'sinWt + y'cosWt
dx = dx'cosWt - x'sinWtdt - dy'sinWt - y'cosWtdt
dy = dx'sinWt + x'cosWt + dy'cosWt - y'sinWtdt
ds^ = [1 - W^2(x'^2 + y'^2)/c^2](cdt)^2 + (2Wy'/c)dx'(cdt) -(2Wx'/c)dy'(cdt) - dx'^2 - dy'^2 - dz'^2
Note the inhomogeneous transverse gravimagnetism here from the physical rotation of the noninertial frame. That is
Bx'(y') = 2Wy'/c
By'(x') = 2Wx'/c
The gravimagnetic 3-vector B = goi points in the plane perpendicular to the axis of rotation. See Ray Chiao's "Gravity Radio" A(em).B(gravity) interaction Hamiltonian papers online for the application of gravimagnetism in rotating superconductors for the high efficiency transduction between gravity waves and electromagnetic waves with application to submarine warfare C^3 and a host of other applications to the cosmology of the Big Bang if it can be achieved. Note the "Stern-Gerlach" inhomogeneous gravimagnetic "potential well" for gyroscopes in this solution. What happens in a medium c/n when n >> 1. Will that enhance the strength of gravimagnetism?
Space Force Academy
Mission Objective:
The whole idea of the practical metric engineering the fabric of spacetime to reverse engineer the alleged alien time travel ships from our future through the Star Gate(s) is to control the flying saucer's timelike geodesic from inside the saucer using tiny amounts of power.
"The Question is: What is The Question?" John Archibald Wheeler
Metric Engineering Investigations 1.5
Again L&L here only address Galilean relativity and Newton's gravity force theory in global inertial and noninertial frames. Much of this, but not all, will carry over to Einstein's geometrodynamics in which there are no global frames, except in very special solutions like the large-scale FRW cosmology solution. Local physics is in terms of COINCIDENT LIFs and LNIFs and there is NO GRAVITY FORCE in the sense of Newton. Newton's gravity force in flat 3D space is eliminated, replaced by WEIGHTLESS geodesic inertial motion in curved 4D space-time. John Wheeler's extended idea of geometrodynamics is that all elementary particles are sourceless curved vacuum solution with non-trivial topology and trapped quantized gauge force fluxes like vortices in superfluids and superconductors. These tiny wormholes would be Bohm's hidden variables guided by micro-quantum qubit pilot waves. They would exist both transiently off-mass-shell and more stably on-mass-shell in sense of micro-quantum field theory. Therefore, dynamical topology change is required. One also needs a strong short scale gravity field G* ~ 10^40G on scale of 1 fermi and smaller. This may be provided by the dark energy residue not absorbed into the Higgs-Goldstone macro-quantum vacuum coherence local field whose single-valuedness requires topological defects like are seen in the Pioneer Anomaly a_g = -cH back to Sun starting at 20 AU that fits the hedgehog configuration where the vacuum coherence is in a S^2 order parameter space. Various species of topological defects in the vacuum coherence would be essential to galaxy and star formation. It may be that ALL STARS have this hedgehog topological defect that the NASA Pioneers are showing for our Sun? Why should the Sun be special in that particular? See David Thouless's book on topological quantum theory for more on the hedgehog that fits the NASA Pioneer data perfectly with a plausible dark energy field between the two spherical boundaries concentric with the Sun. Where the outer boundary ends has not been measured yet. The inner boundary starts at 20 AU out from the Sun.
"Actual" vs "Fake" Gravity in Newton's Theory.
Therefore, in the flat 3D-space Galilean relativity with absolute simultaneity and the instant action at a distance of Newton's non-retarded gravity Force theory pre-1905 special relativity:
"However, the fields to which noninertial reference systems are equivalent are not completely identical with 'actual' gravitational fields which occur also in inertial frames. For there is a very essential difference with respect to their behavior at infinity. At infinite distances from the bodies producing the field, 'actual' gravitational fields always go to zero. Contrary to this, the fields to which noninertial frames are equivalent increase without limit at infinity, or, in any event, remain finite ... for example, the centrifugal force which appears in a rotating frame of reference increases without limit as we move away from the axis of rotation; the field to which a reference frame in accelerated linear motion is equivalent is the same over all space and also at infinity. The fields to which noninertial systems are equivalent vanish as soon as we transform to an inertial system. In contrast to this, 'actual' gravitational fields (existing also in an inertial reference frame) cannot be eliminated by any choice of reference system. This is already clear from what has been said above concerning the difference in conditions at infinity between 'actual' gravitational fields and fields to which noninertial systems are equivalent; since the latter do not approach zero at infinity, it is clear that it is impossible by any choice of reference frame to eliminate an 'actual' field since it vanishes at infinity."
Note how L&L hammer on this point of the importance of the global boundary condition in addition to the LOCAL Galilean equivalence principle. The elimination of Newton's actual gravity force in an inertial frame over a small region is done by switching to a local noninertial frame. This is completely opposite to Einstein's general relativity of course. So keep that cognitive dissonance, that "creative tension" (Ray Chiao) of the "identity of opposites" in mind.
"All that can be done by a suitable choice of reference system is to eliminate the gravitational field in a given region of space, sufficiently small so that the field can be considered uniform over it. This can be done by choosing a system in accelerated motion, the acceleration of which is equal to that which would be acquired by a particle placed in the region of the field which we are considering." p. 227
Now Zielinski is attempting to impose this Newtonian conception on Einstein's and, obviously as will become progressively apparent below, that is impossible and deeply offends Einstein's great Gestalt Shift on the creative tension between "inertial" and "noninertial". Noninertial non-geodesic gravity motion in Newtonian-Galilean flat 3D-space is inertial force-free weightless gravity geodesic motion in curved 4D spacetime. Indeed, Hal Puthoff's PV theory attempting to adapt 1905 Einstein special relativity to gravity, that Einstein soon gave up on 1910 - 1915, suffers from that same defect as Zielinski's doomed attempt.
L&L Then give an example writing the Lagrangian L of a test particle m in an inertial frame
L = mv^2/2 - mV (81.1)
where
dv/dt = - gradV (81.2)
where in a small region of space approximately
V ~ gz
z = height above Earth's surface. Earth's surface is here approximated as a inertial frame neglecting its rotation. Of course one cannot do that in Einstein's general relativity since a fixed point on Earth's surface is on a non-geodesic world line in curved 4D spacetime.
dv/dt ~ -g
"It does not contain the mass or any other constant characterizing the properties of the particle; this is the mathematical expression of the basic property of gravitational fields."
This universality, i.e. the irrelevance of any other constant characterizing the properties of the particle" allowed Einstein to eliminate gravity Force completely by shifting out attention away from flat 3D space to curved 4D spacetime. Now you cannot do that for the electromagnetic force in 4D. You can sort of do it however if you add one additional space dimension (Kaluza-Klein) which is the precursor of string theory now called M-Theory still highly speculative and possibly not falsifiable making it into a Cargo Cult pseudo-physics of seductive mathematical beauty no doubt, but not even wrong all the same. However The Fat Lady has not yet sung on M-Theory.
Mission Objective:
The whole idea of the practical metric engineering the fabric of spacetime to reverse engineer the alleged alien time travel ships from our future through the Star Gate(s) is to control the flying saucer's timelike geodesic from inside the saucer using tiny amounts of power.
"The Question is: What is The Question?" John Archibald Wheeler
Metric Engineering Investigations 1.5
Again L&L here only address Galilean relativity and Newton's gravity force theory in global inertial and noninertial frames. Much of this, but not all, will carry over to Einstein's geometrodynamics in which there are no global frames, except in very special solutions like the large-scale FRW cosmology solution. Local physics is in terms of COINCIDENT LIFs and LNIFs and there is NO GRAVITY FORCE in the sense of Newton. Newton's gravity force in flat 3D space is eliminated, replaced by WEIGHTLESS geodesic inertial motion in curved 4D space-time. John Wheeler's extended idea of geometrodynamics is that all elementary particles are sourceless curved vacuum solution with non-trivial topology and trapped quantized gauge force fluxes like vortices in superfluids and superconductors. These tiny wormholes would be Bohm's hidden variables guided by micro-quantum qubit pilot waves. They would exist both transiently off-mass-shell and more stably on-mass-shell in sense of micro-quantum field theory. Therefore, dynamical topology change is required. One also needs a strong short scale gravity field G* ~ 10^40G on scale of 1 fermi and smaller. This may be provided by the dark energy residue not absorbed into the Higgs-Goldstone macro-quantum vacuum coherence local field whose single-valuedness requires topological defects like are seen in the Pioneer Anomaly a_g = -cH back to Sun starting at 20 AU that fits the hedgehog configuration where the vacuum coherence is in a S^2 order parameter space. Various species of topological defects in the vacuum coherence would be essential to galaxy and star formation. It may be that ALL STARS have this hedgehog topological defect that the NASA Pioneers are showing for our Sun? Why should the Sun be special in that particular? See David Thouless's book on topological quantum theory for more on the hedgehog that fits the NASA Pioneer data perfectly with a plausible dark energy field between the two spherical boundaries concentric with the Sun. Where the outer boundary ends has not been measured yet. The inner boundary starts at 20 AU out from the Sun.
"Actual" vs "Fake" Gravity in Newton's Theory.
Therefore, in the flat 3D-space Galilean relativity with absolute simultaneity and the instant action at a distance of Newton's non-retarded gravity Force theory pre-1905 special relativity:
"However, the fields to which noninertial reference systems are equivalent are not completely identical with 'actual' gravitational fields which occur also in inertial frames. For there is a very essential difference with respect to their behavior at infinity. At infinite distances from the bodies producing the field, 'actual' gravitational fields always go to zero. Contrary to this, the fields to which noninertial frames are equivalent increase without limit at infinity, or, in any event, remain finite ... for example, the centrifugal force which appears in a rotating frame of reference increases without limit as we move away from the axis of rotation; the field to which a reference frame in accelerated linear motion is equivalent is the same over all space and also at infinity. The fields to which noninertial systems are equivalent vanish as soon as we transform to an inertial system. In contrast to this, 'actual' gravitational fields (existing also in an inertial reference frame) cannot be eliminated by any choice of reference system. This is already clear from what has been said above concerning the difference in conditions at infinity between 'actual' gravitational fields and fields to which noninertial systems are equivalent; since the latter do not approach zero at infinity, it is clear that it is impossible by any choice of reference frame to eliminate an 'actual' field since it vanishes at infinity."
Note how L&L hammer on this point of the importance of the global boundary condition in addition to the LOCAL Galilean equivalence principle. The elimination of Newton's actual gravity force in an inertial frame over a small region is done by switching to a local noninertial frame. This is completely opposite to Einstein's general relativity of course. So keep that cognitive dissonance, that "creative tension" (Ray Chiao) of the "identity of opposites" in mind.
"All that can be done by a suitable choice of reference system is to eliminate the gravitational field in a given region of space, sufficiently small so that the field can be considered uniform over it. This can be done by choosing a system in accelerated motion, the acceleration of which is equal to that which would be acquired by a particle placed in the region of the field which we are considering." p. 227
Now Zielinski is attempting to impose this Newtonian conception on Einstein's and, obviously as will become progressively apparent below, that is impossible and deeply offends Einstein's great Gestalt Shift on the creative tension between "inertial" and "noninertial". Noninertial non-geodesic gravity motion in Newtonian-Galilean flat 3D-space is inertial force-free weightless gravity geodesic motion in curved 4D spacetime. Indeed, Hal Puthoff's PV theory attempting to adapt 1905 Einstein special relativity to gravity, that Einstein soon gave up on 1910 - 1915, suffers from that same defect as Zielinski's doomed attempt.
L&L Then give an example writing the Lagrangian L of a test particle m in an inertial frame
L = mv^2/2 - mV (81.1)
where
dv/dt = - gradV (81.2)
where in a small region of space approximately
V ~ gz
z = height above Earth's surface. Earth's surface is here approximated as a inertial frame neglecting its rotation. Of course one cannot do that in Einstein's general relativity since a fixed point on Earth's surface is on a non-geodesic world line in curved 4D spacetime.
dv/dt ~ -g
"It does not contain the mass or any other constant characterizing the properties of the particle; this is the mathematical expression of the basic property of gravitational fields."
This universality, i.e. the irrelevance of any other constant characterizing the properties of the particle" allowed Einstein to eliminate gravity Force completely by shifting out attention away from flat 3D space to curved 4D spacetime. Now you cannot do that for the electromagnetic force in 4D. You can sort of do it however if you add one additional space dimension (Kaluza-Klein) which is the precursor of string theory now called M-Theory still highly speculative and possibly not falsifiable making it into a Cargo Cult pseudo-physics of seductive mathematical beauty no doubt, but not even wrong all the same. However The Fat Lady has not yet sung on M-Theory.
Tuesday, December 28, 2004
Pionner Anomaly a_g = cH
On Dec 28, 2004, at 3:13 PM, iksnileiz@earthlink.net wrote:
Z: OK. So we do now have a local standard of absolute acceleration and velocity with respect to the physical vacuum?
Velocity and time and place for sure. Acceleration also from CBB anisotropy changes. You can also get jerk, snap and pop etc.
The Hubble flow in FRW R(t) where H(t) = R(t)^-1dR/dt is a good definition of state of rest for vacuum I would think. Remember this is spontaneously broken vacuum symmetry like a ferromagnet. The lowest energy state only has a symmetry subgroup H of the larger symmetry group G of the dynamical action and its Euler-Lagrange local GCT tensor field equations
Guv + /\guv = 0
My original conjecture that stellar formation and also galaxy formation seeds require localized topological defects in the world hologram vacuum coherence that absorbs most of the random zero point energy is an unanticipated exciting idea that Pioneer data suggests.
Z: Motion with respect to this cosmic background is something that can be detected by local experiments, correct?
J: These are actual routine facts now.
On Dec 28, 2004, at 3:13 PM, iksnileiz@earthlink.net wrote:
Z: OK. So we do now have a local standard of absolute acceleration and velocity with respect to the physical vacuum?
Velocity and time and place for sure. Acceleration also from CBB anisotropy changes. You can also get jerk, snap and pop etc.
The Hubble flow in FRW R(t) where H(t) = R(t)^-1dR/dt is a good definition of state of rest for vacuum I would think. Remember this is spontaneously broken vacuum symmetry like a ferromagnet. The lowest energy state only has a symmetry subgroup H of the larger symmetry group G of the dynamical action and its Euler-Lagrange local GCT tensor field equations
Guv + /\guv = 0
My original conjecture that stellar formation and also galaxy formation seeds require localized topological defects in the world hologram vacuum coherence that absorbs most of the random zero point energy is an unanticipated exciting idea that Pioneer data suggests.
Z: Motion with respect to this cosmic background is something that can be detected by local experiments, correct?
J: These are actual routine facts now.
Metric Engineering Investigations 1.4
Limitations of the Galilean-Newtonian equivalence principle
"However, the fields to which noninertial reference systems are equivalent are not completely identical with 'actual' gravitational fields which occur also in inertial frames." L & L "Classical Theory of Fields"
Again there are no 'actual" gravitational "force" fields in the local inertial frames (LIFs) of Einstein's general relativity as there are in the GIFs of Newton's gravity theory. There are tidal stretch-squeeze curvature effects in both theories. The term "curvature" is not used in Newton's theory. "Inhomogeneous gravity field" is used instead. Einstein's theory is the better theory including phenomena like gravimagnetism not found in Newton's theory but now actually observed. Gravimagnetism is not in Hal Puthoff's PV over-simplified theory either.
Section 81 of L&L is only about Newton's theory and care must be taken not to import it unthinkingly into Einstein's theory. However, the following is also true in Einstein's GR, actual classical gravitational fields must come from localized sources and, therefore, must be asymptotically flat. This precludes, for example, an exact uniform constant gravitational field over the entire universe. One possible exception to this is the quantum zero point dark energy which at large scales > 10^26 cm has average energy density (10^19Gev/10^-33cm)10^-56 cm^-2 ~ 10^(19 + 33 - 56) Gev/cm^2 ~ 10^-4 Gev/cm^3 = 10^5ev/cm^3 ~ 10^-7 ergs/cm^3 ~ 10^-14 Joules/cm^3.
The equivalence principle is only a local metric principle not a global principle.
""However, the fields to which noninertial reference systems are equivalent are not completely identical with 'actual' gravitational fields which occur also in inertial frames. For there is a very essential difference with respect to their behavior at infinity. At infinite distances from the bodies producing the field, 'actual' gravitational fields always go to zero. Contrary to this, the fields to which noninertial frames are equivalent increase without limit at infinity, or, in any event, remain finite ... for example, the centrifugal force which appears in a rotating frame of reference increases without limit as we move away from the axis of rotation."
Let S' be rotating noninertial frame with angular velocity pseudovector W , S the non-rotating inertial frame whose z axis is the axis of rotation of S'. S & S' share common origin. Think of S roughly as a Foucault Pendulum at rotational North Pole with Earth as S' In Galilean relativity v/c << 1,
a = dv/dt
W,t = dW/dt
a' = a + 2Wxv + WxWxr + W,txr
2Wxv = Coriolis inertial force in the noninertial frame
WxWxr = centrifugal inertial force
W,txr = torque inertial force
dL/dt = torque = (applied force) x r
L = angular momentum
From rotation inertial forces to translational inertial forces
"the field to which a reference frame in accelerated linear motion is equivalent is the same over all space and also at infinity." p. 226
Remember this is only for Galilean relativity of Newton's gravity force theory and it is clearly only a unrealistic idealization not permitted in Einstein's theory.
Note that Einstein's gravity field with GCT comes from locally gauging the 4-parameter translational symmetry group. In addition, if we locally gauge the 6-parameter Lorentz group we get new torsion field dynamical degrees of freedom in addition to the 4 translational degrees of freedom.
Translational gravity field on a point test particle is locally equivalent to a translational inertial force.
Rotational torsion field on an extended gyroscope is locally equivalent to rotational inertial force. (G. Shipov)
This is in addition to the Lense-Thirring gravimagnetic frame drag with happen in zero torsion field.
Torsion means that infinitesimal parallelograms of parallel transport fail to close in second order rather than in third order as in zero torsion 1916 GR.
* Is there a natural split of the Einstein metric tensor field into purely curvilinear coordinate and intrinsic geometry pieces? Even if there is, it will not carry over to the connection field that is NONLINEAR quadratic in the metric tensor components and its first order partial derivatives.
That is even if we have the LOCAL METRIC property
(metric) = (metric)intrinsic + (metric)coordinate
(LC) = (LC)intrinsic + (LC)coordinate + (LC)intrinsic-coordinate
Note that (LC) is a 1-form
Curvature = d(LC) is a 2-form
Obviously if
(LC) = Exact 1-form + Non-Exact 1-form
The Exact 1-form does not contribute to the tidal curvature.
Exactness is a GLOBAL topological property (allowing Vilenken-Taub "curvature without curvature" for a thin unstable wall of dark energy) not to be garbled with the above LOCAL metric property.
Note that
ds^2 = guvdx^u/\dx^v
appears to be a 2-form.
its exterior Cartan derivative is then a 3-form, therefore the connection 1-form (LC) must be *DUAL to the 3-form in 4D spacetime.
guv = (Minkowski)uv + (1/2) Lp^2(Macro-Quantum Coherent Goldstone Phase)(,u ,v)
where ,u is the ordinary partial derivative in the holonomic coordinate basis.
Note that the separate terms on RHS are not GCT tensors individually only their sum is. Also there is no perturbation theory here. The second term on RHS is not "small" compared to first.
* The intrinsic curved spacetime geometry must obviously originate in the post-inflationary world hologram Macro-Quantum Coherent Goldstone Phase second order partial derivatives.
Note that there are 16 second-order partial derivatives. The mixed partials {01, 02, 03, 12, 13, 23} need not commute when there is a torsion field, but if they do, that leaves 10 independent parameters. There are 64 = 4^3 3rd-order partial derivatives for the connection non-tensor and torsion tensor fields and there are 4^4 = 256 4th-order partials for the tidal curvature subject to constraints that cut that down to 20 in the presence of matter and 10 in the classical /\zpf = 0 non-gravitating non-exotic vacuum with zero dark energy/matter density for the conformal curvature tensor. Also the partial derivatives of the macro-quantum Goldstone phase of the Higgs field may be minimally coupled gauge covariant relative to U(1)xSU(2)xSU(3). Note that the virtual electron-positron PV condensate has U(1)xSU(2) coupling even though the net charge is zero.
A GCT comes from the generating function of a canonical transformation overlap transition function, which is obviously the guv|coordinate piece that Z is looking for.
Chi(x^u,x^u') where x^u and x^u' are two local coordinate charts with common support in a neighborhood of physical event P.
The Jacobian matrix of the GCT at P is
Xu'^u(P) = Lp^2Chi,u'^u(P)
The mixed partials here, one from each chart, obviously DO NOT COMMUTE.
Also
Xu^v'Xv'^w = Kronecker Deltau^w etc.
gu'v'(P) = Xu'^u(P)Xv'^v(P)guv(P)
= Xu'^u(P)Xv'^v(P)[(Minkowski)uv + (1/2) Lp^2(Coherent Goldstone Phase)(,u ,v)]
= (Minkowski)u'v' + (1/2) Lp^2(Coherent Goldstone Phase)(,u' ,v')]
(1/2) Lp^2(Coherent Goldstone Phase)(,u' ,v')
= Xu'^u(P)Xv'^v(P)(Minkowski)uv - (Minkowski)u'v'
+ Xu'^u(P)Xv'^v(P)(1/2) Lp^2(Coherent Goldstone Phase)(,u ,v)]
with
Coherent Goldstone Phase (x^u') = Coherent Goldstone Phase(x^u) + Chi(x^u,x^u')
Is this consistent?
That is
Xu'^u(P)Xv'^v(P)(Minkowski)uv = gu'v'|coordinate =/= (Minkowski)u'v'
Xu'^u(P)Xv'^v(P)(1/2) Lp^2(Coherent Goldstone Phase)(,u ,v) = gu'v'|intrinsic
Metric Engineering Investigations 1.3
On Dec 26, 2004, at 7:19 PM, Jack Sarfatti wrote:
Paul
Look at sections 84 & 97 in Landau & Lifshitz Classical Theory of Fields.
More anon.
BTW Landau & Lifshitz from the Soviet Era, in spite of Lysenko, is still probably the best course, even today in 2004, in "classical" theoretical physics including the earlier versions of quantum field theory. Better than Weinberg because it is closer to observation and even in the English translations the explanations are generally very clear. Better than MTW in some respects.
Also in 82:
"in the general theory of relativity it is impossible in general to have a system of bodies which are fixed relative to one another. This result essentially changes the very concept of a system of reference in the general theory of relativity, as compared to its meaning in the special theory. In the latter we meant by a reference system a set of bodies at rest relative to one another in unchanging relative positions. Such systems of bodies do not exist in the presence of a variable gravitational field, and for the exact determination of the position of a particle in space we must, strictly speaking, have an infinite number of bodies which fill all space like some sort of 'medium'. Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity."
Paul you cannot improve on Landau & Lifshitz, they have covered all the real ground on the classical foundations.
"In connection with the arbitrariness of the choice of reference system, the laws of nature must be written in the general theory of relativity in a form which is appropriate to any four dimensional system of coordinates (or, as one says, in 'covariant' form). This, of course, does not imply the physical equivalence of all these reference systems (like the physical equivalence of all inertial reference systems in the special theory)."
Now Hal Puthoff's PV is not generally covariant in this sense as Hal's coworker at IAS Austin, Michael Ibison, has written explicitly. This is basically why Hal's theory is almost universally rejected by all the Top Guns in spacetime physics down to almost the last man standing. Hal is basically using the fixed background of special relativity in his action principle with the variable "dielectric" whose controlled variation is much too small for the practical metric engineering of Warp, Wormhole and Weapon (from C^3 to W^3) anyway. That is not how to Make Star Trek real. PV is not the "Right Stuff" to reach for the stars and beyond.
Note BTW that Landau & Lifshitz explicitly say that non-Cartesian CS are only for "noninertial reference frames" on p. 228 "Thus, in a noninertial system of reference, the square of the interval appears as a quadratic form of a general type in the coordinate differentials, that is, it has the form
ds^2 = gikdx^idx^k
... Thus, when we use a noninertial system, the four-dimensional coordinate system ... is curvilinear."
Now there is no way to make such a non-inertial system of reference as an approximation to "an infinite number of bodies which fill all space like some sort of 'medium'. Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity" without non-gravity forces. That is, in a purely gravitational universe noninertial systems of reference do not exist! Actually gravity is not a fundamental field at all. It cannot exist without the electromagnetic field to give it its light cones. Indeed it also needs quantum theory to emerge as a macro-quantum coherence effect.
Metric Engineering Investigations 1.2
On Dec 27, 2004, at 2:45 PM, iksnileiz@earthlink.net wrote:
OK, then can you explain your understanding of their distinction between "physical equivalence of all these reference frames", on the one hand, and general covariance of the laws, on the other? In your own words?
I have many times now. The distributions of detectors connected by GCTs - different regions of extended phase space of the detector distributions connected by the GCTs in same region of configuration space (within a given neighborhood of event P)
A set of Alice detectors Ai (i = 1 to N) and a set of Bob Bj (j = 1 to N) detectors pepper a neighborhood of even P in the overlap region that is the domain of the GCT transition function. Each set of detectors is a point in extended phase space including the (LC) at the position of each detector. Although the A & B detectors occupy the same point in configuration space, they occupy different points in phase space. Indeed, let the physical event be P with coincident (Ai,Bi) detectors be located at common events Pi close to P. We want to plot (LCA)i and (LCB)i as degrees of freedom, we can also plot the "velocities" relative to the micro-wave background in principle. This gives an extended "phase space" for the two detector configurations in a small neighborhood of the event/process P that is being simultaneously measured. Note that N = 1 is good enough. The GCT connects physically distinct reference frames in this sense.
Obviously the two configurations of detectors measuring the same event P are physically distinct. The GCT connecting these different points in phase space that share a common point in configuration space enable the objective comparison of raw data taken independently by both of them. Of course, this does not extend to quantum measurements on the same micro-quantum system. One can work with ensembles of course.
Limitations of the Galilean-Newtonian equivalence principle
"However, the fields to which noninertial reference systems are equivalent are not completely identical with 'actual' gravitational fields which occur also in inertial frames." L & L "Classical Theory of Fields"
Again there are no 'actual" gravitational "force" fields in the local inertial frames (LIFs) of Einstein's general relativity as there are in the GIFs of Newton's gravity theory. There are tidal stretch-squeeze curvature effects in both theories. The term "curvature" is not used in Newton's theory. "Inhomogeneous gravity field" is used instead. Einstein's theory is the better theory including phenomena like gravimagnetism not found in Newton's theory but now actually observed. Gravimagnetism is not in Hal Puthoff's PV over-simplified theory either.
Section 81 of L&L is only about Newton's theory and care must be taken not to import it unthinkingly into Einstein's theory. However, the following is also true in Einstein's GR, actual classical gravitational fields must come from localized sources and, therefore, must be asymptotically flat. This precludes, for example, an exact uniform constant gravitational field over the entire universe. One possible exception to this is the quantum zero point dark energy which at large scales > 10^26 cm has average energy density (10^19Gev/10^-33cm)10^-56 cm^-2 ~ 10^(19 + 33 - 56) Gev/cm^2 ~ 10^-4 Gev/cm^3 = 10^5ev/cm^3 ~ 10^-7 ergs/cm^3 ~ 10^-14 Joules/cm^3.
The equivalence principle is only a local metric principle not a global principle.
""However, the fields to which noninertial reference systems are equivalent are not completely identical with 'actual' gravitational fields which occur also in inertial frames. For there is a very essential difference with respect to their behavior at infinity. At infinite distances from the bodies producing the field, 'actual' gravitational fields always go to zero. Contrary to this, the fields to which noninertial frames are equivalent increase without limit at infinity, or, in any event, remain finite ... for example, the centrifugal force which appears in a rotating frame of reference increases without limit as we move away from the axis of rotation."
Let S' be rotating noninertial frame with angular velocity pseudovector W , S the non-rotating inertial frame whose z axis is the axis of rotation of S'. S & S' share common origin. Think of S roughly as a Foucault Pendulum at rotational North Pole with Earth as S' In Galilean relativity v/c << 1,
a = dv/dt
W,t = dW/dt
a' = a + 2Wxv + WxWxr + W,txr
2Wxv = Coriolis inertial force in the noninertial frame
WxWxr = centrifugal inertial force
W,txr = torque inertial force
dL/dt = torque = (applied force) x r
L = angular momentum
From rotation inertial forces to translational inertial forces
"the field to which a reference frame in accelerated linear motion is equivalent is the same over all space and also at infinity." p. 226
Remember this is only for Galilean relativity of Newton's gravity force theory and it is clearly only a unrealistic idealization not permitted in Einstein's theory.
Note that Einstein's gravity field with GCT comes from locally gauging the 4-parameter translational symmetry group. In addition, if we locally gauge the 6-parameter Lorentz group we get new torsion field dynamical degrees of freedom in addition to the 4 translational degrees of freedom.
Translational gravity field on a point test particle is locally equivalent to a translational inertial force.
Rotational torsion field on an extended gyroscope is locally equivalent to rotational inertial force. (G. Shipov)
This is in addition to the Lense-Thirring gravimagnetic frame drag with happen in zero torsion field.
Torsion means that infinitesimal parallelograms of parallel transport fail to close in second order rather than in third order as in zero torsion 1916 GR.
* Is there a natural split of the Einstein metric tensor field into purely curvilinear coordinate and intrinsic geometry pieces? Even if there is, it will not carry over to the connection field that is NONLINEAR quadratic in the metric tensor components and its first order partial derivatives.
That is even if we have the LOCAL METRIC property
(metric) = (metric)intrinsic + (metric)coordinate
(LC) = (LC)intrinsic + (LC)coordinate + (LC)intrinsic-coordinate
Note that (LC) is a 1-form
Curvature = d(LC) is a 2-form
Obviously if
(LC) = Exact 1-form + Non-Exact 1-form
The Exact 1-form does not contribute to the tidal curvature.
Exactness is a GLOBAL topological property (allowing Vilenken-Taub "curvature without curvature" for a thin unstable wall of dark energy) not to be garbled with the above LOCAL metric property.
Note that
ds^2 = guvdx^u/\dx^v
appears to be a 2-form.
its exterior Cartan derivative is then a 3-form, therefore the connection 1-form (LC) must be *DUAL to the 3-form in 4D spacetime.
guv = (Minkowski)uv + (1/2) Lp^2(Macro-Quantum Coherent Goldstone Phase)(,u ,v)
where ,u is the ordinary partial derivative in the holonomic coordinate basis.
Note that the separate terms on RHS are not GCT tensors individually only their sum is. Also there is no perturbation theory here. The second term on RHS is not "small" compared to first.
* The intrinsic curved spacetime geometry must obviously originate in the post-inflationary world hologram Macro-Quantum Coherent Goldstone Phase second order partial derivatives.
Note that there are 16 second-order partial derivatives. The mixed partials {01, 02, 03, 12, 13, 23} need not commute when there is a torsion field, but if they do, that leaves 10 independent parameters. There are 64 = 4^3 3rd-order partial derivatives for the connection non-tensor and torsion tensor fields and there are 4^4 = 256 4th-order partials for the tidal curvature subject to constraints that cut that down to 20 in the presence of matter and 10 in the classical /\zpf = 0 non-gravitating non-exotic vacuum with zero dark energy/matter density for the conformal curvature tensor. Also the partial derivatives of the macro-quantum Goldstone phase of the Higgs field may be minimally coupled gauge covariant relative to U(1)xSU(2)xSU(3). Note that the virtual electron-positron PV condensate has U(1)xSU(2) coupling even though the net charge is zero.
A GCT comes from the generating function of a canonical transformation overlap transition function, which is obviously the guv|coordinate piece that Z is looking for.
Chi(x^u,x^u') where x^u and x^u' are two local coordinate charts with common support in a neighborhood of physical event P.
The Jacobian matrix of the GCT at P is
Xu'^u(P) = Lp^2Chi,u'^u(P)
The mixed partials here, one from each chart, obviously DO NOT COMMUTE.
Also
Xu^v'Xv'^w = Kronecker Deltau^w etc.
gu'v'(P) = Xu'^u(P)Xv'^v(P)guv(P)
= Xu'^u(P)Xv'^v(P)[(Minkowski)uv + (1/2) Lp^2(Coherent Goldstone Phase)(,u ,v)]
= (Minkowski)u'v' + (1/2) Lp^2(Coherent Goldstone Phase)(,u' ,v')]
(1/2) Lp^2(Coherent Goldstone Phase)(,u' ,v')
= Xu'^u(P)Xv'^v(P)(Minkowski)uv - (Minkowski)u'v'
+ Xu'^u(P)Xv'^v(P)(1/2) Lp^2(Coherent Goldstone Phase)(,u ,v)]
with
Coherent Goldstone Phase (x^u') = Coherent Goldstone Phase(x^u) + Chi(x^u,x^u')
Is this consistent?
That is
Xu'^u(P)Xv'^v(P)(Minkowski)uv = gu'v'|coordinate =/= (Minkowski)u'v'
Xu'^u(P)Xv'^v(P)(1/2) Lp^2(Coherent Goldstone Phase)(,u ,v) = gu'v'|intrinsic
Metric Engineering Investigations 1.3
On Dec 26, 2004, at 7:19 PM, Jack Sarfatti wrote:
Paul
Look at sections 84 & 97 in Landau & Lifshitz Classical Theory of Fields.
More anon.
BTW Landau & Lifshitz from the Soviet Era, in spite of Lysenko, is still probably the best course, even today in 2004, in "classical" theoretical physics including the earlier versions of quantum field theory. Better than Weinberg because it is closer to observation and even in the English translations the explanations are generally very clear. Better than MTW in some respects.
Also in 82:
"in the general theory of relativity it is impossible in general to have a system of bodies which are fixed relative to one another. This result essentially changes the very concept of a system of reference in the general theory of relativity, as compared to its meaning in the special theory. In the latter we meant by a reference system a set of bodies at rest relative to one another in unchanging relative positions. Such systems of bodies do not exist in the presence of a variable gravitational field, and for the exact determination of the position of a particle in space we must, strictly speaking, have an infinite number of bodies which fill all space like some sort of 'medium'. Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity."
Paul you cannot improve on Landau & Lifshitz, they have covered all the real ground on the classical foundations.
"In connection with the arbitrariness of the choice of reference system, the laws of nature must be written in the general theory of relativity in a form which is appropriate to any four dimensional system of coordinates (or, as one says, in 'covariant' form). This, of course, does not imply the physical equivalence of all these reference systems (like the physical equivalence of all inertial reference systems in the special theory)."
Now Hal Puthoff's PV is not generally covariant in this sense as Hal's coworker at IAS Austin, Michael Ibison, has written explicitly. This is basically why Hal's theory is almost universally rejected by all the Top Guns in spacetime physics down to almost the last man standing. Hal is basically using the fixed background of special relativity in his action principle with the variable "dielectric" whose controlled variation is much too small for the practical metric engineering of Warp, Wormhole and Weapon (from C^3 to W^3) anyway. That is not how to Make Star Trek real. PV is not the "Right Stuff" to reach for the stars and beyond.
Note BTW that Landau & Lifshitz explicitly say that non-Cartesian CS are only for "noninertial reference frames" on p. 228 "Thus, in a noninertial system of reference, the square of the interval appears as a quadratic form of a general type in the coordinate differentials, that is, it has the form
ds^2 = gikdx^idx^k
... Thus, when we use a noninertial system, the four-dimensional coordinate system ... is curvilinear."
Now there is no way to make such a non-inertial system of reference as an approximation to "an infinite number of bodies which fill all space like some sort of 'medium'. Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity" without non-gravity forces. That is, in a purely gravitational universe noninertial systems of reference do not exist! Actually gravity is not a fundamental field at all. It cannot exist without the electromagnetic field to give it its light cones. Indeed it also needs quantum theory to emerge as a macro-quantum coherence effect.
Metric Engineering Investigations 1.2
On Dec 27, 2004, at 2:45 PM, iksnileiz@earthlink.net wrote:
OK, then can you explain your understanding of their distinction between "physical equivalence of all these reference frames", on the one hand, and general covariance of the laws, on the other? In your own words?
I have many times now. The distributions of detectors connected by GCTs - different regions of extended phase space of the detector distributions connected by the GCTs in same region of configuration space (within a given neighborhood of event P)
A set of Alice detectors Ai (i = 1 to N) and a set of Bob Bj (j = 1 to N) detectors pepper a neighborhood of even P in the overlap region that is the domain of the GCT transition function. Each set of detectors is a point in extended phase space including the (LC) at the position of each detector. Although the A & B detectors occupy the same point in configuration space, they occupy different points in phase space. Indeed, let the physical event be P with coincident (Ai,Bi) detectors be located at common events Pi close to P. We want to plot (LCA)i and (LCB)i as degrees of freedom, we can also plot the "velocities" relative to the micro-wave background in principle. This gives an extended "phase space" for the two detector configurations in a small neighborhood of the event/process P that is being simultaneously measured. Note that N = 1 is good enough. The GCT connects physically distinct reference frames in this sense.
Obviously the two configurations of detectors measuring the same event P are physically distinct. The GCT connecting these different points in phase space that share a common point in configuration space enable the objective comparison of raw data taken independently by both of them. Of course, this does not extend to quantum measurements on the same micro-quantum system. One can work with ensembles of course.
NASA PIONEER ANOMALY
Also the Pioneer anomaly measures H from the Hubble flow, i.e. a_g = - cH back to Sun from the dark energy hollow halo hedgehog topological defect in the single-valued vacuum coherence starting at 20AU from the Sun. Such defects should be found in all stars as a remnant of how stars form in the first place. This same vacuum coherence explains how Einstein's cosmological constant is small.
On Dec 28, 2004, at 1:40 PM, Jack Sarfatti wrote:
On Dec 28, 2004, at 1:32 PM, iksnileiz@earthlink.net wrote:
What part of "uniformity of space" do you still not understand? If an observer can locally detect a certain type of motion of himself and his co-moving measuring instruments through the vacuum, then space is not "uniform" with respect to that type of motion.
J: With that definition space is not "uniform" because of the Hubble flow and the cosmic black body radiation. This is now a well-established fact. The dipole Doppler shift anisotropy "velocity" of Earth through CBB is well measured as is the temperature giving a time from the Big Bang hot plasma.
Our galaxy is moving at 454 km/sec mean with error +- 125 km/sec relative to Hubble flow of the expansion of space in the direct l = 163 deg +- 15 deg , b = -11 deg +- 14 deg p. 301 Narlikar Cosmology (Cambridge, 1983)
Fig 1.20 (b) p. 22 explains convention for Galactic longitude l and latitude b on the Celestial Sphere. These measurements are 20 years old. Probably better ones now.
J: Fock may be clear in his text that you have pulled out of context.
Z: I gave you the context. You have his book.
J: Major time required.
Z: I never said or implied that space was actually uniform in the sense of actual "general relativity". In fact I say it is not. Apparently you agree with this.
J: Facts cannot be refuted.
J: How would I measure if space were "physically uniform"? What does it mean?
Z: This is a stupid question that of course is impossible to answer.
J: Wrong. See above.
Also the Pioneer anomaly measures H from the Hubble flow, i.e. a_g = - cH back to Sun from the dark energy hollow halo hedgehog topological defect in the single-valued vacuum coherence starting at 20AU from the Sun. Such defects should be found in all stars as a remnant of how stars form in the first place. This same vacuum coherence explains how Einstein's cosmological constant is small.
On Dec 28, 2004, at 1:40 PM, Jack Sarfatti wrote:
On Dec 28, 2004, at 1:32 PM, iksnileiz@earthlink.net wrote:
What part of "uniformity of space" do you still not understand? If an observer can locally detect a certain type of motion of himself and his co-moving measuring instruments through the vacuum, then space is not "uniform" with respect to that type of motion.
J: With that definition space is not "uniform" because of the Hubble flow and the cosmic black body radiation. This is now a well-established fact. The dipole Doppler shift anisotropy "velocity" of Earth through CBB is well measured as is the temperature giving a time from the Big Bang hot plasma.
Our galaxy is moving at 454 km/sec mean with error +- 125 km/sec relative to Hubble flow of the expansion of space in the direct l = 163 deg +- 15 deg , b = -11 deg +- 14 deg p. 301 Narlikar Cosmology (Cambridge, 1983)
Fig 1.20 (b) p. 22 explains convention for Galactic longitude l and latitude b on the Celestial Sphere. These measurements are 20 years old. Probably better ones now.
J: Fock may be clear in his text that you have pulled out of context.
Z: I gave you the context. You have his book.
J: Major time required.
Z: I never said or implied that space was actually uniform in the sense of actual "general relativity". In fact I say it is not. Apparently you agree with this.
J: Facts cannot be refuted.
J: How would I measure if space were "physically uniform"? What does it mean?
Z: This is a stupid question that of course is impossible to answer.
J: Wrong. See above.
World Hologram 1
On Dec 28, 2004, at 11:36 AM, Jack Sarfatti wrote:
gu'v'|coordinate = Xu'^u(P)Xv'^v(P)(Minkowski)uv =/= (Minkowski)u'v'
gu'v'|intrinsic = Xu'^u(P)Xv'^v(P)(1/2) Lp^2(Coherent Goldstone Phase)(,u ,v)
????
Where in my theory
guv = (Minkowski)uv + (1/2) Lp^2(Macro-Quantum Coherent Goldstone Phase)(,u ,v)
where ,u is the ordinary partial derivative in the holonomic coordinate basis.
Note that the separate terms on RHS are not GCT tensors individually only their sum is. Also there is no perturbation theory here. The second term on RHS is not "small" compared to first.
*The intrinsic curved spacetime geometry must obviously originate in the post-inflationary world hologram Macro-Quantum Coherent Goldstone Phase second order partial derivatives.
Note that there are 16 second-order partial derivatives. The mixed partials {01, 02, 03, 12, 13, 23} need not commute when there is a torsion field, but if they do, that leaves 10 independent parameters. There are 64 = 4^3 3rd-order partial derivatives for the connection non-tensor and torsion tensor fields and there are 4^4 = 256 4th-order partials for the tidal curvature subject to constraints that cut that down to 20 in the presence of matter and 10 in the classical /\zpf = 0 non-gravitating non-exotic vacuum with zero dark energy/matter density for the conformal curvature tensor. Also the partial derivatives of the macro-quantum Goldstone phase of the Higgs field may be minimally coupled gauge covariant relative to U(1)xSU(2)xSU(3). Note that the virtual electron-positron PV condensate has U(1)xSU(2) coupling even though the net charge is zero.
A GCT comes from the generating function of a canonical transformation overlap transition function, which is obviously the guv|coordinate piece that Z is looking for.
Chi(x^u,x^u') where x^u and x^u' are two local coordinate charts with common support in a neighborhood of physical event P.
The Jacobian matrix of the GCT at P is
Xu'^u(P) = Lp^2Chi,u'^u(P)
The mixed partials here, one from each chart, obviously DO NOT COMMUTE.
Also
Xu^v'Xv'^w = Kronecker Deltau^w etc.
gu'v'(P) = Xu'^u(P)Xv'^v(P)guv(P)
= Xu'^u(P)Xv'^v(P)[(Minkowski)uv + (1/2) Lp^2(Coherent Goldstone Phase)(,u ,v)]
= (Minkowski)u'v' + (1/2) Lp^2(Coherent Goldstone Phase)(,u' ,v')]
(1/2) Lp^2(Coherent Goldstone Phase)(,u' ,v')
= Xu'^u(P)Xv'^v(P)(Minkowski)uv - (Minkowski)u'v'
+ Xu'^u(P)Xv'^v(P)(1/2) Lp^2(Coherent Goldstone Phase)(,u ,v)]
with
Coherent Goldstone Phase (x^u') = Coherent Goldstone Phase(x^u) + Chi(x^u,x^u')
Is this consistent?
That is
Xu'^u(P)Xv'^v(P)(Minkowski)uv = gu'v'|coordinate =/= (Minkowski)u'v'
Xu'^u(P)Xv'^v(P)(1/2) Lp^2(Coherent Goldstone Phase)(,u ,v) = gu'v'|intrinsic
On Dec 28, 2004, at 11:36 AM, Jack Sarfatti wrote:
gu'v'|coordinate = Xu'^u(P)Xv'^v(P)(Minkowski)uv =/= (Minkowski)u'v'
gu'v'|intrinsic = Xu'^u(P)Xv'^v(P)(1/2) Lp^2(Coherent Goldstone Phase)(,u ,v)
????
Where in my theory
guv = (Minkowski)uv + (1/2) Lp^2(Macro-Quantum Coherent Goldstone Phase)(,u ,v)
where ,u is the ordinary partial derivative in the holonomic coordinate basis.
Note that the separate terms on RHS are not GCT tensors individually only their sum is. Also there is no perturbation theory here. The second term on RHS is not "small" compared to first.
*The intrinsic curved spacetime geometry must obviously originate in the post-inflationary world hologram Macro-Quantum Coherent Goldstone Phase second order partial derivatives.
Note that there are 16 second-order partial derivatives. The mixed partials {01, 02, 03, 12, 13, 23} need not commute when there is a torsion field, but if they do, that leaves 10 independent parameters. There are 64 = 4^3 3rd-order partial derivatives for the connection non-tensor and torsion tensor fields and there are 4^4 = 256 4th-order partials for the tidal curvature subject to constraints that cut that down to 20 in the presence of matter and 10 in the classical /\zpf = 0 non-gravitating non-exotic vacuum with zero dark energy/matter density for the conformal curvature tensor. Also the partial derivatives of the macro-quantum Goldstone phase of the Higgs field may be minimally coupled gauge covariant relative to U(1)xSU(2)xSU(3). Note that the virtual electron-positron PV condensate has U(1)xSU(2) coupling even though the net charge is zero.
A GCT comes from the generating function of a canonical transformation overlap transition function, which is obviously the guv|coordinate piece that Z is looking for.
Chi(x^u,x^u') where x^u and x^u' are two local coordinate charts with common support in a neighborhood of physical event P.
The Jacobian matrix of the GCT at P is
Xu'^u(P) = Lp^2Chi,u'^u(P)
The mixed partials here, one from each chart, obviously DO NOT COMMUTE.
Also
Xu^v'Xv'^w = Kronecker Deltau^w etc.
gu'v'(P) = Xu'^u(P)Xv'^v(P)guv(P)
= Xu'^u(P)Xv'^v(P)[(Minkowski)uv + (1/2) Lp^2(Coherent Goldstone Phase)(,u ,v)]
= (Minkowski)u'v' + (1/2) Lp^2(Coherent Goldstone Phase)(,u' ,v')]
(1/2) Lp^2(Coherent Goldstone Phase)(,u' ,v')
= Xu'^u(P)Xv'^v(P)(Minkowski)uv - (Minkowski)u'v'
+ Xu'^u(P)Xv'^v(P)(1/2) Lp^2(Coherent Goldstone Phase)(,u ,v)]
with
Coherent Goldstone Phase (x^u') = Coherent Goldstone Phase(x^u) + Chi(x^u,x^u')
Is this consistent?
That is
Xu'^u(P)Xv'^v(P)(Minkowski)uv = gu'v'|coordinate =/= (Minkowski)u'v'
Xu'^u(P)Xv'^v(P)(1/2) Lp^2(Coherent Goldstone Phase)(,u ,v) = gu'v'|intrinsic
Monday, December 27, 2004
Metric Engineering Investigations 1.2
Metric Engineering Investigations 1.2
On Dec 27, 2004, at 2:45 PM, iksnileiz@earthlink.net wrote:
OK, then can you explain your understanding of their distinction between "physical equivalence of all these reference frames", on the one hand, and general covariance of the laws, on the other? In your own words?
I have many times now. The distributions of detectors connected by GCTs - different regions of extended phase space of the detector distributions connected by the GCTs in same region of configuration space (within a given neighborhood of event P)
A set of Alice detectors Ai (i = 1 to N) and a set of Bob Bj (j = 1 to N) detectors pepper a neighborhood of even P in the overlap region that is the domain of the GCT transition function. Each set of detectors is a point in extended phase space including the (LC) at the position of each detector. Although the A & B detectors occupy the same point in configuration space, they occupy different points in phase space. Indeed, let the physical event be P with coincident (Ai,Bi) detectors be located at common events Pi close to P. We want to plot (LCA)i and (LCB)i as degrees of freedom, we can also plot the "velocities" relative to the micro-wave background in principle. This gives an extended "phase space" for the two detector configurations in a small neighborhood of the event/process P that is being simultaneously measured. Note that N = 1 is good enough. The GCT connects physically distinct reference frames in this sense.
Obviously the two configurations of detectors measuring the same event P are physically distinct. The GCT connecting these different points in phase space that share a common point in configuration space enable the objective comparison of raw data taken independently by both of them. Of course, this does not extend to quantum measurements on the same micro-quantum system. One can work with ensembles of course.
Back to 1.1 Newton's theory, L&L explicitly say that the idea of a uniform gravitational field is only an approximation in a limited region of space.
"The field of gravity of the Earth (over small regions, where the field can be considered uniform). Thus a uniformly accelerated system of reference is equivalent to a constant, uniform external field." p. 226 The sentence following that is garbled and incorrect, i.e. the incorrect sentence in my 4th revised English Edition is
"a nonuniformly accelerated linear motion of the reference system is clearly equivalent to a uniform but gravitational field" should be
"a nonuniformly accelerated linear motion of the reference system is clearly equivalent to a nonuniform gravitational field"
Limitations of the Galilean-Newtonian equivalence principle
"However, the fields to which noninertial reference systems are equivalent are not completely identical with 'actual' gravitational fields which occur also in inertial frames."
Note that this is only true in the global inertial frames of Newton's theory. It is not at all true in the local inertial frames of Einstein's theory where the "gravitational field" as "Levi-Civita" connection vanishes. This is analogous to an electromagnetic gauge transformation at FIXED x
A^i(x) -> Ai'(x) = A^i(x) - Grad^if(x) = 0
Note that the "curvature" Curl A^i(x) is not affected since CurlGradf(x) = 0.
However, the global Wilson loop phase factor e^[i(e/hc)Closed Loop Integral A.dl] is the Bohm-Aharonov-Josephson nonlocal quantum observable, e.g. fringe shift at crossing point of an electron interferometer where the electrons only travel through regions where CurlA = 0. There should be a gravity analog to this in Einstein's general relativity where Closed Loop integral (LC)ds is a dimensionless quantum phase difference over two alternative indistinguishable Feynman micro-quantum histories. There may also be a GIANT QUANTUM EFFECT since the local vacuum coherence order parameter must also be single-valued, i.e. path independent state function mod winding number N, i.e. 2piN. That is, Lp^2(Macro-Quantum Coherent Goldstone Phase),u is the distortion of the fabric of spacetime whose strain tensor is the curved part of the Einstein metric tensor in a given local coordinate patch, i.e.
guv = (Minkowski)uv + (1/2) Lp^2(Macro-Quantum Coherent Goldstone Phase)(,u ,v)
where ,u is the ordinary partial derivative in the holonomic coordinate basis.
Note that the separate terms on RHS are not GCT tensors individually only their sum is. Also there is no perturbation theory here. The second term on RHS is not "small" compared to first.
A GCT comes from the generating function of a canonical transformation overlap transition function
Chi(x^u,x^u') where x^u and x^u' are two local coordinate charts with common support in a neighborhood of physical event P.
The Jacobian matrix of the GCT at P is
Xu'^u(P) = Lp^2Chi,u'^u(P)
The mixed partials here, one from each chart, obviously DO NOT COMMUTE.
Also
Xu^v'Xv'^w = Kronecker Deltau^w etc.
gu'v'(P) = Xu'^u(P)Xv'^v(P)guv(P)
= Xu'^u(P)Xv'^v(P)[(Minkowski)uv + (1/2) Lp^2(Coherent Goldstone Phase)(,u ,v)]
= (Minkowski)u'v' + (1/2) Lp^2(Coherent Goldstone Phase)(,u' ,v')]
(1/2) Lp^2(Coherent Goldstone Phase)'(,u' ,v')
= Xu'^u(P)Xv'^v(P)(Minkowski)uv - (Minkowski)u'v'
+ Xu'^u(P)Xv'^v(P)(1/2) Lp^2(Coherent Goldstone Phase)(,u ,v)]
with
Coherent Goldstone Phase (x^u') = Coherent Goldstone Phase(x^u) + Chi(x^u,x^u')
Is this consistent?
Metric Engineering Investigations 1.2
On Dec 27, 2004, at 2:45 PM, iksnileiz@earthlink.net wrote:
OK, then can you explain your understanding of their distinction between "physical equivalence of all these reference frames", on the one hand, and general covariance of the laws, on the other? In your own words?
I have many times now. The distributions of detectors connected by GCTs - different regions of extended phase space of the detector distributions connected by the GCTs in same region of configuration space (within a given neighborhood of event P)
A set of Alice detectors Ai (i = 1 to N) and a set of Bob Bj (j = 1 to N) detectors pepper a neighborhood of even P in the overlap region that is the domain of the GCT transition function. Each set of detectors is a point in extended phase space including the (LC) at the position of each detector. Although the A & B detectors occupy the same point in configuration space, they occupy different points in phase space. Indeed, let the physical event be P with coincident (Ai,Bi) detectors be located at common events Pi close to P. We want to plot (LCA)i and (LCB)i as degrees of freedom, we can also plot the "velocities" relative to the micro-wave background in principle. This gives an extended "phase space" for the two detector configurations in a small neighborhood of the event/process P that is being simultaneously measured. Note that N = 1 is good enough. The GCT connects physically distinct reference frames in this sense.
Obviously the two configurations of detectors measuring the same event P are physically distinct. The GCT connecting these different points in phase space that share a common point in configuration space enable the objective comparison of raw data taken independently by both of them. Of course, this does not extend to quantum measurements on the same micro-quantum system. One can work with ensembles of course.
Back to 1.1 Newton's theory, L&L explicitly say that the idea of a uniform gravitational field is only an approximation in a limited region of space.
"The field of gravity of the Earth (over small regions, where the field can be considered uniform). Thus a uniformly accelerated system of reference is equivalent to a constant, uniform external field." p. 226 The sentence following that is garbled and incorrect, i.e. the incorrect sentence in my 4th revised English Edition is
"a nonuniformly accelerated linear motion of the reference system is clearly equivalent to a uniform but gravitational field" should be
"a nonuniformly accelerated linear motion of the reference system is clearly equivalent to a nonuniform gravitational field"
Limitations of the Galilean-Newtonian equivalence principle
"However, the fields to which noninertial reference systems are equivalent are not completely identical with 'actual' gravitational fields which occur also in inertial frames."
Note that this is only true in the global inertial frames of Newton's theory. It is not at all true in the local inertial frames of Einstein's theory where the "gravitational field" as "Levi-Civita" connection vanishes. This is analogous to an electromagnetic gauge transformation at FIXED x
A^i(x) -> Ai'(x) = A^i(x) - Grad^if(x) = 0
Note that the "curvature" Curl A^i(x) is not affected since CurlGradf(x) = 0.
However, the global Wilson loop phase factor e^[i(e/hc)Closed Loop Integral A.dl] is the Bohm-Aharonov-Josephson nonlocal quantum observable, e.g. fringe shift at crossing point of an electron interferometer where the electrons only travel through regions where CurlA = 0. There should be a gravity analog to this in Einstein's general relativity where Closed Loop integral (LC)ds is a dimensionless quantum phase difference over two alternative indistinguishable Feynman micro-quantum histories. There may also be a GIANT QUANTUM EFFECT since the local vacuum coherence order parameter must also be single-valued, i.e. path independent state function mod winding number N, i.e. 2piN. That is, Lp^2(Macro-Quantum Coherent Goldstone Phase),u is the distortion of the fabric of spacetime whose strain tensor is the curved part of the Einstein metric tensor in a given local coordinate patch, i.e.
guv = (Minkowski)uv + (1/2) Lp^2(Macro-Quantum Coherent Goldstone Phase)(,u ,v)
where ,u is the ordinary partial derivative in the holonomic coordinate basis.
Note that the separate terms on RHS are not GCT tensors individually only their sum is. Also there is no perturbation theory here. The second term on RHS is not "small" compared to first.
A GCT comes from the generating function of a canonical transformation overlap transition function
Chi(x^u,x^u') where x^u and x^u' are two local coordinate charts with common support in a neighborhood of physical event P.
The Jacobian matrix of the GCT at P is
Xu'^u(P) = Lp^2Chi,u'^u(P)
The mixed partials here, one from each chart, obviously DO NOT COMMUTE.
Also
Xu^v'Xv'^w = Kronecker Deltau^w etc.
gu'v'(P) = Xu'^u(P)Xv'^v(P)guv(P)
= Xu'^u(P)Xv'^v(P)[(Minkowski)uv + (1/2) Lp^2(Coherent Goldstone Phase)(,u ,v)]
= (Minkowski)u'v' + (1/2) Lp^2(Coherent Goldstone Phase)(,u' ,v')]
(1/2) Lp^2(Coherent Goldstone Phase)'(,u' ,v')
= Xu'^u(P)Xv'^v(P)(Minkowski)uv - (Minkowski)u'v'
+ Xu'^u(P)Xv'^v(P)(1/2) Lp^2(Coherent Goldstone Phase)(,u ,v)]
with
Coherent Goldstone Phase (x^u') = Coherent Goldstone Phase(x^u) + Chi(x^u,x^u')
Is this consistent?
Lectures on Relativity 1.1
1.1 Newton's force theory of gravity using Galilean relativity v/c << 1 and weak gravity fields where GM/c^2r << 1
The mass m of the point test particle in the external gravity force field in flat 3D Euclidean space in a global inertial frame (AKA GIF good over a large region of space) cancels out of the solution for its motion. This is the Galilean principle of equivalence. If the test particle is an extended rigid body then we are talking about the motion of the center of mass. "Test particle" means we ignore the gravitational field of the test particle itself in the given external gravitational force field written in the global inertial frame.
More on the Galilean relativity equivalence principle, which is "an analogy between the motion of a body in a gravitational field and the motion of a body not located in any external field, but which is considered from the point of view of a noninertial system of reference" (#81 p. 226, Classical Theory of Fields. 1975 Pergamon ed M. Hammermesh).
The force-free geodesic equation for inertial motion of a point test particle that is not "oriented" in G. Shipov's sense, is
d^2x^i/dt^2 = 0
i = 1,23
The gravity force in Newton's theory is on same footing as the electrical force - not so in Einstein's theory where gravity force is eliminated by curved 4D space-time i.e. "force without Force". This causes confusion, e.g. Z's thesis & Hal Puthoff's "PV" theory. There are also confusions from common sense Aristotelian relativity notions discussed by Roger Penrose in "The Road to Reality" Ch 17 Aristotle -> Galileo -> Newton-Cartan -> Einstein SR -> Einstein GR.
A "noninertial frame" is accelerated in some way including rotation. Think of a spaceship firing it's rocket engine. That is a local noninertial frame or LNIF. You only feel weight AKA g-force in noninertial rest frames. In contrast, an advanced alien flying saucer with "acceleration field" "G-Engine" "Geodesic Glider" "Vacuum Propeller" "Warp Drive" is NOT an non-inertial frame but is a free-float weightless local inertial frame LIF. This is just like the astronauts in weightless orbit around the Earth with the big difference that the alleged alien pilots are able to metric engineer their local geodesic into any direction they desire. They will also be able to manufacture Star Gates for quick passage to distant parts of our universe as well as parallel universes next door across hyperspace. With old enough Star Gates they can also, most likely, time travel to the past of their starting point.
It is important to see how great minds in theoretical physics think, so you can tell the difference between skim milk and real cream, between pseudo-physics (all the "free energy schemes" and most, but not all, of the "physics" in say Nick Cook's "The Hunt for Zero Point" on flying saucer propulsion schemes) and physics:
"in an inertial reference system, the free motion of all bodies is uniform and rectilinear, and if, say, at the initial time their velocities are the same, they will be the same for all times. Clearly, therefore, if we consider this motion in a given noninertial system, then relative to this system all the bodies will move in the same way. Thus the properties of the motion in a noninertial system are the same as those in an inertial system in the presence of a gravitational field."
Note, that this last sentence breaks down in the passage from Newton's force theory in flat space to Einstein's geometrodynamics in curved spacetime. Hal Puthoff's PV theory is wrong http://www.earthtech.org/ , as is Yilmaz's and Z's, because they all fail to notice this subtle conceptual shift in paradigm. In Einstein's theory the GIFs are replaced by LIFs in which the gravitational field on the center of mass motion on the test body is zero even though the tidal stretch-squeeze on extended test bodies is generally not zero. We need to distinguish REST FRAMES of the test body, that can be either LIF or LNIF, from EXTERNAL FRAMES, again either LIF or LNIF that are tracking the motion of the test body. It is important to realize that in Einstein's general relativity the equations are LOCAL which means that the LIFs and the LNIFs are "coincident" with the test body. Of course one can track distant objects but then one is extrapolating back along the null geodesics of light signals from the distant objects. "Coincident" in GR means in a neighborhood of an event P that is small compared to the locally variable radii of curvature of the fabric of spacetime, which is now a dynamical variable rather than a fixed RIGID background of some sort as it is even in Einstein's 1905 Global Special Relativity. Special relativity is now demoted to only a local approximation, a fact not recognized in its fullness by Z, Puthoff, Yilmaz & Co.
For example in Einstein's GR the geodesic equation in a locally coincident LIF is simply
c^2d^2x^u/ds^2 = 0
Here we need to specify initial positions and velocities. If the LIF is, in addition, the REST FRAME of the point test body then the initial velocity is zero, also by convention take the initial position as zero.
This same equation in a locally coincident LNIF is
c^2d^2x^u/ds^2 + (LC)^uvw(dx^v/ds)(dx^w/ds) = 0
Where (LC)^uvw(dx^v/ds)(dx^w/ds) are the inertial forces in the LNIF. They are caused by some non-gravity force F^u acting on the LNIF that is not acting on the test body. For example, the LNIF may be a rocket ship in space firing its engine ejecting propellant.
If there is an external electrical force f^u per unit charge on the test body with charge q forcing it off a geodesic, then in an EXTERNAL LIF
mc^2d^2x^u/ds^2 = qf^u
This same equation in a coincident LNIF is
mc^2d^2x^u/ds^2 + mc^2(LC)^uvw(dx^v/ds)(dx^w/ds) = qf^u
Further if we go to the REST LNIF of this test body
dx^i/ds = 0 for i = 1,2,3
dx^0/ds = 1
d^2x^u/ds^2 = 0
Therefore
mc^2(LC)^i00 = f^i
Weight = mc^2(LC)^i00
Note that (LC) has dimensions 1/Length, where metric tensor guv is dimensionless.
For neutral bodies there will be mutually induced electric dipole-dipole Van Der Waal's-Polder forces that are related to the Casimir forces, which, according to Ian Peterson at University of Coventry UK, cannot extract energy from the virtual transverse polarized random zero point photons at all. The only energy they can extract is from the weak electrostatic forces of the dipoles that come from longitudinal virtual photons in macro-quantum coherent states that are not random zero point vacuum fluctuations. Puthoff, according to Peterson, has given a misleading impression on this point in popular media like Nick Cook's "Zero Point" book and Aviation Week's "To The Stars" as well as Eric Davis's USAF funded teleportation study http://www.fas.org/sgp/eprint/teleport.pdf .
On Dec 26, 2004, at 7:59 PM, Jack Sarfatti wrote:
On Dec 26, 2004, at 7:19 PM, Jack Sarfatti wrote:
Paul
Look at sections 84 & 97 in Landau & Lifshitz Classical Theory of Fields.
More anon.
BTW Landau & Lifshitz from the Soviet Era, in spite of Lysenko, is still probably the best course, even today in 2004, in "classical" theoretical physics including the earlier versions of quantum field theory. Better than Weinberg because it is closer to observation and even in the English translations the explanations are generally very clear. Better than MTW in some respects.
Also in 82:
"in the general theory of relativity it is impossible in general to have a system of bodies which are fixed relative to one another. This result essentially changes the very concept of a system of reference in the general theory of relativity, as compared to its meaning in the special theory. In the latter we meant by a reference system a set of bodies at rest relative to one another in unchanging relative positions. Such systems of bodies do not exist in the presence of a variable gravitational field, and for the exact determination of the position of a particle in space we must, strictly speaking, have an infinite number of bodies which fill all space like some sort of 'medium'. Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity."
Paul you cannot improve on Landau & Lifshitz, they have covered all the real ground on the classical foundations.
"In connection with the arbitrariness of the choice of reference system, the laws of nature must be written in the general theory of relativity in a form which is appropriate to any four dimensional system of coordinates (or, as one says, in 'covariant' form). This, of course, does not imply the physical equivalence of all these reference systems (like the physical equivalence of all inertial reference systems in the special theory)."
Now Hal Puthoff's PV is not generally covariant in this sense as Hal's coworker at IAS Austin, Michael Ibison, has written explicitly. This is basically why Hal's theory is almost universally rejected by all the Top Guns in spacetime physics down to almost the last man standing. Hal is basically using the fixed background of special relativity in his action principle with the variable "dielectric" whose controlled variation is much too small for the practical metric engineering of Warp, Wormhole and Weapon (from C^3 to W^3) anyway. That is not how to Make Star Trek real. PV is not the "Right Stuff" to reach for the stars and beyond.
Note BTW that Landau & Lifshitz explicitly say that non-Cartesian CS are only for "noninertial reference frames" on p. 228 "Thus, in a noninertial system of reference, the square of the interval appears as a quadratic form of a general type in the coordinate differentials, that is, it has the form
ds^2 = gikdx^idx^k
... Thus, when we use a noninertial system, the four-dimensional coordinate system ... is curvilinear."
Now there is no way to make such a non-inertial system of reference as an approximation to "an infinite number of bodies which fill all space like some sort of 'medium'. Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity" without non-gravity forces. That is, in a purely gravitational universe noninertial systems of reference do not exist! Actually gravity is not a fundamental field at all. It cannot exist without the electromagnetic field to give it its light cones. Indeed it also needs quantum theory to emerge as a macro-quantum coherence effect.
1.1 Newton's force theory of gravity using Galilean relativity v/c << 1 and weak gravity fields where GM/c^2r << 1
The mass m of the point test particle in the external gravity force field in flat 3D Euclidean space in a global inertial frame (AKA GIF good over a large region of space) cancels out of the solution for its motion. This is the Galilean principle of equivalence. If the test particle is an extended rigid body then we are talking about the motion of the center of mass. "Test particle" means we ignore the gravitational field of the test particle itself in the given external gravitational force field written in the global inertial frame.
More on the Galilean relativity equivalence principle, which is "an analogy between the motion of a body in a gravitational field and the motion of a body not located in any external field, but which is considered from the point of view of a noninertial system of reference" (#81 p. 226, Classical Theory of Fields. 1975 Pergamon ed M. Hammermesh).
The force-free geodesic equation for inertial motion of a point test particle that is not "oriented" in G. Shipov's sense, is
d^2x^i/dt^2 = 0
i = 1,23
The gravity force in Newton's theory is on same footing as the electrical force - not so in Einstein's theory where gravity force is eliminated by curved 4D space-time i.e. "force without Force". This causes confusion, e.g. Z's thesis & Hal Puthoff's "PV" theory. There are also confusions from common sense Aristotelian relativity notions discussed by Roger Penrose in "The Road to Reality" Ch 17 Aristotle -> Galileo -> Newton-Cartan -> Einstein SR -> Einstein GR.
A "noninertial frame" is accelerated in some way including rotation. Think of a spaceship firing it's rocket engine. That is a local noninertial frame or LNIF. You only feel weight AKA g-force in noninertial rest frames. In contrast, an advanced alien flying saucer with "acceleration field" "G-Engine" "Geodesic Glider" "Vacuum Propeller" "Warp Drive" is NOT an non-inertial frame but is a free-float weightless local inertial frame LIF. This is just like the astronauts in weightless orbit around the Earth with the big difference that the alleged alien pilots are able to metric engineer their local geodesic into any direction they desire. They will also be able to manufacture Star Gates for quick passage to distant parts of our universe as well as parallel universes next door across hyperspace. With old enough Star Gates they can also, most likely, time travel to the past of their starting point.
It is important to see how great minds in theoretical physics think, so you can tell the difference between skim milk and real cream, between pseudo-physics (all the "free energy schemes" and most, but not all, of the "physics" in say Nick Cook's "The Hunt for Zero Point" on flying saucer propulsion schemes) and physics:
"in an inertial reference system, the free motion of all bodies is uniform and rectilinear, and if, say, at the initial time their velocities are the same, they will be the same for all times. Clearly, therefore, if we consider this motion in a given noninertial system, then relative to this system all the bodies will move in the same way. Thus the properties of the motion in a noninertial system are the same as those in an inertial system in the presence of a gravitational field."
Note, that this last sentence breaks down in the passage from Newton's force theory in flat space to Einstein's geometrodynamics in curved spacetime. Hal Puthoff's PV theory is wrong http://www.earthtech.org/ , as is Yilmaz's and Z's, because they all fail to notice this subtle conceptual shift in paradigm. In Einstein's theory the GIFs are replaced by LIFs in which the gravitational field on the center of mass motion on the test body is zero even though the tidal stretch-squeeze on extended test bodies is generally not zero. We need to distinguish REST FRAMES of the test body, that can be either LIF or LNIF, from EXTERNAL FRAMES, again either LIF or LNIF that are tracking the motion of the test body. It is important to realize that in Einstein's general relativity the equations are LOCAL which means that the LIFs and the LNIFs are "coincident" with the test body. Of course one can track distant objects but then one is extrapolating back along the null geodesics of light signals from the distant objects. "Coincident" in GR means in a neighborhood of an event P that is small compared to the locally variable radii of curvature of the fabric of spacetime, which is now a dynamical variable rather than a fixed RIGID background of some sort as it is even in Einstein's 1905 Global Special Relativity. Special relativity is now demoted to only a local approximation, a fact not recognized in its fullness by Z, Puthoff, Yilmaz & Co.
For example in Einstein's GR the geodesic equation in a locally coincident LIF is simply
c^2d^2x^u/ds^2 = 0
Here we need to specify initial positions and velocities. If the LIF is, in addition, the REST FRAME of the point test body then the initial velocity is zero, also by convention take the initial position as zero.
This same equation in a locally coincident LNIF is
c^2d^2x^u/ds^2 + (LC)^uvw(dx^v/ds)(dx^w/ds) = 0
Where (LC)^uvw(dx^v/ds)(dx^w/ds) are the inertial forces in the LNIF. They are caused by some non-gravity force F^u acting on the LNIF that is not acting on the test body. For example, the LNIF may be a rocket ship in space firing its engine ejecting propellant.
If there is an external electrical force f^u per unit charge on the test body with charge q forcing it off a geodesic, then in an EXTERNAL LIF
mc^2d^2x^u/ds^2 = qf^u
This same equation in a coincident LNIF is
mc^2d^2x^u/ds^2 + mc^2(LC)^uvw(dx^v/ds)(dx^w/ds) = qf^u
Further if we go to the REST LNIF of this test body
dx^i/ds = 0 for i = 1,2,3
dx^0/ds = 1
d^2x^u/ds^2 = 0
Therefore
mc^2(LC)^i00 = f^i
Weight = mc^2(LC)^i00
Note that (LC) has dimensions 1/Length, where metric tensor guv is dimensionless.
For neutral bodies there will be mutually induced electric dipole-dipole Van Der Waal's-Polder forces that are related to the Casimir forces, which, according to Ian Peterson at University of Coventry UK, cannot extract energy from the virtual transverse polarized random zero point photons at all. The only energy they can extract is from the weak electrostatic forces of the dipoles that come from longitudinal virtual photons in macro-quantum coherent states that are not random zero point vacuum fluctuations. Puthoff, according to Peterson, has given a misleading impression on this point in popular media like Nick Cook's "Zero Point" book and Aviation Week's "To The Stars" as well as Eric Davis's USAF funded teleportation study http://www.fas.org/sgp/eprint/teleport.pdf .
On Dec 26, 2004, at 7:59 PM, Jack Sarfatti wrote:
On Dec 26, 2004, at 7:19 PM, Jack Sarfatti wrote:
Paul
Look at sections 84 & 97 in Landau & Lifshitz Classical Theory of Fields.
More anon.
BTW Landau & Lifshitz from the Soviet Era, in spite of Lysenko, is still probably the best course, even today in 2004, in "classical" theoretical physics including the earlier versions of quantum field theory. Better than Weinberg because it is closer to observation and even in the English translations the explanations are generally very clear. Better than MTW in some respects.
Also in 82:
"in the general theory of relativity it is impossible in general to have a system of bodies which are fixed relative to one another. This result essentially changes the very concept of a system of reference in the general theory of relativity, as compared to its meaning in the special theory. In the latter we meant by a reference system a set of bodies at rest relative to one another in unchanging relative positions. Such systems of bodies do not exist in the presence of a variable gravitational field, and for the exact determination of the position of a particle in space we must, strictly speaking, have an infinite number of bodies which fill all space like some sort of 'medium'. Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity."
Paul you cannot improve on Landau & Lifshitz, they have covered all the real ground on the classical foundations.
"In connection with the arbitrariness of the choice of reference system, the laws of nature must be written in the general theory of relativity in a form which is appropriate to any four dimensional system of coordinates (or, as one says, in 'covariant' form). This, of course, does not imply the physical equivalence of all these reference systems (like the physical equivalence of all inertial reference systems in the special theory)."
Now Hal Puthoff's PV is not generally covariant in this sense as Hal's coworker at IAS Austin, Michael Ibison, has written explicitly. This is basically why Hal's theory is almost universally rejected by all the Top Guns in spacetime physics down to almost the last man standing. Hal is basically using the fixed background of special relativity in his action principle with the variable "dielectric" whose controlled variation is much too small for the practical metric engineering of Warp, Wormhole and Weapon (from C^3 to W^3) anyway. That is not how to Make Star Trek real. PV is not the "Right Stuff" to reach for the stars and beyond.
Note BTW that Landau & Lifshitz explicitly say that non-Cartesian CS are only for "noninertial reference frames" on p. 228 "Thus, in a noninertial system of reference, the square of the interval appears as a quadratic form of a general type in the coordinate differentials, that is, it has the form
ds^2 = gikdx^idx^k
... Thus, when we use a noninertial system, the four-dimensional coordinate system ... is curvilinear."
Now there is no way to make such a non-inertial system of reference as an approximation to "an infinite number of bodies which fill all space like some sort of 'medium'. Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity" without non-gravity forces. That is, in a purely gravitational universe noninertial systems of reference do not exist! Actually gravity is not a fundamental field at all. It cannot exist without the electromagnetic field to give it its light cones. Indeed it also needs quantum theory to emerge as a macro-quantum coherence effect.
Sunday, December 26, 2004
Soviet Physics: Landau & Lifshitz on General Relativity
On Dec 26, 2004, at 7:19 PM, Jack Sarfatti wrote:
Paul
Look at sections 84 & 97 in Landau & Lifshitz Classical Theory of Fields.
More anon.
BTW Landau & Lifshitz from the Soviet Era, in spite of Lysenko, is still probably the best course, even today in 2004, in "classical" theoretical physics including the earlier versions of quantum field theory. Better than Weinberg because it is closer to observation and even in the English translations the explanations are generally very clear. Better than MTW in some respects.
Also in 82:
"in the general theory of relativity it is impossible in general to have a system of bodies which are fixed relative to one another. This result essentially changes the very concept of a system of reference in the general theory of relativity, as compared to its meaning in the special theory. In the latter we meant by a reference system a set of bodies at rest relative to one another in unchanging relative positions. Such systems of bodies do not exist in the presence of a variable gravitational field, and for the exact determination of the position of a particle in space we must, strictly speaking, have an infinite number of bodies which fill all space like some sort of 'medium'. Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity."
Paul you cannot improve on Landau & Lifshitz, they have covered all the real ground on the classical foundations.
"In connection with the arbitrariness of the choice of reference system, the laws of nature must be written in the general theory of relativity in a form which is appropriate to any four dimensional system of coordinates (or, as one says, in 'covariant' form). This, of course, does not imply the physical equivalence of all these reference systems (like the physical equivalence of all inertial reference systems in the special theory)."
Now Hal Puthoff's PV is not generally covariant in this sense as Hal's coworker at IAS Austin, Michael Ibison, has written explicitly. This is basically why Hal's theory is almost universally rejected by all the Top Guns in spacetime physics down to almost the last man standing. Hal is basically using the fixed background of special relativity in his action principle with the variable "dielectric" whose controlled variation is much too small for the practical metric engineering of Warp, Wormhole and Weapon (from C^3 to W^3) anyway. That is not how to Make Star Trek real. PV is not the "Right Stuff" to reach for the stars and beyond.
Note BTW that Landau & Lifshitz explicitly say that non-Cartesian CS are only for "noninertial reference frames" on p. 228 "Thus, in a noninertial system of reference, the square of the interval appears as a quadratic form of a general type in the coordinate differentials, that is, it has the form
ds^2 = gikdx^idx^k
... Thus, when we use a noninertial system, the four-dimensional coordinate system ... is curvilinear."
Now there is no way to make such a non-inertial system of reference as an approximation to "an infinite number of bodies which fill all space like some sort of 'medium'. Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity" without non-gravity forces. That is, in a purely gravitational universe noninertial systems of reference do not exist! Actually gravity is not a fundamental field at all. It cannot exist without the electromagnetic field to give it its light cones. Indeed it also needs quantum theory to emerge as a macro-quantum coherence effect.
On Dec 26, 2004, at 7:19 PM, Jack Sarfatti wrote:
Paul
Look at sections 84 & 97 in Landau & Lifshitz Classical Theory of Fields.
More anon.
BTW Landau & Lifshitz from the Soviet Era, in spite of Lysenko, is still probably the best course, even today in 2004, in "classical" theoretical physics including the earlier versions of quantum field theory. Better than Weinberg because it is closer to observation and even in the English translations the explanations are generally very clear. Better than MTW in some respects.
Also in 82:
"in the general theory of relativity it is impossible in general to have a system of bodies which are fixed relative to one another. This result essentially changes the very concept of a system of reference in the general theory of relativity, as compared to its meaning in the special theory. In the latter we meant by a reference system a set of bodies at rest relative to one another in unchanging relative positions. Such systems of bodies do not exist in the presence of a variable gravitational field, and for the exact determination of the position of a particle in space we must, strictly speaking, have an infinite number of bodies which fill all space like some sort of 'medium'. Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity."
Paul you cannot improve on Landau & Lifshitz, they have covered all the real ground on the classical foundations.
"In connection with the arbitrariness of the choice of reference system, the laws of nature must be written in the general theory of relativity in a form which is appropriate to any four dimensional system of coordinates (or, as one says, in 'covariant' form). This, of course, does not imply the physical equivalence of all these reference systems (like the physical equivalence of all inertial reference systems in the special theory)."
Now Hal Puthoff's PV is not generally covariant in this sense as Hal's coworker at IAS Austin, Michael Ibison, has written explicitly. This is basically why Hal's theory is almost universally rejected by all the Top Guns in spacetime physics down to almost the last man standing. Hal is basically using the fixed background of special relativity in his action principle with the variable "dielectric" whose controlled variation is much too small for the practical metric engineering of Warp, Wormhole and Weapon (from C^3 to W^3) anyway. That is not how to Make Star Trek real. PV is not the "Right Stuff" to reach for the stars and beyond.
Note BTW that Landau & Lifshitz explicitly say that non-Cartesian CS are only for "noninertial reference frames" on p. 228 "Thus, in a noninertial system of reference, the square of the interval appears as a quadratic form of a general type in the coordinate differentials, that is, it has the form
ds^2 = gikdx^idx^k
... Thus, when we use a noninertial system, the four-dimensional coordinate system ... is curvilinear."
Now there is no way to make such a non-inertial system of reference as an approximation to "an infinite number of bodies which fill all space like some sort of 'medium'. Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity" without non-gravity forces. That is, in a purely gravitational universe noninertial systems of reference do not exist! Actually gravity is not a fundamental field at all. It cannot exist without the electromagnetic field to give it its light cones. Indeed it also needs quantum theory to emerge as a macro-quantum coherence effect.
Friday, December 24, 2004
Pioneer Anomaly as Hedgehog Vacuum Coherence Defect
In a message dated 12/24/2004 11:22:01 PM Romance Standard Time, sarfatti@pacbell.net writes:
Cartan's forms and the Hodge-DeRham integrals seem a good
formalism to use here.
On 12/24/04 R. Kiehn
"Try reading
http://www22.pair.com/csdc/download/topthermo69f.pdf
especially Chapter 6.
There are 3 dimensional period integrals A^F and A^G
which go beyond the 1 dimensional period integrals
of Bohm-Aharanov theory and Berry phase, as they are defects
that can form in non-equilibrium systems via dissipative processes.
These irreducibly 3D defects are "excited" states with (relatively) long lifetimes
(in fact the dynamics can have a Hamiltonian representation) but they are
not uniquely integrable configurations (which are only found in spaces
of Pfaff topological dimension 2 or less)."
OK after I get Penrose under my belt, then I will be in a better position to get all this. What happens in 4-spacetime? Are the 3D integrals dual in some sense to the 1D integrals?
**
"Also see Example 10a in chapter 3 section 4 of
http://www22.pair.com/csdc/download/plasmas69d.pdf
for a Hedge Hog B field (equivalent to fluid vorticity)
that produces zero A^F but non zero A^G, and a convective force
that causes attraction towards a rotational center. The defect structure
occurs when d(A^G) is zero ( a 3 D period integral).
The superposition of a closed but not exact term is that which produces
a "hole" in the system topology. In a sense the turbulent "vacuum" condenses
into topological defects, leaving a defect "hole" in the otherwise dissipative
medium. It would seem that the dissipation implies a "force" towards the hole."
Thanks. The NASA Pioneer data is almost a textbook example of the Hedge Hog. I get it from the vacuum coherence defect. The two concentric spherical boundaries - inner boundary starting at 20 AU are filled with a small amount of dark zero point vacuum energy whose effective Newtonian limit gravity g-force per unit test mass is a_g = - cH = 10^-7 cm/sec^2 to Sun. Galactic halo has a similar explanation though its not a hedge hog. Turbulence in the local macro-quantum vacuum coherence is an interesting idea. Basically the fabric of spacetime is coming from the phase factor of this GIANT QUANTUM LOCAL WAVE FUNCTION that lives on different scales and different regions in order parameter spaces that we can directly infer from data. Key as always is single-valuedness of the, in this case giant quantum vacuum wave. It's like superfluid, or, rather super-solid physics. The point is that ALL the dark energy/dark matter data can be understood in terms of topological defects of the vacuum coherence of the fabric of spacetime.
On Dec 23, 2004, at 2:30 PM, Jack Sarfatti wrote:
Thanks SP & M and Happy Holidays! :-)
On Dec 23, 2004, at 2:06 PM, Saul-Paul & Mary-Minn Sirag wrote:
Jack,
I made another trip to the Science Library here, and found three more papers of yours.
J. Sarfatt, "On the 'Type II Superconductor' Model of Self-Trapped Laser Filaments," Physics Letters Vol. 26A, No. 2, PP. 88-89 (18 Dec. 1967).
Yes, this is the one Ray Chiao mentioned to Chas Townes at UCB that was useful to Ray when he started his experiments in late 60's.
J.Sarfatt, "Destruction of Superflow in Unsaturated 4He Films and the Prediction of a new Crystalline Phase of 4He3 With Bose-Einstein Condensation," Physics Letters Vol. 30A, No. 5, pp 300-301 (3 Nov. 1969).
Yes, this idea was 30 years ahead of the curve.
J. Sarfatt, "Quantum-Mechanical Correlation Theory of Electromagnetic Fields," Nuovo Cimento Vol. XXVII, N. 5, pp. 1119-1129 (1 Marzo 1963).
[Note: this paper as well as the paper by Suskind and Glogower is cited in Michael Martin Nieto's paper "Quantum Phase and Quantum Phase Operators: Some Physics and Some History" arXiv:hep-th/9304036 v1 8 Apr 93.]
I will put all 8 of these papers in the mail to your Stockton Street address today.
Hey thanks Santa. It will be in my Stockton. :-)
Nuff said ;-)
Saul-Paul
On Dec 14, 2004, at 12:40 PM, Jack Sarfatti wrote:
Yes, but before you do see if you can find in same time period same Physics Letters something like
Laser Self-Trapped Filaments and Landau-Ginzburg Equation? Thanks. :-)
There is also one about ODLRO in a quantum solid - but maybe it's one of those below.
On Dec 14, 2004, at 12:04 PM, Saul-Paul & Mary-Minn Sirag wrote:
Jack,
I have the following papers. Should I mail them to your Stockton Street address?
Leonard Susskind & Jonathan Glogower, "Quantum Mechanical Phase and Time Operator", Physics Vol.1,No.1,
pp. 49-61, 1964.
J. Sarfatt & A.M. Stoneham, "The Goldstone theorem and the Jahn-Teller effect", Proc. Phys. Soc.., Vol. 91, pp
pp. 214-221, 1967.
J. Sarfatt, "On the Nature of the Superfluid Critical Velocity", Physics Letters, Vol. 24A, No. 5, pp. 287-288,
27 Feb. 1967.
J. Sarfatt, "A New Theory of the Superfluid Vortex Phenomenon", Physics Letters, Vol. 24A, No.7, pp. 399-400,
27 Mar. 1967.
J. Sarfatt, "Local Gauge Invariance and Broken Symmetry in Superfluid Helium", Physics Letters, Vol.25A, No.9,
pp. 642-643, 6 Nov. 1967.
All for now ;-)
Saul-Paul
**
"I have not been able to get to Tony Smith's web site
Is there some problem again?"
I don't know.
*
feliz navidad
RMK
In a message dated 12/24/2004 11:22:01 PM Romance Standard Time, sarfatti@pacbell.net writes:
Cartan's forms and the Hodge-DeRham integrals seem a good
formalism to use here.
On 12/24/04 R. Kiehn
"Try reading
http://www22.pair.com/csdc/download/topthermo69f.pdf
especially Chapter 6.
There are 3 dimensional period integrals A^F and A^G
which go beyond the 1 dimensional period integrals
of Bohm-Aharanov theory and Berry phase, as they are defects
that can form in non-equilibrium systems via dissipative processes.
These irreducibly 3D defects are "excited" states with (relatively) long lifetimes
(in fact the dynamics can have a Hamiltonian representation) but they are
not uniquely integrable configurations (which are only found in spaces
of Pfaff topological dimension 2 or less)."
OK after I get Penrose under my belt, then I will be in a better position to get all this. What happens in 4-spacetime? Are the 3D integrals dual in some sense to the 1D integrals?
**
"Also see Example 10a in chapter 3 section 4 of
http://www22.pair.com/csdc/download/plasmas69d.pdf
for a Hedge Hog B field (equivalent to fluid vorticity)
that produces zero A^F but non zero A^G, and a convective force
that causes attraction towards a rotational center. The defect structure
occurs when d(A^G) is zero ( a 3 D period integral).
The superposition of a closed but not exact term is that which produces
a "hole" in the system topology. In a sense the turbulent "vacuum" condenses
into topological defects, leaving a defect "hole" in the otherwise dissipative
medium. It would seem that the dissipation implies a "force" towards the hole."
Thanks. The NASA Pioneer data is almost a textbook example of the Hedge Hog. I get it from the vacuum coherence defect. The two concentric spherical boundaries - inner boundary starting at 20 AU are filled with a small amount of dark zero point vacuum energy whose effective Newtonian limit gravity g-force per unit test mass is a_g = - cH = 10^-7 cm/sec^2 to Sun. Galactic halo has a similar explanation though its not a hedge hog. Turbulence in the local macro-quantum vacuum coherence is an interesting idea. Basically the fabric of spacetime is coming from the phase factor of this GIANT QUANTUM LOCAL WAVE FUNCTION that lives on different scales and different regions in order parameter spaces that we can directly infer from data. Key as always is single-valuedness of the, in this case giant quantum vacuum wave. It's like superfluid, or, rather super-solid physics. The point is that ALL the dark energy/dark matter data can be understood in terms of topological defects of the vacuum coherence of the fabric of spacetime.
On Dec 23, 2004, at 2:30 PM, Jack Sarfatti wrote:
Thanks SP & M and Happy Holidays! :-)
On Dec 23, 2004, at 2:06 PM, Saul-Paul & Mary-Minn Sirag wrote:
Jack,
I made another trip to the Science Library here, and found three more papers of yours.
J. Sarfatt, "On the 'Type II Superconductor' Model of Self-Trapped Laser Filaments," Physics Letters Vol. 26A, No. 2, PP. 88-89 (18 Dec. 1967).
Yes, this is the one Ray Chiao mentioned to Chas Townes at UCB that was useful to Ray when he started his experiments in late 60's.
J.Sarfatt, "Destruction of Superflow in Unsaturated 4He Films and the Prediction of a new Crystalline Phase of 4He3 With Bose-Einstein Condensation," Physics Letters Vol. 30A, No. 5, pp 300-301 (3 Nov. 1969).
Yes, this idea was 30 years ahead of the curve.
J. Sarfatt, "Quantum-Mechanical Correlation Theory of Electromagnetic Fields," Nuovo Cimento Vol. XXVII, N. 5, pp. 1119-1129 (1 Marzo 1963).
[Note: this paper as well as the paper by Suskind and Glogower is cited in Michael Martin Nieto's paper "Quantum Phase and Quantum Phase Operators: Some Physics and Some History" arXiv:hep-th/9304036 v1 8 Apr 93.]
I will put all 8 of these papers in the mail to your Stockton Street address today.
Hey thanks Santa. It will be in my Stockton. :-)
Nuff said ;-)
Saul-Paul
On Dec 14, 2004, at 12:40 PM, Jack Sarfatti wrote:
Yes, but before you do see if you can find in same time period same Physics Letters something like
Laser Self-Trapped Filaments and Landau-Ginzburg Equation? Thanks. :-)
There is also one about ODLRO in a quantum solid - but maybe it's one of those below.
On Dec 14, 2004, at 12:04 PM, Saul-Paul & Mary-Minn Sirag wrote:
Jack,
I have the following papers. Should I mail them to your Stockton Street address?
Leonard Susskind & Jonathan Glogower, "Quantum Mechanical Phase and Time Operator", Physics Vol.1,No.1,
pp. 49-61, 1964.
J. Sarfatt & A.M. Stoneham, "The Goldstone theorem and the Jahn-Teller effect", Proc. Phys. Soc.., Vol. 91, pp
pp. 214-221, 1967.
J. Sarfatt, "On the Nature of the Superfluid Critical Velocity", Physics Letters, Vol. 24A, No. 5, pp. 287-288,
27 Feb. 1967.
J. Sarfatt, "A New Theory of the Superfluid Vortex Phenomenon", Physics Letters, Vol. 24A, No.7, pp. 399-400,
27 Mar. 1967.
J. Sarfatt, "Local Gauge Invariance and Broken Symmetry in Superfluid Helium", Physics Letters, Vol.25A, No.9,
pp. 642-643, 6 Nov. 1967.
All for now ;-)
Saul-Paul
**
"I have not been able to get to Tony Smith's web site
Is there some problem again?"
I don't know.
*
feliz navidad
RMK
Loops, Knots, Gravity
On Dec 24, 2004, at 1:06 PM, Alex Poltorak wrote:
Jack,
It is only natural that gauge connections used in quantum gravity and
other unification theories are dynamic, while my affine connection is
not.
At some point Alex next year I will read your ontology paper to see what you are really doing. I need to understand the physical motivation before I look at the math closely. What is clear to me is that there is no real relation between what you are doing and what Paul is doing.
The reason is very simple -- the connections describing physical
fields must be dynamic while my affine connection used in Part I of my
GR17 paper DOES NOT describe any physical field whatsoever.
Yes, and this is what I do not understand.
It is a
mathematical device used to separate the physical information about
gravitational field per se from the information about a coordinate
system, in which this field is described.
Again I do not understand how this is possible unless it fits my
LC connection 1-form = Exact 1-form + Non-Exact 1-form
idea because from Penrose I see how to connect that kind of mathematics directly to Einstein's 1916 GR
The info in the coordinate system would then be the Exact 1-form part that makes no contribution to the intrinsic tidal curvature. There is no reason to expect the Non-Exact 1-form to be a 3rd rank GCT tensor relative to the (LC) connection ;w itself as Paul wishes. That is a false idea I think. The only GCT tensor there is guv;w = 0.
It has no more meaning than
the base 2 in binary numbers. Just as any number may be represented in
a binary form, any L-C connection may be represented as the difference
between an arbitrary affine connection and its tensor of nonmetricity.
This I do not understand.
This is just a fact of differential geometry.
Reference?
My use if this
mathematical device to reformulate the classical GR does not change the
theory any more than its reformulation in terms of tetrads. However,
just as in the case of tetrad reformulation, it sheds some interesting
light on the nature of the gravitational field and its conservation
laws.
The only real modification of the classical GR is described in Part III
of my paper, where the affine connection is no longer arbitrary as in
Part I, but is defined by the choice of the RF and the theory describes
the gravitational field in a NIFR having GR as a special case of the
gravity in IFR. If you read the paper instead of judging it based on
the hearsay, you will see it very clearly, just as you easily followed
my logic during our delightful conversation in Dublin.
I have told Paul repeatedly that I have no present opinion on your work until I have a chance to really think about it. What I object to is Paul's invoking your work as a justification for his when he admits he cannot explain your work in his own words. I do not understand yet how you mean "RF", "NIFR" and "IFR". Paul seems to use these terms in the GLOBAL special relativity sense whereas in GR only LOCAL FRAMES have meaning. :-)
The modern Ashtekar approach does start with connections as the fundamental dynamical variables. I am beginning to suspect that the nonlocality of gravity energy and the curvature without curvature of Vilenken-Taub's thin wall of dark energy are related to the global holonomy of the Wilson loops that we see in Berry phases, in Bohm-Aharonov-Josephson effects - all coming from the topological defects in the SINGLE-VALUED vacuum coherence local order parameter which is how I realize Andrei Sakharov's 1967 idea of Einstein's curved spacetime as emergent METRIC ELASTICITY = PW Anderson's "phase rigidity". Cartan's forms and the Hodge-DeRham integrals seem a good formalism to use here. The "loop-knot" formalism may be the most powerful technique. I don't know.
I get a "area" quantum from my Bohmian method that is characteristic of loop quantum gravity. It is also equivalent to a string tension. I get all that trivially without extra dimensions that may be unstable (Penrose).
Best regards,
Alex
-----Original Message-----
From: Jack Sarfatti [mailto:sarfatti@pacbell.net]
Sent: Thursday, December 23, 2004 5:21 PM
To: Paul Zielinski; SarfattiScienceSeminars@YahooGroups. com;
Sarfatti_Physics_Seminars
Cc: Alex Poltorak; ItalianPhysicsCenter
Subject: Loops, Connections & Knots
Paul
There is a unified view of electroweak-strong and gravity as
renormalizable Wilson holonomy loop local gauge theories where the
connections for parallel transport of physical fields in internal spaces
and spacetime respectively are the fundamental background independent
non-perturbative NON-RIGID dynamical variables.
This is why Alex's "RIGID AFFINE CONNECTION" is not a proper way to look
at General Relativity. The issue here is the physical idea. The essence
of Einstein's relativity is the RENUNCIATION of all absolute RIGID
background dependent connection fields.
"The space of loops offers a natural arena for the quantum theories of
connections." A. Ashtekar
I add ODLRO macro-quantum "BCS" theory to that.
There is a "loop transform" is it related to a "wavelet transform"?
Quantum gravity states depend on knot generalizations of Loops. How does
this carry over to ODLRO spontaneous breakdown of vacuum symmetry in pre
-> post inflationary phase transition?
On Dec 24, 2004, at 1:06 PM, Alex Poltorak wrote:
Jack,
It is only natural that gauge connections used in quantum gravity and
other unification theories are dynamic, while my affine connection is
not.
At some point Alex next year I will read your ontology paper to see what you are really doing. I need to understand the physical motivation before I look at the math closely. What is clear to me is that there is no real relation between what you are doing and what Paul is doing.
The reason is very simple -- the connections describing physical
fields must be dynamic while my affine connection used in Part I of my
GR17 paper DOES NOT describe any physical field whatsoever.
Yes, and this is what I do not understand.
It is a
mathematical device used to separate the physical information about
gravitational field per se from the information about a coordinate
system, in which this field is described.
Again I do not understand how this is possible unless it fits my
LC connection 1-form = Exact 1-form + Non-Exact 1-form
idea because from Penrose I see how to connect that kind of mathematics directly to Einstein's 1916 GR
The info in the coordinate system would then be the Exact 1-form part that makes no contribution to the intrinsic tidal curvature. There is no reason to expect the Non-Exact 1-form to be a 3rd rank GCT tensor relative to the (LC) connection ;w itself as Paul wishes. That is a false idea I think. The only GCT tensor there is guv;w = 0.
It has no more meaning than
the base 2 in binary numbers. Just as any number may be represented in
a binary form, any L-C connection may be represented as the difference
between an arbitrary affine connection and its tensor of nonmetricity.
This I do not understand.
This is just a fact of differential geometry.
Reference?
My use if this
mathematical device to reformulate the classical GR does not change the
theory any more than its reformulation in terms of tetrads. However,
just as in the case of tetrad reformulation, it sheds some interesting
light on the nature of the gravitational field and its conservation
laws.
The only real modification of the classical GR is described in Part III
of my paper, where the affine connection is no longer arbitrary as in
Part I, but is defined by the choice of the RF and the theory describes
the gravitational field in a NIFR having GR as a special case of the
gravity in IFR. If you read the paper instead of judging it based on
the hearsay, you will see it very clearly, just as you easily followed
my logic during our delightful conversation in Dublin.
I have told Paul repeatedly that I have no present opinion on your work until I have a chance to really think about it. What I object to is Paul's invoking your work as a justification for his when he admits he cannot explain your work in his own words. I do not understand yet how you mean "RF", "NIFR" and "IFR". Paul seems to use these terms in the GLOBAL special relativity sense whereas in GR only LOCAL FRAMES have meaning. :-)
The modern Ashtekar approach does start with connections as the fundamental dynamical variables. I am beginning to suspect that the nonlocality of gravity energy and the curvature without curvature of Vilenken-Taub's thin wall of dark energy are related to the global holonomy of the Wilson loops that we see in Berry phases, in Bohm-Aharonov-Josephson effects - all coming from the topological defects in the SINGLE-VALUED vacuum coherence local order parameter which is how I realize Andrei Sakharov's 1967 idea of Einstein's curved spacetime as emergent METRIC ELASTICITY = PW Anderson's "phase rigidity". Cartan's forms and the Hodge-DeRham integrals seem a good formalism to use here. The "loop-knot" formalism may be the most powerful technique. I don't know.
I get a "area" quantum from my Bohmian method that is characteristic of loop quantum gravity. It is also equivalent to a string tension. I get all that trivially without extra dimensions that may be unstable (Penrose).
Best regards,
Alex
-----Original Message-----
From: Jack Sarfatti [mailto:sarfatti@pacbell.net]
Sent: Thursday, December 23, 2004 5:21 PM
To: Paul Zielinski; SarfattiScienceSeminars@YahooGroups. com;
Sarfatti_Physics_Seminars
Cc: Alex Poltorak; ItalianPhysicsCenter
Subject: Loops, Connections & Knots
Paul
There is a unified view of electroweak-strong and gravity as
renormalizable Wilson holonomy loop local gauge theories where the
connections for parallel transport of physical fields in internal spaces
and spacetime respectively are the fundamental background independent
non-perturbative NON-RIGID dynamical variables.
This is why Alex's "RIGID AFFINE CONNECTION" is not a proper way to look
at General Relativity. The issue here is the physical idea. The essence
of Einstein's relativity is the RENUNCIATION of all absolute RIGID
background dependent connection fields.
"The space of loops offers a natural arena for the quantum theories of
connections." A. Ashtekar
I add ODLRO macro-quantum "BCS" theory to that.
There is a "loop transform" is it related to a "wavelet transform"?
Quantum gravity states depend on knot generalizations of Loops. How does
this carry over to ODLRO spontaneous breakdown of vacuum symmetry in pre
-> post inflationary phase transition?
On the meaning of Einstein's General Relativity of the Gravitational Field
On Dec 23, 2004, at 9:41 PM, iksnileiz@earthlink.net wrote:
J: The ORTHOGONAL LOCAL CS basis set in this particular representation is
(cdt)et
drer
rdthetaetheta
rsinthetadphiephu
The eu are orthonormal basis vectors at each point (r,theta,phi,t).
Z: OK, so how does this CS relate to the corresponding global Cartesian CS that represents the *same* observer FR in Minkowski spacetime when the permanent field is switched off?
J: Meaningless question apart from obvious formal limit M -> 0. You are not asking a proper physics question if it's not this trivial one.
Z: It's really a mathematical question about the relationship between the two LNIF CSs.
J: Again that's a trivial question with an easy answer.
Xu^u' Jacobian matrix of the GCT goes from LNIF to LNIF' AT SAME EVENT E.
Z: In some sense the flat-manifold LNIF CS and corresponding curved-manifold LNIF CS must converge locally around the point of interest. I am now thinking about this in terms of a local projection of the global Cartesian CS defined over the tangent space onto the curved manifold in the neighborhood of a point.
J: Learn the math Paul. These are all novice level questions. READ PENROSE.
Lim M -> 0 SSS GR vacuum solution = Minkowski space
M -> 0 above is Minkowski metric. What's your problem?
Z: But this doesn't answer my question.
J: There is no real question here.
Z: Or that you cannot answer it.
J: Of course I cannot answer something that is not even there.
"The Question is: What is The Question?" (Wheeler)
Z: Obviously when M -> 0 you are in Minkowski spacetime and you can have a global Cartesian CS. But what is the exact mathematical relationship between the local CS on the curved manifold and the global CS on the flat manifold?
J: Meaningless.
Z: How can you say this question is meaningless?
J: Because it is.
The trivial answer IS THAT THEY ARE FORMALLY IDENTICAL!
That's if your question is intelligible. I am guessing at what you are incoherently stumbling about for.
Z: If the the global Cartesian CS is defined on the flat tangent plane intersecting at the point of interest on the curved manifold, what is the geometrical relationship between the two CSs?
J: "The tangent space Tp to an n-manifold M at a point p may be intuitively understood as the limiting space, when smaller and smaller neighborhoods of p in M are examined at correspondingly greater and greater magnifications. The resulting Tp is flat: an n-dimensional vector space." Penrose p. 224
Note wavelet transform ZOOM feature built into the concept.
Also there is no implication that any curvature tensor is zero because
Curvature(abcd LIF Tangent Space) = ea^ueb^vec^wed^lCurvature(uvwl LNIF Manifold)
ea^u = tetrad components
This is EEP tensor/tetrad structure built into (embedded) the differential geometry at its core. This structure cannot be fragmented/unraveled as you, and also Puthoff in a different way, try to do.
The tangent space idea extends to infinity but of course breaks down on small scale. However, we cannot quantize guv top -> down because it is an emergent bottom-up "More is different" ODLRO effective c-number macro-quantum field theory from the micro-quantum globally flat (no gravity no inertia) pre-inflationary false vacuum substratum. The coherence in the inflationary phase transition to the Big Bang is only PARTIAL leaving dark energy/matter exotic vacuum fragments at different scales as seen in the "cosmological constant", "galactic halo", Pioneer anomaly and other phenomena. The "dark energy" partial incoherence is like the "normal fluid" component in the "two-fluid model". Even in superfluid helium T = 0 ground state the zero momentum BEC condensate is only ~ 10% of total particle number.
Z: How, mathematically speaking, can they both represent the same observer FR?
J: THEY DON'T ! If you mean the LIF?
Z: I don't.
J: You don't know what you mean.
the LIF observer Alice is on a geodesic, the LNIF observer Bob is on a non-geodesic because of a non-gravity force that he feels as "weight". Alice and Bob can reach out and touch hands momentarily - figuratively - they are coincident.
Z: I meant the stationary LNIF when the permanent field is on (curved manifold), and when the field is switched off (flat tangent manifold).
That's the TETRAD relation
guv(curved LNIF) = eu^aev^bnab(LIF ~ FLAT TO FIRST ORDER ONLY)
You keep asking novice questions. None of your concerns here are publishable in a top journal - except for AJP on how to teach elementary concepts in relativity! That's fine, but you think you have come up with a deep new alternative to Riemann and Einstein that will make Wheeler, Thorne and Misner et-al look like Dummies. You have said as much many times now.
Z: Same observer FR, two different CS representations. How does this work, mathematically? In what sense are the two CSs the same in the immediate neighborhood of the tangent point?
It should be possible to smoothly stretch-deform one into the other.
J: Yeah M -> 0 TRIVIAL.
This is hardly a problem.
Z: OK. So you can stretch-deform the Riemann surface
J: Malapropism. "Riemann surface" is inappropriate here.
Z: Are we not dealing with Riemann manifolds? They are abstract any-dimensional analogs of Gaussian *surfaces*. So they are "hypersurfaces" or just "surfaces" for short.
J: Look up "Riemann surface" - it's not used in the context you just described. It's a standard term in functions w = f(z) of complex variables w, z for multi-valued functions - e.g.
z = re^itheta
w = Logz = logr + itheta
Every 2pi rotation takes you to a new sheet of a "spiral winding space" Riemann surface z-domain of f(z) with a branch point (here at z = 0) on which logz is single-valued.
Fig. 8.1 p. 135 Penrose
Z: from curved to flat, and *locally* there is no change to the CS around any given point? The homeomorphic mapping of spacetime points to and from R^4 is locally invariant under the deformation within an infinitesimal neighborhood around the point of interest?
J: It's called geodesic normal coordinates.
Z: I'm talking about the CSs that represent the stationary LNIF. These have nothing to do with normal coordinates.
J: WORK OUT AN EXAMPLE.
That CS is unique mod static rotations, e.g. in SSS
gtt = (1 - 2GM/c^2r) = - 1/grr gtheta,theta = gphi,phi = -1
You have unique local spherical polar Cartan triad of unit vectors er,etheta,ephi mod choice of "celestial sphere" lattitudes and longitudes.
I use basis convention
dx^1 = dr along er
dx^2 = rdtheta along etheta
dx^3 = rsinthetadphi along ephi
dx^0 = cdt
to keep guv DIMENSIONLESS with derivative operators like
(1/rsintheta)d/dphi = d/dx^3
(1/r)d/dtheta = d/dx^2
d/dr = d/dx^1
(1/c)d/dt = d/dx^0
ds^2 = guvdx^udx^v = local differential frame invariant interval
guv(LNIF) = eu^aev^bnab(LIF)
Z: Then it follows from this that if around any given point on the flat manifold, the curved-coordinate contributions to the g_uv, w are zero for the Cartesian CS, then they are also zero for a corresponding local Cartesian CS on the curved manifold, right?
J: Why are we going over this AGAIN! It's the Taylor series I wrote.
Z: But since as we know there is no curved-coordinate correction to the g_uv, w, and thus to (LC), in the global flat-manifold Cartesian CS, we must conclude that there is also no local curved-coordinate correction to these quantities in the corresponding curved-manifold local CS.
J: Hogwash. False reasoning.
Z: OK, why? Don't the two CSs in some sense locally converge in the infinitesimal neighborhood of a tangent point?
J: Meaningless question.
Z: In which case, any residual g_uv, w =/= 0 must be purely geometric in origin (meaning that they reflect the true variation of the g_uv along the manifold), since the curved-coordinate corrections to the g_uv, w (as incorporated into (LC)) all vanish in both CSs.
J: Drivel - angels on head of pin.
Z: Do you think there is a true variation of the g_uv along the manifold around a given point that reflects the intrinsic geometry? Does this concept have any meaning to you?
Paul, again all answers are here if you ask a sensible question:
guv(P') = guv(P) + guv,w(P'- P)^w + (1/2)guv,w,l(P' - P)^w(P' - P)^l + ...
INTRINSIC GEOMETRY OBVIOUSLY BEGINS WITH THE THIRD TIDAL TERM IN THE TAYLOR SERIES
The EINSTEIN EQUIVALENCE PRINCIPLE IS FROM THE ACTION PRINCIPLE
EXTREMUM OF POINT TEST PARTICLE DYNAMICAL ACTION IS A CRITICAL POINT in the space of worldline alternative micro-quantum histories
guv,w = 0
Like dy/dx = 0 in calculus!
The STABILITY physics is in d^2y/dx^2
Therefore, when you and Puthoff wish to eschew the equivalence principle and tensors you also renounce differential geometry and the action principle. This is TOO MUCH to renounce.
Both you and Puthoff in PV, also Haisch in ZPF/SED "inertia" violate Einstein's Golden Rule, you are attempting to make physics simpler than IS possible!
PS Also you can expand the Jacobian matrices of the GCT's Xu^u' as Taylor series.
PPS Your conjecture
Levi-Civita metric connection = GCT Tensor + Non-Tensor
Is wrongly posed.
The only possible GCT 3rd rank tensor is the non-metricity tensor guv;w = 0 with ;w the LC covariant derivative.
The more interesting conjecture is in terms of Cartan's METRIC-INDEPENDENT exterior calculus of GLOBAL topology
Levi-Civita Connection 1-form = Exact 1-form + Non-Exact 1-form
In that case
Curvature 2-form = d(Levi-Civita Connection 1-form) = d(Non-Exact 1-form)
Then your "coordinate part" from Minkowski space is in the Exact 1-form (analog to static electric field potential in EM for Gauss's law and to irrotational fluid flow and defect free lattice).
The intrinsic geometry is in the Non-Exact 1-form! (analog to magnetic field vector potential in EM for Ampere's law).
This also clearly allows for NONLOCAL GLOBAL TOPOLOGY EFFECTS such as
1. NONLOCALITY OF THE PURE GRAVITY ENERGY (e.g. Penrose "Road to Reality")
2. CURVATURE WITHOUT CURVATURE as in Vilenken's thin dark energy wall solution as pointed out by Taub
3. Pioneer Anomaly a_g = -cH pointed back to Sun as a hedgehog topological defect in the post-inflationary macro-quantum vacuum coherence Higgs-Goldstone field.
On Dec 23, 2004, at 9:41 PM, iksnileiz@earthlink.net wrote:
J: The ORTHOGONAL LOCAL CS basis set in this particular representation is
(cdt)et
drer
rdthetaetheta
rsinthetadphiephu
The eu are orthonormal basis vectors at each point (r,theta,phi,t).
Z: OK, so how does this CS relate to the corresponding global Cartesian CS that represents the *same* observer FR in Minkowski spacetime when the permanent field is switched off?
J: Meaningless question apart from obvious formal limit M -> 0. You are not asking a proper physics question if it's not this trivial one.
Z: It's really a mathematical question about the relationship between the two LNIF CSs.
J: Again that's a trivial question with an easy answer.
Xu^u' Jacobian matrix of the GCT goes from LNIF to LNIF' AT SAME EVENT E.
Z: In some sense the flat-manifold LNIF CS and corresponding curved-manifold LNIF CS must converge locally around the point of interest. I am now thinking about this in terms of a local projection of the global Cartesian CS defined over the tangent space onto the curved manifold in the neighborhood of a point.
J: Learn the math Paul. These are all novice level questions. READ PENROSE.
Lim M -> 0 SSS GR vacuum solution = Minkowski space
M -> 0 above is Minkowski metric. What's your problem?
Z: But this doesn't answer my question.
J: There is no real question here.
Z: Or that you cannot answer it.
J: Of course I cannot answer something that is not even there.
"The Question is: What is The Question?" (Wheeler)
Z: Obviously when M -> 0 you are in Minkowski spacetime and you can have a global Cartesian CS. But what is the exact mathematical relationship between the local CS on the curved manifold and the global CS on the flat manifold?
J: Meaningless.
Z: How can you say this question is meaningless?
J: Because it is.
The trivial answer IS THAT THEY ARE FORMALLY IDENTICAL!
That's if your question is intelligible. I am guessing at what you are incoherently stumbling about for.
Z: If the the global Cartesian CS is defined on the flat tangent plane intersecting at the point of interest on the curved manifold, what is the geometrical relationship between the two CSs?
J: "The tangent space Tp to an n-manifold M at a point p may be intuitively understood as the limiting space, when smaller and smaller neighborhoods of p in M are examined at correspondingly greater and greater magnifications. The resulting Tp is flat: an n-dimensional vector space." Penrose p. 224
Note wavelet transform ZOOM feature built into the concept.
Also there is no implication that any curvature tensor is zero because
Curvature(abcd LIF Tangent Space) = ea^ueb^vec^wed^lCurvature(uvwl LNIF Manifold)
ea^u = tetrad components
This is EEP tensor/tetrad structure built into (embedded) the differential geometry at its core. This structure cannot be fragmented/unraveled as you, and also Puthoff in a different way, try to do.
The tangent space idea extends to infinity but of course breaks down on small scale. However, we cannot quantize guv top -> down because it is an emergent bottom-up "More is different" ODLRO effective c-number macro-quantum field theory from the micro-quantum globally flat (no gravity no inertia) pre-inflationary false vacuum substratum. The coherence in the inflationary phase transition to the Big Bang is only PARTIAL leaving dark energy/matter exotic vacuum fragments at different scales as seen in the "cosmological constant", "galactic halo", Pioneer anomaly and other phenomena. The "dark energy" partial incoherence is like the "normal fluid" component in the "two-fluid model". Even in superfluid helium T = 0 ground state the zero momentum BEC condensate is only ~ 10% of total particle number.
Z: How, mathematically speaking, can they both represent the same observer FR?
J: THEY DON'T ! If you mean the LIF?
Z: I don't.
J: You don't know what you mean.
the LIF observer Alice is on a geodesic, the LNIF observer Bob is on a non-geodesic because of a non-gravity force that he feels as "weight". Alice and Bob can reach out and touch hands momentarily - figuratively - they are coincident.
Z: I meant the stationary LNIF when the permanent field is on (curved manifold), and when the field is switched off (flat tangent manifold).
That's the TETRAD relation
guv(curved LNIF) = eu^aev^bnab(LIF ~ FLAT TO FIRST ORDER ONLY)
You keep asking novice questions. None of your concerns here are publishable in a top journal - except for AJP on how to teach elementary concepts in relativity! That's fine, but you think you have come up with a deep new alternative to Riemann and Einstein that will make Wheeler, Thorne and Misner et-al look like Dummies. You have said as much many times now.
Z: Same observer FR, two different CS representations. How does this work, mathematically? In what sense are the two CSs the same in the immediate neighborhood of the tangent point?
It should be possible to smoothly stretch-deform one into the other.
J: Yeah M -> 0 TRIVIAL.
This is hardly a problem.
Z: OK. So you can stretch-deform the Riemann surface
J: Malapropism. "Riemann surface" is inappropriate here.
Z: Are we not dealing with Riemann manifolds? They are abstract any-dimensional analogs of Gaussian *surfaces*. So they are "hypersurfaces" or just "surfaces" for short.
J: Look up "Riemann surface" - it's not used in the context you just described. It's a standard term in functions w = f(z) of complex variables w, z for multi-valued functions - e.g.
z = re^itheta
w = Logz = logr + itheta
Every 2pi rotation takes you to a new sheet of a "spiral winding space" Riemann surface z-domain of f(z) with a branch point (here at z = 0) on which logz is single-valued.
Fig. 8.1 p. 135 Penrose
Z: from curved to flat, and *locally* there is no change to the CS around any given point? The homeomorphic mapping of spacetime points to and from R^4 is locally invariant under the deformation within an infinitesimal neighborhood around the point of interest?
J: It's called geodesic normal coordinates.
Z: I'm talking about the CSs that represent the stationary LNIF. These have nothing to do with normal coordinates.
J: WORK OUT AN EXAMPLE.
That CS is unique mod static rotations, e.g. in SSS
gtt = (1 - 2GM/c^2r) = - 1/grr gtheta,theta = gphi,phi = -1
You have unique local spherical polar Cartan triad of unit vectors er,etheta,ephi mod choice of "celestial sphere" lattitudes and longitudes.
I use basis convention
dx^1 = dr along er
dx^2 = rdtheta along etheta
dx^3 = rsinthetadphi along ephi
dx^0 = cdt
to keep guv DIMENSIONLESS with derivative operators like
(1/rsintheta)d/dphi = d/dx^3
(1/r)d/dtheta = d/dx^2
d/dr = d/dx^1
(1/c)d/dt = d/dx^0
ds^2 = guvdx^udx^v = local differential frame invariant interval
guv(LNIF) = eu^aev^bnab(LIF)
Z: Then it follows from this that if around any given point on the flat manifold, the curved-coordinate contributions to the g_uv, w are zero for the Cartesian CS, then they are also zero for a corresponding local Cartesian CS on the curved manifold, right?
J: Why are we going over this AGAIN! It's the Taylor series I wrote.
Z: But since as we know there is no curved-coordinate correction to the g_uv, w, and thus to (LC), in the global flat-manifold Cartesian CS, we must conclude that there is also no local curved-coordinate correction to these quantities in the corresponding curved-manifold local CS.
J: Hogwash. False reasoning.
Z: OK, why? Don't the two CSs in some sense locally converge in the infinitesimal neighborhood of a tangent point?
J: Meaningless question.
Z: In which case, any residual g_uv, w =/= 0 must be purely geometric in origin (meaning that they reflect the true variation of the g_uv along the manifold), since the curved-coordinate corrections to the g_uv, w (as incorporated into (LC)) all vanish in both CSs.
J: Drivel - angels on head of pin.
Z: Do you think there is a true variation of the g_uv along the manifold around a given point that reflects the intrinsic geometry? Does this concept have any meaning to you?
Paul, again all answers are here if you ask a sensible question:
guv(P') = guv(P) + guv,w(P'- P)^w + (1/2)guv,w,l(P' - P)^w(P' - P)^l + ...
INTRINSIC GEOMETRY OBVIOUSLY BEGINS WITH THE THIRD TIDAL TERM IN THE TAYLOR SERIES
The EINSTEIN EQUIVALENCE PRINCIPLE IS FROM THE ACTION PRINCIPLE
EXTREMUM OF POINT TEST PARTICLE DYNAMICAL ACTION IS A CRITICAL POINT in the space of worldline alternative micro-quantum histories
guv,w = 0
Like dy/dx = 0 in calculus!
The STABILITY physics is in d^2y/dx^2
Therefore, when you and Puthoff wish to eschew the equivalence principle and tensors you also renounce differential geometry and the action principle. This is TOO MUCH to renounce.
Both you and Puthoff in PV, also Haisch in ZPF/SED "inertia" violate Einstein's Golden Rule, you are attempting to make physics simpler than IS possible!
PS Also you can expand the Jacobian matrices of the GCT's Xu^u' as Taylor series.
PPS Your conjecture
Levi-Civita metric connection = GCT Tensor + Non-Tensor
Is wrongly posed.
The only possible GCT 3rd rank tensor is the non-metricity tensor guv;w = 0 with ;w the LC covariant derivative.
The more interesting conjecture is in terms of Cartan's METRIC-INDEPENDENT exterior calculus of GLOBAL topology
Levi-Civita Connection 1-form = Exact 1-form + Non-Exact 1-form
In that case
Curvature 2-form = d(Levi-Civita Connection 1-form) = d(Non-Exact 1-form)
Then your "coordinate part" from Minkowski space is in the Exact 1-form (analog to static electric field potential in EM for Gauss's law and to irrotational fluid flow and defect free lattice).
The intrinsic geometry is in the Non-Exact 1-form! (analog to magnetic field vector potential in EM for Ampere's law).
This also clearly allows for NONLOCAL GLOBAL TOPOLOGY EFFECTS such as
1. NONLOCALITY OF THE PURE GRAVITY ENERGY (e.g. Penrose "Road to Reality")
2. CURVATURE WITHOUT CURVATURE as in Vilenken's thin dark energy wall solution as pointed out by Taub
3. Pioneer Anomaly a_g = -cH pointed back to Sun as a hedgehog topological defect in the post-inflationary macro-quantum vacuum coherence Higgs-Goldstone field.
Thursday, December 23, 2004
Meaning of Einstein's Gravity
Paul
Wake up and smell the coffee. General relativity is a mature well understood theory. There are important issues today with interesting questions and you are not asking them. I started out on this with you very open-minded. Basically you are garbling Newton's force idea with Einstein's geometrical ideas. Your idea is "not even wrong". You cannot compute anything. You cannot explain anything that needs explaining. You have not asked the right question.
"The Question is: What is The Question?" (Wheeler)
Your entire thesis is ill-posed.
There are 3 analogies of some interest (electromagnetic, fluid & elastic) none of which you have even thought of.
The GR (LC) connection is analogous to the EM vector potential connection.
The curvature is the NONLINEAR NON-ABELIAN (LC) curl of itself analogous to the ABELIAN magnetic field or the circulation of a fluid.
Any 3D-vector field A has a gradient (divergence) part and a curl (circulation part).
Roughly and non-rigorously to start (Kiehn can say it better more rigorously)
A = GradF + eCurlB
CurlGrad = 0
DivCurl = 0
In the language of Cartan forms
1-form = exact 1-form + non-exact 1-form
Here we are in 3-space.
The Cartan exterior derivative on a 0-form is a one-form.
Neglecting anholonomic singularities
d^2 = 0
Exact 1-form = dF
F = 0 form scalar
B is a 1 form
curlB = dB is a 2-form
e converts the 2-form to a dual 1 form in 3-space or a dual 2-form in 4-space of GR depending on the context.
The point is that CURVATURE is a 2-form like EM field tensor, and like VORTICITY in a fluid, and like disclination topological defect densities in a crystal lattice (H. Kleinert).
Your COORDINATE PART is the analog to the DIVERGENCE or GRAD exact 1-form part.
The INTRINSIC PART is the analog to the Curl or non-EXACT 1-form part.
However, your ERROR is to think that the non-EXACT 1-form part is a GCT tensor T ALL BY ITSELF.
What we have in fact is
(LC) 1-form = exact 1-form + non-exact 1-form = N NOT A GCT TENSOR in the metric space.
Curvature 2-form = d(LC) = d(non-exact 1-form)
Now in Minkowski space
(LC) = exact 1-form
Note that even in this case, you can still get the Vilenken-Taub "curvature without curvature" effect from global non-trivial topology like the quantized vortices in the superfluid irrotational flow and in Type II superconductors. The dark energy is the "normal fluid" of the macro-quantum coherent holographic vacuum.
Bottom line, the only GCT tensor you can get in 1916 GR (no torsion) from the metric tensor is its (LC) covariant derivative which is the ZERO nonmetricity tensor. (LC) is always a non-GCT tensor that under the GCT X transforms as
(LC) = N -> (LC)' = XXX(LC) + XY = N'
The only possible tensor T here is guv;w = 0 where ;w is the (LC) covariant derivative.
Y is a derivative of X so (LC) is a quasi-tensor under the limited group of linear transformations, i.e. a subgroup of GCT.
The pure gravity energy pseudo-tensor has this same kind of property as (LC). We want gravity energy to be nonlocal. That is a good thing not a bad thing to be eliminated with Rube Goldberg devices.
Curvature is like the breakdown of irrotational flow in a superfluid. There is an analogy to Bohm-Aharonov effect because of the local quantum vacuum coherence that is a giant single-valued wave function. Indeed, this explains the Pioneer 10-11 anomaly as a hedgehog exotic vacuum zero point dark energy topological defect centered at the Sun and maybe ALL stars, the Galactic Halo and other interesting observations that appear mysterious.
PS
Note in the Newtonian mechanics of rotating non-inertial frames
,t' (non inertial) = ,t(inertial) + Wx = Galilean relativity analog to the (LC) covariant derivative of GR
W = instantaneous rotation axial vector of the non-inertial frame (common origin with inertial frame)
For a test particle at displacement r
r' = r
v' = v + Wxr
a' = a + Wxv + W,txr + Wxv + WxWxr
= a + 2Wxv + WxWxr + W,txr
2Wxv = INERTIAL CORIOLIS acceleration of the non-inertial rotating frame
WxWxr = INERTIAL CENTRIFUGAL acceleration ...
W,txr = INERTIAL TORQUE acceleration
In Gennady's Shipov's torsion theory extension of Einstein's 1916 GR this is part of a TORSION FIELD .
Note that there are no-translational inertial accelerations in this particular problem.
If P is fixed to the rotating frame S' then in this REST rotating non-inertial frame, obviously
v' = 0
therefore
v = -Wxr
a' = 0
0 = a + WxWxr + W,txr
When W,t = 0, conservation of angular momentum L
a + WxWxr = 0
But
F = ma
F/m + WxWxr = 0
is compensation of the centrifugal inertial force in the rest rotating non-inertial force by the applied force F measured in the inertial frame.
In Newton's theory, unlike Einstein's, gravity is an external force in the inertial frame
F = GM/r^2
Giving, for a circular orbit Kepler's law
GM = r^3/T^2
T = period of orbit
r = radius of orbit
in flat Euclidean space where the geodesics are traced out by point test particles at constant speed in straight lines.
In contrast, the Newtonian non-inertial non-geodesic motion of this test particle is an inertial geodesic motion in Einstein's curved spacetime in which there is no gravity force.
That is, in Einstein's theory
F = 0 and W = 0 in the above problem.
What we have instead is the geodesic equation
D^2x^u/ds^2 = 0
In a non-inertial LNIF this becomes
d^2x^u/ds^2 + (LC)^uvw(dx^v/ds)(dx^w/ds) = 0
Where (LC) is computed from guv
e.g.
gtt = (1 - 2GM/c^2r) = - grr^-1, gtheta,theta = gphi,phi = -1
in local orthogonal basis (cdt)et, drer, rdthetaetheta, rsinthetadphiephi
r >> GM/c^2 gives same answers as Newton's theory.
This particular REPRESENTATION guv of a CURVED SPACE-TIME is only for REST LNIF observers at fixed r because of some non-gravity force!
In a coincident LIF the EEP gives (LC) = 0 and d^2x^u/dt^2 = 0.
In general for COINCIDENT LNIF & LNIF'
gu'v' = Xu'^uXv'^vguv
Where X is the Jacobian matrix for an element of the GCT symmetry group.
EEP mean the existence of tetrads Eu^a and their inverses where
guv(LNIF) = Eu^anabEv^b
nab = Minkowski metric
Where LNIF and LIF are COINCIDENT at same physical event E.
"Physics is simple, when it is local." Wheeler
* Curved space-time physics is local because Einstein's gravity is "More is different" emergence of the LOCAL macro-quantum cohering of the nonlocal micro-quantum zero point pre-inflationary false vacuum fluctuations. The cohering is only partial so there is some dark energy/matter remnants in our post-inflationary universe.
Cartan Tetrad 1-form ~ (1 + Lp^2(Goldstone Phase),u) dx^u
Vacuum Coherence = (Higgs Field)e^i(Goldstone Phase)
G/c^4 comes from "Goldstone Phase Rigidity" (P.W. Anderson)
X Jacobian matrix of GCT comes from canonical transformation generating function THETA(x^u,x^u')
Where Goldstone Phase(x^u) -> Goldstone Phase(x^u) + THETA(x^u,x^u')
That is Einstein's GR is really a local gauge theory from a post-inflationary macro-quantum LOCAL vacuum coherence ODLRO parameter of zero entropy setting the direction of the Arrow of Time of The Second Law of Thermodynamics in the same direction as the accelerating expansion of the universe.
Note that
Curvature (tidal stretch-squeeze) = (LC) Covariant Curl of (LC) = Ricci Part + Conformal Part
Guv = Ruv - (1/2)Rguv = (8piG/c^4)Tuv(non-gravity source)
When Tuv = 0 Ricci Part of curvature = 0
In "classical vacuum" there is only conformal curvature with 10 independent parameters at each point.
Obviously if Cuvwl =/= 0 in any LNIF, then Cabcd =/= 0 in any COINCIDENT LIF.
Paul
Wake up and smell the coffee. General relativity is a mature well understood theory. There are important issues today with interesting questions and you are not asking them. I started out on this with you very open-minded. Basically you are garbling Newton's force idea with Einstein's geometrical ideas. Your idea is "not even wrong". You cannot compute anything. You cannot explain anything that needs explaining. You have not asked the right question.
"The Question is: What is The Question?" (Wheeler)
Your entire thesis is ill-posed.
There are 3 analogies of some interest (electromagnetic, fluid & elastic) none of which you have even thought of.
The GR (LC) connection is analogous to the EM vector potential connection.
The curvature is the NONLINEAR NON-ABELIAN (LC) curl of itself analogous to the ABELIAN magnetic field or the circulation of a fluid.
Any 3D-vector field A has a gradient (divergence) part and a curl (circulation part).
Roughly and non-rigorously to start (Kiehn can say it better more rigorously)
A = GradF + eCurlB
CurlGrad = 0
DivCurl = 0
In the language of Cartan forms
1-form = exact 1-form + non-exact 1-form
Here we are in 3-space.
The Cartan exterior derivative on a 0-form is a one-form.
Neglecting anholonomic singularities
d^2 = 0
Exact 1-form = dF
F = 0 form scalar
B is a 1 form
curlB = dB is a 2-form
e converts the 2-form to a dual 1 form in 3-space or a dual 2-form in 4-space of GR depending on the context.
The point is that CURVATURE is a 2-form like EM field tensor, and like VORTICITY in a fluid, and like disclination topological defect densities in a crystal lattice (H. Kleinert).
Your COORDINATE PART is the analog to the DIVERGENCE or GRAD exact 1-form part.
The INTRINSIC PART is the analog to the Curl or non-EXACT 1-form part.
However, your ERROR is to think that the non-EXACT 1-form part is a GCT tensor T ALL BY ITSELF.
What we have in fact is
(LC) 1-form = exact 1-form + non-exact 1-form = N NOT A GCT TENSOR in the metric space.
Curvature 2-form = d(LC) = d(non-exact 1-form)
Now in Minkowski space
(LC) = exact 1-form
Note that even in this case, you can still get the Vilenken-Taub "curvature without curvature" effect from global non-trivial topology like the quantized vortices in the superfluid irrotational flow and in Type II superconductors. The dark energy is the "normal fluid" of the macro-quantum coherent holographic vacuum.
Bottom line, the only GCT tensor you can get in 1916 GR (no torsion) from the metric tensor is its (LC) covariant derivative which is the ZERO nonmetricity tensor. (LC) is always a non-GCT tensor that under the GCT X transforms as
(LC) = N -> (LC)' = XXX(LC) + XY = N'
The only possible tensor T here is guv;w = 0 where ;w is the (LC) covariant derivative.
Y is a derivative of X so (LC) is a quasi-tensor under the limited group of linear transformations, i.e. a subgroup of GCT.
The pure gravity energy pseudo-tensor has this same kind of property as (LC). We want gravity energy to be nonlocal. That is a good thing not a bad thing to be eliminated with Rube Goldberg devices.
Curvature is like the breakdown of irrotational flow in a superfluid. There is an analogy to Bohm-Aharonov effect because of the local quantum vacuum coherence that is a giant single-valued wave function. Indeed, this explains the Pioneer 10-11 anomaly as a hedgehog exotic vacuum zero point dark energy topological defect centered at the Sun and maybe ALL stars, the Galactic Halo and other interesting observations that appear mysterious.
PS
Note in the Newtonian mechanics of rotating non-inertial frames
,t' (non inertial) = ,t(inertial) + Wx = Galilean relativity analog to the (LC) covariant derivative of GR
W = instantaneous rotation axial vector of the non-inertial frame (common origin with inertial frame)
For a test particle at displacement r
r' = r
v' = v + Wxr
a' = a + Wxv + W,txr + Wxv + WxWxr
= a + 2Wxv + WxWxr + W,txr
2Wxv = INERTIAL CORIOLIS acceleration of the non-inertial rotating frame
WxWxr = INERTIAL CENTRIFUGAL acceleration ...
W,txr = INERTIAL TORQUE acceleration
In Gennady's Shipov's torsion theory extension of Einstein's 1916 GR this is part of a TORSION FIELD .
Note that there are no-translational inertial accelerations in this particular problem.
If P is fixed to the rotating frame S' then in this REST rotating non-inertial frame, obviously
v' = 0
therefore
v = -Wxr
a' = 0
0 = a + WxWxr + W,txr
When W,t = 0, conservation of angular momentum L
a + WxWxr = 0
But
F = ma
F/m + WxWxr = 0
is compensation of the centrifugal inertial force in the rest rotating non-inertial force by the applied force F measured in the inertial frame.
In Newton's theory, unlike Einstein's, gravity is an external force in the inertial frame
F = GM/r^2
Giving, for a circular orbit Kepler's law
GM = r^3/T^2
T = period of orbit
r = radius of orbit
in flat Euclidean space where the geodesics are traced out by point test particles at constant speed in straight lines.
In contrast, the Newtonian non-inertial non-geodesic motion of this test particle is an inertial geodesic motion in Einstein's curved spacetime in which there is no gravity force.
That is, in Einstein's theory
F = 0 and W = 0 in the above problem.
What we have instead is the geodesic equation
D^2x^u/ds^2 = 0
In a non-inertial LNIF this becomes
d^2x^u/ds^2 + (LC)^uvw(dx^v/ds)(dx^w/ds) = 0
Where (LC) is computed from guv
e.g.
gtt = (1 - 2GM/c^2r) = - grr^-1, gtheta,theta = gphi,phi = -1
in local orthogonal basis (cdt)et, drer, rdthetaetheta, rsinthetadphiephi
r >> GM/c^2 gives same answers as Newton's theory.
This particular REPRESENTATION guv of a CURVED SPACE-TIME is only for REST LNIF observers at fixed r because of some non-gravity force!
In a coincident LIF the EEP gives (LC) = 0 and d^2x^u/dt^2 = 0.
In general for COINCIDENT LNIF & LNIF'
gu'v' = Xu'^uXv'^vguv
Where X is the Jacobian matrix for an element of the GCT symmetry group.
EEP mean the existence of tetrads Eu^a and their inverses where
guv(LNIF) = Eu^anabEv^b
nab = Minkowski metric
Where LNIF and LIF are COINCIDENT at same physical event E.
"Physics is simple, when it is local." Wheeler
* Curved space-time physics is local because Einstein's gravity is "More is different" emergence of the LOCAL macro-quantum cohering of the nonlocal micro-quantum zero point pre-inflationary false vacuum fluctuations. The cohering is only partial so there is some dark energy/matter remnants in our post-inflationary universe.
Cartan Tetrad 1-form ~ (1 + Lp^2(Goldstone Phase),u) dx^u
Vacuum Coherence = (Higgs Field)e^i(Goldstone Phase)
G/c^4 comes from "Goldstone Phase Rigidity" (P.W. Anderson)
X Jacobian matrix of GCT comes from canonical transformation generating function THETA(x^u,x^u')
Where Goldstone Phase(x^u) -> Goldstone Phase(x^u) + THETA(x^u,x^u')
That is Einstein's GR is really a local gauge theory from a post-inflationary macro-quantum LOCAL vacuum coherence ODLRO parameter of zero entropy setting the direction of the Arrow of Time of The Second Law of Thermodynamics in the same direction as the accelerating expansion of the universe.
Note that
Curvature (tidal stretch-squeeze) = (LC) Covariant Curl of (LC) = Ricci Part + Conformal Part
Guv = Ruv - (1/2)Rguv = (8piG/c^4)Tuv(non-gravity source)
When Tuv = 0 Ricci Part of curvature = 0
In "classical vacuum" there is only conformal curvature with 10 independent parameters at each point.
Obviously if Cuvwl =/= 0 in any LNIF, then Cabcd =/= 0 in any COINCIDENT LIF.
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