Monday, January 15, 2007

Differential Geometry in Physics 1-4

Check out this on-line course in differential geometry and Einstein's gravity

Vectors, Co-Vectors, Tensors
"The procedure is ... to cover the manifold with a patchwork of overlapping coordinate systems so that each point lies in at least one patch. Some points will then lie in more than one patch and are correspondingly assigned more than one set of coordinates, so coordinate transformations must then be presented." Roger Penrose

There is, however, a lot of redundancy. The physics is only concerned with local observers. Each local observer is equipped with detectors and can communicate with other local observers by electromagnetic signals. However, there are an infinity of static 3D coordinate transformations as well as local recalibrations of the clock that need to be factored out in the sense of being put into the same equivalence class. The physical observer is, formally abstractly, the equivalence class of these redundant local coordinate patches.

For example transforming from 3D Cartesian coordinates to 3D spherical polar coordinates should not change the local observer. However

z' = z - vt

v/c << 1

is a relationship between two inertial geodesic observers in flat spacetime in small relative speed to each other.

z' = z - (1/2)gt^2

gt/c << 1

is the relationship between weightless inertial geodesic observer Alice (z) free floating in space and the non-inertial non-geodesic observer Bob firing his rocket along the z direction and feeling "weight" g-force g (per unit mass).

All of the physics is in the relationships of different local observers looking at the same happenings and each computing local invariants for them that all agree. Those observers in local coincidence are the most important. Widely separated observers are need to use light cone limited signals to talk to each other with local time delays - unless there are retrocausal advanced signals of course as in the Feynman zig-zag e.g. John Cramer's "transaction" quantum theory. Also, we assume that there are no Heisenberg incompatibility issues when Alice and Bob look at the same process. In all of this, the size of the neighborhoods of the local observers must be small compared to the local ambient radii of curvature. This is not a big problem on Earth where the ambient radius of curvature is ~ 100,000,000 miles! Note that the corresponding time is only 8 minutes and that the 100% inertial force "weight" we feel on our non-geodesic paths fixed to hard Earth is determined by that 8 minute limit. Alice is, technically, only the same local frame for about 8 minutes. This relationship is only contingent, i.e. it is convenient for us because it is actually the closest abstraction of the theory to our direct experience. All theories are incomplete maps of our experience in which our detectors are extensions of our senses getting down in the end to qualia - the stream of consciousness that IS who we are - The Post-Quantum Ghost, The Soul In The Machine. Welcome to The Machine (Pink Floyd).

"The coordinate system is itself something quite arbitrary ... Sometimes ...particular types of coordinates suggest themselves as more 'natural' than the general curvilinear type ... But when the manifold M is curved in an arbitrary way it tends to be inconvenient to try to single out special coordinate systems ... Instead, one asks for propertied of the manifold M which can be ascertained equally well whichever system of coordinates is selected. Indeed this is the advantage of using local coordinate-invariant Eli Cartan's (1869 - 1951) exterior differential forms based on the use of totally antisymmetric tensor fields - more on that later.

"The most primitive objects (after the points ...) are the scalar fields, these being simply smooth functions on (manifold) M ... a function on M is then deemed smooth, if when expressed in terms of the coordinates, is differentiable (in the ordinary calculus sense) as a function of these coordinates. Next, objects called vector fields, covector fields, tensor fields, and connections may be introduced in turn.

The contravariant tangent vector fields at a point p on the manifold form a tangent vector space Tp.

A tangent vector field V(p) of tiny arrows tangent to the manifold at variable p can be split into a basis set {V^i,i = 1 ... N}. For us positive integer N = 4 unless otherwise stated. No fractal manifolds as yet where N is replaced by N/N' rational numbers.

V is a linear differential operator related to the total linear 4-momentum Hermitian operator P^u on complex-numbered qubit waves in "Hilbert space" in quantum theory that generate the globally rigid T4 group of Einstein's 1905 special relativity that, when locally gauged the same way (equivalence principle) on all non-gravity source field globally invariant actions, gives the curvature gravity field of Einstein's 1915 GR. It took Einstein 10 years to go from SR to GR. He did not do it using the local gauge principle however. The local gauge principle from Emmy Noether's theorem connecting conservation laws to continuous symmetries of the field actions, was not well understood back then. In fact she announced it in 1915 much too late for Einstein to use in his thinking from 1905 to 1915.

Let S be a scalar field on the manifold M.

S(scalar) ---> V(S) also a scalar

Then V(S(p)) is the directional derivative rate of change of S along the direction V on M. Using a local coordinate patch (x^u(p)) at p

V(S) = V^1dS/dx^1 + V^2/dx^2 + ... = (V^ud/dx^u)S

V(S +S') = V(S) + V(S')

V(SS') = SV(S') + S'V(S)

*Now here is an important trick from Penrose:

Let p have local coordinates x^u(p). Consider an infinitesimally close neighboring point p' with local coordinates (in same patch of course) x^u(p') = x^u(p) + @V^u where @ << 1. Dilate this by 1/@ >> 1 You now have a finite displacement in the dilated tangent space @^-1Tp, where V has components V^u. However, you cannot use the simple Pythagorean theorem in Cartesian coordinates because that measure of distance is not invariant under coordinate transformations. For that you need the curvilinear metric guv, even when there is no intrinsic curvature because physically inequivalent coordinate patches describe locally coincident generally non-geodesic observers looking at the same local process.

A dual co(variant)vector field U is a map of a vector field V to a scalar field S. It can be pictured as (N-1)-dim plane elements. In spacelike 3D the vector V is a Bohm hidden variable point particle path (center of mass CM of extended particle) and the dual covector U is a piece of quantum deBroglie iso-phase wavefront.

U(V) = 0

if U is dual to V.

"The plane element at p contains all the directions of all the arrows representing vectors V at p for which U(V) = 0. This enables us to proceed to the general concept of a tensor field on M, which can be viewed as a ... multi-linear map from collections of vectors and covectors to scalars (or else to vectors, covectors, or other tensors if desired)"

The symmetric metric tensor field is g(...) operating on two contravariant vector fields V & V' such that we get a locally invariant scalar field, the same number for all possible local coordinate transformations among the overlapping patches at p:

g(V,V') = g(V',V) = guvV^uV'v = g11V^1V'1 + g12V^1V'^2 + ... g1NV^1V'^N

+ g21V^2V^1 + g22V^2V'^2 + ... g2NV^2V'^N


+gN1V^NV'1 + gN2V^NV'^2 + ... gNNV^NV'^N

= guvV^uV^v

*This is in the dilated tangent space @^-1Tp.

Shrinking it back down again like Alice in Wonderland eating the LSD-laced cookies

ds^2 = guvdx^udx^v = e^aea

for the special case of the fundamental infinitesimal spacetime separation ds between p and p'.

Next time "connections" and "curvature" in more detail from Roger Penrose's keen mind that is both intuitive for physicists as well as riigorous without being "rigor-mortified."

Half-Baked Note 4-1: Einstein's "GCTs" (General Coordinate Transformations) from localizing the globally rigid 4-parameter spacetime symmetry group T4 are examples of the overlapping local patches. The four Einstein-Cartan geometrodynamic tetrad 1-forms are then the compensating gauge potential connection fields A^a analogous to the potential 1-form A of the electromagnetic field.

A^a(gravity-curvature) = A^audx^u from locally gauging T4

A(electromagnetism) = Audx^u from locally gauging U(1)

Beyond Einstein's 1915 GR are the dynamically independent

S^a^b(torsion) = -S^b^a = S^a^budx^u from locally gauging O(1,3).

Note, however, that the geometrodynamic connection field is not A^a but is

C^luv = e^la(d/dx^u)e^av

e^la = I^la + A^la

I^la is the globally flat tetrad that is an arbitrary curvilinear function for non-geodesic local observers with the proviso that the geodesic deviation curvature 2-form vanishes. The geometrodynamic covariant partial derivative on a first-rank world tensor is

DTv/dx^u = dTv/dx^u + C^luvTl

What about spinors?

Are spinors BIT with tensors IT in John Wheeler's


The spinor-tensor coupling coefficients in flat spacetime are $l^Z^Z', where l = 0,1,2,3
Z = 0,1, Z' = 0',1'

(we will come back to spinors later)

Begin forwarded message:

From: Jack Sarfatti
Date: January 14, 2007 7:43:19 PM PST
To: Sarfatti_Physics_Seminars
Subject: [Sarfatti_Physics_Seminars] Fwd: Differential Geometry in a Nutshell - from Star Gate 3

The first idea in differential geometry is that of the "manifold." The manifold is a set of "points" with a "local structure." The "point" need not be a tiny infinitesimal. It can be an entire pocket universe in the megaverse on Leonard Susskind's populated "cosmic landscape" in which we are here by random Darwinian selection. That's only one possibility of course. It might be wrong. "Locality" means being "close together" and "continuous" but there is still at this primordial stage no emergent concept of "metric" nothing like a "distance between two points" needed in Einstein's historic 1915 theory of gravity that replaced Newton's 17th century model of "gravity force" acting "at-a-distance" like Haitian voodoo instantly faster-than the-speeding-photon with absolute simultaneity. Time, like Ol' Man River, it just keeps roll'n along. This first level of manifold topology does not tell us if the curves or surfaces or hypersurfaces imbedded inside the manifold are smooth or kinky. That's the next level of differential structure. Beyond that is "size" AKA "metric" and in-between is conformal, projective and affine. I am being sloppy here this first time round about Felix Klein's "Erlangen Program" of 1872 where different kinds of nested geometries like Russian Dolls and Ezekial's "wheels within wheels" are defined in the non-Boolean quantum lattice of invariant groups, semi-groups to categories and functors with equivalent representations. Of great importance to metric engineering super cosmos by men like Gods is the affine level of self-parallel transport of objects along paths in the manifold. This requires the idea of the connection field that is also the origin of the electromagnetic, weak and strong forces among the quarks and leptons as well as the gravity and torsion fields of the fabric of 4D space-time and possibly even extra-dimensional "hyperspace" including fermionic anticommuting dimensions.

Patchwork Quilt

"A topological space is called an n-dimensional manifold if the topology in the neighborhood of any point is the same as in an n-dimensional Euclidean space E^n. ... Thus, we can envision piecing together a number of local portions of a Euclidean n-space with regions of overlap specified to form a manifold M. ... The rules for how this piecing together is to be achieved in any particular case can be given in an entirely intrinsic manner without reference to any embedding space ... One has to be careful ... when specifying how these "gluings" are to be arranged, that the resulting space does not possess leaves or branches." Roger Penrose

However the quantum bit information manifold of the many worlds interpretation must obviously be non-Hausdorff in order for the world to split or for worlds to merge into alternate timelines for time travellers. See papers by David Deutsch from Oxford, where Penrose also is, for details.

Einstein's general coordinate transformations GCT's Xu^u'(P) connecting coordinate patch x^u(P) with overlapping coordinate patch x^u'(P) in the same "hood" of objective happening P are the "gluings" of Penrose above. Ideally each patch is an observer or local frame of reference. However, there is a gauge freedom redundancy that must be factored out by forming equivalence classes of different coordinate patches that correspond to the same objective local observer-detector or local frame. The mathematics here is woefully redundant with inessential formal distinctions not relevant to the physical observations. There be weeds here.

Next lecture, tensors, spinors, local-frame invariant objective intrinsic Cartan forms like Plato's Ideas free from its shadowy fetters grim inside The Cave. Cornell Savoyards 1964 (use powered speakers)

Begin forwarded message:

From: Jack Sarfatti
Date: January 14, 2007 5:05:58 PM PST
To: Sarfatti_Physics_Seminars
Subject: Differential Geometry in a Nutshell - from Star Gate 2

Curved non-Euclidean geometries have a natural scale of length built in unlike globally flat Euclidean geometry. For example, our dark energy deSitter universe has an intrinsic length scale that is the reciprocal square root of the observed Einstein cosmological constant from Type Ia supernovae standard candles. This corresponds to our universe as a retro-causal advanced wave back from the future cosmic computer worth ~ 10^120 bits.

For example the area of a small triangle on an arbitrarily curved 2D surface is proportional to the deviation of the sum of its angles away from pi.

To get an idea of the strength of Newton's gravity on the large-scale, it may change on the small scale so that the original 1Gev hadronic string theory may be OK, the radius of curvature created by the Earth's mass of 6x10^24 kilograms at its surface at 6x10^3km is about 1.6x10^8 km ~ 1AU = Earth-Sun mean distance. For a neutron star at its surface divide by a million. The the entire visible universe from our vantage point multiply by a hundred trillion 10^14. The larger the radius of curvature the weaker is the intrinsic gravity geodesic deviation curvature field that has no fundamental relation to the 100% inertial non-geodesic g-forces. However, if you want to stand still in a curvature field then there is such a contingent relationship. However, if you are inside the event horizon of a non-rotating black hole then you cannot stand still. Space itself collapses to the spacelike infinite curvature singularity that is a finite proper time t measured by the falling observer relative to passing through the lightlike event horizon.

A key Jungian-Pauli archetype for all mathematics is the Athenian Plato's Allegory of The Cave in Book 7 of The Republic that in modern parlance is Invariant -> Representation rather than Light -> Shadow. There are many points of view POV on the same objective invariant "truth" or "Platonic Idea" or "Form."

Subject: Differential Geometry in a Nutshell - from Star Gate 1
The Michelangelo Code

Working on this as a chapter for my next book Star Gate following Roger Penrose here.

Metric engineering dark zero point energy for warp and wormhole pictured above is how Michio Kaku's Type IV advanced time travelers get to Earth from our future and interfere in our history. Indeed, they seem to have genetically engineered us explaining the legend of the Garden of Eden. Warning: There are mainstream physics papers that argue such things are impossible. However, actual facts show that these theoretical papers are all wrong in my opinion. We have a problem of denial of evidence by some of the "experts." Damn the gamma burst torpedos full warp ahead. Engage!

Physics is not mathematics. Einstein said that there is a creative tension a complementarity between mathematics and physics.

"As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality."

Mathematics is the language of physics. However, physics is not about proving theorems. Of course a mathematical statement in physics must be self-consistent and therefore physical theories are always incomplete maps of reality. That's a theorem of Kurt Godel's. Physics is about understanding phenomena. We know that we have an understanding of some phenomenon when we can control it and also predict new unobserved phenomena. For example, an unusual gamma burst has been observed in empty space without any ordinary matter. I attempt to explain it as the implosion of two concentric spherical shells of zero point dark energy. The outer shell has repulsive negative pressure and the inner shell has attractive positive pressure. If my model is correct there will be a blueshift - redshift doublet in the spectroscopy. Therefore, I have predicted something that can be tested.

to be continued

Jack Sarfatti
"If we knew what it was we were doing, it would not be called research, would it?"
- Albert Einstein

No comments: