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
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:
Look at sections 84 & 97 in Landau & Lifshitz Classical Theory of Fields.
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.