Now that I read Sciama 1952 & 1969 I understand what Sciama means by "origin of inertia". He simply means deriving the geodesic structure of the entire spacetime history. Geodesics describe the universal (independent of rest mass) zero g-force "inertial motion" of neutral test particles - the very beginning of Sciama's 1952 paper. Sciama never claims to be able to derive the rest mass of the electron to be 10^-27 grams for example.

Let me explain how Mach's Principle is irrelevant in the modern local gauge theory.

Einstein's 1916 GR is simply 1905 SR when the rigid 4 parameter translation group T4 is locally gauged to T4(x). These are the general coordinate transformations, which are gauge transformations just like in electromagnetism. All solutions of the field equations connected by local gauge transformation represent the same physical space-time! The same physics. All the solutions are on the same gauge orbit correspond to what different sets of detectors measure when each is in an arbitrary world line that does not need to be a timelike geodesic. They can be in rocket ships in space firing their engines. See Rovelli Ch 2 Quantum Gravity free online.

The four LIF Cartan tetrad 1-forms and their six spin-connection 1-forms are induced by the local gauging and restore the symmetry to the enlarged system of matter + gravity. It's very beautiful and same idea as in the other electromagnetic - weak - strong forces. This is a real conceptual unification. The local gauge principle is simply Einstein's locality i.e. no signals outside the light cone. All of this ignores quantum theory entanglement of course - it's classical. With signal locality there is no conflict with quantum theory - what Abner Shimony calls "passion at a distance."

The equivalence principle simply means universal minimal coupling of the gravity field tetrads to all the prior matter fields.

That said, it's all "interior bulk physics" in the sense of the hologram virtual universe conjecture where we are retrocausal computations of the VALIS computer (see Seth Lloyd on event horizons as computers) at the future edge of time - the dark energy de Sitter future horizon. This brings back a super-strong Mach's principle using the ideas of Wheeler-Feynman -> Hoyle-Narlikar -> Cramer's transaction -> Aharonov's Destiny Vector. But this is really speculative. Will say no more about it here.

For example, to couple the electromagnetic vector potential AI(LIF) from 1905 SR to 1916 GR with the 16 tetrad components (gravity field LNIF)

Au(LNIF) = eu^I(tetrad)AI(LIF)

For the Dirac spinor electron the universal EEP minimal coupling of say the electron to gravity is

Du(LNIF) = eu^IPI(LIF) + (Spin Connection)u^I^JPIJ(LIF)

= generally covariant partial derivative matrix operator on the spinor Psi multi-component column vector

There are 24 spin connection components connecting the LIF to the coincident LNIF in accord with the EEP.

Where PIJ is the appropriate matrix representation of the 6-parameter Lorentz group Lie algebra acting on the spinor quantized field operator & PI are the Lie algebra of T4 (energy-momentum) together they are the Lie algebra of the Poincare group that form the global symmetries of 1905 SR.

I've been trying to keep up with the conversation since being in transit here and there for the last day or two. I'd like to try to facilitate things in two ways. One is to provide (attached) a stripped down version of "Flux Capacitors and the Origin of Inertia" that only has the formal derivation of Mach effects so that all of the math is there, without other stuff, to be evaluated. As an aside, I should mention that this has been through peer review. Foundations of Physics requests suggested referees. I volunteered six names (and suggested that the ms be sent to all of them). Were I to break confidence and tell you the names I submitted, you would likely instantly recognize at least two or three, and some of you would recognize four or five. In addition, a number of you have been through this derivation for yourselves. And I took it to Ron Crowley and Stephen Goode (both excellent general relativists) before publishing it.

I think I can be more helpful, though, by taking up where Jack said he "lost" me several emails ago. He asserted that the definition of inertial mass, in concert with the definition championed by Frank Wilczek in his Physics Today articles, lectures, and book, should be m = E/c^2, or, as Wilczek calls it, Einstein's "second law" (the first being E = mc^2). There is an important caveat to be added to this definition: E does NOT include gravitational potential energies. Why? Because the Einstein Equivalence Principle (EEP) prohibits the "localization" of gravitational [potential] energy. Why? Because if gravitational potential energy is localizable, accelerated reference frames can be distinguished from gravity fields, and the geometrization of gravity field in general relativity fails. (This was shown by Carl Brans in 1962.)

The second assertion is that the "origin" of inertial mass is the action of gravity. This follows from the first version of Mach's principle -- the gravitational origin of inertial reaction force -- in a simple way.

[Jack interjects: please define how you mean "inertial mass" - what is that? Is the rest mass of the electron 10^-27 grams "inertial mass"? Do you mean to say that one needs Mach's principle to compute that number?

Also define what you mean by "inertial reaction force" in terms of the equations.

Here is what I mean:

For a weightless zero g-force massive test particle on a timelike geodesic (curvature is irrelevant) Newton's first law of mechanical motion in any local frame of reference is

d^2x^u/ds^2 + {Levi-Civita Connection}^uvw(dx^v/ds)(dx^w/ds) = 0 geodesic equation has no inertial mass m in it at all!

The second term on LHS is zero in a LIF it's only non-zero in a LNIF locally coincident with the LIF (Sciama's "rest frame" of the test particle)

Do you mean the 2nd term on the LHS is the inertial reaction force?

That's not what I mean by the same term. What I mean is this:

Use Newton's 2nd Law of mechanical motion

d^2x^u/ds^2 + {Levi-Civita Connection}^uvw(dx^v/ds)(dx^w/ds) = F^u/m

where F^u is an applied translational non-gravity force e.g. for a charge q

F^u = qF^uv(dx^v/ds)

Fuv - EM field tensor - neglecting Dirac's radiation reaction "jerk" (third order partial differential equation with future light cone nonlocal solutions - Wheeler-Feyman future absorber/Cramer transaction)

The rest mass m of the test particle appears in Newton's second law but not in Newton's first law!

In the LNIF "rest frame" of the test particle m,q in this case under the action of the external non-gravity force F^u

d^2x^i/ds^2 = 0 , i = 1,2,3 (space axes)

dx^i/ds = 0 Therefore

{Levi-Civita Connection}^i00 = F^i/m

i = 1,2,3

For example, for us standing on surface of Earth, the quantum electrical reaction force pushing us off a timelike geodesic of the Earth's total mass M is the term F^i/m. The weight shown by standing on a scale is simply F^i ]

[JS: Perhaps you show this in your attachment I have not read yet. I am very concerned about what the "vector approximation to GRT" means because of John Norton's paper on the subject. But maybe - final test is experiment and you have experimental data.]

[JS: This is something I yet do not understand. The whole idea of the scalar potential is questionable. In Einstein's GR the scalar potential of Newton is subsumed in the metric tensor field guv for static LNIF test particle observers in the simplest case of a spherical static source mass M.]

[JS; Very dubious in the light of modern precision cosmology since our universe is accelerating from dark energy's anti-gravity field never anticipated by Sciama.]

Now, the non-localizability of gravitational potential energy [EEP], the locally measured invariance of phi, and the fact that phi equals c^2, all different facets of the same jewel, have serious consequences for the origin of inertial mass. Take the Sarfatti/Wilczek/Cambier assertion about the "origin" of inertial mass, expressed through Einstein's second law:

m = E/c^2

where E is the total NON-GRAVITATIONAL energy in some local region of spacetime, m the inertial mass possessed by that energy, and c the vacuum speed of light. By the way, I am NOT trying to pick a fight here. This is a special relativity definition. But when we allow for general relativity, we know that c^2 is identically equal to phi via the argument above. So we can replace c^2 with phi and do a little rearranging to get:

E = m phi

[JS: I don't understand this either yet. What about Maxwell's discovery that c^2 = (electrical permittivity x magnetic permeability)^-1 ?]

If you read through the derivation attached, you will find that E = m phi (expressed as densities) is the formulation of Mach's principle that allows you to "separate variables" to recover Mach effects in the relativistic Newtonian order approximation to GRT. So Mach effects really are general relativistic effects -- but much larger than the effects usually attributed to GRT. :-)

For those interested: A lab update. I've succeeded in confirming John's "decline effect" prediction. The peak voltage in the runs done the other day were about 530 counts, as opposed to 575 counts for the 1 uN result obtained a couple of months ago. More telling is the stack accelerometer result: 580 versus 700 counts. Now, i suppose you can do a scaling calculation with these numbers. But I wouldn't. Having scratched a lot of itches, it's time to move on to Bruce's dual resonance matcher. That's first up next trip. I've also got a stack in preparation using the new crystals. It probably won't get run next trip. But perhaps the trip after.

May you all have a good weekend,

Jim

Category: MyBlog

Published on Saturday, 06 August 2011 17:09