[Mike Gale (OP)]

Everything else changes but the photon stays the same. The conditions in the local frame at the time of emission are important.

Correct. Photons don't age. (I assume the conditions you refer to are the relative clock speeds.)

[evan_au]

There is red-shift in that case, but it's due to time dilation, not variable speed of light.

Since speed = distance/time, if there is a difference in the rate of time due to gravitational time dilation, then there will be a difference in the speed of light due to gravitational time dilation.

So you need a correction to the Doppler effect for relativistic effects. It is so small that we normally ignore it on Earth, but it becomes important for velocities near c and gravitational fields near a black hole.

You only measure light to travel at c if you measure it in your laboratory, in your gravitational field.

In someone else's laboratory, in a very different gravitational field, you will observe their speed of light to be different from c (because their clock is ticking at a different rate).

[Mike Gale (OP)]

It occurs to me that there are actually two versions of the SC metric: one with a signature of (-1,-1,-1,+1) and another with (+1,+1,+1,-1). Proper time is imaginary in one case and real in the other. It doesn't matter which version you use to compare reference frames as long as you are consistent. However, when you cross the event horizon in the new metric, you have to use the alternate version to conserve proper time.

[Mike Gale (OP)]

In someone else's laboratory, in a very different gravitational field, you will observe their speed of light to be different from c (because their clock is ticking at a different rate).

That's one way to think about it, but if you mess around with the speed of light in that manner, you will change v/c and that will break SR.

[Mike Gale (OP)]

With the alternate version of the metric, we can track the test mass all the way to the singularity.

[evan_au]

if you mess around with the speed of light in that manner, you will change v/c and that will break SR.

To an observer with a particle accelerator in their lab, once particles get close to c, it is hard to get them going much faster, because of the Lorentz factor: SQRT(1-v²/c²).

To an observer at a much lower gravitational potential, where time goes (say) twice as fast, he will see that in the lab near a black hole, all the clocks are going at half speed, light is crossing the other lab at half speed, and the particles in the accelerator only seem to be traveling at half of the speed he could achieve in his own lab. But if he calls the speed of light in the other lab C, and the velocity of the particles in the other lab V, the results are fully explainable in terms of the Lorentz factor: SQRT(1-V²/C²).

I don't think that the time dilation of GR totally breaks SR, provided you are conscious of the very limited domain of SR, which was really derived for areas outside a gravitational well. ie nowhere in our current universe (but it's still a very useful approximation...).

[Mike Gale (OP)]

I presented my case to a colleague who has a formal education in this field. He was unconvinced that we have found the holy grail because he couldn't tell by inspection that this isn't just another coordinate transformation like Eddington–Finkelstein or Kruskal-Szekeres. He was also unfamiliar with the application of Newtonian dynamics to define the scaling distance and had a vague recollection that it emerges from GR mathematics independently of Newton. I'm not sure about that, but the derivation on Wikipedia certainly invokes Newton. Does anyone know of one that doesn't?

[Mike Gale (OP)]

I don't think that the time dilation of GR totally breaks SR, provided you are conscious of the very limited domain of SR, which was really derived for areas outside a gravitational well. ie nowhere in our current universe (but it's still a very useful approximation...).

Consider two local reference frames in a SC potential. Each perceives the other's clock to run slower. You can interpret that as different light speeds, but the clock rate interpretation is more consistent with SR concepts.

[Mike Gale (OP)]

I don't think that the time dilation of GR totally breaks SR, provided you are conscious of the very limited domain of SR, which was really derived for areas outside a gravitational well. ie nowhere in our current universe (but it's still a very useful approximation...).

Consider two local reference frames in a SC potential. Each perceives the other's clock to run slower. You can interpret that as different light speeds, but the clock rate interpretation is more consistent with SR concepts.

The infinitely removed observer perceives different light speeds at different locations, but all free-fall observers (including the infinitely removed one) measure that same light speed at their respective locations.

[Mike Gale (OP)]

I don't think that the time dilation of GR totally breaks SR, provided you are conscious of the very limited domain of SR, which was really derived for areas outside a gravitational well. ie nowhere in our current universe (but it's still a very useful approximation...).

Consider two local reference frames in a SC potential. Each perceives the other's clock to run slower. You can interpret that as different light speeds, but the clock rate interpretation is more consistent with SR concepts.

The infinitely removed observer perceives different light speeds at different locations, but all free-fall observers (including the infinitely removed one) measure that same light speed at their respective locations.

This is not true of stationary observers, who are held in place by rocket engines or planetary crusts.

[Mike Gale (OP)]

Proper time still stalls at the event horizon with either metric, but with the new one, it resumes on the other side. Even though it takes an infinite amount of (proper or coordinate) time for any classical object to reach the boundary from the outside, a quantum mechanical wave function propagates to all points in space instantaneously. Since the square of the wave amplitude equates to a manifestation probability, the associated particle can manifest anywhere on either side. That is presumably the principle behind Hawking radiation. Given enough time, a particle on the inside can end up on the outside or vice versa. In fact, it could even exist on both sides simultaneously because the two domains are hopelessly separated. The flight path of the interior particle is nothing like that of its exterior counterpart unless you turn the sphere inside out. It takes a lot of twisting and stretching to accomplish that, but it can be done without tearing the spacetime manifold (

The net effect is the center becomes a surface at infinity so gravity becomes repulsive.

[Mike Gale (OP)]

So I guess that's the answer to my original question. Using relativistic conservation of energy to calibrate the SC metric may be a good idea, but it doesn't change the fact that everything, including light, stalls at  the event horizon, whether you're trying to get in or out. My new question is, why are we using the classical version when the relativistic one seems to work so much better?

[Mike Gale (OP)]

Wilson, my local expert, thinks the new metric doesn't satisfy Einstein's field equations, but I don't fully understand the significance of that. He maintains that "Newtonian gravity is a weak-field limit of general relativity, rather than the other way around." The distinction is subtle and most textbooks (e.g. Hartle) tend to gloss over that part. He says the derivation in Wald (section 6)  backs up his claim, but it's far too technical for my little mind and he hasn't figured out how to dumb it down for me yet.

[jeffreyH]

I think somewhere you have doubled up on relativistic terms although I must admit I haven't given this rigorous scrutiny. The fact that you are thinking these ideas through in the first place is praiseworthy in and of itself. Keep it up.

[Mike Gale (OP)]

I think I'm starting to get clued in. His point is that, although the Ricci scalar vanishes, the Ricci tensor does not. Viascience explains the Ricci tensor as a test of conservation of volume. He cites an example where a ring of dust of negligible mass density collapses around a gravitating mass. The inner surface falls faster than the outer one, but if the Ricci tensor is null, the volume will remain constant all the way up to the event horizon. All bets are off at that point because the model fails. My point is that it should start to fail much sooner than that because, in its local reference frame, the inner surface is approaching light speed.

[Mike Gale (OP)]

If the Ricci scalar and cosmological constant are both zero then the Ricci tensor is directly proportional to the stress-energy tensor in Einstein's Field Equations. Both are identical zero for a vacuum solution (e.g. Schwarzschild.) Otherwise all reference frames are accelerated to some extent. That is the case for the new metric, but the Ricci tensor is approximately zero for weak fields and the extra term (rs/2r)^2 has the same effect as the cosmological constant.

[Mike Gale (OP)]

I think I'm starting to get clued in. His point is that, although the Ricci scalar vanishes, the Ricci tensor does not. Viascience explains the Ricci tensor as a test of conservation of volume. He cites an example where a ring of dust of negligible mass density collapses around a gravitating mass. The inner surface falls faster than the outer one, but if the Ricci tensor is null, the volume will remain constant all the way up to the event horizon. All bets are off at that point because the model fails. My point is that it should start to fail much sooner than that because, in its local reference frame, the inner surface is approaching light speed.

A spherical layer of dust, not a ring.

[Mike Gale (OP)]

I think somewhere you have doubled up on relativistic terms although I must admit I haven't given this rigorous scrutiny. The fact that you are thinking these ideas through in the first place is praiseworthy in and of itself. Keep it up.

Thanks. I think I know what you mean. GR certainly incorporates SR, but the Schwarzschild solution is based on classical energy (KE+PE).

[JohnDuffield]

Mike, as you said proper time still stalls at the event horizon with either metric. Resuming on the other side means it's a "non-real solution". Don't be distracted by claims about wavefunction propagating to all points in space instantaneously. That's a "non-real solution" too. As to what's really going on, you were right earlier. See what Einstein said in the second paragraph here:  http://einsteinpapers.press.princeton.edu/vol7-trans/156

[jeffreyH]

What evidence do you have that proper time stalls at the event horizon?