[Mike Gale (OP)]

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

Resume was a bad choice of terms. I meant to say that the speed of light is non-zero on the other side, not that light rays penetrate the horizon and carry on. It's admittedly naive to think about wave functions propagating instantaneously through space, but that is essentially the Copenhagen interpretation. I can't remember the name, but there's another interpretation, which I find more plausible, where the wave function has two parts: one that propagates forwards in time at the speed of light and a complementary one that propagates backwards in time at the same speed. Either way, it's a bit off topic and I probably should have kept those musings to myself. The point I was trying to make is that, although the inner and outer domains are hopelessly separated, QM does allow for some coupling. Hawking radiation for example.

[Mike Gale (OP)]

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

Resume was a bad choice of terms. I meant to say that the speed of light is non-zero on the other side, not that light rays penetrate the horizon and carry on. It's admittedly naive to think about wave functions propagating instantaneously through space, but that is essentially the Copenhagen interpretation. I can't remember the name, but there's another interpretation, which I find more plausible, where the wave function has two parts: one that propagates forwards in time at the speed of light and a complementary one that propagates backwards in time at the same speed. Either way, it's a bit off topic and I probably should have kept those musings to myself. The point I was trying to make is that, although the inner and outer domains are hopelessly separated, QM does allow for some coupling. Hawking radiation for example.

I think it was called transactional QM.

[Mike Gale (OP)]

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

See the link provided by JohnDuffield above. If you set dr=d(angle)=0 and r=rs, the SC metric reduces to (ds)^2 = 0 * (c * dt)^2 - 0^2/0 - rs^2 * 0 = 0.

[Mike Gale (OP)]

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

Resume was a bad choice of terms. I meant to say that the speed of light is non-zero on the other side, not that light rays penetrate the horizon and carry on. It's admittedly naive to think about wave functions propagating instantaneously through space, but that is essentially the Copenhagen interpretation. I can't remember the name, but there's another interpretation, which I find more plausible, where the wave function has two parts: one that propagates forwards in time at the speed of light and a complementary one that propagates backwards in time at the same speed. Either way, it's a bit off topic and I probably should have kept those musings to myself. The point I was trying to make is that, although the inner and outer domains are hopelessly separated, QM does allow for some coupling. Hawking radiation for example.

I think it was called transactional QM.

It takes an infinite amount of time for anything (including light) to reach the event horizon from the outside, but if you're willing (and able) to wait that long, the new metric says that things will eventually emerge on the other side.

[PmbPhy]

It has always struck me as odd that light can't penetrate the event horizon because, although
proper time stalls at that location, it was already stalled in the case of light. The problem seems
to be that, because the scaling factor in the Schwarzschild solution (1-2GM/rc^2) is based on
the weak field approximation, it is not valid at such extremes.

The derivation of this conclusion is not based on the weak field approximation. The reason that light doesn't penetrate the event horizon is as follows; first off, whether light penetrates the event horizon depends only on the observers point of view. From the point of view of observers who are at rest outside the event horizon the speed of light is not constant but is a function of the gravitational potential. According to those observers light slows down as it approaches the event horizon, slowing in a way such it never reaches it.

The scaling distance 2GM/c^2 comes from conservation of classical energy mv^2/2=GM/r, but
it should be based on the relativistically correct version:

mc^2/sqrt(1-v^2/c^2) – mc^2 = GMm/r/sqrt(1-v^2/c^2).

This reduces to (1-v^2/c^2) = (1-GM/rc^2)^2 = 1 – 2GM/rc^2 + (GM/rc^2)^2, which is
approximately equal to the Schwarzschild scaling factor for large r. The scaling factor
(1-GM/rc^2)^2 produces a metric that is both Ricci flat and well behaved beyond the
event horizon.

Am I wrong?

Yes. You're using a n incorrect formula for the value of the energy of a particle in a gravitational field. In  GR the total energy of a particle is a function of both the gravitational energy, kinetic energy and rest energy. In fact it reduces to the sum of those values in a certain approximation. The exact value is the time component of the energy-momentum 1-form (the sign of which depends on one's choice of the sign convention used to define the metric.

[jeffreyH]

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

See the link provided by JohnDuffield above. If you set dr=d(angle)=0 and r=rs, the SC metric reduces to (ds)^2 = 0 * (c * dt)^2 - 0^2/0 - rs^2 * 0 = 0.

So you get an invalid result with an infinity. What about it?

[JohnDuffield]

It takes an infinite amount of time for anything (including light) to reach the event horizon from the outside, but if you're willing (and able) to wait that long, the new metric says that things will eventually emerge on the other side.

The new metric is wrong Mike. The infinite time means it never ever happens. Here's the page from MTW:



The plot on the left depicts an infalling body going to the end of time and back and being in two places at once like you were saying, like  Susskind's elephant. The plot on the right tries to smooth this over, but like I said, it's a non-real solution. You can't change your coordinate system to get rid of eternity. See what Pmb said and note the Wikipedia article propagation of light in non-inertial reference frames: "at the event horizon of a black hole the coordinate speed of light is zero...The coordinate speed of light (both instantaneous and average) is slowed in the presence of gravitational fields". The article also says the local instantaneous proper speed of light is always c, but at the event horizon gravitational time dilation is infinite, so you can't measure the local speed of light. It takes forever to do so. So if you started to measure it, you haven't finished doing so yet, and you never ever will. You will never measure the local instantaneous proper speed of light to be c. The infall stops at the event horizon, and a black hole grows like a hailstone as per Oppenheimer's frozen-star black hole.   

[jeffreyH]

It takes an infinite amount of time for anything (including light) to reach the event horizon from the outside, but if you're willing (and able) to wait that long, the new metric says that things will eventually emerge on the other side.

The new metric is wrong Mike. The infinite time means it never ever happens. Here's the page from MTW:



The plot on the left depicts an infalling body going to the end of time and back and being in two places at once like you were saying, like  Susskind's elephant. The plot on the right tries to smooth this over, but like I said, it's a non-real solution. You can't change your coordinate system to get rid of eternity. See what Pmb said and note the Wikipedia article propagation of light in non-inertial reference frames: "at the event horizon of a black hole the coordinate speed of light is zero...The coordinate speed of light (both instantaneous and average) is slowed in the presence of gravitational fields". The article also says the local instantaneous proper speed of light is always c, but at the event horizon gravitational time dilation is infinite, so you can't measure the local speed of light. It takes forever to do so. So if you started to measure it, you haven't finished doing so yet, and you never ever will. You will never measure the local instantaneous proper speed of light to be c. The infall stops at the event horizon, and a black hole grows like a hailstone as per Oppenheimer's frozen-star black hole.

Prove it.

[Mike Gale (OP)]

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

See the link provided by JohnDuffield above. If you set dr=d(angle)=0 and r=rs, the SC metric reduces to (ds)^2 = 0 * (c * dt)^2 - 0^2/0 - rs^2 * 0 = 0.

So you get an invalid result with an infinity. What about it?

I was using the old metric, but I suppose you're right. The rate of change of proper time with respect to coordinate time doesn't stall at the event horizon, it diverges to infinity. The speed of light as perceived by a distant observer goes to zero at that location though.

[Mike Gale (OP)]

It takes an infinite amount of time for anything (including light) to reach the event horizon from the outside, but if you're willing (and able) to wait that long, the new metric says that things will eventually emerge on the other side.

The new metric is wrong Mike. The infinite time means it never ever happens. Here's the page from MTW:



The plot on the left depicts an infalling body going to the end of time and back and being in two places at once like you were saying, like  Susskind's elephant. The plot on the right tries to smooth this over, but like I said, it's a non-real solution. You can't change your coordinate system to get rid of eternity. See what Pmb said and note the Wikipedia article propagation of light in non-inertial reference frames: "at the event horizon of a black hole the coordinate speed of light is zero...The coordinate speed of light (both instantaneous and average) is slowed in the presence of gravitational fields". The article also says the local instantaneous proper speed of light is always c, but at the event horizon gravitational time dilation is infinite, so you can't measure the local speed of light. It takes forever to do so. So if you started to measure it, you haven't finished doing so yet, and you never ever will. You will never measure the local instantaneous proper speed of light to be c. The infall stops at the event horizon, and a black hole grows like a hailstone as per Oppenheimer's frozen-star black hole.   

I don't think this disproves the new metric. The location of the coordinate singularity changes (to 1 instead of 2) and you don't have to wave your hands to explain what happens to proper time on the other side. (The author of that chart has obviously dropped a factor of i on one side of the horizon.) The picture is otherwise the same for both metrics except that the new one may have the light cones rotating inside the horizon rather than on the horizon. (I'm not sure about that.) The main difference between the two is that the new one is not a vacuum solution. It contains a cosmological variable (within the metric tensor), as opposed to an ad hoc cosmological constant.

[Mike Gale (OP)]

The derivation of this conclusion is not based on the weak field approximation.

I beg to differ. The form of the SC metric certainly emerges from GR when you constrain it to spherically-symmetric vacuum solutions, but the value of the scaling distance only emerges when you apply the weak field constraint. That value cannot (or rather should not) be used to extrapolate the solution to strong fields.

[Mike Gale (OP)]

The reason that light doesn't penetrate the event horizon is as follows; first off, whether light penetrates the event horizon depends only on the observers point of view.

I disagree. Nothing penetrates the horizon, not even light. The local reference frame is completely decoupled from the external one at the event horizon. From a local point of view at that location, the gravitating mass is infinitely far away.

[Mike Gale (OP)]

From the point of view of observers who are at rest outside the event horizon the speed of light is not constant but is a function of the gravitational potential. According to those observers light slows down as it approaches the event horizon, slowing in a way such it never reaches it.

Agreed.

[Mike Gale (OP)]

You're using an incorrect formula for the value of the energy of a particle in a gravitational field. In GR, the total energy of a particle is a function of the gravitational energy, kinetic energy and rest energy. In fact it reduces to the sum of those values in a certain approximation.

The sum of classical energy components comes from the weak field approximation. My contention is that a solution based on that approximation cannot (or rather should not) be extrapolated to domains where the approximation is invalid. In other words, a vacuum solution is dubious for strong fields. Einstein was chipping away at that problem with his cosmological constant. I think the new metric is a better approach because it has a similar effect and it's based on relativistic energy rather than an ad hoc constant.

[Mike Gale (OP)]

The exact value is the time component of the energy-momentum 1-form (the sign of which depends on one's choice of the sign convention used to define the metric.

One's choice of sign conventions makes no never-mind in the external domain of the old metric. One will produce imaginary proper time. The other will produce real proper time. The same is true of the new metric, but to make sense of the interior region, you have to switch conventions when the magnitude of the temporal component exceeds that of the spatial components.

[Mike Gale (OP)]

Wilson, my local expert, remains skeptical of the new metric for the following reasons:
1) Aside from relocating the event horizon, the physics of the external domain are essentially unchanged. He suspects that the new metric reduces to the old one in some coordinate transformation. I have no response to that critique except to say that the two metrics are equivalent in the weak field approximation so it seems unlikely that a coordinate transformation is going to make them look the same.
2) He points out that relativistic mass is an obsolete concept and modern GR theorists prefer to use 4-momentum and rest mass. I don't know how (or if) that changes anything in this discussion.
3) He is under the impression that objects do penetrate the horizon in their local reference frame. I don't see how that can happen, but he's not alone in his belief. The theory of collapsing stars forming black holes presumably necessitates that behavior for example. I've seen videos of Susskind lectures where he eludes to that idea, too. And Viascience gives an example in which a test mass hits the horizon and seems to want to carry on even though proper time is ill defined. I think he's using approximate equations of motion, but he doesn't hesitate to draw conclusions from his calculations and one tends to trust his judgment in these matters because he's obviously nobody's fool.
4) He objects to the Viascience explanation of the Ricci tensor as a test of conservation of volume because the concept of volume is observer-dependent. He says the Raychaudhuri equation and focusing theorem is the best way to see if matter converges or diverges over time because it's independent of your choice of coordinates. I don't know anything about it though.

Does anyone have any insight into any of these points?

[Mike Gale (OP)]

Here's a good account of relativistic escape velocity:
http://www.mrelativity.net/MBriefs/Relativistic%20Escape%20Velocity%20using%20Special%20Relativity.htm

[Mike Gale (OP)]

Penrose interprets gravity by applying SR to snapshots of a falling body:

This is essentially how I arrived at the new metric. The distance vt is constant in any given snapshot so the Lorentz transform gives dr'=dr*gamma. The time transform at the origin of the moving reference frame (x=vt) gives dt'=dt/gamma. Conservation of relativistic energy gives an expression of gamma in terms of radial distance and voila, you have the new metric.

[jeffreyH]

I am more interested in Penrose's entropy reset between eons in universal evolution.

[jeffreyH]

Here's a good account of relativistic escape velocity:
http://www.mrelativity.net/MBriefs/Relativistic%20Escape%20Velocity%20using%20Special%20Relativity.htm

I'm not sure of that site. It seems to be promoting a personal theory stating that relativity is wrong.