[Mike Gale (OP)]

The Schwarzschild scaling distance can be interpreted as the distance at which the Newtonian escape velocity is light speed. However, one would expect it to be the distance at which the relativistic escape velocity is light speed. The relativistic distance is exactly one half the classical value, but in order to maintain compatibility with SR in the free fall case (https://youtu.be/1ogiQ2E6n0U), it is necessary to revise the scaling factor from (1-rs/r) to (1-rs/2r)^2. This follows by recognizing that rs/r=v^2/c^2 in the radial free fall case and (1-v^2/c^2)=(1-GM/rc^2)^2 if you balance kinetic energy against the Newtonian potential using relativistic mass as described here: http://www.mrelativity.net/MBriefs/Relativistic%20Escape%20Velocity%20using%20Special%20Relativity.htm
The scaling factor is then larger by (rs/2r)^2. The resulting metric is not a vacuum solution, but it is approximately so in the weak field limit. Aside from relocating the event horizon and ensuring that the scaling factor does not change sign (thus preserving causality in strong fields), the main consequence of this change is that proper time is well defined beyond the horizon, provided that you switch to the alternate metric signature when the magnitude of the spatial displacement exceeds that of the temporal displacement. This implies a new object, which is complex spacetime - something Penrose has been advocating in the context of his Twistor theory (https://youtu.be/hAWyex1GKRU.)

This discussion thread is a follow on from Why can't light penetrate the event horizon?

[Mike Gale (OP)]

To get the discussion started, allow me to relay some responses from a GR expert I consulted on this idea:
1) Aside from relocating the event horizon, the physics of the external domain are essentially unchanged. It is possible that the new metric reduces to the old one in some coordinate transformation.
2) Relativistic mass, which is the basis of relativistic escape velocity, is a somewhat obsolete concept and modern GR theorists prefer to think in terms of 4-momentum and rest mass. I don't know if that changes anything.
3) Many people I've spoken to profess the opinion that objects with mass can penetrate the event horizon. If not in the coordinate reference frame, then at least from the perspective of a local reference frame. I maintain that penetration is impossible without QM, but I am at a loss to find any consensus of the issue.
4) Although the Ricci scalar is identical to zero for the both metrics, the Ricci tensor for the new one only vanishes in the weak field limit. The meaning of the Ricci tensor is hard to pin down, but the YouTube Viascience channel (which I highly recommend if your GR or QM skills are at all rusty) describes it in terms of conservation of volume. In that context, I would contest the idea that volume must be conserved in strong fields.

[Mike Gale (OP)]

A related link with a reply from a respected expert: http://physics.stackexchange.com/questions/246133/deriving-a-schwarzschild-radius-using-relativistic-mass.
(No comment on the new metric yet, but he argues that the relationship between the SC scaling distance and Newtonian escape velocity is purely coincidental. He bases that judgement on the premise that the Newtonian radial coordinate is different from the one in the SC metric. I'm not so sure about that because GR would be pointless if it was formulated in anything other than Newtonian coordinates. It's true that the local reference frame is foreign to Newton, but the distant one is most certainly the one on which his equations are based.)

[Mike Gale (OP)]

Another controversial subject related to this matter is the variability of the speed of light in a gravitational field from the perspective of a distant observer. Many people seem to balk at this idea because the mantra from SR is that light speed is invariant. Although GR maintains that constraint in local reference frames, it allows the speed of light between distant points to be variable. Einstein's own treatment of this subject can be found here: http://einsteinpapers.press.princeton.edu/vol7-trans/156?highlightText=%22spatially%20variable%22

[Mike Gale (OP)]

On the subject of SR compatibility, the SC metric (old or new) for free fall (radial or orbital) can be expressed as:
ds^2 = ( c dt / gamma )^2 - ( dr gamma )^2 - ( r d(angle) )^2
Where (Lorentz's) gamma is defined in terms of escape velocity (classical or relativistic, depending on which metric you subscribe to) and proper light speed (c.) Note that escape velocity is equal to coordinate velocity (dr/dt) in the radial free fall case, but not the orbital one. Set ds=0 to find the coordinate velocity of light.

[Mike Gale (OP)]

Another argument I've fielded is the idea that the SC solution is sacrosanct because the scaling distance emerges from the bowels of GR when you constrain it to radially symmetric vacuum solutions. From what I can tell, the form of the SC metric certainly emerges from that constraint, but the magnitude of the scaling distance comes from classical theory (i.e. the weak field limit) and the metrics are equivalent in that domain.

[jeffreyH]

On the subject of SR compatibility, the SC metric (old or new) for free fall (radial or orbital) can be expressed as:
ds^2 = ( c dt / gamma )^2 - ( dr gamma )^2 - ( r d(angle) )^2
Where (Lorentz's) gamma is defined in terms of escape velocity (classical or relativistic, depending on which metric you subscribe to) and proper light speed (c.) Note that escape velocity is equal to coordinate velocity (dr/dt) in the radial free fall case, but not the orbital one. Set ds=0 to find the coordinate velocity of light.

So with ds set to zero we end up with ( c dt / gamma )^2 = ( dr gamma )^2 + ( r d(angle) )^2. We are back to Pythagoras.

[timey]

I'm pretty certain that my diagram, that places my model's variable 'times' for the speed of light into a Newtonian geometric, also results in Pythagoras.

[jeffreyH]

I'm pretty certain that my diagram, that places my model's variable 'times' for the speed of light into a Newtonian geometric, also results in Pythagoras.

Well with ( c dt / gamma )^2 = ( dr gamma )^2 + ( r d(angle) )^2 we end up with a distance divided by gamma on the left side. On the right is a distance multiplied by gamma. As one expands the other contracts. It's bigger on the inside.

[jeffreyH]

What do you think that means for the holographic principle and Beckenstein's entropy?

[Mike Gale (OP)]

On the subject of SR compatibility, the SC metric (old or new) for free fall (radial or orbital) can be expressed as:
ds^2 = ( c dt / gamma )^2 - ( dr gamma )^2 - ( r d(angle) )^2
Where (Lorentz's) gamma is defined in terms of escape velocity (classical or relativistic, depending on which metric you subscribe to) and proper light speed (c.) Note that escape velocity is equal to coordinate velocity (dr/dt) in the radial free fall case, but not the orbital one. Set ds=0 to find the coordinate velocity of light.

So with ds set to zero we end up with ( c dt / gamma )^2 = ( dr gamma )^2 + ( r d(angle) )^2. We are back to Pythagoras.

That's right. Just as in SR, you can recover flat space with a coordinate transform. But it's different from a Lorentz transform because the GR coordinate systems are not in motion with respect to one another. Spacetime dilation in GR is not the same as spacetime dilation in SR. One holds c constant and attributes dilation to relative motion. The other has no relative motion so either dilation is due to variable light speed or variable light speed is due to dilation. Both views are valid. The observables are the same in either case. Whichever view gets you through the night, causality must be preserved if we are going to make any sense of it.
 

[Mike Gale (OP)]

I'm pretty certain that my diagram, that places my model's variable 'times' for the speed of light into a Newtonian geometric, also results in Pythagoras.

I'm still digesting your model, but now that you mention it, a cross reference is in order:
Beginning:
https://www.thenakedscientists.com/forum/index.php?topic=69032.0
Middle:
https://www.thenakedscientists.com/forum/index.php?topic=69585.0
Latest:
https://www.thenakedscientists.com/forum/index.php?topic=69592.0

[Mike Gale (OP)]

I'm pretty certain that my diagram, that places my model's variable 'times' for the speed of light into a Newtonian geometric, also results in Pythagoras.

Well with ( c dt / gamma )^2 = ( dr gamma )^2 + ( r d(angle) )^2 we end up with a distance divided by gamma on the left side. On the right is a distance multiplied by gamma. As one expands the other contracts. It's bigger on the inside.

The same is true of the Minkowski metric if you set x'=vt.

[Mike Gale (OP)]

What do you think that means for the holographic principle and Beckenstein's entropy?

Both should hold in either case because all things (including light) stall at the event horizon.

[jeffreyH]

If we look at kinetic energy we can say that


1/2 gamma m dv^2 - (GM/dr) (gamma m) = 0


is of interest. If we set a value for the right hand side in a strong field that would be interesting.

[Mike Gale (OP)]

If we look at kinetic energy we can say that


1/2 gamma m dv^2 - (GM/dr) (gamma m) = 0


is of interest. If we set a value for the right hand side in a strong field that would be interesting.

I think you meant: 1/2 gamma m v^2 - (GM/r) (gamma m) = 0
That's not quite right. The correct form is:
gamma * mc^2 - mc^2 = (GM/r) * gamma * m
This reduces to your equation in the classical limit (where v<<c.)

[jeffreyH]

No that is not what I meant. If you progress through the equation you end up with 1/2 gamma m a^2 - v^2/2 (gamma m). So that the particle has a force moving it but the gravitational field appears to have no such force since it reduces to a kinetic energy term. Which is why people convince themselves that the gravitational force is fictitious.

[Mike Gale (OP)]

No that is not what I meant. If you progress through the equation you end up with 1/2 gamma m a^2 - v^2/2 (gamma m). So that the particle has a force moving it but the gravitational field appears to have no such force since it reduces to a kinetic energy term. Which is why people convince themselves that the gravitational force is fictitious.

Ficticious in the sense of the equivalence principle, but point taken. I still think your equation is flawed because gamma is a function of velocity. You're missing the d(gamma) term.

[jeffreyH]

Yes in fact it is completely wrong. At the moment I am dealing with lots of stuff so I am not entirely concentrating properly. We currently have no kitchen and chaos reigns. Builders don't appreciate the disruption they cause. My apologies.

[timey]

I'm pretty certain that my diagram, that places my model's variable 'times' for the speed of light into a Newtonian geometric, also results in Pythagoras.

Well with ( c dt / gamma )^2 = ( dr gamma )^2 + ( r d(angle) )^2 we end up with a distance divided by gamma on the left side. On the right is a distance multiplied by gamma. As one expands the other contracts. It's bigger on the inside.

Yes - understood.

...and my model's introduction of an additional gravitational time dilation, in relation to GR gravitational time dilation, could, (I think...scratches head), be described with such maths viewed from an alternate perspective that ensures that the distance of a metre remains constant.

Thanks Mike:  I haven't published my diagram online, but if it became of interest for you to see it as you further digest, I can pm it to you.