[Mike Gale (OP)]

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 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?

[jeffreyH]

I would have to review my notes to formulate an answer. Tonight is not the time for that though.

[evan_au]

It has always struck me as odd that light can't penetrate the event horizon

In my primitive understanding, light should have no problem penetrating the event horizon.

From the viewpoint of someone falling directly into a black hole from a great distance, their velocity increases to above 10% of c before they reach the event horizon. Laser pulses emitted by this observer towards the black hole seem to be traveling at c away from the observer, as the observer falls all the way to the event horizon. Photons that strike the event horizon don't come back.

From the viewpoint of an observer at a large distance from the black hole, they would see the above observer plunging into the black hole, and emitting laser pulses. Photons that strike the event horizon get swallowed by the black hole. They may see some delayed photons which have skimmed the edge of the event horizon, travelling around the black hole one or more times before escaping back to the distant observer.

So light should have no problem being absorbed by the event horizon.
Light just has a problem subsequently exiting the event horizon (which it does in an incredibly diffuse form, as Hawking radiation).

[yor_on]

Quite interesting idea Mike. But isn't that a question of the geodesics allowed by gravity, presenting you with a 'apparent event horizon'? I still would like a updated two way mirror experiment, taking in account gravitational time dilations though, now as NIST clocks are becoming so good.
==

I'm sort of stuck on your formulation at first there btw

"although proper time stalls at that location, it was already stalled in the case of light."
What do you mean by that? The' proper time' as defined from a local observer at the BH, or as defined by a far away observer? The only proper time I know of is the local one, aka my wristwatch. And how did light become 'already stalled'?

[jeffreyH]

Firstly why apply a gamma correction?
Do you know how the r parameter is derived and why?
If instead of r we used an area this may relate better to entropy.
That is the only advantageous change that I can see.

[Mike Gale (OP)]

Firstly why apply a gamma correction?
Do you know how the r parameter is derived and why?
If instead of r we used an area this may relate better to entropy.
That is the only advantageous change that I can see.

Gamma correction is for relativistic mass. It comes from E^2=m^2c^4+p^2c^2.
r is the radius as perceived by an infinitely removed observer.
Entropy is a statistical concept. Not applicable for 2-body problem except perhaps when special relativity breaks down in the new metric at r=GM/2c^2.

[Mike Gale (OP)]

I'm sort of stuck on your formulation at first there btw

"although proper time stalls at that location, it was already stalled in the case of light."
What do you mean by that? The' proper time' as defined from a local observer at the BH, or as defined by a far away observer? The only proper time I know of is the local one, aka my wristwatch. And how did light become 'already stalled'?

Proper time is always equal to zero in the reference frame of a photon.

[jeffreyH]

Firstly why apply a gamma correction?
Do you know how the r parameter is derived and why?
If instead of r we used an area this may relate better to entropy.
That is the only advantageous change that I can see.

Gamma correction is for relativistic mass. It comes from E^2=m^2c^4+p^2c^2.
r is the radius as perceived by an infinitely removed observer.
Entropy is a statistical concept. Not applicable for 2-body problem except perhaps when special relativity breaks down in the new metric at r=GM/2c^2.

I'm still thinking about the kinetic and potential energies. That would have been a big oversight.
It has to be something to do with the equivalence principle. Have you asked elsewhere?

[jeffreyH]

Your photon sphere is going to be out as 1/2r_s instead of 1.5r_s.

Since this is all theoretical anyway I don't suppose it's a concern.

[yor_on]

Ok Mike, thanks and I see how you thought there now. When it comes to time dilations
I don't see that as being what will prevent light from leaving a BH though. To me it's the geodesics imposed on the propagation that takes care of that, no matter from where you study that 'event horizon'.
=

A time dilation is a comparison between frames of reference (my local clock being 'proper time' and reference frame relative any comparison).  While the geodesics near a event horizon will exist/stay for both frames of reference (presuming we can 'see' them) the time dilation experimentally will disappear as soon as you join whatever frame of reference you earlier compared your local clock to.

[Mike Gale (OP)]

It has always struck me as odd that light can't penetrate the event horizon

In my primitive understanding, light should have no problem penetrating the event horizon.

From the viewpoint of someone falling directly into a black hole from a great distance, their velocity increases to above 10% of c before they reach the event horizon. Laser pulses emitted by this observer towards the black hole seem to be traveling at c away from the observer, as the observer falls all the way to the event horizon. Photons that strike the event horizon don't come back.

From the viewpoint of an observer at a large distance from the black hole, they would see the above observer plunging into the black hole, and emitting laser pulses. Photons that strike the event horizon get swallowed by the black hole. They may see some delayed photons which have skimmed the edge of the event horizon, travelling around the black hole one or more times before escaping back to the distant observer.

So light should have no problem being absorbed by the event horizon.
Light just has a problem subsequently exiting the event horizon (which it does in an incredibly diffuse form, as Hawking radiation).

Light speed as perceived by an infinitely removed observer decreases asymptotically towards zero at the event horizon due to compression of space and dilation of time. Proper time in the standard Schwarzschild metric is ill defined beyond that point so light is thought to be trapped at that location. The revised metric allows it to carry on, but its speed starts to exceed c at r=GM/2c^2. Special relativity breaks down in that domain.

[jeffreyH]

You need to read this.

https://www.math.ku.edu/~lerner/GR/Schwarzschild.pdf

Relativistic corrections should already be taken into account.

[yor_on]

You know, it sounds good but I think you need to break it down a little for us.

Whatever do you mean by "light speed as perceived by an infinitely removed observer decreases asymptotically towards zero at the event horizon due to compression of space and dilation of time." Are you suggesting that if we made 'two way mirror' experiments outside the event horizon we should measure light 'slowing down'? Depending on the mirrors distance from an 'event horizon', to then 'stop' at the event horizon?

As far as I get it 'reflected' light will red shift for the far away observer, not change its 'speed', and for light at the event horizon, if it now could escape, the redshift should be so extreme that we probably wouldn't even be able to measure it. I don't really expect you to mean it as above but it sure sounds that way. As for 'space' getting compressed in the vicinity of a black hole? When it comes to time dilations there are two ways to see it, either you define a gravitational potential as a 'field(s)' of sorts, in where both 'time' as well as distance change, For the time to 'slow down' inside such a patch of  'field' you then also need to slow down the propagation of light, alternatively shrink this patch of space so that it unnoticeable for the observer will present you 'c'. The alternative to this is to presume a speed faster than 'c' as time slowed down but 'c' kept some universal speed. The other way, that I use, is to define it as a constant. And in that case it doesn't matter how you expect it to be 'universally' as everything you measure is a local definition, in the same exact way that 'physics' are expected to be the same, no matter where you go, or at what time. the later definition gave you the computer you wrote on.

[Mike Gale (OP)]

I was hoping to focus this discussion on the validity of the Schwarzschild metric near the event horizon rather than the consequences of warped spacetime.

[Mike Gale (OP)]

I'm still thinking about the kinetic and potential energies. That would have been a big oversight.
It has to be something to do with the equivalence principle. Have you asked elsewhere?

I posed the question to viascience.com a few days ago but he hasn't noticed yet.

[yor_on]

well, we have Pete here at times. You could PM him with a link, or the whole idea and see what he thinks about it.

[Mike Gale (OP)]

Whatever do you mean by "light speed as perceived by an infinitely removed observer decreases asymptotically towards zero at the event horizon due to compression of space and dilation of time."

If you set the Schwarzschild metric equal to zero, dr/dt is the speed of light as perceived by an infinitely removed observer. Light speed in a free-fall reference frame is always c.. There is no red-shift between free-falling reference frames. Have a look at appendix video 11d at viascience.com

[yor_on]

Will do

Actually, thinking about it as a 'slow time field' of sorts with our observer inside it, doing this 'two way mirror experiment'. How should I go about shrinking this patch of field, without shrinking the observer too? It reminds me of that alternative offered to a 'accelerating expansion' of the universe. A 'accelerating shrinking' of everything inside it instead, so to speak

But if it shrinks then....

[Mike Gale (OP)]

You need to read this.

This is indeed the derivation I'm talking about. The equation on page 2 for conservation of energy is:
E=1+v^2/2-m/r+smaller terms
And it is the smaller terms that I call into question.

[Mike Gale (OP)]

Will do

Actually, thinking about it as a 'slow time field' of sorts with our observer inside it, doing this 'two way mirror experiment'. How should I go about shrinking this patch of field, without shrinking the observer too? It reminds me of that alternative offered to a 'accelerating expansion' of the universe. A 'accelerating shrinking' of everything inside it instead, so to speak

But if it shrinks then....

Yes. The mind wobbles as Kelly on Married with Children once observed. But keep in mind that the Schwarzschild metric does not permit any energy beyond that which is provided by the gravitating mass.