[jeffreyH (OP)]

Taking the equation ve = √((2GM)/r) we can rearrange as r = (2GM)/v^2 so we can now associate a radial distance with a field potential.

[Bill S]

Thanks Evan.  I wouldn't know if you murdered the maths, or not, the essential thing, for me, is that it made sense.

So many Pop Sci explanations give the impression that the entire accretion history of a black hole, since the formation of its event horizon, should be visible to any observer whose technology allows him or her to manoeuvre into the right position. 

When I was thinking about this originally, I came up with a possible alternative that seemed reasonable to me.

I saw it as an example of asymptotic decay, in which the infalling object was not simply stuck for ever in the same state, but was gradually vanishing, with its progress being recorded by an asymptotic curve,  in theory, it would never actually vanish, but in reality, it would come to a conclusion.  In other words, it would vanish.  This seemed to be the simplest explanation, but yours has the benefit of mathematical backing.         

[Bill S]

Taking the equation ve = √((2GM)/r) we can rearrange as r = (2GM)/v^2 so we can now associate a radial distance with a field potential.

Surprise!  I got the maths, but you lost me after that.

[jeffreyH (OP)]

Taking the equation ve = √((2GM)/r) we can rearrange as r = (2GM)/v^2 so we can now associate a radial distance with a field potential.

Surprise!  I got the maths, but you lost me after that.

Look at the calculation for the Schwarzschild radius near the top of the page and compare.
https://en.m.wikipedia.org/wiki/Schwarzschild_radius

[evan_au]

How would we calculate the time left to contact with an EH?

To a distant observer, the location of the event horizon r_s  is where the escape velocity v_e = c.

However, to a closer observer - say, one about 53.6km from the center of the Cygnus-X1 black hole (10km outside the Schwarzchild radius r_s), the escape velocity from r_s to her current position is not c, but something lower.

Like most people, I have trouble visualising the distortions of space and time in relativistic environments (and even more trouble calculating them!).

So let's say that a space probe is now 53.6km from the center of the black hole, and 10km outside the Schwarzchild radius.
- To the person inside the space probe, the speed of light inside the space probe seems "normal"=c.
- I estimate the escape velocity from r_s out to 53.6km to be around  29,000km/s. This is a lot less than c
- So the observer in the space probe should be able to see light from inside r_s?
- For example, if another space probe was dropped just before her, and this probe had a bright beacon
- I estimate that the person in this space probe should be able to see light emitted from a radius > 12km from the center of the black hole.
- It is as if this observer sees a smaller event horizon than the more distant observer?
- If the observer in the space probe sees the black hole approaching at (say) c/3, then they would extrapolate impact with this smaller event horizon (41km away) in around 400μs (linear extrapolation)

So, my estimates are crude, but to the observer in the space probe, does the size of the event horizon shrink as they get closer?

Of course, if they blinked they would miss it! (A typical human blink takes around 200,000μs...)

[jeffreyH (OP)]

Exactly at the event horizon light has just the right speed to escape to infinity. Inside the event horizon it is not possible for light to escape to infinity since it would require a speed greater than that of light. If photons cannot achieve light speed at all between the event horizon and infinity then they cannot leave. Otherwise they would exist in a state in the external universe that violates the constancy of light speed. This viewpoint may well be very wrong. The wavelength/frequency of light at the event horizon is of interest. What would these values be? How could they be determined?

[PmbPhy]

If we are approaching the event horizon how can we determine how much time we have left before reaching it. Length contraction will be increasing but our proper time will be of no use in such circumstances.

I haven't done those calculations in many years so I can't say off hand. I suggest looking at Taylor's website for the new version of Exploring Black Holes. I think he does it in there.

The basic idea is to first note that there are two answers depending on whose point of view/frame of reference you want to consider. If you're considering the time as measured by an outside observer who is at rest wrt the BH then that time is infinite since that time is infinite. From the in-falling observers frame then one uses that observers proper time and that will be finite. I don't suspect that its too difficult to calculate though. I'm just old and my memory is old too.

[jeffreyH (OP)]

You've been away too long Pete. Welcome back.

[PmbPhy]

You've been away too long Pete. Welcome back.

Thanks, Jeff. Every year I became .5% more tolerant of certain personality types than the previous year so I'm more tolerant than I was last year. Lol!