[jeffreyH (OP)]

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.

[Bill S]

My first question would have to be: How do we calculate our distance from the EH?

[Bill S]

This may constitute thread drift (sorry Jeffrey), but I came across it in trying to answer my own question.

The proper length is simply the proper time converted to a length by multiplying by c.  Its physical meaning is that it is the arc length measured along a geodesic……….. (the proper length is of course always zero for photons).

How relevant is this to the idea that photons do not "experience" distance?

[jeffreyH (OP)]

It isn't a drift at all. If our proper time interval is 1 second then our length becomes 1 light second. Of course the line element is an infinitesimal in calculations. This makes the line element Minkowskian. We need something different.

[jeffreyH (OP)]

How relevant is this to the idea that photons do not "experience" distance?

You cannot assume the position of a photon experiencing anything. The photon simply propagates.

[Bill S]

You cannot assume the position of a photon experiencing anything. The photon simply propagates.

Which seems to support my view that the answer to the frequently asked questions about photons “experiencing” zero time/distance, must be: “Don’t know”.

OK, I'll stop drifting now. 

[jeffreyH (OP)]

Drift away Bill. If we want to know how far we are away from the event horizon we must first know the mass of the black hole. From this we can calculate the radial distance out to the horizon. We then need a reliable way of determining our initial distance from the horizon. We can't bounce light off the black hole so another method must be formulated.

[evan_au]

We then need a reliable way of determining our initial distance from the horizon.

What about looking at the distortion of the distant starfield caused by the gravitational field of the black hole?

If there were an accretion disk, you could use interferometry to measure the distance to the accretion disk.

If there was no accretion disk, you could fire a laser tangentially at the black hole event horizon, and measure the delay before the light appears on the opposite side of the black hole, after being bent in a U-Turn around the black hole. Then do a lot of complex maths that is beyond me....

[jeffreyH (OP)]

We then need a reliable way of determining our initial distance from the horizon.

What about looking at the distortion of the distant starfield caused by the gravitational field of the black hole?

If there were an accretion disk, you could use interferometry to measure the distance to the accretion disk.

If there was no accretion disk, you could fire a laser tangentially at the black hole event horizon, and measure the delay before the light appears on the opposite side of the black hole, after being bent in a U-Turn around the black hole. Then do a lot of complex maths that is beyond me....

They are all very good suggestions. I had initially thought about lensing effects. I am thinking about the required maths at the moment. I am delving into differential equations and tensors. It's a hard slog.

[jeffreyH (OP)]

If we are in free fall how do we determine g at any position along our path? This is a related but equally tricky question.

[alancalverd]

....or any other single point in space?

Until the invention of radar, the only means of ranging was by visual or acoustic triangulation from a known baseline. So you can use two optical telescopes to fix on the absence of starlight from behind the black hole, or two x-ray telescopes to triangulate the Hawking radiation from the event horizon. As the baseline is perpendicular to the direction of travel, you won't need any correction for velocity.

[evan_au]

Perhaps another way to measure your distance from a black hole, if you are in a circular orbit around it:
- Monitor the pulses from a number of millisecond pulsars, in different directions on the sky.
- By monitoring over many orbits, you should be able to determine the diameter of your orbit, and the mass of the black hole
- The center of the black hole is at half of the diameter
- You can work out the size of the event horizon from theory

If you were in an elliptical orbit, the maths gets more complicated, but still feasible.

Of course, if you are in a stable orbit, you will never reach the event horizon (until collisions with an accretion disk or gravitational waves steal your angular momentum...).

And the Delta-V to reach a black hole from a circular orbit would be... astronomical!

[Bill S]

Jeffrey, it seems you have a lot of calculating to do before you address the time question directly.  My suspicion is that you would have done most, if not all, of that before you asked the question. 

Alan's suggestions must (?) involve information from external sources.  If you have access to such information, is it possible that a remote observer could tell you how far you were from the EH?

[jeffreyH (OP)]

Jeffrey, it seems you have a lot of calculating to do before you address the time question directly.  My suspicion is that you would have done most, if not all, of that before you asked the question. 

Alan's suggestions must (?) involve information from external sources.  If you have access to such information, is it possible that a remote observer could tell you how far you were from the EH?

I originally looked at ways of determining final velocity at the horizon. Then lots of other questions came up. I am still taking on board the points made by Alan and Evan.

[Bored chemist]

From my point of view, outside the EH, you never reach it.
No calculation needed.

[jeffreyH (OP)]

From my point of view, outside the EH, you never reach it.
No calculation needed.

Do you have observational evidence to back up the assertion?

[evan_au]

From my point of view, outside the EH, you never reach it.

Any light emitted by a probe plunging into the black hole would get red-shifted into oblivion by the extreme time dilation close to the event horizon.

So a distant observer would never see you actually reach the event horizon - but it would look very much like you had hit the event horizon and disappeared from sight.

To an observer plunging into the black hole, this time dilation would not be apparent, and you would reach the event horizon in a finite time, and traveling at a significant fraction of c.

[Bill S]

From my point of view, outside the EH, you never reach it.

Would that prevent you from measuring my distance from the EH, when I was still at a considerable distance?

[Bill S]

So a distant observer would never see you actually reach the event horizon - but it would look very much like you had hit the event horizon and disappeared from sight.

Are you saying that assertions such as: "To an outside observer any object approaching the Schwarzschild radius appears to take an infinite time to penetrate the event horizon", are incorrect, because the object would vanish before, in the observer's RF, it reached the EH?

[evan_au]

Are you saying that assertions such as: "To an outside observer any object approaching the Schwarzschild radius appears to take an infinite time to penetrate the event horizon", are incorrect, because the object would vanish before, in the observer's RF, it reached the EH?

Yes.

I hope I don't murder the maths too much, but:
- Let's take Kryptid's calculation of the escape velocity of a black hole at a certain radius: v_e = √((2GM)/r)
- If we are looking at a space probe falling "from infinity", it's velocity at radius r from the center of the black hole will equal the escape velocity given by this equation.
- Plug in the measured mass of the Cygnus X-1 black hole of 14.8 solar masses
- The event horizon is at a radius of r_s=43.6km

For an observer seeing the space probe fall away from him towards the black hole, there would be considerable relativistic doppler redshift, even without gravitational time dilation.
https://en.wikipedia.org/wiki/Relativistic_Doppler_effect#Motion_along_the_line_of_sight

But let's ignore Doppler shift, and put the observer at right angles to the infalling probe, so there is very little Doppler shift.
So now the main source of frequency change is due to gravitational time dilation. This frequency ratio is sqrt(1-r_s/r)

If the distant observer looks at the speed of the infalling space probe, and does a linear extrapolation of the time to impact with the event horizon:
D: Distance to event horizon, in meters
v/c: Velocity, as a fraction of the speed of light
T: Extrapolated time to impact event horizon (in seconds)
TD: Gravitational Time Dilation ratio = ratio of emitted frequency of light to frequency as seen by the distant observer


D(m)     v/c    T(s)   TD
10000   0.902   4E-05   0.432
 1000   0.989   3E-06   0.150
  100   0.999   3E-07   0.048
   10   1.000-   3E-08   0.015
    1   1.000-   3E-09   0.005


If the infalling space probe had a bright beacon on it,
- at 10km its frequency would be halved, and you would predict impact in about 40μs.
- at 100m, it's frequency would be reduced by a factor of 20, and you would expect impact in 0.3μs.
- at 1m, it's frequency would be reduced by a factor of 200, the received power would be reduced by a factor of 200, and you would expect impact in 3ns

You could extend the visibility for a nanosecond or two by using an X-ray beacon, or even setting off a gamma-ray burst with an atomic bomb.

I am sure there is some factor I have missed, but I think the general conclusion is valid - a remote observer would see the space probe approaching the event horizon at a relativistic speed, and around the expected time of impact, the frequency of the beacon would be reduced to undetectability within a very short time (effectively zero frequency and zero emitted power), so it would look as if the space probe had crashed into the black hole and disappeared
See: https://en.wikipedia.org/wiki/Gravitational_time_dilation#Outside_a_non-rotating_sphere