# How can we watch BHs collide in a finite time?

Paige_Turner
TL;DR Summary
A BH can't touch another hole's horizon...
A BH can't touch another hole's horizon for the same reason nothing else can: time drags the object to a halt for a distant observer.

Right?

Manifestly not. So are we wrong about gravitational time dilation, or what?

Gold Member
Summary:: A BH can't touch another hole's horizon...

A BH can't touch another hole's horizon for the same reason nothing else can:
The event horizon isn't an object.

Homework Helper
Gold Member
2022 Award
Summary:: A BH can't touch another hole's horizon...

A BH can't touch another hole's horizon for the same reason nothing else can: time drags the object to a halt for a distant observer.

Right?

Manifestly not. So are we wrong about gravitational time dilation, or what?
Gravitational time dilation in general is not a particularly useful concept in GR. If two black holes collide and merge, then there is a much more complex spacetime and time dilation is not really well defined.

https://www.physicsforums.com/threa...les-in-a-finite-time-in-Earth's-frame.997342/

Mentor
So are we wrong about gravitational time dilation, or what?
No, but you may be misunderstanding what it means here.

I’ll start with the quick and sloppy answer (and it would be a good exercise to identify the sloppiness):
Suppose I drop a clock into a black hole. It falls towards and through the event horizon just like anything else; gravitational time dilation just means that it’s running slow when it does.

Next, a slightly less sloppy answer: instead of an ordinary clock displaying the time, we start with a strobe light designed to flash once per second. We’re holding it in our hands, and a flash of light arrives at our eyes once every second - no time dilation. Then we let go so it starts falling towards the horizon. When we do, we find that subsequent flashes arrive at our eyes more than a second after the previous one - that’s time dilation, the strobe rate is slowing down relative to us. But that doesn’t mean the strobe is slowing as it approaches the horizon, it just means that its flashes are reaching our eyes more than one second apart.
Also note that there is a “last” flash. The strobe light flashes then passes through the event horizon before the next flash and that will be the last flash that ever reaches our eyes. Thus we can justifiably say that the strobe light really does pass through the horizon even though we can’t see it happen, time dilation or no.

Be aware that there are still some horrible sloppinesses here, including glossing over the Doppler effect which also affects the time between receiving flashes.

FactChecker
Paige_Turner
sloppinesses here, including glossing over the Doppler effect which also affects the time between receiving flashes.
Ok. But isn't the sloppiness that you say the Doppler shift here reduces the frequency (red/blue) but not the frequency (of strobe pulses)? Light is always c.

Mentor
Ok. But isn't the sloppiness that you say the Doppler shift here reduces the frequency (red/blue) but not the frequency (of strobe pulses)?
Doppler affects both the spacing between the flashes and the frequency of the light within each flash. When the source and the receiver are moving apart each successive flash has to travel a longer distance than its predecessor, so spends a bit longer in flight.

FactChecker
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But isn't the sloppiness that you say the Doppler shift here reduces the frequency (red/blue) but not the frequency (of strobe pulses)?
Imagine I shine two identical lasers at you. One I leave continuous and the other I block and un-block for 1,000 wave cycles each time. If the frequency you receive is lower than the one I emit then the 1,000 cycle pulses must be lengthened. Also the gaps must be longer, since you need to receive 1,000 cycles of the red shifted continuous laser in the gap between the pulses.

So all frequencies are affected by Doppler, including the light frequency and any modulation of the light.

Gold Member
Next, a slightly less sloppy answer: instead of an ordinary clock displaying the time, we start with a strobe light designed to flash once per second. We’re holding it in our hands, and a flash of light arrives at our eyes once every second - no time dilation. Then we let go so it starts falling towards the horizon. When we do, we find that subsequent flashes arrive at our eyes more than a second after the previous one - that’s time dilation, the strobe rate is slowing down relative to us. ...

Be aware that there are still some horrible sloppinesses here, including glossing over the Doppler effect which also affects the time between receiving flashes.
My try for still minimizing sloppiness: To identify "purely" gravitational time dilation would require to measure the elapsed time of a clock stationary close to the black hole (eventually by means of a long rope for bringing the clock down and up again for comparison with our clock and thereby neglecting the elapsed time for this).

PeroK
Homework Helper
My try for still minimizing sloppiness: To identify "purely" gravitational time dilation would require to measure the elapsed time of a clock stationary close to the black hole (eventually by means of a long rope for bringing the clock down and up again for comparison with our clock and thereby neglecting the elapsed time for this).
That sounds like "slow clock transport". It is one way to synchronize clocks at a distance (as long as you can hold those clocks "stationary"). Unless my intuition is failing me badly, it is a method that coincides with Schwarzschild coordinates.

Schwarzschild coordinates are a fairly natural choice for factoring the observed red shift from a falling clock into a gravitational component between "hovering" clocks and a separate kinematic component applicable to clocks moving past the hovering clocks. But it's still just a coordinate system, not a source of ultimate truth.

timmdeeg
2022 Award
To identify "purely" gravitational time dilation would require to measure the elapsed time of a clock stationary close to the black hole (eventually by means of a long rope for bringing the clock down and up again for comparison with our clock and thereby neglecting the elapsed time for this).
You don't have to do anything so complex with the ropes. You can just have clocks at fixed altitudes use radar to confirm that they are in a fixed altitude relationship and simply watch each other's clocks through telescopes. They'll find that the time dilation each one sees is the reciprocal of the time dilation the other sees.

timmdeeg
Gold Member
Thanks @jbriggs444 and @Ibix for your comments. I must confess that regarding the rope I was influenced by Robert Geroch's "General Relativity fro A to B" where he explains several situations around Black holes with sketches showing ropes.
You don't have to do anything so complex with the ropes. You can just have clocks at fixed altitudes use radar to confirm that they are in a fixed altitude relationship and simply watch each other's clocks through telescopes. They'll find that the time dilation each one sees is the reciprocal of the time dilation the other sees.
But isn't the rate of the clock tick at constant ##r## distorted (smudged?) if we watch it with the telescope, as the light loses energy during climbing up?

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2022 Award
Thanks @jbriggs444 and @Ibix for your comments. I must confess that regarding the rope I was influenced by Robert Geroch's "General Relativity fro A to B" where he explains several situations around Black holes with sketches showing ropes.

But isn't the rate of the clock tick at constant ##r## distorted (smudged?) if we watch it with the telescope, as the light looses energy during climbing up?
Don't see why. And if there is some error ##\Delta t## in your timing of a tick of my clock I would not expect that error to increase with repeated pulses. So you can always time (say) 100 ticks of my clock and get an error of ##\Delta t/100## on my tick duration.

timmdeeg
Homework Helper
Thanks @jbriggs444 and @Ibix for your comments. I must confess that regarding the rope I was influenced by Robert Geroch's "General Relativity fro A to B" where he explains several situations around Black holes with sketches showing ropes.

But isn't the rate of the clock tick at constant ##r## distorted (smudged?) if we watch it with the telescope, as the light loses energy during climbing up?
The tick rate is subject to what we could refer to as gravitational time dilation. Yes. If both sending and receiving clocks are hovering there is no "smudging". One receives a perfectly regular dilated tick rate on the high end. Or a perfectly regular increased tick rate on the bottom end.

Once the two ends have agreed upon the time ratio, they can adjust their clocks to match rates. From there, they can perform Einstein synchronization as usual: The time there upon receipt of a signal is halfway between the time here of transmission and the time here of receipt of the echo.

I am not clear how one would watch light lose energy through a telescope. You see the light that reaches you when it reaches you. You cannot directly look at light beam from the side. Even if you could (smoky room), what you see depends on how the light from there gets to you and on the interaction with the smoke. There are no magic means to remotely sense some underlying truth of the matter. [Which in a very important sense means that there is no underlying truth of the matter].

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timmdeeg
Gold Member
The tick rate is subject to what we could refer to as gravitational time dilation. Yes. If both sending and receiving clocks are hovering there is no "smudging". One receives a perfectly regular dilated tick rate on the high end. Or a perfectly regular increased tick rate on the bottom end.
Yes. If the ticks are light pulses then each tick arrives redshifted at the high end but that doesn't affect the tick rate. I have been misled initially by imagining the clock to be replaced by a monochromatic light source and came up wrongly with "loses energy during climbing up".

Homework Helper
Yes. If the ticks are light pulses then each tick arrives redshifted at the high end but that doesn't affect the tick rate.
It does affect the tick rate.

If the sender at the bottom is sending signals with a 400 nm wavelength and a 0.1 second duration at 1 second intervals then the receiver at the top might be picking up signals with an 800 nm wavelength and a 0.2 second duration at 2 second intervals.

Gold Member
It does affect the tick rate.

If the sender at the bottom is sending signals with a 400 nm wavelength and a 0.1 second duration at 1 second intervals then the receiver at the top might be picking up signals with an 800 nm wavelength and a 0.2 second duration at 2 second intervals.
Ok let's stay with this example. Idealized now, if one could bring up this clock extremely fast it would show 2 second intervals. This tickrate (2 seconds per interval) is due to the stay at the bottom. That's what I tried to express.

Homework Helper
Ok let's stay with this example. Idealized now, if one could bring up this clock extremely fast it would show 2 second intervals. This tickrate (2 seconds per interval) is due to the stay at the bottom. That's what I tried to express.
I am not following.

So we have this clock at the bottom. It is sending out a signal at 1 second intervals (local proper time at the bottom of the rope). These pulses are received at 2 second intervals (local proper time at the top of the rope).

Immediately after this clock sends a pulse, the rope is locally felt to tighten and the clock is yanked to the top at the maximum possible speed, just short of the speed of light. It arrives at the top and slams to a stop. It still has one second (roughly) to go before its next tick. Its tick rate is now 1 second per interval at the top of the rope.

timmdeeg and Ibix
Gold Member
I am not following.
My fault.
So we have this clock at the bottom. It is sending out a signal at 1 second intervals (local proper time at the bottom of the rope). These pulses are received at 2 second intervals (local proper time at the top of the rope).
Yes.
Immediately after this clock sends a pulse, the rope is locally felt to tighten and the clock is yanked to the top at the maximum possible speed, just short of the speed of light. It arrives at the top and slams to a stop. It still has one second (roughly) to go before its next tick. Its tick rate is now 1 second per interval at the top of the rope.
Yes, at the same place both clocks tick at the same rate. What I meant but didn't express unfortunately is that the clock after being brought up very fast shows less elapsed time compared with the top clock due to its stay at the bottom.