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If so, how do we know it speeds up? Doesn't time slow down as gravity increases? If time slows down around a black hole, is it possible matter doesn't actually speed up?

DBWeinstein
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    Additionally to RobJeffries's answer, if you're very interested, familiar with General Relativity and have a bit of time leftover, I can recommend https://www.youtube.com/watch?v=BdYtfYkdGDk this video lecture on how black hole physics works. The speed-up and slow-down is discussed there as well. – AtmosphericPrisonEscape May 01 '19 at 20:52
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    It depends from which frame of reference we are measuring the object velocity – Donald Duck May 02 '19 at 10:23

2 Answers2

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The answer is neither yes or no or possibly both.

Take a simple example. If something falls freely towards a black hole along a radial path, and is observed by someone who is far from the black hole, its velocity (according to the distant observer) is given by $$v = -\left(1 - \frac{r_s}{r}\right)\left(\frac{r_s}{r}\right)^{1/2}c\, ,$$ (e.g. see chapter 6 of Exploring Black Holes by Taylor, Wheeler & Bertschinger - freely available) where $r_s$ is the Schwarzschild radius and the negative sign just indicates an inward velocity with $r$ decreasing.

If you plot this function (see Fig.2 in Ch.6 of Taylor et al. - freely available) you will see that initially the magnitude of the velocity increases as $r$ decreases, but as $r\rightarrow r_s$ then $v \rightarrow 0$ and the falling object appears to come to a standstill (actually, because the light from the object is gravitationally redshifted, this may not actually be observed). However, if the velocity first increases and then slows to a standstill, then it must go through a maximum!

The maximum observed speed in this scenario is achieved at $r=3r_s$ and is $0.384c$.

Of course this story is different for different observers. If you are the falling object then your speed just keeps increasing through the event horizon and towards the singularity. On the other hand, an observer who was somehow able to hover just above the event horizon would measure the falling object's speed as just below $c$ as it passed.

ProfRob
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  • I take it this is for falling from infinity? – Acccumulation May 01 '19 at 21:27
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    For this question, in the absence of a yes or no there is little to no validity in the details. – John May 01 '19 at 21:38
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    @John what does your comment mean? There is no yes/no answer without specifying frames of reference and according to whose measurements. Welcome to GR. – ProfRob May 01 '19 at 22:50
  • @Acccumulation Freefall from infinity. If you start from somewhere else then the precise details are different but the qualitative answer is the same. – ProfRob May 01 '19 at 22:51
  • You mean, nothing ever falls into a black hole (from our point of view)? (But instead all the matter of a black hole accumulates at the Schwarzschild radius? Hey, that would explain why the entropy grows with its surface!) That's news to me ;-). If you have time, would you mind to explain in a few sentences how this formula is arrived at? – Peter - Reinstate Monica May 02 '19 at 05:55
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    @RobJeffries The question is asked in a yes or no format; details are important though they should follow the most concise and direct opening: yes or no [...]. – John May 02 '19 at 06:07
  • @John This would be the perfect opportunity to go: "Short answer: No. Longer answer: Yes." – JollyJoker May 02 '19 at 13:02
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    @John Black and white answers for a black and white world? The answer is neither yes or no. I've now edited that in at the top, but it hardly seems necessary for a 15 line answer. – ProfRob May 02 '19 at 13:33
  • Thanks for what seems to be an excellent answer. I’m not nearly sophisticated enough to understand the math. If we we observing a black hole from earth and we could see an object just outside the event Horizon, and that object we’re traveling at x speed, what speed would we observe? I.e. would the object appear to slow way down? – DBWeinstein May 02 '19 at 15:02
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    @dwstein Yes. If the black hole had a Schwarzschild radius of 1km, for instance, we would see it accelerate to 38.4% of lightspeed when it was 2km above the event horizon. It would then appear to slow to a stop as it approached the event horizon, but also become more and more red-shifted and darkened. On average we would see our last photon from it after a fairly short time. – Steve Linton May 02 '19 at 15:45
  • @SteveLinton fascinating. Is there a black hole mass at which the object would appear to stop entirely? – DBWeinstein May 02 '19 at 15:46
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    @dwstein the mass of the black hole doesn't matter. As a falling object approaches any black hole event horizon it appears (to a distant observer) to slow down as it is going to come to a stop at the event horizon. Red shift and time dilation mean that we never quite see this happen (it emits no photons that get to us from exactly the event horizon). Also, as you add matter or energy to the black hole, its event horizone grows, so the object kind of gets encompassed, despite "stopping". – Steve Linton May 02 '19 at 21:13
  • So, have “we” observed very slow moving matter just beyond the event Horizon? – DBWeinstein May 02 '19 at 21:16
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    @dwstein That could be a good question. We have observed gravitational time dilation using very accurate clocks on Earth. We have observed subtle patterns in the X-ray, radio and light emmissions from black holes which show subtle patterns which line up with the predictions of GR, but I'm not aware of anything more direct. – Steve Linton May 03 '19 at 08:13
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    @SteveLinton Maybe beyond the scope of this question, but the case of black hole mergers may be interesting here. We do see those happening without one BH being frozen smeared over the event horizon of the other. – JollyJoker May 03 '19 at 11:02
  • @JollyJoker if you look at the simulations at https://www.youtube.com/user/SXSCollaboration what happens is that the two black holes approach closely enough and then a new event horizon snaps into existence around both of them. – Steve Linton May 03 '19 at 12:12
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    @SteveLinton Thanks for the link! Should I be seeing the slowdown after 3r in those simulations and how? Presumably not in rotation speed, but how fast they get closer? – JollyJoker May 03 '19 at 12:26
  • @JollyJoker Not sure what time coordinates they use for their simulations. They do also sometimes reduce the time step for the most interesting bits, so watch the details. – Steve Linton May 03 '19 at 12:34
  • I was watched the simulation. What perspective is that from. For a certain distance, would all that gravity make it appear as through the BH’s just freeze? – DBWeinstein May 04 '19 at 03:01
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    @SteveLinton your example of $r_s$ = 1km shows just how extraordinary this all is: at 2km from EH the object is travelling at over 100,000 km/s so in "ordinary" spacetime it would cover the remaining 2km in less than 20 millionths of a second. At 1km above EH it's still travelling at 35% of c; at 10m above EH it has "slowed down" to just 1% of c; even at 1m above EH it's still doing 300 km/s, and it's only in the last 1mm above EH that it slows to the speed of a jet airliner at full throttle! – Chappo Hasn't Forgotten May 07 '19 at 00:42
  • @Chappo That's how it looks from far away -- assuming any photons actually make it out from that close to the event horizon, which they probably don't (time dilation means that the distant observer will see only very few photons emitted by the falling object, as well as them being red-shifted). An observer falling with the object sees it hit the event horizon at close to light-speed. – Steve Linton May 07 '19 at 06:23
  • @SteveLinton Since max observed speed in free fall occurs at $3 r_s$ and the photon sphere is at $\frac{3}{2} r_s$, it kinda makes sense that fewer photons make it back to us from the second half of that final free fall. I was just surprised to find how fast the object is travelling at only 1mm above the horizon - I guess I'd assumed that "slowing to a standstill" would take years, or perhaps days, rather than <0.001 of a second. It's mind-blowing! – Chappo Hasn't Forgotten May 07 '19 at 12:01
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The dilation of time is only relevant from the perspective of someone far away from the black hole. Close to the black hole time is still progressing forward at what would appear to be a normal rate to someone who is close to the black hole. The movie Interstellar had a great depiction this phenomenon, with the astronauts Copper and Brand on Miller's planet, near the black hole, spending only a few hours, but the astronaut Romilly aging decades as he remained far from the planet. Copper and Brand didn't experience any change in the passage of time, from their perspective.

Matter falling into a black hole would not experience any change in its perspective of time, so would not appear to change speed, other than what would be expected by the gravitational attraction.

Bob516
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