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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?

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    $\begingroup$ Additionally to RobJeffries's answer, if you're very interested, familiar with General Relativity and have a bit of time leftover, I can recommend 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. $\endgroup$ – AtmosphericPrisonEscape May 1 at 20:52
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    $\begingroup$ It depends from which frame of reference we are measuring the object velocity $\endgroup$ – Donald Duck May 2 at 10:23
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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.

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    $\begingroup$ For this question, in the absence of a yes or no there is little to no validity in the details. $\endgroup$ – John May 1 at 21:38
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    $\begingroup$ @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. $\endgroup$ – Rob Jeffries May 1 at 22:50
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    $\begingroup$ @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 [...]. $\endgroup$ – John May 2 at 6:07
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    $\begingroup$ @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. $\endgroup$ – Rob Jeffries May 2 at 13:33
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    $\begingroup$ @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. $\endgroup$ – Steve Linton May 2 at 15:45
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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.

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