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$$v_\mathrm e=\sqrt{\frac{2GM}r}$$

If we understand the escape velocity to be the speed needed to escape from the 'surface' of a gravitational object described by the above equation.

It is equally true that the escape velocity is the speed an object pulled from rest will reach when it hits a certain point in the gravitational field.

Since the event horizon is defined as when the escape velocity reaches the speed of light for a black hole, would matter accelerate near the speed of light as it approaches the singularity? Accretion disks are known to move at relativistic speeds near the event horizon, but perhaps oddities occur beyond this point?

The event horizon can be derived from setting the escape velocity of an object to the speed of light, it is a point of no return. I understand there are better definitions, but this is not really germane to the question at hand. How about from the perspective of another in-falling observer?

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    $\begingroup$ "Since the event horizon is defined as when the escape velocity reaches the speed of light ". No, it isn't. That is merely a coincidence. Your question needs to specify from who's point of view. $\endgroup$ – Rob Jeffries Mar 13 at 7:35
  • $\begingroup$ @RobJeffries I thought that was a fair statement and not "merely a coincidence". This question and answer seems to agree. physics.stackexchange.com/questions/33916/… I think it's unique that from all points of view, the escape velocity at the event horizon = c, a rare thing that all points of view agree on, but maybe I'm missing something. Of course, escape velocity doesn't equal in-falling velocity for a falling object because a falling velocity of c is impossible. $\endgroup$ – userLTK Mar 13 at 16:53
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    $\begingroup$ @userLTK You are missing that in Newtonian physics (that cannot be used near a black hole) it is the escape speed; a body does not have to travel radially at the escape speed to escape from a massive object in Newtonian physics, it can travel more slowly if a force is applied, but in GR no escape is possible from within a black hole event horizon at any speed or with any force applied. $\endgroup$ – Rob Jeffries Mar 13 at 17:23
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    $\begingroup$ @userLTK Even outside the event horizon, an object would escape in Newtonian physics if its speed was > than the escape speed, that isn't so in GR. Something travelling at $c$ (i.e. photons) can orbit a black hole at $1.5r_s$, any closer and the velocity must be directed outwards to escape and radially at the event horizon - again, unlike Newtonian physics. In fact anywhere but at the event horizon, there is no coincidence between a Newtonian escape speed and a radially directed GR escape speed. That's why I claim it is a coincidence. Anyway, it doesn't materially affect the question. $\endgroup$ – Rob Jeffries Mar 13 at 17:25
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There is indeed a sense that infalling objects accelerate to the speed of light as it reaches the event horizon, but you need to be careful what you mean by the speed of the object because the speed is observer dependent.

This is explained in detail on the Physics site in my answer to Will an object always fall at an infinite speed in a black hole? An observer watching from far from the black hole would see the falling object initially accelerate towards the black hole but then decelerate to a halt at the event horizon. However an observer hovering a distance $d$ above the event horizon would see the falling object pass them at a speed:

$$ v = c \sqrt{\frac{r_s}{r_s + d}} \tag{1} $$

where $r_s$ is the radius of the event horizon. As the distance above the horizon $d$ goes to zero the speed calculated from equation (1) goes to the speed of light $c$.

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    $\begingroup$ This site really could use your help in answering some of the unanswered. Great answers BTW. $\endgroup$ – Muze the good Troll. Apr 16 at 16:31
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Does matter accelerate to the speed of light as it approaches the singularity?

In a way. But let's get rid of that word "singularity", and replace it with "black hole". Yes, falling bodies fall faster and faster. As you know, falling bodies do not slow down. Taken to the limit, the falling body would be falling at the speed of light.

Since the event horizon is defined as when the escape velocity reaches the speed of light for a black hole

Where did you get that definition from? I don't recognise it. Let's just agree that the event horizon is the location from which the upward light beam cannot escape.

Would matter accelerate near the speed of light as it approaches the singularity?

Yes, but again let's replace singularity with black hole. As the falling body approaches the black hole, it falls faster and faster. Eventually it approaches the speed of light. But there's a catch, and it is a whopper.

Accretion disks are known to move t relativistic speeds near the event horizon, but perhaps oddities occur beyond this point?

Perhaps. However we don't know what happens on the other side of the event horizon. But we do know that oddities happen on this side of the event horizon. Such as gamma ray bursts.

The event horizon can be derived from setting the escape velocity of an object to the speed of light, it is a point of no return...I understand there are better definitions, but this is not really germane to the question at hand. How about from the perspective of another in-falling observer?

See what Einstein said in 1920: “Second, this consequence shows that the law of the constancy of the speed of light no longer holds, according to the general theory of relativity, in spaces that have gravitational fields. As a simple geometric consideration shows, the curvature of light rays occurs only in spaces where the speed of light is spatially variable”. The falling body is falling into the black hole because the speed of light is reducing. It falls faster and faster, until at some point it's falling at the local speed of light. Then it erupts into a gamma ray burst.

For some strange reason not many people know about this. But see the 2013 AMPS paper an apologia for firewalls. Tucked away in the conclusion is footnote 31, containing a reference 87 to Friedwardt Winterberg’s 2001 paper gamma ray bursters and Lorentzian relativity. Winterberg talks about the direct conversion of an entire stellar rest mass into gamma ray energy. See the Wikipedia gamma ray burst article and note that “a typical burst releases as much energy in a few seconds as the Sun will in its entire 10-billion-year lifetime”. I think this is the hard scientific evidence that Winterburg is correct, it ties in with what Einstein said about the speed of light, and it clears up the issue wherein falling bodies are said to accelerate to the speed of light and decelerate to a halt.

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    $\begingroup$ Crackpot "theories". $\endgroup$ – Florin Andrei Apr 17 at 18:12
  • $\begingroup$ @Florin Andrei : Einstein said what he said, gamma ray bursts do happen, and Winterberg was the guy who came up with the idea for GPS. So my answer isn't the crackpot theory. The crackpot theory is the one that says says an infalling object accelerates to the speed of light as it reaches the event horizon AND decelerates to a halt at the event horizon. $\endgroup$ – John Duffield Apr 18 at 10:54

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