# Will free-fall object into black hole exceed speed of light $c$ before hitting black hole surface?

In Newtonian mechanics, if we throw an object in against direction of gravity with speed $v$ and it achieve max height of $h$. Now if we allow object to fall from that height $h$, it will eventually attain speed $v$ when it reach position where we launch it.

Now applying same idea to a black hole in general relativity. Speed require to escape black hole gravity is greater than $c$, so if we throw something into black hole with almost the speed of light, the object speed will exceed speed of light $c$ before hitting black hole surface! How relativity explain this? Can space-time curvature reduce speed of this freely falling object from attaining speed of light?

• Your analogy is based on Newtonian mechanics, which is not applicable to strong gravitation field around black holes. Apr 24, 2012 at 6:01

To answer your question you need to be clear what coordinates you're using. If you use coordinates that are co-moving with the rock falling into the black hole then the rock will see the event horizon pass at the speed of light.

External observers, using Schwarzchild coordinates, will see the rock slow down as it approaches the horizon, and if you wait an infinite time you'll see it stop.

External observers obviously can't comment on the speed of the rock after it has passed the event horizon because it takes longer than an infinite time to get there. If you use the rock co-moving coordinates then you can ask what speed you hit the singularity and ... actually I'm not sure what the answer is. I'll have to go away and think about it.

Incidentally http://jila.colorado.edu/~ajsh/insidebh/schw.html is a fun site describing what happens when you fall into a black hole.

• Although I +1'd this at a quick glance, on second reading, you absolutely need to say that the "slowing down" is a coordinate effect, that you don't actually slow down relative to an outside observer, since the coordinate distance you travel per unit coordinate time (scaled by the metric) is going to c even for the external observer, it's just that the speed c in external coordinates is frozen at the horizon, since the external coordinates are symmetric between white hole and black hole, they do not distinguish the sense of the horizon, so there is no crossing of light through the horizon. May 1, 2012 at 21:36
• @Ron Maimon: So if they don't "actually" slow down, does that mean they even pass the horizon? If so, at what speed are they going then? Nov 20, 2014 at 0:26
• @mike4ty4: The coordinates where the objects freeze are degenerate on the horizon, there is no paradox. The objects freeze because the coordinate time stops, not because their intrinsic velocity is slow. It requires knowing the metric form at a horizon, something you can work out for Rindler space easily, because it's just flat Minkowski space in disguise. Nov 21, 2014 at 4:15
• Has (almost) a decade been enough time to think about it? :) Do you think you'd hit the singularity at a speed that's higher than $c$ but the same speed at which light would hit it? Also, how would you respond to Ron's comment now? Is there any chance that he was right about the speed "going to $c$ even for the external observer"? Apr 3 at 8:24
• @GumbyTheGreen the speed any point that is distant from you is coordinate dependent. In your rest frame at your position you can approximate the spacetime as flat so speed has a nice clear definition. But as soon as you consider points that aren't at your position you need to consider the curvature and then the number you get for the speed depends on what coordinates you choose. I don't think Ron is correct, though this is a somewhat philosophical perspective because we probably just differ in what we mean by speed. Apr 3 at 8:46

Nope, it will just fall in a reasonable amount of time (if you go with it, but watch out for tidal forces!), or take forever to fall in (if you are watching from outside).

Also, if I may be so bold as to suggest doing some quantum mechanics instead of kinematics while you are there, you could probably lock down some funding no problem.

• +1 for you could probably lock down some funding :) Apr 24, 2012 at 7:18
• ohhh yes, that make sense, gravity make time slower, so it would take infinite time for object to hit the black hole. thanks. and well am novice to quantum mechanics. will explore it for more :) Apr 24, 2012 at 8:31

It will reach the speed of light exactly at the black hole surface.

• What surface? I assume you mean the event horizon, but that's just a mathematical entity, not a physical thing. Jan 28, 2018 at 16:51
• I meant event horizon. Whether ir is physical or not is a matter of opinion/phylosophy. Jan 28, 2018 at 17:07
• Sorry, but I can't see "surface" as ever being anything but an abstraction, which leaves this answer just like John Rennie's, except that he stresses the importance of coordinate systems, GR being local. Jun 10, 2019 at 16:40

if we throw something into black hole with almost the speed of light, the object speed will exceed speed of light c before hitting black hole surface! How relativity explain this? Can space-time curvature reduce speed of this freely falling object from attaining speed of light?

If you throw (or drop someting for that matter) radially inwards towards a black hole the gravitational acceleration in coordinate time will go as:

$$\frac{d\bar{v}}{dt}=-\frac{GM}{r^2}\hat{r}(1-2\frac{v^2}{c^2(1-\frac{2GM}{rc^2})}-\frac{v^2}{c^2(1-\frac{2GM}{rc^2})^2})$$

The two extra terms will prevent any object moving radially inwards from reaching the speed of light. You can only reach the Schwarzschild radius of $$r=2GM/c^2$$ if you move infinitely slow. Usually you do not talk of the Schwarzshild radius as the "black hole surface", but I guess that is what you mean.

• Does this imply that the escape velocity at the event horizon is not actually the speed of light but is rather less than it? If so, how can the event horizon be the point beyond which light can't escape? Apr 3 at 8:13
• @GumbyTheGreen well you sort of get infinite redshift at the event horizon as viewed by a distant observer and also the speed of light as viewed by a distant observer would be infinitely slow but for a person at the event horizon time also goes infinitely slow compared to a distant observer leading him to believe the speed of light is normal. I do not know if you for sure can know what is going on beyond the event horizon. Apr 20 at 10:08