# Does matter break the light speed barrier after the event horizon?

This is a question someone asked me today and it's bugging me allot. If the acceleration caused by gravity is greater than the speed of light at a black hole event horizon then does this mean that the matter is falling at faster than light speed?

## Answers and Replies

Dale
Mentor
2020 Award
No. Matter inside the event horizon is still timelike. FTL is spacelike.

This is a question someone asked me today and it's bugging me allot. If the acceleration caused by gravity is greater than the speed of light at a black hole event horizon then does this mean that the matter is falling at faster than light speed?
A similar question comes up in an expanding universe, do some objects retreat faster than the speed of light that are passed the cosmic horizon?

It all depends on how you define velocity at a distance in curved spacetime.

Locally it is certainly not falling faster than the speed of light.

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bcrowell
Staff Emeritus
Gold Member
If the acceleration caused by gravity is greater than the speed of light at a black hole event horizon[...]

You can't compare acceleration and speed. They have different units.

When you talk about the speed of the infalling object, what is it relative to?

Note that general relativity doesn't offern any meaningful definition of the velocity of an object with respect to a second, *distant* object.

My understanding of gravity in the context of GR is that mass warps space-time so that forward motion through time naturally results in special displacement. Think of yourself in a 3 dimensional coordinate system with the familiar X, Y, and Z axes. Now imagine a 4'th axis perpendicular to all 3 which represents a time coordinate, call it T. You are moving along a line within this 4 dimensional space-time at a constant rate according to your own watch, that is, each tick of your watch represents the same distance along that line regardless of which way the line is angled. This is why increasing your velocity along the X, Y, or Z axes reduces the distance you travel along the T axis. The presence of mass/energy/momentum distorts your coordinate system so that the X, Y, Z, and T axes are no longer all straight and perpendicular. Now obviously if there are no forces acting on you, then you will continue to travel in a "straight" line, but traveling in a "straight" line through a curvy coordinate system will result in changes to your X, Y, Z, and T coordinates which appear to describe a curved line. This curved line is the path of an object in free-fall.

Now back to the original question. At the event horizon of a black hole the coordinates are curved so that the T axis points toward the center of the black hole. At this point asking how fast an object is falling is kind of weird because you are asking about a velocity. Velocity has units of distance/time. An object falling through the event horizon is moving along its T axis so this "motion" would have units of time/time, and so is not really motion at all. Therefore a question about the velocity of an object falling into a black hole is a nonsensical question. It's like asking what the color 9 smells like.

PAllen
Now back to the original question. At the event horizon of a black hole the coordinates are curved so that the T axis points toward the center of the black hole. At this point asking how fast an object is falling is kind of weird because you are asking about a velocity. Velocity has units of distance/time. An object falling through the event horizon is moving along its T axis so this "motion" would have units of time/time, and so is not really motion at all. Therefore a question about the velocity of an object falling into a black hole is a nonsensical question. It's like asking what the color 9 smells like.

This last part of your discussion is a coordinate artifact and is not correct. Inside the event horizon, a small region of spacetime is still almost like flat spacetime, with 3 possible orthonormal spatial directions and a time direction that may be defined by the 4-velocity of the infaller (which is still timelike inside the horizon as well as outside and everywhere except being undefined at the singularity). Thus, from the point of view of one infaller, another nearby infaller is moving with a well defined spatial velocity (that can be in any direction), and all sufficiently local physics remains identical to SR.

Another way of looking at this is simply different coordinates. Lemaitre coordinates (as well as Kruskal) have a time coordinate that is everywhere timelike (both inside, on, and outside the horizon), and spatial coordinates that are everywhere spacelike.

What is true is that for a SC geometry, the singularity is spacelike, and all world lines inside the event horizon end on it. For a a rotating BH, neither of these is true - in a Kerr solution there are even stable orbits inside one of the two EH that never reach the singularity.

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...If the acceleration caused by gravity is greater than the speed of light at a black hole event horizon...

A free falling observer cannot even detect the event horizon as it is approached and passed...and for such an observer there is nothing unusual at the Schwarzschild radius.

Such an observer continues on their way undisturbed toward the singularity....For a supermassive BH, which may be billions of solar masses in size, the event horizon may be thousands of light years in diameter....In other words, gravity there is rather weak.

Rather weak but still strong enough to keep whatever fell in inside and not let it escape.

A free falling observer cannot even detect the event horizon as it is approached and passed...and for such an observer there is nothing unusual at the Schwarzschild radius.
If the observer free falls from infinity the light from the stars behind him have a frequency of exactly 50%.

Bill_K