# Object's velocity in black hole

What would the final velocity of an object be as it falls into a black hole? I assume you could calculate the gravitational field strength of the black hole to determine the acceleration of the object and then calculate the final velocity with the equation

V(final) = gt + v(initial)

Orodruin
Staff Emeritus
Homework Helper
Gold Member
2021 Award
You cannot use Newtonian physics to describe a black hole and you cannot define "the" velocity without specifying a proper local inertial frame.

jbriggs444
Homework Helper
What would the final velocity of an object be as it falls into a black hole? I assume you could calculate the gravitational field strength of the black hole to determine the acceleration of the object and then calculate the final velocity with the equation

V(final) = gt + v(initial)
That equation models gravity as a force that produces an acceleration which can be measured against an inertial coordinate system. In general relativity, gravity is not a force. Worse, there is no such thing as an inertial coordinate system that works in the neighborhood of a black hole. [Edit: you can define a local coordinate system that is approximately inertial, but you cannot extend it far enough to cover the entire fall while still remaining approximately inertial]. You can define non-inertial coordinate systems, but any velocity that you come up with will be as much a function of the coordinate system that you choose as a function of the object that you are considering.

Drat, beaten by Orodruin.

Last edited:
without specifying a proper local inertial frame.
How would one do that?

jbriggs444
Homework Helper
How would one do that?
Pick, for instance, a coordinate system in which the object is at rest. Then its velocity is easy to compute.

Ibix
How would one do that?
The problem is that you have to ask "velocity relative to what?". In special relativity and Newtonian physics you can pick an object, any object, and measure the velocity of any other object with respect to that one. In general realtivity, however, you can't do that in general because curved spacetime makes the process of comparing the velocities of two objects that aren't in the same place an ambiguous process.

So you need to pick an object - perhaps one hovering just above the event horizon on a sufficiently powerful rocket. The black hole itself won't work (the event horizon isn't an object). Then you can ask what the velocity of an object free-falling into the hole is with respect to that object.

Einstein's Cat
The problem is that you have to ask "velocity relative to what?". In special relativity and Newtonian physics you can pick an object, any object, and measure the velocity of any other object with respect to that one. In general realtivity, however, you can't do that in general because curved spacetime makes the process of comparing the velocities of two objects that aren't in the same place an ambiguous process.

So you need to pick an object - perhaps one hovering just above the event horizon on a sufficiently powerful rocket. The black hole itself won't work (the event horizon isn't an object). Then you can ask what the velocity of an object free-falling into the hole is with respect to that object.
Please... What would the free- falling object's velocity be? Would it not exceed c due to special theory of relativity?

Ibix
Please... What would the free- falling object's velocity be? Would it not exceed c due to special theory of relativity?
Not a clue. Still learning this stuff myself. It cannot exceed the speed of light, though, and special relativity isn't applicable anywhere gravity is relevant.

Einstein's Cat and Binky
jbriggs444
Homework Helper
Not a clue. Still learning this stuff myself. It cannot exceed the speed of light, though, and special relativity isn't applicable anywhere gravity is relevant.
It cannot exceed the speed of light in any locally inertial frame. It can exceed the speed of light if one measures against a non-inertial frame. But that measured speed is largely meaningless.

Einstein's Cat
Orodruin
Staff Emeritus
Homework Helper
Gold Member
2021 Award
Please... What would the free- falling object's velocity be? Would it not exceed c due to special theory of relativity?
You also cannot apply the special theory of relativity to a black hole, other than in a small local region - a local inertial frame. There is no global velocity which the object will have "relative to the black hole".

You also cannot apply the special theory of relativity to a black hole, other than in a small local region - a local inertial frame. There is no global velocity which the object will have "relative to the black hole".
I see; may I inquire what the object's velocity would be?

Orodruin
Staff Emeritus
Homework Helper
Gold Member
2021 Award
I see; may I inquire what the object's velocity would be?
I just told you there is no such thing as a uniquely definable velocity which would make physical sense. The entire point of the issue is that the space-time is so curved around the black hole that you cannot define a local frame covering the entire black hole and the object. You simply cannot define the object's velocity in a relevant way.

I just told you there is no such thing as a uniquely definable velocity which would make physical sense. The entire point of the issue is that the space-time is so curved around the black hole that you cannot define a local frame covering the entire black hole and the object. You simply cannot define the object's velocity in a relevant way.
I apologise for my ignorance; and thank you