Object's Velocity in Black Hole: Calculate Final Vel.

[Einstein's Cat]

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]

You cannot use Newtonian physics to describe a black hole and you cannot define "the" velocity without specifying a proper local inertial frame.

 

[jbriggs444]

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.

 

[Einstein's Cat]

without specifying a proper local inertial frame.

How would one do that?

 

[jbriggs444]

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.

 

[jbriggs444]

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.

 

[Orodruin]

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".

 

[Einstein's Cat]

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]

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.

 

[Einstein's Cat]

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
for all your help

 

[Ibix]

I don't think there's a straightforward answer. There isn't a single answer to "how fast does an object hit the floor" in Newtonian physics - it depends on how high it falls from and how fast it was pushed as well as whether it has any tangential velocity. So you need to specify an initial position and state of motion.

Relativity just makes things worse by insisting that you specify a set of coordinates, which is equivalent to specifying who is doing the observing of the fall. At that point, you can get a physically meaningful answer - but because it's specified relative to a particular observer it's only meaningful to that observer. Other observers at the same place would say the speed was different, while observers in other places would say the speed isn't well defined from their perspective.

Edit: I think...