The discussion centers around the final velocity of an object as it falls into a black hole, exploring the implications of general relativity on this scenario. Participants examine the challenges of defining velocity in the context of curved spacetime and the limitations of Newtonian physics in such extreme environments.
Participants generally disagree on how to define and calculate the final velocity of an object falling into a black hole, with multiple competing views and no consensus reached on the matter.
The discussion reveals limitations in defining velocity due to the curvature of spacetime around black holes, and the necessity of specifying reference frames and initial conditions for meaningful comparisons.
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.Einstein's Cat said: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)
How would one do that?Orodruin said:without specifying a proper local inertial frame.
Pick, for instance, a coordinate system in which the object is at rest. Then its velocity is easy to compute.Einstein's Cat said: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.Einstein's Cat said:How would one do that?
Please... What would the free- falling object's velocity be? Would it not exceed c due to special theory of relativity?Ibix said: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.
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 said:Please... What would the free- falling object's velocity be? Would it not exceed c due to special theory of relativity?
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.Ibix said: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.
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 said:Please... What would the free- falling object's velocity be? Would it not exceed c due to special theory of relativity?
I see; may I inquire what the object's velocity would be?Orodruin said: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 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 said:I see; may I inquire what the object's velocity would be?
I apologise for my ignorance; and thank youOrodruin said: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.