Gravitational acceleration of relativistic objects.

[jwes44]

If you have an object (e.g. an alpha particle) traveling at nearly the speed of light towards another massive object (e.g. a black hole) from a large distance, how would you calculate the gravitational acceleration on that object from the viewpoint of an external observer? Obviously, the Newtonian formula does not work as it would have the object exceed the speed of light before it reached the event horizon.

 

[ZikZak]

In GR, the equations of motion of a freely falling body are the geodesic equations $$\frac{d^2 x^\alpha}{d\tau^2} + \Gamma^\alpha_{\beta\gamma} \frac{dx^\beta}{d\tau}\frac{dx^\gamma}{d\tau}=0,$$ where $x^\alpha$ is the position 4-vector of the particle and $\Gamma^\alpha_{\beta\gamma}$ is the Christoffel symbol, which is pretty easily derived from the metric and contains the curvature terms. There's a book by Taylor & Wheeler that does a really good job of solving the equations for a body falling into a black hole and then actually telling you what the interpretations are.

 

[A.T.]

Have a look a this thread:
https://www.physicsforums.com/showthread.php?t=310397

 

[HallsofIvy]

If you have an object (e.g. an alpha particle) traveling at nearly the speed of light towards another massive object (e.g. a black hole) from a large distance, how would you calculate the gravitational acceleration on that object from the viewpoint of an external observer? Obviously, the Newtonian formula does not work as it would have the object exceed the speed of light before it reached the event horizon.

How do you arrive at that "obvious" conclusion? As the object's speed increases toward c, its mass also increases, so that the acceleration, for the same force, is decreases. I don't see how it could possibly accelerate past the speed of light if its mass is increasing without bound at the same time.

 

[Jonathan Scott]

How do you arrive at that "obvious" conclusion? As the object's speed increases toward c, its mass also increases, so that the acceleration, for the same force, is decreases. I don't see how it could possibly accelerate past the speed of light if its mass is increasing without bound at the same time.

In Newtonian theory, the force is proportional to the mass so the acceleration is unaffected by the increase in mass.