Calculating proper time falling toward a black hole

[joshyp93]

Hello everyone,

I have a homework question for general relativity that is driving me nuts. It goes like this:

An observer falls from rest at radius 10GM in the spacetime of a black-hole of mass M (in natural units). What time does it take for them to travel from a radius of 6GM to 4GM, according to them? You may assume the Geodesic equations, rather than derive them. (Hint: introduce X = r/(10GM) and y = sqrt(X/(1-X)) to perform the integration. Also note that ∫1/(1+u^2)^2 du = 0.5 ( u/(1+u^2) + arctan(u)).

I have used the Schwarzschild metric to find the geodesic equations for t and r. Since it is falling radially I have ignored the θ and φ terms. I know how to get the Christoffel symbols for both the t and r equations. I have read in some places that we must first find dt/dτ from the time equation and then substitute it back into the line element equation to find dr/dτ. Then once we have this, we can integrate to find the proper time it takes for the particle falling into the black hole to go from 6GM to 4GM.

I have tried countless times using different methods and the 'hints' given in the question, but I can't seem to get an integral of the form ∫1/(1+u^2)^2 du = 0.5 ( u/(1+u^2) + arctan(u)) like I should.

I just need to know in what order I must do these things. I have never even seen an example of this so I don't know where to start. I would prefer if the G's and c's were kept in the equations for now since it helps me understand where they come from.

Thanks a lot
Josh

 

[TSny]

I have used the Schwarzschild metric to find the geodesic equations for t and r. Since it is falling radially I have ignored the θ and φ terms. I know how to get the Christoffel symbols for both the t and r equations. I have read in some places that we must first find dt/dτ from the time equation and then substitute it back into the line element equation to find dr/dτ. Then once we have this, we can integrate to find the proper time it takes for the particle falling into the black hole to go from 6GM to 4GM.

This approach should lead to the answer. You'll need to show your work in order for us to provide specific help.