- #1
joshyp93
- 5
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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
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