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2) [15 points, 3 parts, 5 points each] Life beyond the horizon. Consider a massive particle (not necessarily on a geodesic) that has fallen inside the event horizon, r < 2GM, of an uncharged, unrotating black hole. Use the ordinary Schwarzschild coordinates (t,r,Î¸) with the usual metric. Even though they break down at the event horizon, they are well-behaved between the horizon and the singularity, which is good enough for this calculation. a) Use the usual normalization condition on the 4-velocity to show that the dr 2GM Bear in mind that the coordinates may do counterintuitive things inside the horizon. b) Calculate the maximum lifetime for a particle along a trajectory from r = 2GM to r = 0. Then express this in seconds for a black hole with mass measured in solar masses. (Be sure to restore c when you plug in numbers: GM - GM/c^2 gives a length, and GM - GM/c^3 gives a time.) c) Show that the maximum proper time is achieved by falling freely (that is, following a geodesic) with conserved energy E = 0.

          2) [15 points, 3 parts, 5 points each] Life beyond the horizon. Consider a massive particle (not necessarily on a geodesic) that has fallen inside the event horizon, r < 2GM, of an uncharged, unrotating black hole. Use the ordinary Schwarzschild coordinates (t,r,Î¸) with the usual metric.

Even though they break down at the event horizon, they are well-behaved between the horizon and the singularity, which is good enough for this calculation.

a) Use the usual normalization condition on the 4-velocity to show that the
dr
2GM

Bear in mind that the coordinates may do counterintuitive things inside the horizon.

b) Calculate the maximum lifetime for a particle along a trajectory from r = 2GM to r = 0. Then express this in seconds for a black hole with mass measured in solar masses. (Be sure to restore c when you plug in numbers: GM - GM/c^2 gives a length, and GM - GM/c^3 gives a time.)

c) Show that the maximum proper time is achieved by falling freely (that is, following a geodesic) with conserved energy E = 0.

2 15 points 3 parts 5 points each life beyond the horizon consider a massive particle not necessarily on a geodesic that has fallen inside the event horizon r 2gm of an uncharged unrotating 89123

Added by Zachary J.

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