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Radial free fall in Schwarztchild metric.

Consider a Schwarzchild black hole of mass M=15Ms where Ms is the mass of the sun. A particle that was initially at rest at r= falls radially inward. Determine the proper time along the particle's worldline for it to reach (a) the event horizon and (b) the singularity, starting from r0 = 5rs.

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Solution Preview

I'll explain how to derive the "free fall formula" for radial coordinate as a function of proper time for a free falling object in the radial direction.

The equation of motion follows from the Lagrangian.

L = 1/2 g_{mu,nu}x^{mu}-dot x^{nu}-dot

where the dot means the derivative w.r.t. proper time.

The action is Integral over tau of L dtau and the Euler Lagrange equations are thus:

d/dtau [dL/dx^{mu}-dot] - dL/dx^{mu} = 0

You can read-off the g_{mu, nu} from the line element:

ds^2 = c^2(1-2m/r)dt^2 ...

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