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Proper time in Schwarzschild metric

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1. Consider a Schwarzchild black hole of mass M=15Ms where Ms is the mass of the sun. Two stationary clocks are on the same radial line, one at r1 = 300GMs and the other at r2 = 10Rs where Rs is the radius of the sun.
(a) If 1000 seconds elapse on the clock at r2, determine the amount of time (in seconds) that will elapse on the clock at r1.
(b) Determine the radial distance between the two clocks. Compare your result to what it would be if the space surrounding this mass were flat.

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Solution Preview

We need the following figures:

M_{solar} = 1.98892*10^(30)kg

R_{solar} = 6.96*10^(8) meters

The mass of the black hole is M = 15 M_{solar}, so the parameter m in the Schwarzschild metric is:

m = M G/c^2 = 22.15*10^(3) meters

The r coordinate of point 1 is :

r1 = 300 m = 6.645*10^(6) meters

The r coordinate of point 2 is 10*R_{solar}:

r2 = 6.96*10^(9) meters.

It is important to realize that r1 and r2 are not physical distances, but just labels to indicate points in space. Distances and time intervals must be derived from the Schwarzschild metric:

ds^2 = c^2(1-2m/r)dt^2 - (1-2m/r)^(-1)dr^2 - r^2 d theta^2 - r^2sin^2(theta)d phi^2

Let's use this to calculate the distance from r1 to r2. You again use the fact that ds^2 is an invariant. At any point you can define local coordinates in which the metric is locally like
c^2 d T^2 - d L^2. Then dT is a time interval measured by the observer and dL a distance measured by ...

Solution Summary

A detailed solution is given that determines the proper time in schwarzschild metric.