- #1

alc95

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## Homework Statement

Hartle, Gravity, P9.8

A spaceship is moving without power in a circular orbit about a black hole of mass M, with Schwarzschild radius 7M.

(a) What is the period as measured by an observer at infinity?

(b) What is the period as measured by a clock on the spaceship?

## Homework Equations

Eqn 9.46 from Hartle:

[tex]

\omega =\sqrt{(M/r^3)}

[/tex]

Proper time:

[tex]

d\tau^2=-ds^2

[/tex]

and the Schwarzschild metric.

## The Attempt at a Solution

(a) This part is fine. Using T=2*pi/omega and substituting r=7M, T=14*pi*sqrt(7)*M.

However, I'm not sure how Eqn 9.46 is derived? Can it be derived using the Schwarzschild metric and noting that the coordinate r is constant?

(b) The clock in the spaceship measures proper time:

[tex]

d\tau=\sqrt{(1-2M/r)dt^2+r^2d\phi^2}=\sqrt{(1-2M/r)+M/r}dt

[/tex]

Here, [tex]d\phi^2=M/r^3dt[/tex]

[tex]

T'=\sqrt{(1-2M/r)+M/r}T

[/tex]

Is my reasoning correct? Assuming this is correct, is all that remains is to substitute r=7M and the period T from part (a)?