Black hole mass as function proper time

[pstq]

Hi there.

1. The problem statement
I am asked to write the equations which give us the mass of a black hole as function the proper time.

Homework Equations

The Schwarzschild metrics is given by
$$ ds^2=-(1- \frac{2GM}{r})dt^2+(1-\frac{2GM}{r})^{-1}dr^2+ r^2(d\theta^2+ \sin^2(\theta) d\phi^2) $$

The Attempt at a Solution

The proper time $\tau$ is related to the metrics by the eq.$ds^2=-d\tau ^2$ hence I need to calculate the following expression $\Delta\tau= - \int \sqrt{ds^2}$ in order to get the proper time, and finally i have to solve for M, (the mass)

Am I right? , any idea?

Thanks

 

[member 553083]

.Yes, your approach is correct. Solving for M in the Schwarzschild metrics equation will give you the mass of a black hole as a function of proper time.