- #1

espen180

- 834

- 2

The metric in spherical coordinates for a point mass at [tex]r=0[/tex] is

[tex]\eta=\left(\begin{matrix}-\left(1-\frac{2GM}{c^2r}\right) & 0 & 0 & 0 \\ 0 & \frac{1}{1-\frac{2GM}{c^2r}} & 0 & 0 \\ 0 & 0 & r^2 & 0 \\ 0 & 0 & 0 & r^2\sin^2\theta \end{matrix}\right)[/tex]

As expected, it is symmetrical wrt angle, and I recognize [tex]\eta_{00}[/tex] (time entry) as the negativa square of the gravitational time dilation constant.

The spatial [tex]\eta_{11}[/tex] entry for radius goes to infinity as r approaches the swartzschild radius. At this radius, photons can orbit the point mass.

Is the a way to compute the curvature of space and of space-time a radius r away from the sphere without computing the entire Riemann tensor?