- #1
espen180
- 831
- 2
I'm just getting a taste of computational GR, and I have a question regarding the metric for the single point mass solution for the einstein equation.
The metric in spherical coordinates for a point mass at r=0 is
\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)
As expected, it is symmetrical wrt angle, and I recognize \eta_{00} (time entry) as the negativa square of the gravitational time dilation constant.
The spatial \eta_{11} 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?
The metric in spherical coordinates for a point mass at r=0 is
\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)
As expected, it is symmetrical wrt angle, and I recognize \eta_{00} (time entry) as the negativa square of the gravitational time dilation constant.
The spatial \eta_{11} 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?
