Einstein equation point mass solution

[espen180]

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?

 

[Nabeshin]

At this radius, photons can orbit the point mass.

Not really, though. At r=2M, photons directed radially outward stay at constant radial coordinate, but I wouldn't call this an orbit. The photon sphere, where photons truly do move in circular orbits, is at r=3M.

 

[Entropia]

Yes, there are ways to compute the curvature at a specific radius without computing the entire Riemann tensor. One way is to use the Ricci scalar, which is a scalar quantity that describes the overall curvature of a space or spacetime. It can be calculated using the metric components and their derivatives at a specific radius. Another method is to use the Kretschmann scalar, which is a scalar quantity that measures the strength of the gravitational field at a specific point. It can also be calculated using the metric components at a specific radius. Additionally, there are other curvature invariants that can be calculated at a specific radius, such as the Weyl tensor. These methods can provide a simpler way to understand the overall curvature at a specific radius without having to compute the entire Riemann tensor.