Monday, 9 March 2020

The Schwarzschild metric

 Event horizon

In this section 5.1 Carroll rediscovers the Schwarzschild metric and I discover a little fib of Carroll's which leads us to an event horizon, which he doesn't even mention. What a cheek!

It might be interesting to some that Schwarzschild was German and in German Schwarz means black and Schild means shield so his name means 'black shield'. Like Rotschild (red shield) the great bankers. Schwarzschild published his metric in the same year that Einstein published his theory of general relativity. Quick work. He died the same year. Perhaps the effort killed him, but I bet he was pleased.

FYI  the Schwarzschild metric  is$${ds}^2=-\left(1-\frac{2GM}{r}\right){dt}^2+\left(1-\frac{2GM}{r}\right)^{-1}{dr}^2+r^2{d\Omega}^2$$where ${d\Omega}^2$ is the metric on a two sphere$${d\Omega}^2={d\theta}^2+\sin^2{\theta}{d\phi}^2$$and, of course, the spatial coordinates are spherical polar, in all they are $\left\{t,r,\theta,\phi\right\}$.

The Schwarzschild radius is given by$$R_S=2GM$$That is also the radius of the event horizon of a black hole of mass $M$ so we are able to calculate how compressed the Sun would have to be to be a black hole: radius 3 km.

Read all about it: Commentary 5.1 The Schwarzschild metric.pdf (6 pages including the Riemann tensor)