Greg Egan on Nostr: It’s a famous result in Newtonian gravity that if you drop an object down a ...

It’s a famous result in Newtonian gravity that if you drop an object down a borehole in a ball of uniform density it will oscillate at the same rate as an orbiting body grazing the surface.

But what happens in General Relativity?

The answer lies in the 2nd Schwarzschild metric!

In 1916, shortly before his death, Schwarzschild published *two* exact solutions to Einstein’s equations for gravity.

The first was for a point mass, and it’s now famous for describing a black hole.

The second was for a solid ball of uniform density.

You can find these two metrics by first writing down what an unchanging, spherically symmetric metric must look like, with two unknown functions.

You then compute the corresponding Einstein tensor, which describes aspects of the space-time curvature that need to be matched by the density and the pressure (in the radial direction, and the directions perpendicular to it) of the matter responsible for that curvature.

Setting density and pressure to zero gives the first Schwarzschild metric.

Constant non-zero density, and equal pressures in the radial and non-radial directions, give the second.