#### shounak

Hello All,

I was going through a video on singularity in YouTube when I came across this equation. I cannot quiet recall what is this equation. Can anyone please explain to me?

delta s^2= - (1-rs/r)delta t^2 + 1/(1-rs/r)delta r ^2

where rs (s is the subscript)

It deals with the schwarzschild radius

#### topsquark

Forum Staff
Hello All,

I was going through a video on singularity in YouTube when I came across this equation. I cannot quiet recall what is this equation. Can anyone please explain to me?

delta s^2= - (1-rs/r)delta t^2 + 1/(1-rs/r)delta r ^2

where rs (s is the subscript)

It deals with the schwarzschild radius
(There is some work being done on the site. Neither LaTeX nor uploading an image seems to work right now.)

This is a shortened form of the Schwarzchild metric where r_s = 2GM is the Schwarzchild radius. (The units here are set to c = 1.) In this case we are restricting motion to moving radially inward or outward. The metric tells you how to define a distance between two points in space-time (technically in relativistic terms it's called the "interval" between two "events" in space-time.) This is a GR version of how t and r will behave to produce such things as time dilation or length contraction and the like.

The most famous use of this is in terms of black holes. If an object lies entirely inside of its Schwarzchild radius it is a black hole and in this case r_s is the distance from the center of the hole where the escape speed is equal to the speed of light, meaning it cannot escape. But if we have an object that doesn't lie entirely inside its Schwarzchild radius we can still define r_s and work with it. In this case r_s doesn't act like a singularity.