How?

Einstein created the equation for gravitational time dilation based on special relativity. Einstein reasoned that because space-time is bent with motion, the same must be with gravitation. The curvature of space-time is described by the metric.

The metric is written as [latex]ds^2[/latex], and this describes the length squared of an infinitesimal curve in space-time. When there is no curvature, [latex]ds^2[/latex] takes the form

[latex]ds^2=-dt^2 + dx^2 + dy^2 + dz^2[/latex]

because of the pythagorean theorem.

The above is also written as

[latex]ds^2=\sum dx^a dx^b[/latex]

When there is curvature, we have to multiply the above by something called the metric tensor, [latex]g_{ab}[/latex]. It is a four by four matrix that corrects the pythagorus in curved space-time.

The metric becomes

[latex]ds^2=g_{ab} \sum dx^a dx^b[/latex]

There are various components of the metric tensor, each corresponding to coordinates. For example, [latex]g_{00}[/latex] is for time, [latex]g_{11}[/latex] is for x, and so on.

Now that you understand the metric tensor, you need to know that when you multiply [latex]dt^2[/latex] by [latex]g_{00}[/latex], you get [latex]d\tau ^2[/latex].Therefore,

[latex]d\tau = \sqrt{g_{00}} \ dt[/latex]

and that is how he found out the equation for time dilation.

Edited February 7, 201411 yr by Endercreeper01

How?

 

He wrote and published papers describing this.

Wikipedia has a good overview: http://en.wikipedia.org/wiki/History_of_general_relativity

Einstein came up with the idea of gravitational slowing of time from his Equivalence Principle and the elevator thought experiment. See, for example:

 

http://eagle.phys.utk.edu/guidry/astro616/lectures/lecture_ch15.pdf