What is Gravitational Time Dilation

Gravitational time dilation is a phenomenon whereby time runs slower when in a higher gravitational potential. Put simply, the closer you are to a large body like the Earth the slower time runs, thus time runs slower for someone on the surface of the earth compared to someone in orbit around the earth.

Whilst the Einstein’s field equation for General relativity is very complicated the equation for time dilation is much simpler and bears certain resemblance to the equation for time dilation for relative motion. The equation is:

In this equation t0 is the proper time for an event, i.e. the time measured when observer and event are in the same gravitational potential and tf is the time as measured when at an infinite distance from any mass. The values G and c are again the Newtonian gravitational potential and speed of light respectively and M and r are the mass of the object you are near and r is your distance from said object respectively.

This equation can be written differently by applying Schwarzschild’s solution to general relativity which gives you the Schwarzschild radius; the radius of an object of mass M, for which if all mass is contained within that radius then the gravitational field is too strong that not even light can escape.

If one therefore applies this formula whose mass is contained within their Schwarzschild radius, i.e. Black Holes, then when something or someone reaches the edge of a black hole when r=rs time for the object/person appears to stop from the point of view of an observer further away. Thus you can never see something fall into a black hole as time stops at the edge of the black hole.

It then gets even stranger once the person has fallen into the black hole (although you will never see it) as the value in the square root operator becomes negative so the answer come back as an imaginary number and no one knows how to interpret an imaginary result to the real world.

1. Equation for gravitational time dilation taken from http://en.wikipedia.org/wiki/Gravitational_time_dilation