General Relativity Worksheets
Worksheet #3: Time

One of the consequences of general relativity is gravitational time dilation.  You are probably familiar with time dilation in special relativity.  In general relativity the time dilation is due to gravity: time is affected by the presence of gravitational fields.  In particular clocks at points with higher gravitational potentials run faster as compared to those at points with lower gravitational potential energy. 

We can see the effect of time by looking at Schwarzschild spacetime via the metric:

dτ² = [1 − 2M/r] dt² − [1 − 2M/r]^-1 dr² − r² dφ² .

Since we are considering clocks that are at a single position, we have that:

dτ² = [1 − 2M/r] dt² ,

and therefore we can compare the proper time interval, dτ, to the Schwarzschild coordinate (bookkeeper) time interval, dt, as

dτ = [1 − 2M/r]^1/2 dt .

We therefore find that, if we wish to compare the proper time intervals on two separate clocks,

dτ₁ = [1 − 2M/r₁]^1/2 dτ₂/[1 − 2M/r₂]^1/2 .