Gravitational time dilation from the metric

[bueller11]

How does one go about finding what the gravitational time dilation is from the metric? Is it simply $t'/t_0=1/\sqrt{g_{tt}}$? It seems that could be true for static metrics, but perhaps not more dynamic ones like the Kerr metric. My confusion on this arises on how to treat the time cross terms (e.g. the $g_{t\phi}$ term in the Kerr metric).

If that simple formula is not a general truth, will someone point me in the direction of some textbooks or papers that describe how gravitational time dilation is derived from a metric?

 

[Bill_K]

Time dilation is the ratio of proper time to coordinate time, dt/dτ, which depends on two things: the world line of the observer, and the choice of time coordinate. In the Kerr metric for a stationary observer (dr = dθ = dφ = 0) you'll have dt/dτ = 1/√g_tt, as you said.