Is time dilated if gravity is pulling equally in all diections?

[CosmicVoyager]

Greetings,

Is time dilated by gravity at the center of the earth where there is no acceleration?

Thanks

[DaleSpam]

Yes.

[bcrowell]

Gravitational time dilation depends on the difference in gravitational potential. It doesn't have anything to do with the gravitational field.

[Pengwuino]

So I suppose to more advanced question might be what is the time dilation between an observer in a spherically symmetric cavity and an observer out at infinity in minkowski space. I know the center of the earth isn't the same as a cavity but the spherical symmetry concept works the same.

[Q-reeus]

So I suppose to more advanced question might be what is the time dilation between an observer in a spherically symmetric cavity and an observer out at infinity in minkowski space. I know the center of the earth isn't the same as a cavity but the spherical symmetry concept works the same.
I take it by cavity you mean the interior region of a gravitating spherical mass shell. In the case of a thin shell, time dilation within the equipotential, Minkowski metric interior region is taken as uniformly just slightly in excess of that of the redshift factor applying at the shell exterior surface - as given at eg http://en.wikipedia.org/wiki/Gravitational_redshift
This really leads to a can of worms imho when boundary matching all components of an exterior Schwarzschild to interior Minkowski metric are considered, not just clock rate.

[D H]

Gravitational time dilation depends on the difference in gravitational potential.
Rhetorical question: the difference between what and what? Elaborating on bcrowell's answer, gravitational time dilation depends on the difference between the gravitational potential at the point of interest and a point infinitely removed from the mass in question.

It doesn't have anything to do with the gravitational field.
However, both gravitational time dilation and the gravitational field are functions of the gravitational potential.

The problem is that gravitational field locally does not contain enough information to determine the potential. For example, the gravitational potential due to the Earth is flat at the center of the Earth and at an infinite distance from the Earth. This means the gravitational acceleration is zero at the center of the Earth and at an infinite distance from the Earth. This does not mean that the gravitational potential at the center of the Earth is the same as the potential at an infinite distance from the Earth.

[bcrowell]

Rhetorical question: the difference between what and what? Elaborating on bcrowell's answer, gravitational time dilation depends on the difference between the gravitational potential at the point of interest and a point infinitely removed from the mass in question.

I'd say that what's relevant is the difference between the gravitational potential at one point and at another point. You don't need an asymptotically flat spacetime in order to talk about gravitational time dilation; all you need is a static spacetime. In a static but not asymptotically flat spacetime, there is no notion of the gravitational time dilation at a point P; there is only the notion of gravitational time dilation of P relative to some other point Q.

[D H]

I'd say that what's relevant is the difference between the gravitational potential at one point and at another point.
Point taken. There is no reason to require that the other be be a point at infinity. GPS is a good example. The GPS satellites are in orbit, so obviously not at infinity, yet we still can talk (must talk!) about both the special and the general relativistic affects on the rates at which the GPS satellite clocks tick.