GR Time Dilation: Does It Occur in Inertial Frames?

[CJames]

I am curious as to just when time dilation plays a part as an effect of GR. I am told that clocks tick slower in a gravitational field. Is this an accurate portrayal, or would it be more accurate to say that time ticks slower in all accelerating reference frames? That is, does GR time dilation occur only when a reference frame is diverted from a geodesic?

I'm asking because I wonder if a body falling in a gravitational field would experience time dilation, since this is evidently an inertial reference frame. My guess is that time dilation only occurs for objects that "feel" acceleration, like us as we sit in our chairs. Of course, SR would dictate that relative velocity due to gravitational acceleration would cause time dilation, but that's a different matter entirely.

Also, with respect to what reference frame is time dialated in an accelerated reference frame? An inertial reference frame of the same instantaneous velocity?

 

[pervect]

Time dilation is not a consequence of simply accelerating. Time dilation is related to acceleration*distance, which can be interpreted in the Newtonian limit as "gravitational potential energy".

There is some discussion of this in

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/gratim.html

but it does not include the interesting case of the accelerated observer.

Gravitational time dilation depends on the coordinate system one uses.

Given a metric in which the lorentz interval ds^2 is given by

ds^2 = g_00 d^t^2 - g_11 dx^2 - g_22 dy^2 - g_33 dz^2

and in a system of units where c=1

gravitational time dilation for an observer stationary in the coordinate system is proportional to the square root of the metric coefficient g_00.

The standard metric associated with an accelerating observer is the so-called Rindler metric

ds^2 = (1 + gz) dt^2 - dx^2 - dy^2 - dz^2

the acceleration is of magnitude g and in the z direction. (Again, c=1).

In this standard coordinate system, at z=0, there is no gravitational time dilation. Clocks "above" the origin of the observer tick faster, clocks "below" the origin of the observer tick slow - i.e. there is apparent time dilation for z<0, apparent time accerlation for z>0.

Note that from an observer who uses inertial coordiantes, there is no apparent gravity, and thus no gravitational time dilation. Here the metric is the standard metric from SR

ds^2 = dt^2 - dx^2 - dy^2 - dz^2

For the gravitational time dilation of a massive body, the usual coordinate system is the Schwarzschild coordiantes, in which

ds^2 = (1-2M/r)dt^2 + functions of dr^2, dtheta^2, dphi^2

so g_00 = sqrt(1-2M/r)

(compare this to the formulas in the hyperphysics link I quoted earlier (which uses more standard units in which c is not necessarily 1).