Time dilation on accelerated reference frames

hover

Hey all,

I made a post earlier today on gravitational time dilation. It got me thinking that in any accelerated frame of reference there must also be a time dilation, due to the equivalence principle. This can simply be a elevator accelerating through space or it can be caused by centripetal force. I was wondering what these equations are. I've tried searching for these equations but every time i searched, i somehow end up with the equation for gravitational time dilation.

What are the equations?

Loren Booda

This reminds me of the twin paradox, which to solve one must somewhere apply acceleration to return to Earth. It requires more than just special relativity to reckon. Acceleration involves general relativity, but some simple solutions are known.

bernhard.rothenstein

Could that be of interest arXiv:physics/0607288v1 [physics.class-ph]
Comments appreciated

SpitfireAce

I don't get the question... according to the equivalence principle an accelerated reference frame is equivalent to a uniform gravitational field... so the equations for time dilation in an accelerated reference frame would be the same as those for the corresponding uniform gravitational field, right?...

hover

The answer of both equations would be the same because both situations are the same/equivalent. But say for example someone is in a box accelerating in space. He knows he is truly accelerating through space because before he got in the box it was already accelerating. Since he knows he is accelerating and not in a gravitational field, he knows that there is no planet with mass pulling him. So no mass and no gravitational constant. In the gravitational time dilation equation, there is variable M for mass and G for gravitational constant-

t'=t/sqrt(1-(2GM/rc^2))

pervect

Here's one simple way to get the answer. Do your work in an inertial frame, then there is no time dilation except for that due to velocity.

The rotating case is particularly well suited to this approach, where you probably have all the dimensions measured in the lab frame anyway.

In an inertial frame, there is no gravity, or gravity like forces, so there is no time dilation. It's important to realize this, to avoid false "double counting" of the effects.

In the elevator, it may be natural to use non-inertial coordinates (much more so in the rotating case).

I'm going to skip over a long discussion of how these non-inertial coordinates are defined, and skip on to the answer using them.

It turns out that the accelerating elevator can be described by the Rindler metric.

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

This metric actually completely defines the coordinate system used (this may not be obvious to you yet, unfortunately).

The result of this metric is that clocks tick at a rate of 1+gz in this coordinate system. You can think of the z coordinate as being the altitude, so "higher" clocks tick faster. Lower clocks tick slower, and eventually stop at some negative value of z. This marks the "rindler horizon", which is a topic in and of itself.

You should think of the coordinate system of an accelerated observer as being only defined for points "close enough" to the observer. Any point that is lower than the Rindler horizon is "too far away".

Note again that the very concept of time dilation depends on the coordinates you use, it's not a "geometric" concept. So in the elevator case, it's only when one uses non-inertial coordinates that one sees anything simlar to gravitational time dialtion, in an inertial coordinate system there is none.