# Does time pass fastest in isolated, resting space?

While it is fairly established that both fast movement and the presence of gravity make time pass slower as compared to a system at rest / free of gravity, does that mean that there is no way for time to pass faster than in vacuum, or does general relativity also have "faster" metrics?

To be more precise, is there any frame into which one could go for a while, and on returning to vacuum less time would have passed there?

You need to be a little careful with your definition of vacuum. For instance inside a spherical shell of matter spacetime is flat, however time still runs more slowly than it does outside the shell. I'm assuming you have no such trickery in mind, and by vacuum you mean the usual concept of far (effectively infintely far) from any matter.

To a good approximation a time interval measured at some other point in the universe is related to the time interval measured by you by:

$$\Delta t_0 = \Delta t \left( 1 + \frac{2 \Phi}{c^2}\right)^{-1/2}$$

where $t_0$ is your time, $t$ is the time at the other point and $\Phi$ is the Newtonian gravitational potential at that point (relative to you). So for example, relative to infinity at some distance $r$ from the Earth the gravitational potential is:

$$\Phi = -\frac{GM}{r}$$

and therefore:

$$\Delta t_0 = \Delta t \left( 1 - \frac{2 GM}{c^2r}\right)^{-1/2}$$

and time runs more slowly as you nearer to the Earth. Relative to the vacuum in the sense you mean it gravitational potentials are (as far as we know) always negative so time always runs slow compared to flat spacetime.

I guess the question is whether there are any special cases e.g. if exotic matter exists or if spacetime has some non-trivial topology. However I know of none.

• "For instance inside a spherical shell of matter spacetime is flat, however time still runs more slowly than it does outside the shell". Why? This may be a new PSE question if you want. – Trimok Oct 9 '13 at 17:54
• @trimok: see physics.stackexchange.com/questions/69043/…. This isn't a hollow shell, but the same principle applies. – John Rennie Oct 10 '13 at 6:08
• Indeed I assume vacuum to be non-cheatable, i.e. there is neither gravity nor anything else. So you're suggesting that time acceleration were only possible if repelling mass existed? - Interestingly enough, this means the decay of such particles (and if there were stable ones we'd probably have discovered some by now, wouldn't we?) would be accelerated even more by their own presence... – Tobias Kienzler Oct 10 '13 at 7:32
• @TobiasKienzler: I'm not sure I want to speculate about what stress-energy tensor would be required to make time run faster than vacuum. I'd need to sit down with a blank sheet of paper and many, many cups of coffee. – John Rennie Oct 10 '13 at 9:41

Yes. It is usually considered that in Casimir vacuum time goes faster.