# Periodic time for a generic soliton solution

1. Jul 23, 2015

### Einj

Hello everyone, I have a question regarding the possible periodicity of time in a generic metric.
Suppose that for some reason I have a solution to Einstein's equations of the kind (in Euclidean time):

$$ds^2_E=+f(r)dt_E^2+\frac{dr^2}{g(r)}+r^2(dx^2+dy^2).$$

Am I always allowed to assign some periodicity to the Eucledean time $t_E$ or is there any restriction?

For example, I know that there is a particular solution called "thermal AdS" which is nothing but the usual AdS metric (i.e. not a black hole with an horizon) to which a periodic time has been assigned.

When can I do that?

Thanks!

2. Jul 23, 2015

### WannabeNewton

A periodic euclidean time is equivalent to gluing the two ends of the path integral that prepares your state. This is then equivalent to preparing a thermal state. So periodic Euclidean time is always going to prepare a thermal state via the path integral. Whether or not it is a black hole depends on the actual path integral.

Note when I say path integral I mean that of the boundary CFT. So yes AdS and thermal AdS are the same as far as space-time geometry is concerned but in the latter the CFT is prepared in a thermal state. Black holes in the bulk are also dual to thermal CFTs (at least in most examples e.g. BTZ).

EDIT: I forgot to mention that the path integrals for thermal AdS and BTZ are closely related. In the former the path integral is a torus with one circle parametrized by Euclidean time and the other by the angular coordinate; the latter can be obtained simply by switching the roles of these two coordinates on the torus.

Last edited: Jul 23, 2015
3. Jul 23, 2015

### Einj

Thanks a lot for your reply! I guess my question was: since periodic Eucledean time always means thermal state for the boundary CFT, does this mean that if I have a generic soliton solution (no horizon) and I impose the time to be periodic this is going to be a thermal state? Do I have any constrain on the solution in order to be allowed to impose periodic time conditions?

Thanks a gain!