Space-Times with non-trivial topology and compact dimensions.

Wormholes pop up in many areas of gravitation. They are certainly important when studying the topology of space or the fluctuations in a quantum theory of gravity. Some theories related to quantum gravity, such as string theory actually require extra dimensions to be consistent. You may ask why we only perceive four dimensions and not more. Well, the most likely answer is that they are Kaluza-Klein compact.

What we have done is study wormhole solutions to Einstein gravity when there are compact extra dimensions present. If you look at my 4D wormhole page you will see that the models there all required matter which violates some energy condition. It is possible, however, if you add extra-dimensions this violation can be minimized or even eliminated. The extra dimensions in our models give an extra parameter, namely the compact scale factor, which may allow for energy condition respecting wormholes.

If the extra dimensions are fixed in size, then one has a direct product manifold and the new parameter does little to change the wormhole from its four dimensional counterpart, save for one interesting point. In four dimensions the size of the wormhole throat is unrestricted even if you wish the weak energy condition (WEC) to be respected at the throat (which leads to a wormhole with minimal energy condition violation). With the extra dimensions, the size of the throat (in the non-compact dimensions) is restricted for WEC respecting throat. Not good news if you want macroscopic wormholes.

If the extra dimensions are dynamic, energy conditions are affected. In fact, they are adversely affected. In the class we studies, the presence of the extra dimensions actually tended to push the solution towards WEC violation. The reason is that the scale factor is not arbitrary. The solution should be patched to vacuum at some point since the matter shouldn't continue indefinitely (all astronomical bodies have some sort of vacuum-matter boundary, even if it's not sharp!). The scale factor is governed by the vacuum so this extra parameter is not completely free. If it were, it would be possible to make WEC respecting static (static in the non-compact dimensions) wormholes.

Nevertheless, if is known that energy condition violating systems do exist. Indirect evidence has even been measured in laboratories. I am referring to the Casimir effect.

NOTE: THIS PAGE IS CURRENTLY UNDER CONSTRUCTION.