A family of timeless six-dimensional universes

This paper highlights a family of timeless universes connected by the same simple change of coordinates which allows to transfer between the two metrics involved in the description of each such universe. The first example can be called the simple-basic timeless universe. This universe allows one to create some other timeless universes, one of them being without matter, one in expansion and another one of the wormhole type. In all these universes, there are regions where classical physical time exists, but a global time coordinate indicating the temporal order of events does not exist. This happens because, in these universes, we can mathematically construct future oriented time-like loops and closed chains of future oriented time-like curves. The example of timeless expanding universe shows that time, in general, is not a consequence of space expansion.