Homogeneity and isotropy of time

The homogeneity and isotropy of space are well-known. The homogeneity of time is partly known but is confused by an “arrow of time” concept that is not applicable to space and time. The isotropy of time is unknown (and usually denied) also because of confusion with an inapplicable “arrow of time” concept.

I previously wrote about the Multiple dimensions of time. As space has three dimensions, so there are three dimensions of time, and they are the same three dimensions.

As space is homogeneous in each dimension, so is time. For example, it does not matter whether an experiment takes place “here” or 10 minutes north and 5 minutes east of “here” (if they are both inertial reference frames). The translational invariance of time is exactly like the translational invariance of space.

As space is isotropic, i.e., the same in all directions, so is time. For example, the duration measured by a clock is the same whether it is facing north, south, east, west, up, or down. And the duration is the same whether it is oriented horizontally, longitudinally, or transversely.

It is said that in classical mechanics time is reversible. This is a confused statement. What can be shown is that if a classical particle moves in one direction, its movement in the opposite (“reverse”) direction is also classical. Since both space and time are directional, that would equally well be true of space as of time but no-one says that space is reversible. It is best to leave questions of (ir)reversibility to thermodynamics, causality, etc.

Noether’s theorem shows that the homogeneity of space leads to the conservation of momentum, the homogeneity of time leads to the conservation of energy, and the isotropy of space leads to the conservation of angular momentum. I haven’t checked it yet but it is natural to expect that the isotropy of time leads to the conservation of rotational energy.