See also the other posts in this category. Reading them in the order posted might be a good idea.
Archimerged finds that it is a bad idea to use well-known terms when subtle changes in the rules invalidate a lot of what people know about the terms. Since he is changing a bunch of rules, he needs some new terms. The systems in an ensemble (i.e., the universes in a multiverse) will be sets of points with certain additional structure.
Ark’s approach starts with a topological space, i.e., a set of points and a topology, which is a set of subsets of the space known as the open sets. He includes a manifold structure so as to be able to label the points. Next, instead of adding a metric, he adds the most abstract form of time and causality. Ark has decided to call a topological space with this added structure a photological space. Photology is an obsolete word for optics, and it seems appropriate.
So, a photological space is a topological space together with a photology, an ordered set (B, H, T, N) of four relations. (A relation is a set of ordered pairs of points. We denote the fact that the pair (p, q) is in the relation R by writing pRq.) The four relations of a are
- the before / after relation denoted bBa or aAb (when point b is before a and a is after b),
- the here relation denoted pHq (when p and q can be considered to be in the same place at different times),
- the there (or elsewhere) relation denoted pTq (when p and q can be considered to be in different places at the same time),
- and the neither here nor there (or null) relation denoted pNq (when p and q can neither be in the same place at different times, nor in different places at the same time, but are on a path light could follow).
Relative to a given point p, all sufficiently nearby points q must satisfy exactly one of pHq, pTq, or pNq. Points which satisfy pHq or pNq must also satisfy either pAq and qBp, or pBq and qAp.
A photology (B, H, T, N) has a close connection to an equivalence class of conformally equivalent metrics, and Ark’s approach uses a manifold together with a photology instead a manifold together with a metric.
This machinery is not really necessary, but it serves to emphasize the difference between this approach and standard general relativity. Ark used to say “all conformally equivalent metrics are really the same thing.” See Wald, General Relativity, p. 445: conformally equivalent spaces have identical causal structure, and their Weyl tensors are equal.