Our universe appears to be a smooth manifold of four real dimensions with an intrinsic length structure; this assigns lengths to trajectories, a.k.a. smooth paths, on the smooth manifold. Some paths, described as light-like, have zero length. For a path of any length, one can scale the length by a positive scalar and get a length; at least for scalings close to 1, there will be paths of the thus-scaled length.
We also find two kinds of paths, time-like and space-like, with the curious property that the square of a time-like length and the square of a space-like length have values with opposite sign. Thus, in principle, lengths must have complex values; space-like lengths with one phase, and time-like lengths whose phase is a quarter-turn different so that their squares' phases will differ by a half turn. We can take one phase an eighth of a turn one way and the other conjugate to it to achieve this effect, we can chose one of space and time to be `real' and make the other `imaginary', we can even chose to make one be a sixth of a turn, the other a twelfth of a turn in the opposite direction. None the less, by adding lengths of two phases a quarter turn apart, we obtain a quadrant of the complex plane, bounded by the rays with the given phases. A general path on the smooth manifold has a length made up of some parts space-like, some time-like (and some light-like, but these don't contribute to its length), yielding a complex value in the quadrant between the phases of space and time. A cuboid of space-time, with three side-directions space-like, the other time-like, will have a 4-volume equal to the product of three space-like lengths and one time-like one; its phase will be four times the space-like phase plus or minus a quarter turn; this is positive real if the real phase is a sixteenth of a turn and the time-like one is three sixteenths in the opposite direction; and stepping both forwards by a quarter turn. Likewise, a space-like 3-cuboid has volume (which it would be nice to be able to describe as real and positive) whose phase is the cube of the space-like phase; so spatial volumes will be positive reals if the space-like phase is zero or a third of a turn (in one sense or the other); this will give 4-volumes the phase of time-like lengths.
and so on. However, crucially, one needs the complex numbers to describe lengths, and that raises a whole lot of interesting possibilities when one tries to describe what's going on. In particular, one's measure (the thing which does integration) is as apt to come out imaginary as to come out real; the phase of a measure combines three space-like phases with one light-like.
From the length structure, one can infer a `metric', which serves to encode the length structure, as follows: at each point on the smooth path, the path has the form (manifold: f(t) ←t :{scalars}) with our point being f(T) for some particular T; the tangent to the path is then f', for which f'(t) is a tangent to the manifold at f(t). The metric provides a scalar value as the length-rate of any given tangent, rate(f'(t)), which is worth thinking of as sqrt(f'(t)*f'(t)); this gives ({scalars}: rate(f'(t)) ← t :{scalars}) which we can integrate (with respect to t) over some range of scalars, corresponding via f with a portion of our trajectory. Now, this portion of path has a length, which doesn't depend on how we parameterise the path; so the intergral of rate(f'(t)) is independent of parameterisation.
A general re-parameterisation of (:f|) is just a monotonic (i.e. always increasing or always decreasing - you traverse a path from one end to the other without going backwards and forwards along it) mapping (f: s :{scalars}). This yields (f&on;s)' = f'&on;s.s' = (: f'(s(t)).s'(t) ← t :(:s|)), wherein s'(t) is a linear map from scalars to scalars - generally encoded as a scalar for some natural reasons propper to scalars - while f'(s(t)) is formally a linear map from scalars to the tangent bundle, turning a small change in s(t) into the change in f which would result; f'(s(t)).s'(t) formally maps a small change in t, via s'(t), to a small change in s(t), which f'(s(t)) duly transforms to the small change in f we wanted. However, a linear map from scalars to anywhere is in practice synonymous with the value it produces when given 1 as input (whence its value for any other scalar input follows directly by scaling 1's answer), so we just deal with s'(t) as a scalar and f'(s(t)) as a tangent.
We'll be integrating rate&on;(f&on;s)' = rate&on;((f'&on;s).s') over some range U, subsumed by (f:s|), and integrating rate&on;f' over (|s:U). If s' were constant, the range (|s:U) would span s' times as much of the scalars as U does; this requires rate((f'&on;s).s') to be equal, when s' is positive, to rate(f'&on;s).s', making rate respect positive scalings; when s' is negative, we need -s'.rate(f'&on;s) in place of s'.rate(f'&on;s) because reversing the direction in which we traverse a path should give the same length, not -1 times it. Thus rate, as a mapping from tangents at points on the smooth manifold to scalars, respects positive scalings and ignores reversal.
A chart-forming system of coordinates, in some neighbourhood, N, on the smooth manifold, is some indexed family x= (: ({scalars}: x(i) |N) ←i |I), for some index set I (archetipycally a natural so that x is a list of scalar fields), for which dx, the correspondingly-indexed family of gradients of these scalar fields, is linearly independent and spans G in our neighbourhood, N. Such systems are presumed to exist - our smooth manifold is presumed finite-dimensional. Now the index set can perfectly readilly be a collection of labels - e.g. I = {future, up, North, East} might be useful in some practical applications - but it's commonplace to use a natural (so as not to get distracted by the semantics of the labels), known as the dimension of the manifold, dim(M), making x = [x(0), x(1), ..., x(dim(M)-1)], with each x(i) being a scalar field on N. At any point, p in N, for each i in I, we can now look at y = (: (: x(j,p) ←j, j is not i :I) ←p :N) which delivers an indexed family of scalar values at each point of N, and ask for {q in N: y(q) = y(p)}. Since our co-ordinates were chart-forming, this must form a line in some neighbourhood of p, with x(i) varying along the line and all other x(j) constant on the line. The resulting line gives us a tangent, bx(i), in the kernel of every dx(j) except dx(i), which maps bx(i) to 1. The tangents (|bx:I) thus obtained provide a basis of the tangent bundle at each point of N; this basis is dual to the basis (:dx:I).
Geometry tells us that certain linear maps on the tangent bundle at each point respect our length-rate function: rate(u(t)) = rate(t) for every tangent t, with u a linear map from tangents to tangents; if rate&on;u = rate, I'll describe u as an isometry. We have already seen that negation, (: -t ←t :{tangents}), is an isometry. Thus for rate&on;f' and rate&on;(f&on;s)' to produce equal integrals, when given ranges of integration for which (|f&on;s:) = (|f:) - specifically, integrate we need to integrate rate&on;f' to be equal to the integral of rate&on;(f&on;s)' which is just rate&on;f'&on;s.s' = (: rate(f'(s(t))).s'(t) ← t :(:s|)), now integrating over the range (:f&on;s|) rather than (:f|). for which some paths' lenghts are positive, some are negative and some are zero; the last are called light-like paths. The paths with positive and negative lengths are known as time-like and space-like,
Written by Eddy.