All physical processes are local

After the geometry axiom, here is my next axiom.

Locality axiom: All physical processes are local.

Locality means that physical processes are only affected by nearby processes. It means that there is no action-at-a-distance. It means that your personality is not determined by your astrological sign. It means that the Moon cannot cause the tides unless some force or energy is transmitted from the Moon to the Earth at some finite speed.

Maxwell's electromagnetism theory of 1861 was local because the effects are transmitted by local fields at the speed of light. Newton's gravity was not. The concept of locality is closely related to the concept of causality. Both require a careful understanding of time. Quantum mechanics is local, if properly interpreted.

Time is very different from space. You can go back and forth in any spatial direction, but you can only go forward in time. If you want to go from position (0,0,0) to (1,1,1), you can go first in the x-direction, and then in the y-direction, or go in any order of directions that you please. Symmetries of Euclidean space guarantee these alternatives. But if you want to go from (0,0,0) at time t=0 to (1,1,1) at t=1, then your options are limited. Locality prohibits you from visiting two spatially distinct points at the same time. Time must always be increasing along your path.

Locality demands that there is some maximum speed for getting from one point to another. Call that speed c, as it will turn out to be the (c for constant) speed of light. No signal or any other causation can go faster than c.

Thus a physical event at a particular point and time can only influence events that are within a radius ct at a time t later. The origin (0,0,0) at t=0 can only influence future events (x,y,z,t) if x²+y²+z² ≤ c²t², t > 0. This is called the forward light cone. Likewise, the origin can only be influenced by past events in the backward light cone, x²+y²+z² ≤ c²t², t < 0. Anything outside those cones is essentially nonexistent to someone at the origin. Such is the causal structure of spacetime.

Since the world is geometrical, there should be some geometry underlying this causal structure. The geometry is not Euclidean. The simplest such geometry to assume a metric where distance squared equals −dx²−dy²−dz²+c²dt². This will be positive inside the light cones. It does not make sense to relate to something outside the light cones anyway. Four-dimensional space with this non-Euclidean geometry is called Minkowski space.

The symmetries of spacetime with this metric may be found by an analogy to Euclidean space. Translations and reflections preserve the metric, just as with Euclidean space. A spatial rotation is a symmetry: (x,y,z,t) → (x cos u − y sin u, x sin u + y cost u, z, t). We can formally find additional symmetries by assuming that time is imaginary, and the metric is a Euclidean one on (x,y,z,ict). Poincare used this trick for an alternate derivation of the Lorentz transformations in 1905. Rotating by an imaginary angle iu, and converting the trigonometric functions to hyperbolic ones:

(x,y,z,ict) → (x, y, z cos iu − ict sin iu, z sin iu + ict cos iu)
    = (x, y, z cosh u + ct sinh u, iz sinh u + ict cosh u)

This is a Lorentz (boost) transformation with rapidity u, or with velocity v = c tanh u. The more familiar Lorentz factor is cosh u = 1/√(1 − v²/c²).

Imaginary time seems like just a sneaky math trick with no physical consequences, but there are some. If a physical theory has functions that are analytic in spacetime variables, then they can be continued to imaginary time, and sometimes this has consequences for real time.

Thus locality has far-reaching consequences. From the principle, it appears that the world must be something like Minkowski space, with Lorentz transformations.

Next, we will combine our two axioms.