The Wave Equation and Permutation of Rays

The usual wave equation in one space and one time dimension is

This same equation applies to spherically symmetrical waves in three-dimensional space if we replace x with the radial distance r from the center of the disturbance, and if we replace the wave function ψ with ϕ = rψ. Therefore, the solution of the above equation is relevant to many important physical phenomena. The general analytical solution is not difficult to find, but we can gain useful insights into wave propagation if we consider this equation in the form of finite differences.

For any point x,t in the medium we can consider the four neighboring points at incremental distances in space and time as shown below.

Expressing the second partial derivative of the wave function with respect to x in terms of finite differences around the central point, we have

Likewise the second partial derivative of the wave function with respect to t can be represented in the form

If we then choose our units of space and time so that dx = c dt, we can substitute these finite difference expressions into the wave equation, simplify, and multiply through by (dt)² to give

Notice that the value of the wave function at the central point drops out, so the finite difference equation operates only on the four corners of the surrounding cell. Thus the wave equation simply expresses the requirement that the sum of the values of the wave function just before and just after a given event equals the sum of the values on either side of the event. (The average of its neighbors in time equals the average of its neighbors in space.) With this simple rule, we can examine how a disturbance propagates. A grid of such cells is shown below.

The values shown in each node of the grid represent the value of the wave function at that point in space and time. A disturbance of magnitude 1 is posited at the origin, with zero specified at every other spatial location at that instant. (Alternately, we can specify zero for every other instant at the origin.) Assuming both spatial and temporal symmetry, the resulting propagation of this disturbance is indicated by the nodes marked with "1/2". Since the space increment dx equals c times the time increment, and since the disturbance propagates at ±dx/dt, this shows that the disturbance propagates with the speed c. Also, it's clear that solutions of the wave equation are linear, in the sense that the sum of any two solutions is another solution.

It's often most useful to express the basic difference equation in one of the two forms

These equations show explicitly that the change in the wave function along one edge of a diamond cell equals the change along the opposite edge, as illustrated below.

By transitivity with adjoining cells, it immediately follows that the change in the wave function is invariant along opposite edges of any rectangular region oriented orthogonally to these cells. Consequently, the sums of the wave function values on opposite vertices of any such rectangular region are equal, as shown below.

In general we have

for any intervals Δt₁ and Δt₂, no matter how large. It's interesting that partial differential equations are often regarded as characteristic of local processes, and the wave equation is the archetype partial differential equation, but we find that the basic relationships can just as well be expressed in non-local form. This illustrates how problematical it is to define a flow of causality for a deterministic process, even when the governing equation can be expressed in the form of a partial differential equation. It also suggests that the "topology of implication" of the wave function is not Euclidean, but is more accurately represented by the indefinite "metric" of Voigt and Lorentz, with singular measures along the diagonals.

Of course, as d'Alembert observed, in terms of the coordinates u = x + t and v = x – t the wave equation (with unit c) reduces to

which implies that ∂ψ/∂u is strictly a function of u, and likewise ∂ψ/∂v is strictly a function of v. Consequently the entire solution can be expressed as the sum of two single-variable functions, ψ(x,t) = f(x+t) + g(x–t). Another way of encoding this solution is to say that to each point x in the one-dimensional space we assign two values, ∂ψ/∂u and ∂ψ/∂v. The entire spacetime solution is projected (along lines of constant u and lines of constant v) from this single time-slice. Also, notice in particular that if ψ(u,v) is a solution of the wave equation, then so is ψ(αu,βv) for any constants α,β.

It's not difficult to show that if ψ(x,t) is a solution of the wave equation in terms of x and t, and if we postulate a linear transformation between x,t and X,T of the form

then ψ(X,T) is a solution of the wave equation in terms of X and T if and only if

From this it follows that

and therefore we have (A+C) = ±(B+D) and (A–C) = ±(B–D). Consequently the transformation from x,t to X,T can be written in the form

which confirms that any re-scaling of the u,v variables preserves the solution.

Although the Voigt-Lorentz transformation is singled out as the continuous linear inevitable transformation that preserves the wave equation, it's obvious from the preceding that the wave equation is actually preserved by a much larger class of transformations. In fact, returning to our finite difference grids, we can see that the wave equation is preserved under any permutation of the constant-u lines, and under any permutation of the constant-v lines (i.e., the lines of constant x+t and of constant x–t). In physical terms, the wave equation if preserved under any permutation of the light rays.

With two spatial dimensions and one time dimension we have a pencil of light rays intersecting at each point, forming forward and backward cones. Again the transverse derivatives are constant along light rays, so the entire spacetime solution can be represented by the projection onto a single time slice, where at each point we have a range of angles from 0 to 2π. In effect, this is a three dimensional space where one of the dimensions is finite and curled up cylindrically. A light ray in spacetime maps to a single point in this projected space, whereas a point (event) in spacetime can be regarded as the entire pencil of rays that intersect at that point, which imply that it maps to thread that wraps around the cylindrical dimension of the projected space as illustrated below.

With three spatial dimensions and one time dimension the rays of light comprise an expanding shell of light converging on and emanating from each point (event). These rays can be projected onto a single 3D time slice with a closed curled-up dimensional spherical surface at each point. In other words, it can be modeled as E₃ x S₂, the Cartesian product of three-dimensional Euclidean space and two-dimensional spherical surface. Again each individual light ray is a point in this projected space, whereas the pencil of rays intersecting at a given event maps to a closed manifold that wraps around the spherical sub-space. (If at each point in the three-dimensional space we have not only the two-dimensional manifold of spatial directions, but also a spin orientation about each direction, then the full six-dimensional space can be represented by the Cartesian product E₃ x S₂ x S₁.)

In full 3+1 dimensional spacetime the spherically symmetrical wave equation can be reduced to the same form as the 1+1 dimensional equation, except that the space wave function is normalized by r, and the parameter x is replaced with the radius r, so each "light ray" actually represents a sequence of expanding and converging shells. The above reasoning shows that we can permute any of these spatially concentric sequences of shells and still preserve the wave equation. Of course, we can also apply a Lorentz transformation and then a permutation, so we can effectively permute any two "pencils" of light shells that are within each others past or forward light cones. In this sense, we could say that two events (associated with their respective light cones) are causally ordered if and only if they can be permuted while preserving the wave equation.

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