A Causal 5D Interval from a Fourier–Green's Function Solution of a Wave Equation
Alexander Harrison
*
School of Engineering, The University of Newcastle, Newcastle, NSW, Australia and Berkeley Vale Computational, NSW 2261, Australia.
*Author to whom correspondence should be addressed.
Abstract
A Fourier transform of a 5D wave equation ◻5 Q(X) = Ĵ(X) in which Q and J are a 5D scalar field and source, respectively, produces a retarded Green’s function G5R(X, X′) which describes how the solution’s value at a point P2 depends on the source term J at another point P1. For point P1, the coordinate X1 = (ṟ1, t1, τ1), where ṟ1 is the 3D coordinate, t1 is time, and τ1 is its 5D axis. Dirac delta functions pin each coordinate of the P1 source exactly at its 5D coordinate X1. Upon activation of the delta functions, the field Q(X2) at location X2 is a function of both the classical source amplitude Ŝ1 localised at X1 and the value of G5R(X2, X1). The 5D function G5R contains a Heaviside function that enforces causality. A 5D squared interval D, derived from the retarded Green’s function, is functionally analogous to the 4D Minkowski interval in special relativity. The interval D is of considerable interest since it provides the bounds for causality while allowing adjustment of D’s components to ensure G5R is always real. Whenever D > 0, the boundary-value problem approaches a light-speed limit and admits a non-zero solution for the field amplitude at X2 that depends on the strength of P1’s classical source amplitude at X1. Other cases for D = 0 or D < 0 are examined.
Keywords: 5D wave equation, scalar field, Fourier transform, retarded Green’s function, causal interval, Minkowski interval, Dirac delta function, Wick rotation, fifth-dimensional space, causal propagation