The wave equation in five dimensions

Abstract
It is well known that the primary object of introducing a 5-dimensional relativity scheme is to enable the paths of all particles, charged or uncharged, to be represented by the geodesics of the space. Having selected suitable forms for the coefficients of the fundamental tensor γμv to satisfy this requirement, we proceed to write down, in the appropriate tensor form, the well known wave equation for a changed particle. Now the 4-dimensional form of this equation is certainly not a simple one, and, if the physical world is really based on a 5-dimensional scheme, we should naturally expect that such a fundamental equation would have a particularly simple mathematical form, and the form which is naturally suggested is div grad ψ = 0. (1) The components of the fundamental tensor can be written γik = gik + Y55 α2 ϕi ϕk; Y5i = αY55ϕi γik = gik; γ5i = — α ϕi; γ55 = α2 ϕi ϕi + 1/γ55 }, (2) with γ55 and α both constant (independent of x) and |γμv | = γ55 |gik|.

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