Is a solution to the Klein-Gordon equation homeomorphic (or even diffeomorphic) to a solution of an equation with a different covariance group?
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Consider some solution ψ(x,t)ψ(x,t) to the linear Klein-Gordon equation: −∂2tψ+∇2ψ=m2ψ−∂t2ψ+∇2ψ=m2ψ. Up to homeomorphism, can ψψ serve as a solution to some other equation that has a different covariance group?
In other words: consider some solution to the linear Klein-Gordon equation, and model that solution by a mapping ψ:M→Rψ:M→R, where M=(R4,η)M=(R4,η) is Minkowski spacetime. Now consider a coordinate transformation---but not necessarily a Lorentz transformation. Instead, consider any coordinate transformation modeled by an autohomeomorphism ϕ:R4→R4ϕ:R4→R4, where R4R4 has the standard induced topology. This gives ψ′(x′,t′):R4→Rψ′(x′,t′):R4→R. The question is this: if one has free choice of metric on the new coordinates established by ϕϕ, is there such a ψ′ψ′ that solves another equation with some different covariance group?
Further, if this is possible for simple, plane-wave solutions, then what are the conditions under which it is no longer possible? What about linear combinations of plane-wave solutions? What about the case of multiple solutions, the values of one being coincident with the values of the other(s) in MM?
(Note: empirical adequacy is irrelevant; this is just a question about the conditions under which the values of a scalar-field solution to a partially-differential equation with some particular covariance group can also act as the solution to another equation with some different covariance group, while preserving their topological (or perhaps even differential) structure.)
In other words: consider some solution to the linear Klein-Gordon equation, and model that solution by a mapping ψ:M→Rψ:M→R, where M=(R4,η)M=(R4,η) is Minkowski spacetime. Now consider a coordinate transformation---but not necessarily a Lorentz transformation. Instead, consider any coordinate transformation modeled by an autohomeomorphism ϕ:R4→R4ϕ:R4→R4, where R4R4 has the standard induced topology. This gives ψ′(x′,t′):R4→Rψ′(x′,t′):R4→R. The question is this: if one has free choice of metric on the new coordinates established by ϕϕ, is there such a ψ′ψ′ that solves another equation with some different covariance group?
Further, if this is possible for simple, plane-wave solutions, then what are the conditions under which it is no longer possible? What about linear combinations of plane-wave solutions? What about the case of multiple solutions, the values of one being coincident with the values of the other(s) in MM?
(Note: empirical adequacy is irrelevant; this is just a question about the conditions under which the values of a scalar-field solution to a partially-differential equation with some particular covariance group can also act as the solution to another equation with some different covariance group, while preserving their topological (or perhaps even differential) structure.)
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