# Is a solution to the Klein-Gordon equation homeomorphic (or even diffeomorphic) to a solution of an equation with a different covariance group?

Consider some solution $\psi(x,t)$ to the linear Klein-Gordon equation: $-\partial^2_t \psi + \nabla^2 \psi = m^2 \psi$. Up to homeomorphism, can $\psi$ 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 $\psi : M \rightarrow \mathbb{R}$, where $M = (\mathbb{R}^4, \eta)$ is Minkowski spacetime. Now consider a coordinate transformation---but not necessarily a Lorentz transformation. Instead, consider any coordinate transformation modeled by an autohomeomorphism $\phi : \mathbb{R}^4 \rightarrow \mathbb{R}^4$, where $\mathbb{R}^4$ has the standard induced topology. This gives $\psi'(x', t') : \mathbb{R}^4 \rightarrow \mathbb{R}$. The question is this: if one has free choice of metric on the new coordinates established by $\phi$, is there such a $\psi'$ 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 $M$?

(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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You might want to look at Morse and Feshbach's Methods of Theoretical Physics -- it was written back in the day when questions like this were important in physics, and actually answers them quite thoroughly! – Alex Nelson Jun 26 '13 at 15:34