Abstract
We describe a procedure naturally associating relativistic Klein-Gordon equations in static curved spacetimes to non-relativistic quantum motion on curved spaces in the presence of a potential. Our procedure is particularly attractive in application to (typically, superintegrable) problems whose energy spectrum is given by a quadratic function of the energy level number, since for such systems the spacetimes one obtains possess evenly spaced, resonant spectra of frequencies for scalar fields of a certain mass. This construction emerges as a generalization of the previously studied correspondence between the Higgs oscillator and Anti-de Sitter spacetime, which has been useful for both understanding weakly nonlinear dynamics in Anti-de Sitter spacetime and algebras of conserved quantities of the Higgs oscillator. Our conversion procedure ("Klein-Gordonization") reduces to a nonlinear elliptic equation closely reminiscent of the one emerging in relation to the celebrated Yamabe problem of differential geometry. As an illustration, we explicitly demonstrate how to apply this procedure to superintegrable Rosochatius systems, resulting in a large family of spacetimes with resonant spectra for massless wave equations.
Highlights
Geometrization of dynamics is a recurrent theme in theoretical physics
Our procedure is attractive in application to problems whose energy spectrum is given by a quadratic function of the energy level number, since for such systems the spacetimes one obtains possess evenly spaced, resonant spectra of frequencies for scalar fields of a certain mass
This construction emerges as a generalization of the previously studied correspondence between the Higgs oscillator and anti–de Sitter spacetime, which has been useful for both understanding weakly nonlinear dynamics in anti–de Sitter spacetime and algebras of conserved quantities of the Higgs oscillator
Summary
Geometrization of dynamics is a recurrent theme in theoretical physics. While it has underlain such fundamental developments as the creation of general relativity and search for unified theories of interactions, it has a more modest (but often fruitful) aspect of reformulating conventional, well-established theories in more geometrical terms, in the hope of elucidating their structure. The reason for our emphasis on systems with quadratic spectra is that, in application to such systems, our geometrization program generates Klein-Gordon equations whose frequency spectra are linear in the frequency level number, and the spectrum is highly resonant (the difference of any two frequencies is integer in appropriate units). It is well known that in the context of weakly nonlinear dynamics, highly resonant spectra have a dramatic impact, as they allow arbitrarily small nonlinear perturbations to produce arbitrarily large effects over long times. This feature has been crucial in the extensive investigations of the AdS stability problem in the literature (for a brief review and references, see [16]). We conclude with a review of the current state of our formalism and open problems
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