The Campbell–magaard Theorem Is Inadequate and Inappropriate as a Protective Theorem for Relativistic Field Equations

Given a particular prescription for the Einstein field equations, it is important to have general protective theorems that lend support to it. The prescription of data on a timelike hypersurface for the (n + 1)-d Einstein field equations arises in ‘noncompact Kaluza–Klein theory’, and in certain kinds of braneworlds and low-energy string theory. The Campbell–Magaard theorem, which asserts local existence and uniqueness of analytic embeddings of completely general n-d manifolds into vacuum (n + 1)-d manifolds, has often recently been invoked as a protective theorem for such prescriptions. But in this paper I argue that there are problems both with the Campbell–Magaard theorem and especially with these proposed applications of it. While I remedy some problems by identifying the required topology, spelling out what ‘local’ means, and offering a new, more robust and covariant proof, other problems remain insurmountable. The theorem lends only inadequate support, both because it offers no guarantee of continuous dependence on the data and because it disregards causality. Furthermore, the theorem is only for the analytic functions which renders it inappropriate for the study of the relativistic field equations of modern physics. Unfortunately, there are no known general theorems that offer adequate protection to the proposed applications’ prescription. I conclude by making some suggestions for more modest progress. PACS 04.50.+h, 04.20.Ex, 04.20.Cv ∗ Next addresses: as of Oct 2004, Peterhouse College Cambridge UK and DAMTP Cambridge UK; additionally, as of Jan 2005, Department of Physics, P-412 Avadh Bhatia Physics Laboratory, University of Alberta, Edmonton, Alberta, Canada. Email addresses: [email protected], [email protected]