Calculation of eigenvalues of a strongly chaotic system using Gaussian wavepacket dynamics

We apply the approximate dynamics derived from the Gaussian time-dependent variational principle to the Hamiltonian $\hat H= {1/2}(\hat p_x ^2+ \hat p_y ^2)+ {1/2}\hat x^2\hat y^2$, which is strongly chaotic in the classical limit. We are able to calculate, essentially analytically, low-lying eigenvalues for this system. These approximate eigenvalues agree within a few percent with the numerical results. We believe that this is the first example of the use of TDVP dynamics to compute individual eigenvalues in a non-trivial system and one of the few such computations in a chaotic system by any method. There is a short self-contained discussion on the validity of Gaussian approximations in the paper.Comment: 19 pages, Revtex + 1 ps fig , Phys. Rev. E, to appear (1997

Null Cones in Lorentz-Covariant General Relativity

The oft-neglected issue of the causal structure in the flat spacetime approach to Einstein's theory of gravity is considered. Consistency requires that the flat metric's null cone be respected, but this does not happen automatically. After reviewing the history of this problem, we introduce a generalized eigenvector formalism to give a kinematic description of the relation between the two null cones, based on the Segre' classification of symmetric rank 2 tensors with respect to a Lorentzian metric. Then we propose a method to enforce special relativistic causality by using the naive gauge freedom to restrict the configuration space suitably. A set of new variables just covers this smaller configuration space and respects the flat metric's null cone automatically. In this smaller space, gauge transformations do not form a group, but only a groupoid. Respecting the flat metric's null cone ensures that the spacetime is globally hyperbolic, indicating that the Hawking black hole information loss paradox does not arise.Comment: groupoid nature of gauge transformations explained; shortened, new references, 102 page

Light Cone Consistency in Bimetric General Relativity

General relativity can be formally derived as a flat spacetime theory, but the consistency of the resulting curved metric's light cone with the flat metric's null cone has not been adequately considered. If the two are inconsistent, then gravity is not just another field in flat spacetime after all. Here we discuss recent progress in describing the conditions for consistency and prospects for satisfying those conditions.Comment: contribution to the Proceedings of the 20th Texas Symposium on Relativistic Astrophysics; 3 pages, 1 figur

Null Cones and Einstein's Equations in Minkowski Spacetime

If Einstein's equations are to describe a field theory of gravity in Minkowski spacetime, then causality requires that the effective curved metric must respect the flat background metric's null cone. The kinematical problem is solved using a generalized eigenvector formalism based on the Segr\'{e} classification of symmetric rank 2 tensors with respect to a Lorentzian metric. Securing the correct relationship between the two null cones dynamically plausibly is achieved using the naive gauge freedom. New variables tied to the generalized eigenvector formalism reduce the configuration space to the causality-respecting part. In this smaller space, gauge transformations do not form a group, but only a groupoid. The flat metric removes the difficulty of defining equal-time commutation relations in quantum gravity and guarantees global hyperbolicity