Dirac equation in exotic spacetimes

We find solutions of the Dirac equation in curved spacetime. In particular, we consider 1+1 dimensional sections of several exotic metrics: the Alcubierre metric, which describes a scenario that allows faster-than-light (FTL) velocity; the G\"odel metric, that describes a universe containing closed timelike curves (CTC); and the Kerr metric, which corresponds to the spacetime of a rotating black hole. Moreover, we also show that the techniques that we use in these cases can be extended to nonstatic metrics.Comment: 7 pages, 3 figure

Collision of plane gravitational and electromagnetic waves in a Minkowski background: solution of the characteristic initial value problem

We consider the collisions of plane gravitational and electromagnetic waves with distinct wavefronts and of arbitrary polarizations in a Minkowski background. We first present a new, completely geometric formulation of the characteristic initial value problem for solutions in the wave interaction region for which initial data are those associated with the approaching waves. We present also a general approach to the solution of this problem which enables us in principle to construct solutions in terms of the specified initial data. This is achieved by re-formulating the nonlinear dynamical equations for waves in terms of an associated linear problem on the spectral plane. A system of linear integral ``evolution'' equations which solve this spectral problem for specified initial data is constructed. It is then demonstrated explicitly how various colliding plane wave space-times can be constructed from given characteristic initial data.Comment: 33 pages, 3 figures, LaTeX. Accepted for publication in Classical and Quantum Gravit

Relative entropy as a measure of inhomogeneity in general relativity

We introduce the notion of relative volume entropy for two spacetimes with preferred compact spacelike foliations. This is accomplished by applying the notion of Kullback-Leibler divergence to the volume elements induced on spacelike slices. The resulting quantity gives a lower bound on the number of bits which are necessary to describe one metric given the other. For illustration, we study some examples, in particular gravitational waves, and conclude that the relative volume entropy is a suitable device for quantitative comparison of the inhomogeneity of two spacetimes.Comment: 15 pages, 7 figure

From Euclidean to Minkowski space with the Cauchy-Riemann equations

We present an elementary method to obtain Green's functions in non-perturbative quantum field theory in Minkowski space from calculated Green's functions in Euclidean space. Since in non-perturbative field theory the analytical structure of amplitudes is many times unknown, especially in the presence of confined fields, dispersive representations suffer from systematic uncertainties. Therefore we suggest to use the Cauchy-Riemann equations, that perform the analytical continuation without assuming global information on the function in the entire complex plane, only in the region through which the equations are solved. We use as example the quark propagator in Landau gauge Quantum Chromodynamics, that is known from lattice and Dyson-Schwinger studies in Euclidean space. The drawback of the method is the instability of the Cauchy-Riemann equations to high-frequency noise, that makes difficult to achieve good accuracy. We also point out a few curiosities related to the Wick rotation.Comment: 12 pages in EPJ double-column format, 16 figures. This version: added paragraph, two reference