Infinite slabs and other weird plane symmetric space-times with constant positive density

We present the exact solution of Einstein's equation corresponding to a static and plane symmetric distribution of matter with constant positive density located below $z=0$. This solution depends essentially on two constants: the density $\rho$ and a parameter $\kappa$. We show that this space-time finishes down below at an inner singularity at finite depth. We match this solution to the vacuum one and compute the external gravitational field in terms of slab's parameters. Depending on the value of $\kappa$, these slabs can be attractive, repulsive or neutral. In the first case, the space-time also finishes up above at another singularity. In the other cases, they turn out to be semi-infinite and asymptotically flat when $z\to\infty$. We also find solutions consisting of joining an attractive slab and a repulsive one, and two neutral ones. We also discuss how to assemble a "gravitational capacitor" by inserting a slice of vacuum between two such slabs.Comment: 8 page

The electromagnetic energy-momentum tensor

We clarify the relation between canonical and metric energy-momentum tensors. In particular, we show that a natural definition arises from Noether's Theorem which directly leads to a symmetric and gauge invariant tensor for electromagnetic field theories on an arbitrary space-time of any dimension

On gauge invariant regularization of fermion currents

We compare Schwinger and complex powers methods to construct regularized fermion currents. We show that although both of them are gauge invariant they not always yield the same result.Comment: 10 pages, 1 figur

An alternative well-posedness property and static spacetimes with naked singularities

In the first part of this paper, we show that the Cauchy problem for wave propagation in some static spacetimes presenting a singular time-like boundary is well posed, if we only demand the waves to have finite energy, although no boundary condition is required. This feature does not come from essential self-adjointness, which is false in these cases, but from a different phenomenon that we call the alternative well-posedness property, whose origin is due to the degeneracy of the metric components near the boundary. Beyond these examples, in the second part, we characterize the type of degeneracy which leads to this phenomenon.Comment: 34 pages, 3 figures. Accepted for publication in Class. Quantum Gra

On the energy-momentum tensor

We clarify the relation among canonical, metric and Belinfante's energy-momentum tensors for general tensor field theories. For any tensor field T, we define a new tensor field \til {\bm T}, in terms of which the Belinfante tensor is readily computed. We show that the latter is the one that arises naturally from Noether Theorem for an arbitrary spacetime and it coincides on-shell with the metric one.Comment: 11 pages, 1 figur

Determinants of Dirac operators with local boundary conditions

We study functional determinants for Dirac operators on manifolds with boundary. We give, for local boundary conditions, an explicit formula relating these determinants to the corresponding Green functions. We finally apply this result to the case of a bidimensional disk under bag-like conditions.Comment: standard LaTeX, 24 pages. To appear in Jour. Math. Phy

On the Initial Singularity Problem in Two Dimensional Quantum Cosmology

The problem of how to put interactions in two-dimensional quantum gravity in the strong coupling regime is studied. It shows that the most general interaction consistent with this symmetry is a Liouville term that contain two parameters $(\alpha, \beta)$ satisfying the algebraic relation $2\beta - \alpha =2$ in order to assure the closure of the diffeomorphism algebra. The model is classically soluble and it contains as general solution the temporal singularity. The theory is quantized and we show that the propagation amplitude fall tozero in $\tau =0$. This result shows that the classical singularities are smoothed by quantum effects and the bing-bang concept could be considered as a classical extrapolation instead of a physical concept.Comment: 9pp, Revtex 3.0. New references added. To appear in Phys. Rev.

On the global version of Euler-Lagrange equations

The introduction of a covariant derivative on the velocity phase space is needed for a global expression of Euler-Lagrange equations. The aim of this paper is to show how its torsion tensor turns out to be involved in such a version.Comment: 5 pages, 1 figur

Symmetry types of hyperelliptic Riemann surfaces

Let $X$ be a compact hyperelliptic Riemann surface which admits anti-analytic involutions (also called symmetries or real structures). For instance, a complex projective plane curve of genus two, defined by an equation with real coefficients, gives rise to such a surface, and complex conjugation is such a symmetry. In this memoir, the real structures $\tau$ of $X$ are classified up to isomorphism (i.e., up to conjugation). This is done as follows: the number of connected components of the set of fixed points of $\tau$ together with the connectedness or disconnectedness of the complementary set in $X$ classifies $\tau$ topologically; they determine the species of $\tau$, which only depends on the conjugacy class of $\tau$ (however, different conjugacy classes may have identical species). On these grounds, for a given genus $g\ge2$, the authors first give a list of all full groups of analytic and anti-analytic automorphisms of genus $g$ compact hyperelliptic Riemann surfaces. For every such group $G$, the authors compute polynomial equations for a surface $X$ having $G$ as full group and then find the number of conjugacy classes containing symmetries; they also compute a representative $\tau$ in every such class. Finally, they compute the species corresponding to such classes. This memoir is an exhaustive piece of work, going through a case-by-case analysis. The problem for general compact Riemann surfaces dates back to 1893, when {\it F. Klein} [Math. Ann. 42, 1--29 (1893)] first studied it. For zero genus, it is easy. For genus one, that is, for elliptic surfaces, it was solved by {\it N. Alling} ["Real elliptic curves" (1981)]. Partial results for hyperelliptic surfaces of genus two were obtained by {\it E. Bujalance} and {\it D. Singerman} [Proc. Lond. Math. Soc. 51, 501--519 (1985)]

Empty singularities in higher-dimensional Gravity

We study the exact solution of Einstein's field equations consisting of a ($n+2$)-dimensional static and hyperplane symmetric thick slice of matter, with constant and positive energy density $\rho$ and thickness $d$, surrounded by two different vacua. We explicitly write down the pressure and the external gravitational fields in terms of $\rho$ and $d$, the pressure is positive and bounded, presenting a maximum at an asymmetrical position. And if $\sqrt{\rho}\,d$ is small enough, the dominant energy condition is satisfied all over the spacetime. We find that this solution presents many interesting features. In particular, it has an empty singular boundary in one of the vacua.Comment: 13 page