Ricardo E. Gamboa Saravi

On the energy?momentum tensor

We clarify the relation among canonical, metric and Belinfantes energy?momentum tensors for general tensor field theories. For any tensor field T, we define a new tensor field in terms of which metric and Belinfantes energy?momentum tensors are readily computed. We show that the latter is the one that arises naturally from Noethers theorem for an arbitrary spacetime and it coincides on-shell with the metric one.

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 ρ and a parameter κ. We show that these space–times finish down below at an inner singularity at finite depth. We show that for κ ≥ 0.3513 . . . the dominant energy condition is satisfied all over the space–time. 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 κ, these slabs can be attractive, repulsive or neutral. In the first case, the space–time also finishes up above at an empty repelling singular boundary. In the other cases, they turn out to be semi-infinite and asymptotically flat when z → ∞. 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.

THE WHITE WALL, A GRAVITATIONAL MIRROR

We describe the exact solution of Einsteins equation corresponding to a static homogenous distribution of matter with plane symmetry lying below z = 0. We study the geodesics in it and we show that this simple space–time exhibits very curious properties. In particular, it has a repelling singular boundary and all geodesics bounce off it.

Static plane symmetric relativistic fluids and empty repelling singular boundaries

We present a detailed analysis of the general exact solution of Einsteins equation corresponding to a static and plane symmetric distribution of matter with density proportional to pressure. We study the geodesics in it and we show that this simple spacetime exhibits very curious properties. In particular, it has a free of matter repelling singular boundary and all geodesics bounce off it.

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 timelike boundary is well-posed, if we only require 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.

On the relation between determinants and green functions of elliptic operators with local boundary conditions

A formula relating quotients of determinants of elliptic differential operators sharing their principal symbol, with local boundary conditions, to the corresponding Green function is given.

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 ρ and thickness d, surrounded by two different vacua. We explicitly write down the pressure and the external gravitational fields in terms of ρ and d, the pressure is positive and bounded, presenting a maximum at an asymmetrical position. And if

On well-posedness of the Cauchy problem for the wave equation in static spherically symmetric spacetimes

\sqrt{\rho} d

Higher-dimensional perfect fluids and empty singular boundaries

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.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 ρ and thickness d, surrounded by two different vacua. We explicitly write down the pressure and the external gravitational fields in terms of ρ 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.

Trends in Theoretical Physics II

We give simple conditions implying the well-posedness of the Cauchy problem for the propagation of classical scalar fields in general (n + 2)-dimensional static and spherically symmetric spacetimes. They are related to the properties of the underlying spatial part of the wave operator, one of which being the standard essentially self-adjointness. However, in many examples the spatial part of the wave operator turns out to be not essentially self-adjoint, but it does satisfy a weaker property that we call here quasi-essentially self-adjointness, which is enough to ensure the desired well-posedness. This is why we also characterize this second property. We state abstract results, then general results for a class of operators encompassing many examples in the literature, and we finish with the explicit analysis of some of them.