Daniel de la Fuente

Radial Solutions of the Dirichlet Problem for the Prescribed Mean Curvature Equation in a Robertson-Walker Spacetime

Abstract We consider the prescribed mean curvature problem of spacelike graphs in Robertson- Walker spacetimes of flat fiber with homogeneous Dirichlet conditions on an Euclidean ball. Under reasonable assumptions, it is shown that every possible solution must be radially symmetric. Besides, an existence result for a singular nonlinear equation is proved by making use of the classical Schauder fixed point Theorem. The results are applied to realistic examples of Robertson-Walker spacetimes.

Entire spherically symmetric spacelike graphs with prescribed mean curvature function in Schwarzschild and Reissner–Nordström spacetimes

For each spacetime of a family of static spacetimes, we prove the existence of entire spherically symmetric spacelike graphs with prescribed mean curvature function. In particular, classical Schwarzschild and Reissner–Nordstrom spacetimes are considered. In both cases, the entire spacelike graph asymptotically approaches the event horizon. Spacelike graphs of constant mean curvature remain as a particular situation in the existence results, obtaining explicit expressions for the solutions. The proof of the results is based on the analysis of the associated homogeneous Dirichlet problem on a Euclidean ball, together with the obtention of a suitable bound for the length of the gradient of a solution which permits the prolongability to the whole space.

Existence and multiplicity of entire radial spacelike graphs with prescribed mean curvature function in certain Friedmann–Lemaître–Robertson–Walker spacetimes

We provide sufficient conditions for the existence of a uniparametric family of entire spacelike graphs with prescribed mean curvature in a Friedmann–Lemaitre–Robertson–Walker spacetime with flat fiber. The proof is based on the analysis of the associated homogeneous Dirichlet problem on a Euclidean ball together with suitable bounds for the gradient which permit the prolongability of the solution to the whole space.

Unchanged direction motion in general relativity: The problems of prescribing acceleration and the extensibility of trajectories

The notion of unchanged direction (UD) motion in general relativity is introduced, extending widely the concept of uniformly accelerated motion. An observer which obeys an UD motion is characterized as a pointing future unit timelike curve with all its curvatures identically zero up to the first one. The initial value problem when the acceleration of the motion is prescribed is analysed. It is also studied that the completeness of inextensible UD motions, that can be physically interpreted saying that the observers which obey an UD motion lives forever. For certain spacetimes with relevant symmetries that includes the generalized Robertson-Walker spacetimes, a geometric approach leads to the completeness. On the other hand, a more analytical approach permits to prove completeness of inextensible UD motions in a plane wave spacetime.

Galilean generalized Robertson-Walker spacetimes: A new family of Galilean geometrical models

We introduce a new family of Galilean spacetimes, the Galilean generalized Robertson-Walker spacetimes. This new family is relevant in the context of a generalized Newton-Cartan theory. We study its geometrical structure and analyse the completeness of its inextensible free falling observers. This sort of spacetimes constitutes the local geometric model of a much wider family of spacetimes admitting certain conformal symmetry. Moreover, we find some sufficient geometric conditions which guarantee a global splitting of a Galilean spacetime as a Galilean generalized Robertson-Walker spacetime.