Self-gravitating Newtonian disks revisited

Recent analytic results concerning stationary, self-gravitating fluids in Newtonian theory are discussed. We give a theorem that forbids infinitely extended fluids, depending on the assumed equation of state and the rotation law. This part extends previous results that have been obtained for static configurations. The second part discusses a Sobolev bound on the mass of the fluid and a rigorous Jeans-type inequality that is valid in the stationary case.Comment: A talk given at the Spanish Relativity Meeting in Portugal 2012. To appear in Progress in Mathematical Relativity, Gravitation and Cosmology, Proceedings of the Spanish Relativity Meeting ERE2012, University of Minho, Guimaraes, Portugal, 3-7 September 2012, Springer Proceedings in Mathematics & Statistics, Vol. 6

The initial singularity of ultrastiff perfect fluid spacetimes without symmetries

We consider the Einstein equations coupled to an ultrastiff perfect fluid and prove the existence of a family of solutions with an initial singularity whose structure is that of explicit isotropic models. This family of solutions is `generic' in the sense that it depends on as many free functions as a general solution, i.e., without imposing any symmetry assumptions, of the Einstein-Euler equations. The method we use is a that of a Fuchsian reduction.Comment: 16 pages, journal versio

(In)finite extent of stationary perfect fluids in Newtonian theory

For stationary, barotropic fluids in Newtonian gravity we give simple criteria on the equation of state and the "law of motion" which guarantee finite or infinite extent of the fluid region (providing a priori estimates for the corresponding stationary Newton-Euler system). Under more restrictive conditions, we can also exclude the presence of "hollow" configurations. Our main result, which does not assume axial symmetry, uses the virial theorem as the key ingredient and generalises a known result in the static case. In the axially symmetric case stronger results are obtained and examples are discussed.Comment: Corrections according to the version accepted by Ann. Henri Poincar

Observers in an accelerated universe

If the current acceleration of our Universe is due to a cosmological constant, then a Coleman-De Luccia bubble will nucleate in our Universe. In this work, we consider that our observations could be likely in this framework, consisting in two infinite spaces, if a foliation by constant mean curvature hypersurfaces is taken to count the events in the spacetime. Thus, we obtain and study a particular foliation, which covers the existence of most observers in our part of spacetime.Comment: revised version, accepted in EPJ

Static perfect fluids with Pant-Sah equations of state

We analyze the 3-parameter family of exact, regular, static, spherically symmetric perfect fluid solutions of Einstein's equations (corresponding to a 2-parameter family of equations of state) due to Pant and Sah and "rediscovered" by Rosquist and the present author. Except for the Buchdahl solutions which are contained as a limiting case, the fluids have finite radius and are physically realistic for suitable parameter ranges. The equations of state can be characterized geometrically by the property that the 3-metric on the static slices, rescaled conformally with the fourth power of any linear function of the norm of the static Killing vector, has constant scalar curvature. This local property does not require spherical symmetry; in fact it simplifies the the proof of spherical symmetry of asymptotically flat solutions which we recall here for the Pant-Sah equations of state. We also consider a model in Newtonian theory with analogous geometric and physical properties, together with a proof of spherical symmetry of the asymptotically flat solutions.Comment: 32 p., Latex, minor changes and correction

Fundamental Solutions for the Klein-Gordon Equation in de Sitter Spacetime

In this article we construct the fundamental solutions for the Klein-Gordon equation in de Sitter spacetime. We use these fundamental solutions to represent solutions of the Cauchy problem and to prove $L^p-L^q$ estimates for the solutions of the equation with and without a source term