Theorems on existence and global dynamics for the Einstein equations

This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local-in-time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symmetry are discussed. A selection of results from Newtonian theory and special relativity that offer useful comparisons is presented. Treatments of global results in the case of small data and results on constructing spacetimes with prescribed singularity structure or late-time asymptotics are given. A conjectural picture of the asymptotic behaviour of general cosmological solutions of the Einstein equations is built up. Some miscellaneous topics connected with the main theme are collected in a separate section.Comment: Submitted to Living Reviews in Relativity, major update of Living Rev. Rel. 5 (2002)

The entropy of black holes: a primer

After recalling the definition of black holes, and reviewing their energetics and their classical thermodynamics, one expounds the conjecture of Bekenstein, attributing an entropy to black holes, and the calculation by Hawking of the semi-classical radiation spectrum of a black hole, involving a thermal (Planckian) factor. One then discusses the attempts to interpret the black-hole entropy as the logarithm of the number of quantum micro-states of a macroscopic black hole, with particular emphasis on results obtained within string theory. After mentioning the (technically cleaner, but conceptually more intricate) case of supersymmetric (BPS) black holes and the corresponding counting of the degeneracy of Dirichlet-brane systems, one discusses in some detail the ``correspondence'' between massive string states and non-supersymmetric Schwarzschild black holes.Comment: 51 pages, 4 figures, talk given at the "Poincare seminar" (Paris, 6 December 2003), to appear in Poincare Seminar 2003 (Birkhauser

Brane-world black holes and the scale of gravity

A particle in four dimensions should behave like a classical black hole if the horizon radius is larger than the Compton wavelength or, equivalently, if its degeneracy (measured by entropy in units of the Planck scale) is large. For spherically symmetric black holes in 4 + d dimensions, both arguments again lead to a mass threshold MC and degeneracy scale Mdeg of the order of the fundamental scale of gravity MG. In the brane-world, deviations from the Schwarzschild metric induced by bulk effects alter the horizon radius and effective four-dimensional Euclidean action in such a way that MC \simeq Mdeg might be either larger or smaller than MG. This opens up the possibility that black holes exist with a mass smaller than MG and might be produced at the LHC even if M>10 TeV, whereas effects due to bulk graviton exchanges remain undetectable because suppressed by inverse powers of MG. Conversely, even if black holes are not found at the LHC, it is still possible that MC>MG and MG \simeq 1TeV.Comment: 4 pages, no figur

Theorems on existence and global dynamics for the Einstein equations

This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local in time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symmetry are discussed. A selection of results from Newtonian theory and special relativity which offer useful comparisons is presented. Treatments of global results in the case of small data and results on constructing spacetimes with prescribed singularity structure are given. A conjectural picture of the asymptotic behaviour of general cosmological solutions of the Einstein equations is built up. Some miscellaneous topics connected with the main theme are collected in a separate section.Comment: 54 pages, submitted to Living Reviews in Relativit

The EROS2 search for microlensing events towards the spiral arms: the complete seven season results

The EROS-2 project has been designed to search for microlensing events towards any dense stellar field. The densest parts of the Galactic spiral arms have been monitored to maximize the microlensing signal expected from the stars of the Galactic disk and bulge. 12.9 million stars have been monitored during 7 seasons towards 4 directions in the Galactic plane, away from the Galactic center. A total of 27 microlensing event candidates have been found. Estimates of the optical depths from the 22 best events are provided. A first order interpretation shows that simple Galactic models with a standard disk and an elongated bulge are in agreement with our observations. We find that the average microlensing optical depth towards the complete EROS-cataloged stars of the spiral arms is $\bar{\tau} =0.51\pm .13\times 10^{-6}$, a number that is stable when the selection criteria are moderately varied. As the EROS catalog is almost complete up to $I_C=18.5$, the optical depth estimated for the sub-sample of bright target stars with $I_C<18.5$ ($\bar{\tau}=0.39\pm >.11\times 10^{-6}$) is easier to interpret. The set of microlensing events that we have observed is consistent with a simple Galactic model. A more precise interpretation would require either a better knowledge of the distance distribution of the target stars, or a simulation based on a Galactic model. For this purpose, we define and discuss the concept of optical depth for a given catalog or for a limiting magnitude.Comment: 22 pages submitted to Astronomy & Astrophysic

Dynamics of Disks and Warps

This chapter reviews theoretical work on the stellar dynamics of galaxy disks. All the known collective global instabilities are identified, and their mechanisms described in terms of local wave mechanics. A detailed discussion of warps and other bending waves is also given. The structure of bars in galaxies, and their effect on galaxy evolution, is now reasonably well understood, but there is still no convincing explanation for their origin and frequency. Spiral patterns have long presented a special challenge, and ideas and recent developments are reviewed. Other topics include scattering of disk stars and the survival of thin disks.Comment: Chapter accepted to appear in Planets, Stars and Stellar Systems, vol 5, ed G. Gilmore. 32 pages, 17 figures. Includes minor corrections made in proofs. Uses emulateapj.st

Sequential formation and resolution of multiple rosettes drive embryo remodelling after implantation

The morphogenetic remodelling of embryo architecture after implantation culminates in pro-amniotic cavity formation. Despite its key importance, how this transformation occurs remains unknown. Here, we apply high-resolution imaging of embryos developing in vivo and in vitro, spatial RNA sequencing and 3D trophoblast stem cell models to determine the sequence and mechanisms of these remodelling events. We show that cavitation of the embryonic tissue is followed by folding of extra-embryonic tissue to mediate the formation of a second extra-embryonic cavity. Concomitantly, at the boundary between embryonic and extra-embryonic tissues, a hybrid 3D rosette forms. Resolution of this rosette enables the embryonic cavity to invade the extra-embryonic tissue. Subsequently, β1-integrin signalling mediates the formation of multiple extra-embryonic 3D rosettes. Podocalyxin exocytosis leads to their polarized resolution, permitting the extension of embryonic and extra-embryonic cavities and their fusion into a unified pro-amniotic cavity. These morphogenetic transformations of embryogenesis reveal a previously unappreciated mechanism for lumen expansion and fusionThe M.Z.G lab is supported by grants from the European Research Council (669198) and the Welcome Trust (098287/Z/12/Z) and the EU Horizon 2020 Marie Sklodowska-Curie actions (ImageInLife,721537). C.K is supported by BBSRC Doctoral training studentship

The Einstein-Vlasov System/Kinetic Theory

The main purpose of this article is to provide a guide to theorems on global properties of solutions to the Einstein--Vlasov system. This system couples Einstein's equations to a kinetic matter model. Kinetic theory has been an important field of research during several decades in which the main focus has been on non-relativistic and special relativistic physics, i.e., to model the dynamics of neutral gases, plasmas, and Newtonian self-gravitating systems. In 1990, Rendall and Rein initiated a mathematical study of the Einstein--Vlasov system. Since then many theorems on global properties of solutions to this system have been established.Comment: Published version http://www.livingreviews.org/lrr-2011-