Ricci flow for homogeneous compact models of the universe

Using quaternions, we give a concise derivation of the Ricci tensor for homogeneous spaces with topology of the 3-dimensional sphere. We derive explicit and numerical solutions for the Ricci flow PDE and discuss their properties. In the collapse (or expansion) of these models, the interplay of the various components of the Ricci tensor are studied. We dedicate this paper to honor the work of Josh Goldberg.Comment: 18 pages, 2 figure

Pressure as a Source of Gravity

The active mass density in Einstein's theory of gravitation in the analog of Poisson's equation in a local inertial system is proportional to $\rho+3p/c^2$. Here $\rho$ is the density of energy and $p$ its pressure for a perfect fluid. By using exact solutions of Einstein's field equations in the static case we study whether the pressure term contributes towards the mass

Interacting Constituents in Cosmology

Universe evolution, as described by Friedmann's equations, is determined by source terms fixed by the choice of pressure $\times$ energy-density equations of state $p(\rho)$. The usual approach in Cosmology considers equations of state accounting only for kinematic terms, ignoring the contribution from the interactions between the particles constituting the source fluid. In this work the importance of these neglected terms is emphasized. A systematic method, based on the Statistical Mechanics of real fluids, is proposed to include them. A toy-model is presented which shows how such interaction terms can engender significant cosmological effects.Comment: 24 pages, 6 figures. It includes results presented in "Cosmic Acceleration from Elementary Interactions" [arXiv:gr-qc/0512135]. Citations added in v.

Shear free solutions in General Relativity Theory

The Goldberg-Sachs theorem is an exact result on shear-free null geodesics in a vacuum spacetime. It is compared and contrasted with an exact result for pressure-free matter: shear-free flows cannot both expand and rotate. In both cases, the shear-free condition restricts the way distant matter can influence the local gravitational field. This leads to intriguing discontinuities in the relation of the General Relativity solutions to Newtonian solutions in the timelike case, and of the full theory to the linearised theory in the null case. It is a pleasure to dedicate this paper to Josh Goldberg.Comment: 17 pages, no figures. For GRG special issue in honor of Josh Goldber

Theorems on shear-free perfect fluids with their Newtonian analogues

In this paper we provide fully covariant proofs of some theorems on shear-free perfect fluids. In particular, we explicitly show that any shear-free perfect fluid with the acceleration proportional to the vorticity vector (including the simpler case of vanishing acceleration) must be either non-expanding or non-rotating. We also show that these results are not necessarily true in the Newtonian case, and present an explicit comparison of shear-free dust in Newtonian and relativistic theories in order to see where and why the differences appear.Comment: 23 pages, LaTeX. Submitted to GR

Effect of inhomogeneity of the Universe on a gravitationally bound local system: A no-go result for explaining the secular increase in the astronomical unit

We will investigate the influence of the inhomogeneity of the universe, especially that of the Lema{\^i}tre-Tolman-Bondi (LTB) model, on a gravitationally bound local system such as the solar system. We concentrate on the dynamical perturbation to the planetary motion and derive the leading order effect generated from the LTB model. It will be shown that there appear not only a well-known cosmological effect arisen from the homogeneous and isotropic model, such as the Robertson-Walker (RW) model, but also the additional terms due to the radial inhomogeneity of the LTB model. We will also apply the obtained results to the problem of secular increase in the astronomical unit, reported by Krasinsky and Brumberg (2004), and imply that the inhomogeneity of the universe cannot have a significant effect for explaining the observed $d{\rm AU}/dt = 15 \pm 4 ~{\rm [m/century]}$.Comment: 12 pages, no figure, accepted for publication in Journal of Astrophysics and Astronom

Gravitational-wave research as an emerging field in the Max Planck Society. The long roots of GEO600 and of the Albert Einstein Institute

On the occasion of the 50th anniversary since the beginning of the search for gravitational waves at the Max Planck Society, and in coincidence with the 25th anniversary of the foundation of the Albert Einstein Institute, we explore the interplay between the renaissance of general relativity and the advent of relativistic astrophysics following the German early involvement in gravitational-wave research, to the point when gravitational-wave detection became established by the appearance of full-scale detectors and international collaborations. On the background of the spectacular astrophysical discoveries of the 1960s and the growing role of relativistic astrophysics, Ludwig Biermann and his collaborators at the Max Planck Institute for Astrophysics in Munich became deeply involved in research related to such new horizons. At the end of the 1960s, Joseph Weber's announcements claiming detection of gravitational waves sparked the decisive entry of this group into the field, in parallel with the appointment of the renowned relativist Juergen Ehlers. The Munich area group of Max Planck institutes provided the fertile ground for acquiring a leading position in the 1970s, facilitating the experimental transition from resonant bars towards laser interferometry and its innovation at increasingly large scales, eventually moving to a dedicated site in Hannover in the early 1990s. The Hannover group emphasized perfecting experimental systems at pilot scales, and never developed a full-sized detector, rather joining the LIGO Scientific Collaboration at the end of the century. In parallel, the Max Planck Institute for Gravitational Physics (Albert Einstein Institute) had been founded in Potsdam, and both sites, in Hannover and Potsdam, became a unified entity in the early 2000s and were central contributors to the first detection of gravitational waves in 2015.Comment: 94 pages. Enlarged version including new results from further archival research. A previous version appears as a chapter in the volume The Renaissance of General Relativity in Context, edited by A. Blum, R. Lalli and J. Renn (Boston: Birkhauser, 2020

Conservation of energy-momentum of matter as the basis for the gauge theory of gravitation

According to Yang \& Mills (1954), a {\it conserved} current and a related rigid (`global') symmetry lie at the foundations of gauge theory. When the rigid symmetry is extended to a {\it local} one, a so-called gauge symmetry, a new interaction emerges as gauge potential $A$; its field strength is $F\sim {\rm curl} A$. In gravity, the conservation of the energy-momentum current of matter and the rigid translation symmetry in the Minkowski space of special relativity lie at the foundations of a gravitational gauge theory. If the translation invariance is made local, a gravitational potential $\vartheta$ arises together with its field strength $T\sim {\rm curl}\,\vartheta$. Thereby the Minkowski space deforms into a Weitzenb\"ock space with nonvanishing torsion $T$ but vanishing curvature. The corresponding theory is reviewed and its equivalence to general relativity pointed out. Since translations form a subgroup of the Poincar\'e group, the group of motion of special relativity, one ought to straightforwardly extend the gauging of the translations to the gauging of full Poincar\'e group thereby also including the conservation law of the {\it angular momentum} current. The emerging Poincar\'e gauge (theory of) gravity, starting from the viable Einstein-Cartan theory of 1961, will be shortly reviewed and its prospects for further developments assessed.Comment: 46 pages, 4 figures, minor corrections, references added, contribution to "One Hundred Years of Gauge Theory" edited by S. De Bianchi and C. Kiefe