Optimal Lβ\mathfrak{L}^{\beta}-Control for the Global Cauchy Problem of the Relativistic Vlasov-Poisson System

Recently, M.K.-H. Kiessling and A.S. Tahvildar-Zadeh proved that a unique global classical solution to the relativistic Vlasov-Poisson system exists whenever the positive, integrable initial datum is spherically symmetric, compactly supported in momentum space, vanishes on characteristics with vanishing angular momentum, and for $\beta \ge 3/2$ has $\mathfrak{L}^{\beta}$-norm strictly below a positive, critical value $\mathcal{C}_{\beta}$. Everything else being equal, data leading to finite time blow-up can be found with $\mathfrak{L}^{\beta}$-norm surpassing $\mathcal{C}_{\beta}$ for any $\beta >1$, with $\mathcal{C}_{\beta}>0$ if and only if $\beta\geq 3/2$. In their paper, the critical value for $\beta = {3}/{2}$ is calculated explicitly while the value for all other $\beta$ is merely characterized as the infimum of a functional over an appropriate function space. In this work, the existence of minimizers is established, and the exact expression of $\mathcal{C}_{\beta}$ is calculated in terms of the famous Lane-Emden functions. Numerical computations of the $\mathcal{C}_{\beta}$ are presented along with some elementary asymptotics near the critical exponent ${3}/{2}$.Comment: 24 pages, 2 figures Refereed and accepted for publication in Transport Theory and Statistical Physic

Algorithmic construction of static perfect fluid spheres

Perfect fluid spheres, both Newtonian and relativistic, have attracted considerable attention as the first step in developing realistic stellar models (or models for fluid planets). Whereas there have been some early hints on how one might find general solutions to the perfect fluid constraint in the absence of a specific equation of state, explicit and fully general solutions of the perfect fluid constraint have only very recently been developed. In this article we present a version of Lake's algorithm [Phys. Rev. D 67 (2003) 104015; gr-qc/0209104] wherein: (1) we re-cast the algorithm in terms of variables with a clear physical meaning -- the average density and the locally measured acceleration due to gravity, (2) we present explicit and fully general formulae for the mass profile and pressure profile, and (3) we present an explicit closed-form expression for the central pressure. Furthermore we can then use the formalism to easily understand the pattern of inter-relationships among many of the previously known exact solutions, and generate several new exact solutions.Comment: Uses revtex4. V2: Minor clarifications, plus an additional section on how to turn the algorithm into a solution generalization technique. This version accepted for publication in Physical Review D. Now 7 page

The Physics and Mathematics of the Second Law of Thermodynamics

The essential postulates of classical thermodynamics are formulated, from which the second law is deduced as the principle of increase of entropy in irreversible adiabatic processes that take one equilibrium state to another. The entropy constructed here is defined only for equilibrium states and no attempt is made to define it otherwise. Statistical mechanics does not enter these considerations. One of the main concepts that makes everything work is the comparison principle (which, in essence, states that given any two states of the same chemical composition at least one is adiabatically accessible from the other) and we show that it can be derived from some assumptions about the pressure and thermal equilibrium. Temperature is derived from entropy, but at the start not even the concept of `hotness' is assumed. Our formulation offers a certain clarity and rigor that goes beyond most textbook discussions of the second law.Comment: 93 pages, TeX, 8 eps figures. Updated, published version. A summary appears in Notices of the Amer. Math. Soc. 45 (1998) 571-581, math-ph/980500

The interior spacetimes of stars in Palatini f(R) gravity

We study the interior spacetimes of stars in the Palatini formalism of f(R) gravity and derive a generalized Tolman-Oppenheimer-Volkoff and mass equation for a static, spherically symmetric star. We show that matching the interior solution with the exterior Schwarzschild-De Sitter solution in general gives a relation between the gravitational mass and the density profile of a star, which is different from the one in General Relativity. These modifications become neglible in models for which $\delta F(R) \equiv \partial f/\partial R - 1$ is a decreasing function of R however. As a result, both Solar System constraints and stellar dynamics are perfectly consistent with $f(R) = R - \mu^4/R$.Comment: Published version, 6 pages, 1 figur

Symmetric hyperbolic systems for a large class of fields in arbitrary dimension

Symmetric hyperbolic systems of equations are explicitly constructed for a general class of tensor fields by considering their structure as r-fold forms. The hyperbolizations depend on 2r-1 arbitrary timelike vectors. The importance of the so-called "superenergy" tensors, which provide the necessary symmetric positive matrices, is emphasized and made explicit. Thereby, a unified treatment of many physical systems is achieved, as well as of the sometimes called "higher order" systems. The characteristics of these symmetric hyperbolic systems are always physical, and directly related to the null directions of the superenergy tensor, which are in particular principal null directions of the tensor field solutions. Generic energy estimates and inequalities are presented too.Comment: 24 pages, no figure

G\"{o}del-type universes in f(R) gravity

The $f(R)$ gravity theories provide an alternative way to explain the current cosmic acceleration without a dark energy matter component. If gravity is governed by a $f(R)$ theory a number of issues should be reexamined in this framework, including the violation of causality problem on nonlocal scale. We examine the question as to whether the $f(R)$ gravity theories permit space-times in which the causality is violated. We show that the field equations of these $f(R)$ gravity theories do not exclude solutions with breakdown of causality for a physically well-motivated perfect-fluid matter content. We demonstrate that every perfect-fluid G\"{o}del-type solution of a generic $f(R)$ gravity satisfying the condition $df/dR > 0$ is necessarily isometric to the G\"odel geometry, and therefore presents violation of causality. This result extends a theorem on G\"{o}del-type models, which has been established in the context of general relativity. We also derive an expression for the critical radius $r_c$ (beyond which the causality is violated) for an arbitrary $f(R)$ theory, making apparent that the violation of causality depends on both the $f(R)$ gravity theory and the matter content. As an illustration, we concretely take a recent $f(R)$ gravity theory that is free from singularities of the Ricci scalar and is cosmologically viable, and show that this theory accommodates noncausal as well as causal G\"odel-type solutions.Comment: 7 pages, V3: Version to appear in Phys. Rev. D (2009), typos corrected, the generality of our main results is emphasized. The illustrative character of a particular theory is also made explici

Galactic rotation curves in modified gravity with non-minimal coupling between matter and geometry

We investigate the possibility that the behavior of the rotational velocities of test particles gravitating around galaxies can be explained in the framework of modified gravity models with non-minimal matter-geometry coupling. Generally, the dynamics of test particles around galaxies, as well as the corresponding mass deficit, is explained by postulating the existence of dark matter. The extra-terms in the gravitational field equations with geometry-matter coupling modify the equations of motion of test particles, and induce a supplementary gravitational interaction. Starting from the variational principle describing the particle motion in the presence of the non-minimal coupling, the expression of the tangential velocity of a test particle, moving in the vacuum on a stable circular orbit in a spherically symmetric geometry, is derived. The tangential velocity depends on the metric tensor components, as well as of the coupling function between matter and geometry. The Doppler velocity shifts are also obtained in terms of the coupling function. If the tangential velocity profile is known, the coupling term between matter and geometry can be obtained explicitly in an analytical form. The functional form of this function is obtained in two cases, for a constant tangential velocity, and for an empirical velocity profile obtained from astronomical observations, respectively. Therefore, these results open the possibility of directly testing the modified gravity models with non-minimal coupling between matter and geometry by using direct astronomical and astrophysical observations at the galactic or extra-galactic scale.Comment: 8 pages, accepted for publication in PR

Quadratic superconducting cosmic strings revisited

It has been shown that 5-dimensional general relativity action extended by appropriate quadratic terms admits a singular superconducting cosmic string solution. We search for cosmic strings endowed with similar and extended physical properties by directly integrating the non-linear matrix field equations thus avoiding the perturbative approach by which we constructed the above-mentioned \textsl{exact} solution. The most general superconducting cosmic string, subject to some constraints, will be derived and shown to be mathematically \textsl{unique} up to linear coordinate transformations mixing its Killing vectors. The most general solution, however, is not globally equivalent to the old one due to the existence of Killing vectors with closed orbits.Comment: 6 page

Some notes on the Kruskal - Szekeres completion

The Kruskal - Szekeres (KS) completion of the Schwarzschild spacetime is open to Synge's methodological criticism that the KS procedure generates "good" coordinates from "bad". This is addressed here in two ways: First I generate the KS coordinates from Israel coordinates, which are also "good", and then I generate the KS coordinates directly from a streamlined integration of the Einstein equations.Comment: One typo correcte