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

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

Relativistic theory for time and frequency transfer to order c^{-3}

This paper is motivated by the current development of several space missions (e.g. ACES on International Space Station) that will fly on Earth orbit laser cooled atomic clocks, providing a time-keeping accuracy of the order of 5~10^{-17} in fractional frequency. We show that to such accuracy, the theory of frequency transfer between Earth and Space must be extended from the currently known relativistic order 1/c^2 (which has been needed in previous space experiments such as GP-A) to the next relativistic correction of order 1/c^3. We find that the frequency transfer includes the first and second-order Doppler contributions, the Einstein gravitational red-shift and, at the order 1/c^3, a mixture of these effects. As for the time transfer, it contains the standard Shapiro time delay, and we present an expression also including the first and second-order Sagnac corrections. Higher-order relativistic corrections, at least O(1/c^4), are numerically negligible for time and frequency transfers in these experiments, being for instance of order 10^{-20} in fractional frequency. Particular attention is paid to the problem of the frequency transfer in the two-way experimental configuration. In this case we find a simple theoretical expression which extends the previous formula (Vessot et al. 1980) to the next order 1/c^3. In the Appendix we present the detailed proofs of all the formulas which will be needed in such experiments.Comment: 11 pages, 2 figures, to appear in Astronomy & Astrophysic

Physical Acceptability of Isolated, Static, Spherically Symmetric, Perfect Fluid Solutions of Einstein's Equations

We ask the following question: Of the exact solutions to Einstein's equations extant in the literature, how many could represent the field associated with an isolated static spherically symmetric perfect fluid source? The candidate solutions were subjected to the following elementary tests: i) isotropy of the pressure, ii) regularity at the origin, iii) positive definiteness of the energy density and pressure at the origin, iv) vanishing of the pressure at some finite radius, v) monotonic decrease of the energy density and pressure with increasing radius, and vi) subluminal sound speed. A total of 127 candidate solutions were found. Only 16 of these passed all the tests. Of these 16, only 9 have a sound speed which monotonically decreases with radius. The analysis was facilitated by use of the computer algebra system GRTensorII.Comment: 25 pages. To appear in Computer Physics Communications Thematic Issue on "Computer Algebra in Physics Research

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

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