962 research outputs found
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
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
Optimal -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 has
-norm strictly below a positive, critical value
. Everything else being equal, data leading to finite time
blow-up can be found with -norm surpassing
for any , with if and
only if . In their paper, the critical value for is calculated explicitly while the value for all other 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 is calculated in terms of the
famous Lane-Emden functions. Numerical computations of the
are presented along with some elementary asymptotics near
the critical exponent .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
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
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 is a decreasing function of R however. As a result, both Solar System
constraints and stellar dynamics are perfectly consistent with .Comment: Published version, 6 pages, 1 figur
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
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