Norma G. Sanchez

Statistical mechanics of the self-gravitating gas: I. Thermodynamic limit and phase diagrams

Abstract We provide a complete picture to the self-gravitating non-relativistic gas at thermal equilibrium using Monte Carlo simulations, analytic mean field methods (MF) and low density expansions. The system is shown to possess an infinite volume limit in the grand canonical (GCE), canonical (CE) and microcanonical (MCE) ensembles when ( N , V )→∞, keeping N / V 1/3 fixed. We compute the equation of state (we do not assume it as is customary), as well as the energy, free energy, entropy, chemical potential, specific heats, compressibilities and speed of sound; we analyze their properties, signs and singularities. All physical quantities turn out to depend on a single variable η≡ Gm 2 N V 1/3 T that is kept fixed in the N →∞ and V →∞ limit. The system is in a gaseous phase for η η T and collapses into a dense object for η > η T in the CE with the pressure becoming large and negative. At η ≃ η T the isothermal compressibility diverges. This gravitational phase transition is associated to the Jeans instability. Our Monte Carlo simulations yield η T ≃1.515. PV /[ NT ]= f ( η ) and all physical magnitudes exhibit a square root branch point at η = η C > η T . The values of η T and η C change by a few percent with the geometry for large N : for spherical symmetry and N =∞ (MF), we find η C =1.561764… while the Monte Carlo simulations for cubic geometry yields η C ≃1.540. In mean field and spherical symmetry c V diverges as ( η C − η ) −1/2 for η ↑ η C while c P and κ T diverge as ( η 0 − η ) −1 for η↑η 0 =1.51024… . The function f ( η ) has a second Riemann sheet which is only physically realized in the MCE. In the MCE, the collapse phase transition takes place in this second sheet near η MC =1.26 and the pressure and temperature are larger in the collapsed phase than in the gaseous phase. Both collapse phase transitions (in the CE and in the MCE) are of zeroth order since the Gibbs free energy has a jump at the transitions. The MF equation of state in a sphere, f ( η ), obeys a first order non-linear differential equation of first kind Abels type. The MF gives an extremely accurate picture in agreement with the MC simulations both in the CE and MCE. Since we perform the MC simulations on a cubic geometry they describe an isothermal cube while the MF calculations describe an isothermal sphere. The local properties of the gas, scaling behaviour of the particle distribution and its fractal (Haussdorfxa0) dimension are investigated in the companion paper quoted as paperxa0II in the text: H.J.xa0de Vega, N. Sanchez, astro-ph/0101567, next paper in this issue.

Semiclassical quantization of circular strings in de Sitter and anti–de Sitter spacetimes

We compute the {ital exact} equation of state of circular strings in the (2+1)--dimensional de Sitter (dS) and anti--de Sitter (AdS) spacetimes, and analyze its properties for the different (oscillating, contracting, and expanding) strings. The string equation of state has the perfect fluid form {ital P}=({gamma}{minus}1){ital E}, with the pressure and energy expressed closely and completely in terms of elliptic functions, the instantaneous coefficient {gamma} depending on the elliptic modulus. We semiclassically quantize the oscillating circular strings. The string mass is {ital m}= {radical}{ital C} /({pi}{ital H}{alpha}{prime}), {ital C} being the Casimir operator, {ital C}={minus}{ital L}{sub {mu}{nu}}{ital L}{sup {mu}{nu}}, of the O(3,1)-dS [O(2,2)-AdS] group, and {ital H} is the Hubble constant. We find {alpha}{prime}{ital m}{sub dS}{sup 2}{approx}4{ital n}{minus}5{ital H}{sup 2}{alpha}{prime}{ital n}{sup 2} ({ital n}{element_of}{ital N}{sub 0}), and a {ital finite} number of states {ital N}{sub dS}{approx}0.34/({ital H}{sup 2}{alpha}{prime}) in de Sitter spacetime; {ital m}{sub AdS}{sup 2}{approx}{ital H}{sup 2}{ital n}{sup 2} (large {ital n}{element_of}{ital N}{sub 0}) and {ital N}{sub AdS}={infinity} in anti--de Sitter spacetime. The level spacing grows with {ital n} in AdS spacetime, while it is approximately constant (although smaller than in Minkowski spacetime and slightly decreasing) in dS spacetime. The massive states in dS spacetime decay through the tunnelmorexa0» effect and the semiclassical decay probability is computed. The semiclassical quantization of {ital exact} (circular) strings and the canonical quantization of generic string perturbations around the string center of mass qualitatively agree.«xa0less

Jeans-like instabilities for strings in cosmological backgrounds

Abstract We study quantum-string propagation in cosmological backgrounds pointing out the possible emergence of Jeans-like instabilities. We also determine under which conditions the universe expands, when distances are measured by stringy rods. Examples include Friedmann-Robertson-Walker (FRW) and de Sitter metrics, as well as a linear-in-time dilaton (constant background charge). A time-dependent dilaton does not give an expanding universe in the above sense, while FRW or de Sitter metrics usually do. For the latter case (inflation), instabilities occur for large enough Hubble constant.

Quantum string scattering in the Aichelburg-Sexl geometry

At energies of the order or larger than the Planck mass, the curved space-time geometry created by the particles dominates their collision process. The so-called Aichelburg-Sexl (AS) metric is relevant in this problem. We find the exact solution to the quantization and scattering of a closed bosonic string in D-dimensional AS geometry. The mass spectrm and critical dimension are the same as in flat space-time but there is non-trivial elastic and inelastic scattering by the shock wave. We find the exact (non-linear) and the Bogoliubov (linear) transformations relating the ingoing and outgoing string mode operators. The transition amplitudes between the internal modes and the total pair-creation rate are computed (this pair mode creation is a genuine string effect which does not exist in the quantum point particle theory). We find the deflection angle and the quantum string corrections to the classical cross section. The ground-state scattering amplitude is analyzed. Comparison with the black-hole results and recent investigations of string collisions in flat space-time are discussed.

Self-sustained inflation and dimensional reduction from fundamental strings

Abstract In a previous paper we have shown that an ideal gas of fundamental strings is not able to sustain, by itself, a phase of isotropic inflation of the Universe. We show here that fundamental strings can sustain, instead, a phase of anisotropic inflation accompanied by the contraction of a sufficient number of internal dimensions. The conditions to be met for the existence of such a solution to the Einstein and string equations are derived, and the possibility of a successful resolution of the standard cosmological problems in the context of this model is discussed.

sinh-Gordon, cosh-Gordon, and Liouville equations for strings and multistrings in constant curvature spacetimes.

We find that the fundamental quadratic form of classical string propagation in (2+1)-dimensional constant curvature spacetimes solves the sinh-Gordon equation, the cosh-Gordon equation, or the Liouville equation. We show that in both de Sitter and anti{endash}de Sitter spacetimes (as well as in the 2+1 black hole anti{endash}de Sitter spacetime), {ital all} three equations must be included to cover the generic string dynamics. The generic properties of the string dynamics are directly extracted from the properties of these three equations and their associated potentials (irrespective of any solution). These results complete and generalize earlier discussions on this topic (until now, only the sinh-Gordon sector in de Sitter spacetime was known). We also construct new classes of multistring solutions, in terms of elliptic functions, to all three equations in both de Sitter and anti{endash}de Sitter spacetimes. Our results can be straightforwardly generalized to constant curvature spacetimes of arbitrary dimension, by replacing the sinh-Gordon equation, the cosh-Gordon equation, and the Liouville equation by their higher dimensional generalizations. {copyright} {ital 1996 The American Physical Society.}

Back reaction effects in black hole spacetimes

Abstract We solve the semiclassical Einstein equations for the static spherically symmetric case. Using expressions for the renormalized 〈Tμv〉, we study the effects of the back reaction on black hole spacetimes at the one-loop level. Two different situtaions appear depending on the graviton-matter balance. If matter is relevant, the temperature is increased and thus the lifetime decreased from their standard values T H = 1 8 πM and τH=CM3. If the graviton is dominant T is smaller than TH and more important, it can have a maximum at M≈Mp and then go to zero. The connection with string theory is discussed.

Statistical mechanics of the self-gravitating gas: II. Local physical magnitudes and fractal structures

We complete our study of the self-gravitating gas by computing the fluctuations around the saddle point solution for the three statistical ensembles (grand canonical, canonical and microcanonical). Although the saddle point is the same for the three ensembles, the fluctuations change from one ensemble to the other. The zeroes of the small fluctuations determinant determine the position of the critical points for each ensemble. This yields the domains of validity of the mean field approach. Only the S-wave determinant exhibits critical points. Closed formulae for the S- and P-wave determinants of fluctuations are derived. The local properties of the self-gravitating gas in thermodynamic equilibrium are studied in detail. The pressure, energy density, particle density and speed of sound are computed and analyzed as functions of the position. The equation of state turns out to be locally p(r→)=TρV(r→) as for the ideal gas. Starting from the partition function of the self-gravitating gas, we prove in this microscopic calculation that the hydrostatic description yielding locally the ideal gas equation of state is exact in the N=∞ limit. The dilute nature of the thermodynamic limit (N∼L→∞ with N/L fixed) together with the long range nature of the gravitational forces play a crucial role in obtaining such ideal gas equation. The self-gravitating gas being inhomogeneous, we have PV/[NT]=f(η)⩽1 for any finite volume V. The inhomogeneous particle distribution in the ground state suggests a fractal distribution with Haussdorf dimension D, D is slowly decreasing with increasing density, 1<D<3. The average distance between particles is computed in Monte Carlo simulations and analytically in the mean field approach. A dramatic drop at the phase transition is exhibited, clearly illustrating the properties of the collapse.

Highly unstable fundamental strings in inflationary cosmologies

We characterize a regime of extreme (Jeans-like) instability, for strings evolving in cosmological backgrounds, in which the string’s proper size grows asymptotically like the scale factor of the expanding universe. We develop a new approximation scheme, based on the asymptotic proportionality of world-sheet and conformal times, for the systematic, quantitative description of such a nonlinear regime. We find that only inflationary geometries (accelerated expansion) are compatible with this instability, and we derive an equation of state for a perfect fluid of unstable strings. The effective pressure is negative, but not large enough to sustain by itself a phase of accelerated expansion.

Quantum dynamics of strings in black hole space times

Abstract We quantize a bosonic string in the D-dimensional Schwarzschild geometry following the method recently proposed by the present authors. This allows us to take into account strong-curvature effects of the black hole. We start from the exact motion of the center of mass of the string, and compute the quantum fluctuations around it to first and second order. This provides the dominant term for physical magnitudes in an expansion in powers of √α / RS (√α = lpl = Planck length, RS = Schwarzschilds radius). The mass spectrum and critical dimension are the same as in flat space-time but there is non-trivial elastic and inelastic scattering by the black hole. Ingoing and outgoing modes are introduced in a light-cone-gauge formalism. A linear transformation relating these modes desribes two main effects: (i) polarization changes and (ii) mixing of the particle and antiparticle modes reversing, at the same time, their right or left character.