Relativistic velocity - potential hydrodynamics and stellar stability

Schutz's Hamiltonian theory of a relativistic perfect fluid, based on the velocity-potential version of classical perfect fluid hydrodynamics as formulated by Seliger and Whitham, is used to derive, in the framework of the Arnowitt, Deser, and Misner (ADM) method, a general partially reduced Hamiltonian for relativistic systems filled with a perfect fluid. The time coordinate is chosen, as in Lund's treatment of collapsing balls of dust, as minus the only velocity potential different from zero in the case of an irrotational and isentropic fluid. A "semi-Dirac" method can be applied to quantize astrophysical and cosmological models in the framework of this partially reduced formalism. If one chooses Taub's adapted comoving coordinate system, it is possible to derive a fully reduced ADM Hamiltonian, which is equal to minus the total baryon number of the fluid, generalizing a result previously obtained by Moncrief in the more particular framework of Taub's variational principle, valid for self-gravitating barotropic relativistic perfect fluids. An unconstrained Hamiltonian density is then explicitly derived for a fluid obeying the equation of state $p=(\ensuremath{\gamma}\ensuremath{-}1)\ensuremath{\rho}$ ($1\ensuremath{\le}\ensuremath{\gamma}\ensuremath{\le}2$), which can adequately describe the phases of very high density attained in a catastrophic collapse or during the early stages of the Universe. This Hamiltonian density, shown to be equivalent to Moncrief's in the particular case of an isentropic fluid, can be simplified for fluid-filled class-$A$ diagonal Bianchi-type cosmological models and appears as a suitable starting point for the study of the canonical quantization of these models.

The conference brought together people working in mathematical general relativity, a field which lies at the interface of analysis, differential geometry and physics. There was a mix of established workers in the subject with participants at the start of their careers and researchers from neighbouring fields. The Einstein equations are at the heart of general relativity theory. They form a system of evolution equations and their solutions are conveniently parametrized by initial data. These initial data are required to satisfy constraint equations and these were the topic of several lectures at the conference. The talks of Corvino and Pollack were concerned with recent progress in methods for constructing solutions of the constraints. An important concept for solutions of the constraints is that of mass, including quasilocal mass. This was a central theme of the talks of Degeratu, Huisken and Shi. Huisken introduced a striking new approach to the definition of mass based on isoperimetric inequalities. Dain presented a variational characterization of the extreme Reissner–Nordström solution, thus relating the study of the constraint equations to the theory of black holes. The talk of Wohlfarth was rather outside the main area of the conference. He described a new concept of geometry motivated by string theory which may come to enrich the circle of ideas within mathematical relativity. Many of the talks at the conference were on evolution equations related to general relativity. Dafermos and Finster presented results related to the dynamical stability of black holes. Dafermos described new results on the rate of decay of solutions of the wave equation on the Schwarzschild spacetime while Finster explained applications of methods of functional analysis to the wave equation on the Kerr spacetime. Struwe's talk concerned uniqueness for supercritical nonlinear wave equations which from the point of view of the Einstein equations are an important example to compare with. The Maxwell equations are another important comparison system and Bauer showed how they can be used to understand more about radiation formulae. Velázquez gave an introduction to singularity formation in the Keller–Segel model, a parabolic system coming from mathematical biology. This is a possible source of insight for obtaining a rigorous understanding of critical collapse in general relativity. The talks of Andréasson, Choptuik and Lindblom dealt with various aspects of the application of numerical techniques to the study of the Einstein evolution equations. Choptuik showed impressive new simulations of coalescing black holes due to Pretorius which could hardly have been imagined just a year ago. Cosmology is at present a very active area of research in general relativity. This is in part due to the challenge of understanding the observed accelerated expansion of our universe. Mathematics is beginning to make its mark in this subject and this was reflected by talks of Heinzle, Rendall and Tod. Mathematical relativity is a meeting point for many ideas and the abstracts which follow give some idea of the variety of the subject. In fact the spectrum of topics discussed by the participants at the conference was much wider than those for which talks could be scheduled. By limiting the number of presentations it was possible to leave plenty of time for people to exchange insights. The lively interactions observed make us hopeful that this conference has given a boost to the development of the subject in the next few years.

The scalar and Einstein field equations are considered in the general symmetric comoving system. The self-gravitational interaction of the scalar field is expressed by identifying the energy momentum tensor with the one canonically associated to the scalar field. It is shown that the scalar field can depend only on the radial and time variables and that the energy momentum tensor is implicitly conserved. Two classes of solutions are then studied. A first class, that contains the static case previously considered and the complete time-dependent case, is determined by a priori assuming the unknown functions to depend on one only variable. This class is meaningful essentially for massless field. A second class of solutions refers to the a priori assumption that the scalar field alone depends on r or on t. In both cases some comments are developed for the massive scalar field and for a large class of massless fields we are able to determine the solutions.

The gravitational interaction of a scalar field, with allowance for the possible influence of the torsional and nonmetric nature of space-time, is investigated within the framework of the affine-metric theory of gravitation. The equations of the theory are derived from the variational principle. It is shown that in an affine-metric space, the combined Lagrangian of the gravitational and scalar fields with conformal coupling is reduced to the Lagrangian of the system of gravitational and axion fields in the general theory of relativity. All of the exact general solutions of the consistent system of equations of gravitational and scalar (massless) fields in the affine-metric space under consideration are obtained for all types of homogeneous Friedmann cosmological models, with the initial singularity being removed from some of them. Homogeneous, anisotropic cosmological models, for which all of the exact general solutions are also obtained, are investigated. Some of these models are nonsingular, and the effect of isotropization due to the torsional and nonmetric nature of space-time occurs for many of them.

Late-time acceleration with a scalar field source: Observational constraints and statefinder diagnostics

view Abstract Citations (1) References (2) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Odd-parity perturbations of spherically symmetric star clusters in general relativity Semenzato, R. ; Ipser, J. R. Abstract A theory of odd-parity nonspherical perturbations of collisionless, isotropic, spherically symmetric star clusters is developed within general relativity for a spherical-harmonic index of not less than 2. A variational principle is derived for the associated normal modes of oscillation, and reveals that an unstable normal mode has a pure exponentially growing time dependence, and hence a normal mode can become unstable in a smooth fashion only through zero frequency. No instabilities can set in through zero-frequency modes along smooth sequences of models with fewer high-energy stars than low-energy stars, and unless unstable normal modes suddenly appear in a nonsmooth fashion, the implication is that these models possess no unstable modes. Publication: The Astrophysical Journal Pub Date: July 1981 DOI: 10.1086/159078 Bibcode: 1981ApJ...247..671S Keywords: Astrophysics; Dynamic Stability; Perturbation Theory; Relativity; Star Clusters; Distribution Functions; Einstein Equations; Spherical Harmonics; Star Distribution; Variational Principles; Astrophysics full text sources ADS |

Conserved vectors are divergencies of superpotentials. In field theory on curved backgrounds, they are useful in calculating global ‘charges’ in arbitrary coordinates and local conserved quantities for small perturbations with specific gauge conditions. Superpotentials are, however, ill–defined. A new criterion of Julia and Silva selects uniquely for Dirichlet boundary conditions the ‘KBL superpotential’ as proposed by Katz, Bičák and Lynden–Bell, which has remarkable properties.Here, we show that a Belinfante–type addition to the KBL superpotential in general relativity gives an expression that is independent of boundary conditions defined by a variational principle. The modified superpotential has the same global properties as the KBL one, except for angular momentum at null infinity, and it does not differ from the KBL superpotential in the linearized theory of gravitation.As an illustration in linearized theory on curved backgrounds, we calculate conserved quantities for small perturbations on a Friedmann–Robertson–Walker spacetime associated with conformal Killing vectors. Our unifying view relates a number of applications in cosmology found in the literature. Globally conserved quantities have simple physical interpretations in the ‘uniform Hubble expansion’ gauge.

We consider a class of one dimensional Vector Nonlocal Non-linear Schrödinger equation (VNNLSE) in an external complex potential with time-modulated Balanced Loss-Gain(BLG) and Linear Coupling(LC) among the components of Schrödinger fields, and space-time dependent nonlinear strength. The system admits Lagrangian and Hamiltonian formulations under certain conditions. It is shown that various dynamical variables like total power,  -symmetric Hamiltonian, width of the wave-packet and its speed of growth, etc are real-valued despite the Hamiltonian density being complex-valued. We study the exact solvability of the generic VNNLSE with or without a Hamiltonian formulation. In the first part, we study time-evolution of moments which are analogous to space-integrals of Stokes variables and find condition for existence of solutions which are bounded in time. In the second part, we use a non-unitary transformation followed by a coordinate transformation to map the VNNLSE to various solvable equations. The coordinate transformation is not required at all for the limiting case when non-unitary transformation reduces to pseudo-unitary transformation. The exact solutions are bounded in time for the same condition which is obtained through the study of time-evolution of moments. Various exact solutions of the VNNLSE are presented.

Stability of a self-gravitating gaseous mass is formulated on the basis of the general variational method developed by PRIGOGINE and GLANSDORFF, for the most general case in which the perturbation is non-adiabatic and the unperturbed configuration may be contracting, rotating differentially, and moving internally. The characteristic equation which determines the complex frequencies of oscillation is derived. For adiabatic oscillations of a steady configuration, this equation reduces to LYNDEN-BELL and OSTRIKER’s equation of the ordinary generalized variational principle. With the help of the characteristic equation, the stability criterion is derived formally for non-adiabatic oscillations of a uniformly rotating equilibrium configuration. The stable cases are classified into three types. As another illustration of the general method, the stability of a homologously contracting star is examined analytically for the case of the homologous oscillations. The result of KATO and UNNO is thus confirmed in a straight-forward analysis. The scale change of the unperturbed configuration is found to be an additional cause of exciting the pulsation other than the gravitational ε-mechanism.

An (n+1) factorization of an (n+1)-dimensional Riemann manifold is performed. For a space permitting a Killing vector, the (n+l)-dimensional Hubert variational principle reduces to the variational principle for the corresponding quantities in an n-dimensional space. Hence, setting n=4 and n=3, versions of a unified theory of gravitational, electromagnetic, and scalar fields and the steady-space theory of general relativity theory, respectively, are constructed. The five-dimensional variational principle for geodesics reduces to the four-dimensional leastaction principle for the test charged particle moving in gravitational, electromagnetic, and scalar fields.

We develop a strictly geometric reformulation of General Relativity (GR) in which a scalar clock field ϕ defines the spacetime foliation through its timelike gradient nµ = -∇µϕ/ -(∇ϕ) 2 . The field ϕ is not a new dynamical degree of freedom and does not enter the action; its sole role is to generate a covariant, hypersurface-orthogonal slicing that renders the ADM lapse, shift, and extrinsic curvature algebraic functionals of (gµν , ϕ).Starting from the pure Einstein-Hilbert action, we carry out a complete 3 + 1 decomposition adapted to the clock-field foliation and derive the Hamiltonian constraint, momentum constraint, and full constraint algebra. All coincide exactly with their canonical GR counterparts, confirming that the theory propagates only the two tensorial graviton modes and contains no additional scalar or vector excitations. The Einstein equations are recovered identically, the ADM algebra remains first class, and both cosmological dynamics and linear perturbations match those of standard GR.The novelty of the construction is conceptual rather than dynamical: ϕ provides a covariant, relational time variable that fixes the foliation non-perturbatively, offering a clean geometric notion of time for canonical quantization and Wheeler-DeWitt evolution. Because the framework introduces no new fields, couplings, or potentials, it avoids the instabilities and extra degrees of freedom typical of scalar-tensor or modified-gravity theories.This yields a mathematically consistent and fully covariant formulation of GR with an internally defined geometric clock, opening new directions for relational observables, deparametrized quantum gravity, and emergent-time frameworks.

The standard model of cosmology with postulated dark energy and dark matter sources may be considered as a fairly successful fitting model to observational data. However, this model leaves the question of the physical origin of these dark components open. Fully relativistic contributions that act like dark energy on large scales and like dark matter on smaller scales can be found through generalization of the standard model by spatially averaging the inhomogeneous Universe within general relativity. The spatially averaged 3 + 1 Einstein equations are effective balance equations that need a closure condition. Heading for closure we here explore topological constraints. Results are straightforwardly obtained for averaged 2 + 1 model universes. For the relevant 3 + 1 case, we employ a method based on the Gauss–Bonnet–Chern theorem generalized to Lorentzian spacetimes and implement a sandwich approach to obtain spatial average properties. The 3 + 1 topological approach supplies us with a new equation linking evolution of scalar invariants of the expansion tensor to the norm of the Weyl tensor. From this we derive general evolution equations for averaged scalar curvature and kinematical backreaction, and we discuss related evolution equations on this level of the hierarchy of averaged equations. We also discuss the relation between topological properties of cosmological manifolds and dynamical topology change, e.g. as resulting from the formation of black holes.

Cosmological perturbations

Plane walls (including plane domain walls) without reflection symmetry are studied in the framework of Einstein's general relativity. Using the distribution theory, all the Einstein field equations and Bianchi identities are split into two groups: one holding in the regions outside of the wall and the other holding at the wall. The Einstein field equations at the wall are found to take a very simple form, and given explicitly in terms of the discontinuities of the metric coefficients and their derivatives. The Bianchi identities at the wall are also given explicitly. Using the latter, the interaction of a plane wall with gravitational waves and some specific matter fields is studied. In particular, it is found that, when a gravitational plane wave passes through a wall, if the wall has no reflection symmetry, the phenomena, such as reflection, stimulation, or absorption, in general, occur. It is also found that, unlike for gravitational waves, a massless scalar wave or an electromagnetic wave continuously passes through a wall without any reflection. The repulsion and attraction of a plane wall are also studied. It is found that the acceleration of an observer who is at rest relative to the wall usually consists of threemore » parts: one is due to the force produced by the wall, the second is due to the force produced by the space-time curvature, which is zero if the wall has reflection symmetry, and the last is due to the accelerated motion of the wall. As a result, a repulsive (attractive) plane wall may not be repulsive (attractive) at all. Finally, the collision and interaction among the walls are studied.« less

An asymptotic approach based upon Lundquist equations is developed for the study of nonlinear wave propagation on a pinched cylindrical plasma. Wave speed and asymptotic evolution equations are derived. A comparison between linear stability criteria and results of the nonlinear theory is made and a set of refined stability criteria is obtained.