Non-equilibrium statistical mechanics in the general theory of relativity: II. Linear fields in a kinetic approximation

The first paper in this series introduced a new, manifestly covariant approach to non-equilibrium statistical mechanics in classical general relativity. The object of this second paper is to apply that formalism to the evolution of a collection of particles that interact via linear fields in a fixed curved background spacetime. Given the viewpoint adopted here, the fundamental objects of the theory are a many-particle distribution function, which lives in a many-particle phase space, and a many-particle conservation equation which this distribution satisfies. By viewing a composite N-particle system as interacting one- and (N ? 1)-particle subsystems, one can derive exact coupled equations for appropriately defined reduced one- and (N ? 1)-particle distribution functions. Alternatively, by treating all the particles on an identical footing, one can extract an exact closed equation involving only the one-particle distribution. The implementation of plausible assumptions, which constitute straightforward generalizations of standard non-relativistic “kinetic approximations”, then permits the formulation of an approximate kinetic equation for the one-particle distribution function. In the obvious non-relativistic limit, one recovers the well-known Vlasov-Landau equation. The explicit form for the relativistic expression is obtained for three concrete examples, namely, interactions via an electromagnetic field, a massive scalar field, and a symmetric second rank tensor field. For a large class of interactions, of which these three examples are representative, the kinetic equation will admit a relativistic Maxwellian distribution as an exact stationary solution; and, for these interactions, an H-theorem may be proved.     

Nonequilibrium statistical mechanics in the general theory of relativity I. A general formalism

This is the first in a series of papers, the overall objective of which is the formulation of a new covariant approach to nonequilibrium statistical mechanics in classical general relativity. The object here is the development of a tractable theory for self-gravitating systems. It is argued that the “state” of an N-particle system may be characterized by an N-particle distribution function, defined in an 8N-dimensional phase space, which satisfies a collection of N conservation equations. by mapping the true physics onto a fictitious “background” spacetime, which may be chosen to satisfy some “average” field equations, one then obtains a useful covariant notion of “evolution” in response to a fluctuating “gravitational force.” For many cases of practical interest, one may suppose (i) that these fluctuating forces satisfy linear field equations and (ii) that they may be modeled by a direct interaction. In this case, one can use a relativistic projection operator formalism to derive exact closed equations for the evolution of such objects as an appropriately defined reduced one-particle distribution function. By capturing, in a natural way, the notion of a dilute gas, or impulse, approximation, one is then led to a comparatively simple equation for the one-particle distribution. If, furthermore, one treats the effects of the fluctuating forces as “localized” in space and time, one obtains a tractable kinetic equation which reduces, in the newtonian limit, to the standard Landau equation.     

On the mechanics of photon rockets in the general theory of relativity

Generally covariant equations of motion of a photon rocket comprising all the basic dynamic and thermodynamic processes are derived and are written for the case of a weak gravitational field. The equations lead to results which agree with those obtained by other means.     

Non-equilibrium statistical mechanics of synchrotron-Cerenkov radiation

The synergic synchrotron-Cerenkov (SC) radiation emitted by a relativistic charged particle under the combined effect of the constant external magnetic field and the collective interactions in the ambient plasma (medium) is given in the framework of the non-equilibrium statistical mechanics developed by Prigogine and his co-workers. Starting from the formal solution of the Liouville equation, the one-particle distribution function is calculated. Restricting the motion of the test particle to a circular orbit in the plane normal to the magnetic field, we use the above distribution function to calculate the power emitted per unit solid angle by the test particle as a function of time. We have thus obtained the time evolution of the synergic SC radiation which in the asymptotic limit reproduces the results of Schwinger and his co-workers. It is also shown that the collective interactions within the system produces a shift in the frequency of the outcoming radiation.     

On the mechanics of a mass point in the general theory of relativity

An investigation is made into the equations of motion for mass points of comparable masses, the energy-momentum tensor of which is a linear function of the-function, in connection with the principle of equivalence. If some conditions are fulfilled the equations of motion for mass points subject to a supplementary non-gravitational force effect can be transformed to equations of motion for free motion of mass points in a certain equivalent gravitional field. The equivalent field, in agreement with the local validity of the principle of equivalence, is defined merely along the trajectories of the mass points. , - -, . , , , . , , .

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The dynamics of charged dust in the general theory of relativity

The spherically symmetric motion of charged dust in its own gravitational field (the analog of Tolman's problem) is investigated. A conclusion is reached about the cessation of the contraction of the charged dust cloud.Translated from Izvestiya VUZ. Fizika, No. 12, pp. 63–67, December, 1973.     

The gauge in general relativity. III

A noncustomary gauge theory of general relativity, developed in detail in the preceding paper (II), is here applied to cosmology. A universe that is homogeneous and isotropic in the customary gauge, is considered-first generally, and then in more detail for the case where the noncustomary universe is matter dominated and static. With a particular choice of equation, this model is solved and a new relation between customary mass density, Hubble constant, and deceleration parameter is found. For a customary deceleration parameter of 1.98, this relation yields a customary mass density of 3.1×10^–31 g/cm³-in good agreement with experiment. Finally, the age of the universe in this model is found to be>6.6× 10⁹ yr, again in agreement with other estimates.     

Nonlinear effects in the relative dynamics of test bodies in the general theory of relativity. III. The case of slowly varying parameters

A solution to the equations of the relative dynamics of test bodies in the framework of general relativity is constructed for the case of slowly varying coefficients (when these last depend on the truncated proper time on a reference trajectory). Nonlinear radial oscillations of a test body in the neighborhood of a point of equilibrium in the Nordström space are considered. Some nonlinear effects of stochastic forces in the relative dynamics of test bodies are considered in a space of constant curvature.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 5, pp. 56–61, May, 1980.     

A new approach to the theory of relativity. II. The general theory of relativity

The considerations of Part I are extended and the experimental data and hypotheses that led to the establishment of the general theory of relativity are analyzed. It is found that one of the fundamental assumptions is that light is propagated homogeneously; i.e., by using arbitrary systems of coordinates, propagation of light can be represented by a homogeneous quadratic form. This is shown to be an assumption that can be verified by experiment, at least in principle. As a result of adding a number of further assumptions to this, the usual formalism of the general theory of relativity can be established. In the above point of view, the general theory of relativity—like any other theory—cannot be built upad hoc, but is built on distinct physical hypotheses, each of which can be subjected to test by experiment.     

Spin dynamics in general relativity

The energy-momentum tensor of a fluid composed of spinning particles is presented in a form that is structurally similar to that of a fluid with the energy flux q^a with respect to the fluid velocity u^a, isotropic pressure p and the trace-free anisotropic pressure π^ab. The effect of spin in the evolution of a shear free cosmological model with irrotational geodesic flow is considered.     

Observables in the general theory of relativity

The theory of finite deviations of geodesics as directly observable variables in general relativity theory is elaborated in the proper coordinates of the reference frame of a solitary observer.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 80–83, July, 1976.     

Non-equilibrium quantum statistical mechanics of a damped harmonic oscillator

we give some general remarks on the quantization of systems where the energy operator is not conserved in time. Explicitly, we consider a damped harmonic oscillator and transform the corresponding Schrödinger-Langevin equation to the dual picture where the observables are time dependent but not the states. In this picture the time evolution will be described by a completely positive dynamical semigroup. At finite temperatures the situation is more complicated.     

Electromagnetic radiation in the general theory of relativity. I

The possibilities are investigated for the joint solution of Einstein's and Maxwell's equations for an isotropic electromagnetic field. Three groups of metric forms of space are shown, in which such a solution is permissible.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 4, pp. 65–69, April, 1973.In conclusion the author thanks Professor V. I. Rodichev and Professor D. D. Ivanenko for an interesting discussion of the problem considered in the paper.     

Gravitational waves in the general theory of relativity. IV

A generalization of the invariant criterion for pure gravitational radiation which was presented earlier is carried out in the tetrad formalism of the general theory of relativity. Its connection with the Petrov algebraic classification and other criteria based on it is established.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 15–20, September, 1972.In conclusion the author would like to thank Rodichev and Ivanenko for their constant interest in his work, and also the members of the gravitational seminars directed by Ivanenko, Levashev, and Sokolik for valuable critical observations on the questions here discussed.     

Black hole dynamics in general relativity

Basic features of dynamical black holes in full, non-linear general relativity are summarized in a pedagogical fashion. Qualitative properties of the evolution of various horizons follow directly from the celebrated Raychaudhuri equation.      

Interacting fields in general relativity theory

The direct interaction of a massless neutral scalar field with an electromagnetic field is investigated with regard for the proper gravitational field. The interaction Lagrangian is chosen in the form L_int=FF, =e–1, where the parameter characterizes the interaction force. Exact static spherically and cylindrically symmetric solutions are obtained. A solution with a finite total field energy is extracted. A comparison is made with the corresponding system in flat space-time. It is concluded that the gravitational field performs a regulatory function.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 25–30, September, 1977.The authors are indebted to Yu. S. Vladimirov for valuable comments.