arXiv subjects

Christian Frønsdal

Publications and source records attributed to Christian Frønsdal.

7 recordsLinked to original sources

arXiv2021

Sources for gravity. The Noetherian field theories

This paper attempts to throw some light on what is the correct choice of sources for Einstein's field equations for the gravitational metric, and on the definition of the Cauchy-Noether energy-momentum tensor of relativistic field theories, the natural source of Einstein's equation. The paper opens with a brief review of the development of an idea first advanced by Maupertui (1741): the Dynamical Action Principle. The story reached a turning point with the creation of Einstein's theory of General Relativity. The associated work of Noether served as an inspiration for particle physics for 100 years. The discovery of gravity waves (LIGO 2016) showed that Gravity is a phenomenon akin to, and part of, particle physics, to be treated as a canonical field theory, and eventually quantized. Here the subject is the theory of the classical metric field in interaction with extended distributions of matter.We determine a class of Noetherian field theories that provide extended sources for Einstein's equations and action integrals for classical hydrodynamics. The intimate relationship between hydrodynamics and General Relativity is emphasized. Results: The 2-form gauge theory of Ogievetskij and Polubarinov (1964) makes a crucial contribution to the matter source of gravity. Multiple, unexpected roles are played by `permittivities'. There is a unique, relativistic hydrodynamics with 4 degrees of freedom that includes vorticity, the equation of continuity and a generalization of the Bernoulli equation. It is based on an action principle and it reduces, in the non relativistic limit, to a simple generalization of Lagrange theory of 1760, with just one free parameter..

physics.gen-ph↗

arXiv2016

Relativistic Thermodynamics, a Lagrangian Field Theory for general flows including rotation

The formulation of a dynamical theory of General Relativity, including matter, is viewed as a problem of coupling Einstein's theory of pure gravity, formulated as an action principle, to an independently chosen and well defined field theory of matter. It is well known that this is accomplished in a most natural way when the matter theory is formulated as a relativistic, Lagrangian field theory. Special matter models of this type have been available; here a thermodynamical model that allows for general flows is used. A problem that is of even older date, one that was pursued vigorously by leading scientists of the 19'th century, is that of subjecting hydrodynamics and thermodynamics to an Action Principle. A solution to this problem has been known for some time, but only under the strong restriction to potential flows. A variational principle for general flows has now become available. The present paper lifts this theory to the relativistic context, Special Relativity and General Relativity. The energy momentum tensor has a structure that is more general than that postulated by Tolman, and different from proposed generalizations; it appears to be well suited to represent rotational flows in General Relativity. It incorporates a conserved mass current, the relativistic analogue of the hydrodynamical equation of continuity.

gr-qc↗

arXiv2016

Action Principle for Hydrodynamics and Thermodynamics including general, rotational flows

The restriction of hydrodynamics to non-viscous, potential (gradient, irrotational) flows is a theory both simple and elegant; a favorite topic of introductory textbooks. It is known that this theory can be formulated as an action principle and expanded to include thermodynamicics. This paper presents an action principle for hydrodynamics that includes general, rotational flows. The new theory is a combination of Eulerian and Lagrangian hydrodynamics, with an extension to thermodynamics that includes all the elements of the Gibbsean variational principle. In the first place it is an action principle for adiabatic systems, including the usual conservation laws. Viscosity can be introduced in the usual way, by adding a dissipative term to the momentum equation. The equation for energy dissipation then follows. It is an ideal framework for the description of quasi-static processes, including dissipation. It is a major development of the Navier-Stokes-Fourier approach, the principal advantage being a hamiltonian structure with a natural concept of energy in the form of a first integral of the motion, conserved by virtue of the Euler-Lagrange equations.

physics.gen-ph↗

arXiv2011

On entropy in eulerian thermodynamics

To the student of thermodynamics the most difficult subject is entropy. In this paper we examine the actual, practical application of entropy to two simple systems, the homogeneous slab with fixed boundary values of the temperature, and an isolated atmosphere in the presence of the static gravitational field. The first gives valuable insight into the nature of entropy that is subsequently applied to the second system. It is a basic tenet of thermodynamics that the equilibrium of an extended, homogeneous and isolated system is characterized by a uniform temperature distribution and it is a strongly held belief that this remains true in the presence of gravity. We find that this is consistent with the equations of extended thermodynamics but that entropy enters in an essential way. The principle of equivalence takes on a new aspect.

cond-mat.stat-mech↗

arXiv2011

Heat and Gravitation. I. The Action Principle

This first article of a series formulates the thermodynamics of ideal gases in a constant gravitational field in terms of an action principle that is closely integrated with thermodynamics. The theory, in its simplest form, does not deviate from standard practice, but it lays the foundations for a more systematic approach to the various extensions, such as the incorporation of radiation, the consideration of mixtures and the integration with General Relativity. We study the interaction between an ideal gas and the photon gas, and propose a new approach to this problem. We study the propagation of sound in a vertical, isothermal column and are led to suggest that the theory is incomplete, and to ask whether the true equilibrium state of an ideal gas may turn out be adiabatic, in which case the role of solar radiation is merely to compensate for the loss of energy by radiation into the cosmos. An experiment with a centrifuge is proposed, to determine the influence of gravitation on the equilibrium distribution with a very high degree of precision.

physics.gen-ph↗

arXiv2009

Heat and Gravitation. II. Stability

In this second article of a series we propose to base criteria of stability on the hamiltonian functional that is provided by the variational principle, to replace the reliance that has often been placed on {\it ad hoc} definitions of the "energy". We introduce a new virial principle that is formulated entirely within the Eulerian description of hydrodynamics, which allows a simpler derivation of a well known stability criterion for polytropic stellar configurations. Boundary conditions are based entirely on mass conservation.

physics.gen-ph↗

arXiv1996

Quasi Hopf Deformations of Quantum Groups

The search for elliptic quantum groups leads to a modified quantum Yang-Baxter relation and to a special class of quasi-triangular quasi Hopf algebras. This paper calculates deformations of standard quantum groups (with or without spectral parameter) in the category of quasi-Hopf algebras. An earlier investigation of the deformations of quantum groups, in the category of Hopf algebras, showed that quantum groups are generically rigid: Hopf algebra deformations exist only under some restrictions on the parameters. In particular, affine Kac-Moody algebras are more rigid than their loop algebra quotients; and only the latter (in the case of sl(n)) can be deformed to elliptic Hopf algebras. The generalization to quasi-Hopf deformations lifts this restriction. The full elliptic quantum groups (with central extension) associated with $sl(n)$ are thus quasi-Hopf algebras. The universal R-matrices satisfy a modified Yang-Baxter relation and are calculated more or less explicitly. The modified classical Yang-Baxter relation is obtained, and the elliptic solutions are worked out explicitly. The same method is used to construct the Universal R-matrices associated with Felder's quantization of the Knizhnik-Zamolodchikov-Bernard equation, to throw some light on the quasi Hopf structure of conformal field theory and (perhaps) the Calogero-Moser models.

q-alg↗