arXiv subjects

Christian Fronsdal

Publications and source records attributed to Christian Fronsdal.

At least 19 recordsLinked to original sources

arXiv2023

Theory of sounds in He II

A dynamical model for Landau's original approach to superfluid Helium is presented, with two velocities but only one mass density. Second sound is an adiabatic perturbation that involves the temperature and the roton, aka the notoph. The action incorporates all the conservation laws, including the equation of continuity. With only 4 canonical variables it has a higher power of prediction than Landau's later, more complicated model, with its 8 degrees of freedom. The roton is identified with the massless notoph. This theory gives a very satisfactory account of second and fourth sounds. Second sound is an adiabatic oscillation of the temperature and both vector fields, with no net material motion. Fourth sound involves the roton, the temperature and the density. With the experimental confirmation of gravitational waves the relations between Hydrodynamics and Relativity and particle physics have become more clear, and urgent. The appearance of the Newtonian potential in irrotational hydrodynamics comes directly from Einstein's equations for the metric. The density factor $\rho$ is essential; it is time to acknowledge the role that it plays in particle theory. To complete the 2-vector theory we include the massless roton mode. Although this mode too is affected by the mass density, it turns out that the wave function of the unique notoph propagating mode $\mathcal{N}$ satisfies the normal massless wave equation $\Box\mathcal{N}$ = 0; the roton propagates as a free particle in the bulk of the superfluid without meeting resistance. In this circumstance we may have discovered the mechanism that lies behind the flow of He-II through very thin pores.

physics.gen-ph↗

arXiv2020

Stability analysis for cylindrical Couette flow of compressible fluids

A new analysis of basic Couette flow, is based on an Action Principle for compressible fluids, with a Hamiltonian as well as a kinetic potential. An effective criterion for stability recognizes the tensile strength of water. This interpretation relates the problem to capillary action and to metastable configurations (Berthelot's negative pressure experiment of 1850). We calculate the pressure and density profiles and find that the first instability of basic Couette flow is localized near the bubble point. This theoretical prediction has been confirmed by recent experiments. The theory is the result of merging the two versions of classical hydrodynamics, as advocated by Landau for superfluid Helium II, but here applied to fluids in general, in accord with a widely held opinion concerning superfluidity. In this paper two-flow dynamics is created by merging two actions, not by choosing between them, nor by combining the two vector fields as in the Navier-Stokes equation . At rest, as contributions to the mass flow they cancel, but a non-zero kinetic energy and kinetic potential as well as non-zero angular momentum remain, manifest as liquid tension, as is well known to exist by observation of the meniscus and configurations with negative pressure. (Fronsdal 2020b in preparation). This theory gives a very satisfactory characterization of the limit of stability of the most basic Couette flow. The inclusion of a vector field that is not a gradient has the additional affect of introducing spin, which explains a most perplexing experimental discovery: the ability of frozen Helium to remember its angular momentum.

physics.gen-ph↗

arXiv2018

Rotating planets in Newtonian gravity

Variational techniques have been used in applications of hydrodynamics in special cases but an action that is general enough to deal with both potential flows and solid-body flows, such as cylindrical Couette flow and rotating planets, has been proposed only recently. This paper is one of a series that aims to test and develop the new Action Principle. We study a model of rotating planets, a compressible fluid in a stationary state of motion, under the influence of a fixed or mutual gravitational field. The main problem is to account for the shape and the velocity fields, given the size of the equatorial bulges, the angular velocity at equator and the density profiles. The theory is applied to the principal objects in the solar system, from Earth and Mars to Saturn with fine details of its hexagonal flow and to Haumea with its odd shape. With only 2 parameters the model gives a fair fit to the shapes and the angular velocity field near the surface. Planetary rings are an unforeseen, but a natural and inevitable feature of the dynamics; no cataclysmic event need be invoked to justify them. The simple solutions that have been studied so far are most suitable for the hard planets, and for them the predicted density profiles are reasonable. The effect of precession was not taken into account, nor were entropic forces, so far. There has not yet been a systematic search for truly realistic solutions. The intention is to test the versatility of the action principle; the indications are are very encouraging.

physics.gen-ph↗

arXiv2018

A classical mistake and what it tells us. How to do better with an action principle for Hydro and Thermodynamics

Rayleigh's stability analysis of cylindrical Couette flow, of 1889 and 1916, is in contradiction with observation. The analysis is repeated in many textbooks and reviews up to 2017, and its failure to agree with observation was duly noted. More successful approaches have been found, but little was done to discover the weak point of Rayleigh's argument, what is the reason that it fails. This paper identifies the mistake as one that is endemic in the literature. Since the physics of the problem remains poorly understood, a discussion of this paradox should prove useful. Briefly, the argument depends on the Navier-Stokes equation and on the assumption that a certain expression called "energy density" or "kinetic potential" can be interpreted and used as such. It is shown here that %no energy density is compatible with the Navier-Stokes equation, in the context of Couette flow or in general, and that the use of any expression as a kinetic potential is in conflict with the Navier-Stokes equation, in all but a very limited context. An alternative analysis of basic Couette flow, based on an action principle for compressible fluids, provides a Hamiltonian density as well as a kinetic potential. The two are not the same, even in the simplest cases. The action principle provides a kinetic potential; a new criterion for stability recognizes the profound effect of the surface adhesion and the tensile strength of water. It is in full agreement with observation. Several new experiments are suggested.

physics.gen-ph↗

arXiv2012

An equation of state for dark matter in the Milky Way

Dark matter, believed to be present in many galaxies, is interpreted as a hydrodynamical system in interaction with the gravitational field and with nothing else. The gravitational field of our Galaxy can be inferred from observation of orbital velocities of the visible stars, in a first approximation in which the field is taken to be due to the distribution of dark matter only. An equation of state is determined by the gravitational field via the equations of motion. To arrive at an estimate of the distribution of dark matter in our galaxy, and simultaneously learn something about the gravitational field in the inner regions, the following strategy was adopted: 1. The observed rotation curves suggest an expression for the newtonian potential, valid in the outer region. 2. The assumption of a quasi stationary, spherically symmetric distribution of dark matter then leads to a unique equation of state. 3. This equation of state is assumed to be valid all the way to the center (though of course the newtonian approximation is not). 4. Using this equation of state, together with Einstein's equations and the relativistic hydrostatic condition, we calculate the metric and the matter density throughout the galaxy. The solutions are regular all the way to the center; there is no indication of a structure of the type of a Black Hole. The equation of state that is thus determined experimentally is of the type used by Chandrasekhar and others for the degenerate Fermi gas. In the approximation of weak fields the associated "sinh-Emden" equation has a global, nonsingular solution.

gr-qc↗

arXiv2011

An equation of state for dark matter

Dark matter, believed to be present in many galaxies, is interpreted as a hydrodynamical system in interaction with the gravitational field and nothing else. An equation of state determines the mass distribution and the associated gravitational field. Conversely, the gravitational field can be inferred from observation of orbital velocities of stars in the Milky Way, in a first approximation in which the field is mainly due to the distribution of dark matter. In this approximation, the equation of state is determined by the gravitational field via the equations of motion. The resulting equation of state is a simple expression that accounts for the main features of the galactic rotation curve over 6 orders of magnitude.

astro-ph.CO↗

arXiv2010

Heat and Gravitation. III. Mixtures

The standard treatment of relativistic thermodynamics does not allow for a systematic treatment of mixtures. It is proposed that a formulation of thermodynamics as an action principle may be a suitable approach to adopt for a new investigation. This third paper of the series applies the action principle to a study of mixtures of ideal gases. The action for a mixture of ideal gases is the sum of the actions for the components, with an entropy that, in the absence of gravity, is determined by the Gibbs-Dalton hypothesis. Chemical reactions such as hydrogen dissociation are studied, with results that include the Saha equation and that are more complete than traditional treatments, especially so when gravitational effects are included. A mixture of two ideal gases is a system with two degrees of freedom and consequently it exhibits two kinds of sound. In the presence of gravity the Gibbs-Dalton hypothesis is modified to get results that agree with observation. The possibility of a parallel treatment of real gases is illustrated by an application to van der Waals gases. The overall conclusion is that experimental results serve to pin down the lagrangian in a very efficient manner. This leads to a convenient theoretical framework in which many dynamical problems can be studied.

physics.gen-ph↗

arXiv2008

Thermodynamics with an Action Principle, heat and gravitation (2nd version)

Some features of hydro- and thermodynamics, as applied to atmospheres and to stellar structures, are puzzling: 1. The suggestion, first made by Laplace, that our atmosphere has an adiabatic temperature distribution, is confirmed for the lower layers, but the reason why it should be so is understood only qualitatively. 2. Arguments in which a concept of energy plays a role, in the context of hydro-thermodynamical systems and gravitation, are often flawed, and some familiar results concerning the stability of model stellar structures, first advanced at the end of the 19th century and repeated in the most modern textbooks, are less than completely convincing. 3. The standard treatment of relativistic thermodynamics does not allow for a systematic treatment of mixtures, such as the mixture of a perfect gas with radiation. 4. The concept of mass in applications of general relativity to stellar structure is unsatisfactory. It is proposed that a formulation of thermodynamics as an action principle may be a suitable approach to adopt for a new investigation of these matters. We formulate thermodynamics of ideal gases in terms of an action principle and study the interaction between an ideal gas and the photon gas, or heat. The action principle provides a hamiltonian functional, not available in traditional approaches where familiar expressions for the energy have no operative meaning. The usual polytropic atmosphere in an external gravitational field is examined, in order to determine to what extent it is shaped by radiation. A new formulation of the concept of radiative equilibrium is proposed.

physics.flu-dyn↗

arXiv2007

Reissner-Nordstrom and charged gas spheres

The main point of this paper is a suggestion about the proper treatment of the photon gas in a theory of stellar structure and other plasmas. This problem arises in the study of polytropic gas spheres, where we have already introduced some innovations. The main idea, already advanced in the contextof neutral, homogeneous, polytropic stellar models, is to base the theory firmly on a variational principle. Another essential novelty is to let mass distribution extend to infinity, the boundary between bulk and atmosphere being defined by an abrupt change in the polytropic index, triggered by the density. The logical next step in this program is to include the effect of radiation, which is a very significant complication since a full treatment would have to include an account of ionization, thus fieldsrepresenting electrons, ions, photons, gravitons and neutral atoms as well. In way of preparation, we consider models that are charged but homogeneous, involving only gravity, electromagnetism and a single scalar field that represents both the mass and the electric charge; in short, anon-neutral plasma. While this work only represents a stage in the development of a theory of stars, without direct application to physical systems, it does shed some light on the meaning of the Reissner-Nordstrom solution of the modified Einstein-Maxwell equations., with an application to a simple system.

gr-qc↗

arXiv2007

Stability of Polytropes

This paper is an investigation of the stability of some ideal stars. It is in- tended as a study in General Relativity, with emphasis on the coupling to matter, eventually aimed at a better understanding of very strong gravitational fields and Black Holes. The work is based on an action principle for the complete system of metric and matter fields. We propose a complete revision of the treatment of boundary conditions. An ideal star in our terminology has spherical symmetry and an isentropic equation of state. In our first work on this subject it was assumed that the density vanishes beyond a finite distance from the origin. But it is difficult to decide what the proper boundary conditions should be and we are consequently skeptical of the concept of a fixed boundary. In this paper we investigate the double polytrope, characterized by a polytropic index n less than 5 in the bulk of the star and a value larger than 5 in an outer atmosphere that extends to infinity. It has no fixed boundary but a region of critical density where the polytropic index changes from a value that is appropriate for the bulk of the star to a value that provides a crude model for the atmosphere. The existence of a relation between mass and radius is confirmed, as well as an upper limit on the mass. The principal conclusion is that all the static configurations are stable.

gr-qc↗

arXiv2006

Quantization on Curves

Deformation quantization on varieties with singularities offers perspectives that are not found on manifolds. Essential deformations are classified by the Harrison component of Hochschild cohomology, that vanishes on smooth manifolds and reflects information about singularities. The Harrison 2-cochains are symmetric and are interpreted in terms of abelian $*$-products. This paper begins a study of abelian quantization on plane curves over $\Crm$, being algebraic varieties of the form R2/I where I is a polynomial in two variables; that is, abelian deformations of the coordinate algebra C[x,y]/(I). To understand the connection between the singularities of a variety and cohomology we determine the algebraic Hochschild (co-)homology and its Barr-Gerstenhaber-Schack decomposition. Homology is the same for all plane curves C[x,y]/(I), but the cohomology depends on the local algebra of the singularity of I at the origin.

math-ph↗

arXiv2006

Ideal Stars and General Relativity

We study a system of differential equations that governs the distribution of matter in the theory of General Relativity. The new element in this paper is the use of a dynamical action principle that includes all the degrees of freedom, matter as well as metric. The matter lagrangian defines a relativistic version of non-viscous, isentropic hydrodynamics. The matter fields are a scalar density and a velocity potential; the conventional, four-vector velocity field is replaced by the gradient of the potential and its scale is fixed by one of the eulerian equations of motion, an innovation that significantly affects the imposition of boundary conditions. If the density is integrable at infinity, then the metric approaches the Schwarzschild metric at large distances. There are stars without boundary and with finite total mass; the metric shows rapid variation in the neighbourhood of the Schwarzschild radius and there is a very small core where a singularity indicates that the gas laws break down. For stars with boundary there emerges a new, critical relation between the radius and the gravitational mass, a consequence of the stronger boundary conditions. Tentative applications are suggested, to certain Red Giants, and to neutron stars, but the investigation reported here was limited to polytropic equations of state. Comparison with the results of Oppenheimer and Volkoff on neutron cores shows a close agreement of numerical results. However, in the model the boundary of the star is fixed uniquely by the required matching of the interior metric to the external Schwarzschild metric, which is not the case in the traditional approach.

gr-qc↗

arXiv2006

Cosmology with an Action Principle

The Friedman universe is re-examined in a context that is non-standard only in that the properties of matter are postulated in the form of an action principle. Applications to equilibrium configurations of ideal stars have already been reported. In this paper we apply the same theory to a fresh examination of the Friedman universe. The results agree with standard theory in the case of low densities. A suggestion is made to replace "vacuum energy" by "external force".

gr-qc↗

arXiv2005

Deformation Quantization on Singular Coadjoint Orbits

Invariant star products are constructed on minimal coadjoint orbits of all the simple Lie algebras. Explicit expressions are given for the generators of the Joseph ideals and the associated infinitesimal characters.

math.RT↗

arXiv2005

Growth of a Black Hole

This paper studies the interpretation of physics near a Schwarzschild black hole. A scenario for creation and growth is proposed that avoids the conundrum of information loss. In this picture the horizon recedes as it is approached and has no physical reality. Radiation is likely to occur, but it cannot be predicted.

gr-qc↗

arXiv2003

q-Algebras and Arrangements of Hyperplanes

Varchenko's approach to quantum groups, from the theory of arrangements of hyperplanes, can be usefully applied to q-algebras in general, of which quantum groups and quantum (super) Kac-Moody algebras are special cases. New results are obtained on the classification of q-algebras, and of the Serre ideals of generalized quantum (super) Kac-Moody algebras.

math.QA↗

arXiv2001

Retrospective on Quantization

Quantization is still a central problem of modern physics. One example of an unsolved problem is the quantization of Nambu mechanics. After a brief comment on the role of Harrison cohomology, this review concentrates on the central problem of quantization of QCD and, more generally, quark confinement seen as a problem of quantization. Several suggestions are made, some of them rather extravagant.

hep-th↗

arXiv2000

On the classification of q-algebras

The problem is the classification of the ideals of ``free differential algebras", or the associated quotient algebras, the q-algebras; being finitely generated, unital C-algebras with homogeneous relations and a q-differential structure. This family of algebras includes the quantum groups, or at least those that are based on simple (super) Lie or Kac-Moody algebras. Their classification would encompass the so far incompleted classification of quantized (super) Kac-Moody algebras and of the (super) Kac-Moody algebras themselves. These can be defined as singular limits of q-algebras, and it is evident that to deal with the q-algebras in their full generality is more rational than the examination of each singular limit separately. This is not just because quantization unifies algebras and superalgebras, but also because the points "q = 1" and "q = -1" are the most singular points in parameter space. In this paper one of two major hurdles in this classification program has been overcome. Fix a set of integers n_1,...,n_k, and consider the space B_Q of homogeneous polynomials of degree n_1 in the generator e_1, and so on. Assume that there are no constants among the polynomials of lower degree, in any one of the generators; in this case all constants in the space B_Q have been classified. The task that remains, the more formidable one, is to remove the stipulation that there are no constants of lower degree.

math.QA↗