arXiv subjects

Sidney Bludman

Publications and source records attributed to Sidney Bludman.

13 recordsLinked to original sources

arXiv2012

How Scaling Symmetry Solves a Second-Order Differential Equation

While not generally a conservation law, any symmetry of the equations of motion implies a useful reduction of any second-order equationto a first-order equation between invariants, whose solutions (first integrals) can then be integrated by quadrature (Lie's Theorem on the solvability of differential equations). We illustrate this theorem by applying scale invariance to the equations for the hydrostatic equilibrium of stars in local thermodynamic equilibrium: Scaling symmetry reduces the Lane-Emden equation to a first-order equation between scale invariants un; vn, whose phase diagram encapsulates all the properties of index-n polytropes. From this reduced equation, we obtain the regular (Emden) solutions and demonstrate graphically how they transform under scale transformations.

math-ph↗

arXiv2012

Scaling Symmetry and Integrable Spherical Hydrostatics

Any symmetry reduces a second-order differential equation to a first integral: variational symmetries of the action (exemplified by central field dynamics) lead to conservation laws, but symmetries of only the equations of motion (exemplified by scale-invariant hydrostatics) yield first-order {\em non-conservation laws} between invariants. We obtain these non-conservation laws by extending Noether's Theorem to non-variational symmetries and present an innovative variational formulation of spherical adiabatic hydrostatics. For the scale-invariant case, this novel synthesis of group theory, hydrostatics, and astrophysics allows us to recover all the known properties of polytropes and define a {\em core radius}, inside which polytropes of index $n$ share a common core mass density structure, and outside of which their envelopes differ. The Emden solutions (regular solutions of the Lane-Emden equation) are obtained, along with useful approximations. An appendix discusses the $n=3$ polytrope in order to emphasize how the same mechanical structure allows different thermal structures in relativistic degenerate white dwarfs and zero age main sequence stars.

math-ph↗

arXiv2011

Applications of Classical Scaling Symmetry

Any symmetry reduces a second-order differential equation to a first-order equation: variational symmetries of the action (exemplified by central field dynamics) lead to conservation laws, but symmetries of only the equations of motion (exemplified by scale-invariant hydrostatics), yield first-order {\em non-conservation laws} between invariants. We obtain these conservation laws by extending Noether's Theorem to non-variational symmetries, and present a variational formulation of spherical adiabatic hydrostatics. For scale-invariant hydrostatics, we directly recover all the published properties of polytropes and define a {\em core radius}, a new measure of mass concentration in polytropes of index n. The Emden solutions (regular solutions of the Lane-Emden equation) are finally obtained, along with useful approximations. An appendix discusses the special n=3 polytrope, emphasizing how the same mechanical structure allows different {\em thermostatic} structures in relativistic degenerate white dwarfs and and zero age main sequence stars.

math-ph↗

arXiv2010

Scale invariant stellar structure

In scale invariant hydrostatic barotropes, the radial evolutionary equation linearly relates the local gravitational and internal energies. From this first-order equation, directly follow all the properties of polytropes and the important mass-radius relation. Quadrature then leads to the regular Lane-Emden functions and their Picard and Pade approximations, which are useful wherever stars are approximately or exactly polytropic. We illustrate this particularly for the n=3 regular polytrope and obtain analytic approximations to the solution of the Lane-Emden equation, valid over the bulk of relativistic degenerate stars (massive white dwarfs) and chemically homogeneous stars in radiative equilibrium (ZAMS stars).

physics.class-ph↗

arXiv2010

Invariant relationships deriving from classical scaling transformations

Because scaling symmetries of the Euler-Lagrange equations are generally not variational symmetries of the action, they do not lead to conservation laws. Instead, an extension of Noether's theorem reduces the equations of motion to evolutionary laws that prove useful, even if the transformations are not generalized symmetries of the equations of motion. In the case of scaling, symmetry leads to a scaling evolutionary law, a first-order equation in terms of scale invariants, linearly relating kinematic and dynamic degrees of freedom. This scaling evolutionary law appears in dynamical and in static systems. Applied to dynamical central-force systems, the scaling evolutionary equation leads to generalized virial laws, which linearly connect the kinetic and potential energies. Applied to barotropic hydrostatic spheres, the scaling evolutionary equation linearly connects the gravitational and internal energy densities. This implies well-known properties of polytropes, describing degenerate stars and chemically homogeneous non-degenerate stellar cores.

physics.class-ph↗

arXiv2010

Classical Scaling Symmetry Implies Useful Nonconservation Laws

Scaling symmetries of the Euler-Lagrange equations are generally not variational symmetries of the action and do not lead to conservation laws. Nevertheless, by an extension of Noether's theorem, scaling symmetries lead to useful {\em nonconservation} laws, which still reduce the Euler-Lagrange equations to first order in terms of scale invariants. We illustrate scaling symmetry dynamically and statically. Applied dynamically to systems of bodies interacting via central forces, the nonconservation law is Lagrange's identity, leading to generalized virial laws. Applied to self-gravitating spheres in hydrostatic equilibrium, the nonconservation law leads to well-known properties of polytropes describing degenerate stars and chemically homogeneous nondegenerate stellar cores.

physics.class-ph↗

arXiv2008

Gravitational vacuum energy in our recently accelerating universe

We review current observations of the homogeneous cosmological expansion which, because they measure only kinematic variables, cannot determine the dynamics driving the recent accelerated expansion. The minimal fit to the data, the flat $ΛCDM$ model, consisting of cold dark matter and a cosmological constant, interprets $4Λ$ geometrically as a classical spacetime curvature constant of nature, avoiding any reference to quantum vacuum energy. (The observed Uehling and Casimir effects measure forces due to QED vacuum polarization, but not any quantum material vacuum energies.) An Extended Anthropic Principle, that Dark Energy and Dark Gravity be indistinguishable, selects out flat $ΛCDM$. Prospective cosmic shear and galaxy clustering observations of the growth of fluctuations are intended to test whether the 'dark energy' driving the recent cosmological acceleration is static or moderately dynamic. Even if dynamic, observational differences between an additional negative-pressure material component within general relativity (Dark Energy) and low-curvature modifications of general relativity (Dark Gravity) will be extremely small.

astro-ph↗

arXiv2007

What Drives Our Accelerating Universe?

The homogeneous expansion history H(z) of our universe measures only kinematic variables, but cannot fix the underlying dynamics driving the recent acceleration: cosmographic measurements of the homogeneous universe, are consistent with either a static finely-tuned cosmological constant or a dynamic `dark energy' mechanism, which may be material Dark Energy or modified gravity (Dark Gravity). Resolving the composition and foreground noise to reduce their large systematic errors.dynamics of either kind of `dark energy', will require complementing the homogeneous expansion observations with observations of the growth of cosmological fluctuations. Because the 'dark energy' evolution is at least quasi-static, any dynamical effects on the fluctuation growth function g(z) will be minimal. They will be best studied in the weak lensing convergence of light from galaxies at 0<z<5, from neutral hydrogen at 6<z<20, and ultimately from the CMB last scattering surface at z=1089. Galaxy clustering also measures g(z), but requires large corrections for baryonic composition and foreground noise. Projected observations potentially distinguish static from dynamic `dark energy', but distinguishing dynamic Dark Energy from Dark Gravity will require a weak lensing shear survey more ambitious than any now projected. Low-curvature modifications of Einstein gravity are also, in principle, observable in the solar system or in isolated galaxy clusters. The cosmological constant can only be fine-tuned, at present. The Cosmological Coincidence Problem, that we live when the ordinary matter density approximates the 'gravitational vacuum energy', on the other hand, is a material problem, calling for an understanding of the observers' role in cosmology.

astro-ph↗

arXiv2006

Cosmological Acceleration: Dark Energy or Modified Gravity?

We review the evidence for recently accelerating cosmological expansion or "dark energy", either a negative pressure constituent in General Relativity (Dark Energy) or modified gravity (Dark Gravity), without any Dark Energy constituent. If constituent Dark Energy does not exist, so that our universe is now dominated by pressure-free matter, Einstein gravity must be modified at low curvature. The vacuum symmetry of any Robertson-Walker universe then characterizes Dark Gravity as low- or high-curvature modifications of Einstein gravity. The dynamics of either kind of "dark energy" cannot be derived from the homogeneous expansion alone, but requires also observing the growth of inhomogeneities. Present and projected observations are all consistent with a small fine tuned cosmological constant, with nearly static Dark Energy, or with gravity modified at cosmological scales. The growth of cosmological fluctuations will potentially distinguish static "dark energy" from dynamic "dark energy" with equation of state $w(z)$ either changing rapidly or tracking the background matter. But to cosmologically distinguish $Λ$CDM from modified gravity will require a weak lensing shear survey more ambitious than any now projected. Dvali-Gabadadze-Porrati modifications of Einstein gravity may also be detected in refined bservations in the solar system or at the intermediate Vainstein scale. Dark Energy's epicyclic character, failure to explain the original Cosmic Coincidence ("Why now?") without fine tuning, inaccessibility to laboratory or solar system tests, along with braneworld theories, now motivate future precision solar system, Vainstein-scale and cosmological-scale studies of Dark Gravity.

astro-ph↗

arXiv2004

Tracking Quintessence Would Require Two Cosmic Coincidences

Good tracking requires that the quintessence energy fraction slowly increase while the roll $λ\equiv -d\ln V/\varkappa dϕ$ slowly decreases, but is not yet truly slow-rolling. The supernova bound on the present quintessence equation of state requires either (1) a cosmological constant or other fine-tuned "crawling quintessence" or (2) "roll-over quintessence" that tracked until recently, but now became slow rolling, because of a sharp increase in potential curvature. Thus, fine-tuning is required by constant equation of state and inverse power potentials, but can be avoided by the SUGRA and Skordis-Albrecht potentials and other good trackers, provided quintessence energy domination and slow roll {\em both} began only recently. This makes the time in which we live special in {\em two} respects.

astro-ph↗

arXiv2003

What We Already Know About Quintessence

Good tracking requires significant quintessence energy fraction, even in the past, but a potential that is not yet truly slow-rolling. The supernova bound on cosmic acceleration excludes constant equation of state and inverse power potentials, but allows the SUGRA potential and other good trackers, in which quintessence energy domination and kinetic energy suppression {\em both} began only recently. This makes the time in which we live special in {\em two} respects.

astro-ph↗

arXiv2000

Vacuum Energy: Cosmological Constant or Quintessence?

For a flat universe presently dominated by smooth energy, either cosmological constant (LCDM) or quintessence (QCDM), we calculate the asymptotic collapsed mass fraction as function of the present ratio of smooth energy to matter energy $\mathcal R_0$. Identifying the normalized collapsed fraction as a conditional probability for habitable galaxies, we observe that the observed present ratio $\mathcal R_0 \sim 2$ is likely in LCDM, but more likely in QCDM. Inverse application of Bayes' Theorem makes the Anthropic Principle a predictive scientific principle: the data implies that the prior probability for $\mathcal R_0$ must be essentially flat over the anthropically allowed range. Interpreting this prior as a distribution over {\em theories} lets us predict that any future theory of initial conditions must be indifferent to $\mathcal R_0$. This application of the Anthropic Principle does not demand the existence of other universes.

astro-ph↗

arXiv2000

Observed Smooth Energy is Anthropically Even More Likely As Quintessence Than As Cosmological Constant

For a universe presently dominated by static or dynamic vacuum energy, cosmological constant (LCDM) or quintessence (QCDM), we calculate the asymptotic collapsed mass fraction as function of the present ratio of vacuum energy to clustered mass, Ω_{Q0}/Ω_{m0}. Identifying these collapsed fractions as anthropic probabilities, we find the value Ω_{Q0}/Ω_{m0} \sim 2 to be reasonably likely in LCDM, and very likely in QCDM.

astro-ph↗