A relativistic self-gravitating equilibrium system with spherical symmetry as well as with steady energy flow is investigated perturbatively around the hydrostatic limit, where the radial component of the fluid velocity field <italic>u&#956;</italic> is sufficiently small. Each component of vectors and tensors consisting of the system is expanded in different powers, which makes the covariant perturbation approach ineffective. The differential equations to determine the subleading correction of the structure variables are presented. The system retains the current <italic>j&#956;</italic> accounting for the steady flow, which contributes to the entropy current <italic>s&#956;</italic> in such a general covariant form that <math altimg='si1.svg'><mrow><msup><mi>s</mi><mi>&#956;</mi></msup><mo linebreak='goodbreak'>=</mo><mi>a</mi><msup><mi>u</mi><mi>&#956;</mi></msup><mo linebreak='goodbreak'>+</mo><mi>b</mi><msup><mi>j</mi><mi>&#956;</mi></msup></mrow></math> with <italic>a, b</italic> unknown parametric functions. To determine them, a new condition is proposed. This condition imposes the entropy current to be of an unconventional form <math altimg='si2.svg'><mrow><msup><mi>s</mi><mi>&#956;</mi></msup><mo linebreak='goodbreak'>=</mo><mrow><mo>(</mo><mi>s</mi><mo linebreak='badbreak'>&#8722;</mo><mi>b</mi><msup><mi>j</mi><mn>0</mn></msup><mo>)</mo></mrow><msup><mi>u</mi><mi>&#956;</mi></msup><mo linebreak='goodbreak'>/</mo><msup><mi>u</mi><mn>0</mn></msup><mo linebreak='goodbreak'>+</mo><mi>b</mi><msup><mi>j</mi><mi>&#956;</mi></msup></mrow></math>, where <italic>s</italic> is the entropy density. The remaining parameter <italic>b</italic> is fixed by the current conservation equation. The perturbative analysis shows that <italic>b</italic> starts with the quadratic order and its leading term is determined explicitly.

Japan

Shuichi Yokoyama

Self-gravitating equilibrium with slow steady flow and its consistent form of entropy current

Erratum to: “Gravitational Equilibrium with Steady Flow and Relativistic Local Thermodynamics” [Phys. Lett. B 868 (2025) 139740]

A relativistic self-gravitating equilibrium system with steady flow as well as spherical symmetry is discovered. The energy-momentum tensor contains the contribution of a current related to the flow and the metric tensor does an off-diagonal component to balance with the flow momentum. The presence of the off-diagonal component of the metric implies the radial motion of the reference frame, which gives rise to a problem how the relativistic effect is included in thermodynamic observables for such a general relativistic system. This problem is solved by taking an instantaneously rest frame in which geometric thermodynamic observables read as previously and giving them the special relativistic effect emerged from the inverse transformation to the original frame pointwise. The solution of the thermodynamic observables in accord with the laws of thermodynamics and the theory of relativity is presented. Finally the relativistic structure equations for the equilibrium are derived, from which the general relativistic Poisson equation as well as the heat conduction one are developed exactly.

Gravitational equilibrium with steady flow and relativistic local thermodynamics

By refining the method proposed in Aoki et al. (2021) [5], entropy current and entropy density for a relativistic hydrostatic equilibrium system with spherical symmetry are constructed as a non-Nöther conserved charge in the Einstein gravity with cosmological constant. It is shown that the constructed entropy density satisfies both the local Euler relation and the first law of thermodynamics non-perturbatively with respect to the Newton constant. Finally the established relativistic thermodynamics is applied to one of the systems with uniform energy density as a crude model of a degenerate star and its local thermodynamic observables are determined analytically.

Local thermodynamics and entropy for relativistic hydrostatic equilibrium

We investigate gravitational collapse of thick shell of fluid in the isotropic homogeneous universe without radiation described by the Einstein gravity with cosmological constant. We construct analytic solutions of this kind interpolating the Friedmann-Lemaitre-Robertson-Walker (FLRW) metric and the de-Sitter Schwarzschild black hole one by the Gullstrand-Painlevé one. After determining the scale factor for a perfect fluid coexisting with cosmological constant, we determine the orbit of the collapsing shells of fluid from the continuity condition for density and the perpendicular component of pressure at interface. However the continuity condition cannot fix the orbit when there does not exist a perfect fluid inside the shells. In this case we determine the orbit from an equation of state for the fluid consisting a shell near the surface. Finally we confirm that the total energy defined in <inter-ref xmlns:ns0='http://www.w3.org/1999/xlink' ns0:role='http://www.elsevier.com/xml/linking-roles/preprint' ns0:href='arxiv:2005.13233' id='inf0020'>arXiv:2005.13233</inter-ref> is independent of the given time evolution in this system.

Gravitational collapse of spherical shells of fluid in the isotropic homogeneous universe

Abstract We develop a framework for the reconstruction of the bulk theory dual to conformal field theory without any assumption by means of a flow equation. To this end we investigate a minimal extension of the free-flow equation and find that at a special parametrization the conformal transformation for a normalized smeared operator exactly becomes the isometry of anti-de Sitter space (AdS). By employing this special flow equation for O(N) vector models, we explicitly show that the AdS geometry as well as the scalar field satisfying the GKP–Witten relation concurrently emerge in this framework.

Sinya Aoki

Hungary

Janos Balog

Tetsuya Onogi

Special flow equation and the GKP–Witten relation Supersymmetric gauge theories and the AdS/CFT correspondence

We study a method to compute a topological phase factor of partition function for pure Chern-Simons theory incorporating the supersymmetric localization. We develop a regularization preserving supersymmetry and the topological phase appears as a result of the supersymmetric regularization. Applying this method to pure Chern-Simons theory on lens space we compute the background dependent phase factor coming from the Chern-Simons term. We confirm that the partition function computed in this method enjoys a couple of level rank dualities including the one recently proposed in arXiv:1607.07457 for all ranks and levels within our numerical calculation. We also present a phase factor with which the lens space partition function exhibits the perfect match between any level rank dual pair including the total phase.

Naotaka Kubo

Topological phase, spin Chern-Simons theory and level rank duality on lens space

We consider a free theory of multiple scalar fields at finite temperature and study the induced geometry defined through a free flow of the scalar fields, following the method proposed by the present authors as a possible candidate of the constructive approach for AdS/CFT correspondence. We find that the holographic metric has the following properties: i) It is an asymptotic Anti-de Sitter (AdS) black brane metric with some unknown matter contribution. ii) It has no coordinate singularity and milder curvature singularity. iii) Its time component decays exponentially at a certain AdS radial slice. We find that the matter spreads all over the space, which we speculate to be due to thermal excitation of infinitely many massless higher spin fields. We conjecture that the above three are generic features of a black hole holographically realized by the flow equation method.

What does a quantum black hole look like?

The level-k <math><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> Chern-Simons theory is a spin topological quantum field theory for k odd. Its dynamics is captured by the 2d CFT of a compact boson with a certain radius. Recently it was recognized that a dependence on the 2d spin structure can be given to the CFT by modifying it using the so-called Arf invariant. We demonstrate that one can reorganize the torus partition function of the modified CFT into a finite sum involving a finite number of conformal blocks. This allows us to reproduce the modular matrices of the spin theory. We use the modular matrices to calculate the partition function of the spin Chern-Simons theory on the lens space <math><mi>L</mi><mo>(</mo><mi>a</mi><mo>,</mo><mo>±</mo><mn>1</mn><mo>)</mo></math>, and demonstrate the expected dependence on the 3d spin structure.

Takuya Okuda

Koichi Saito

(1) spin Chern-Simons theory and Arf invariants in two dimensions

Abstract A dynamical aspect of quantum gravity on de Sitter spacetime is investigated by holography and the de Sitter/conformal field theory correspondence. We show that de Sitter spacetime emerges from a free Sp() vector model by complexifying the ghost fields and course-graining them by flow equation in parallel to the imaginary axis. We confirm that the emergence of de Sitter spacetime is ensured by conformal symmetry. We also compute the quantum corrections to the cosmological constant up to the next-to-leading order of the  expansion in a proposed holographic approach. As a result the sub-leading corrections have the opposite sign to the classical value. This implies that a quantum gravity on de Sitter spacetime is perturbatively stable and quantum effects make the universe flatter and the cosmological constant smaller.