Generalized gravitational entropy

Conserved vectors are divergencies of superpotentials. In field theory on curved backgrounds, they are useful in calculating global ‘charges’ in arbitrary coordinates and local conserved quantities for small perturbations with specific gauge conditions. Superpotentials are, however, ill–defined. A new criterion of Julia and Silva selects uniquely for Dirichlet boundary conditions the ‘KBL superpotential’ as proposed by Katz, Bičák and Lynden–Bell, which has remarkable properties.Here, we show that a Belinfante–type addition to the KBL superpotential in general relativity gives an expression that is independent of boundary conditions defined by a variational principle. The modified superpotential has the same global properties as the KBL one, except for angular momentum at null infinity, and it does not differ from the KBL superpotential in the linearized theory of gravitation.As an illustration in linearized theory on curved backgrounds, we calculate conserved quantities for small perturbations on a Friedmann–Robertson–Walker spacetime associated with conformal Killing vectors. Our unifying view relates a number of applications in cosmology found in the literature. Globally conserved quantities have simple physical interpretations in the ‘uniform Hubble expansion’ gauge.

The classical gravity approximation is often employed in AdS/CFT to study the dual field theory, as it allows for many computations. A drawback is however the generic presence of singularities in classical gravity, which limits the applicability of AdS/CFT to regimes where the singularities are avoided by bulk probes, or some other form of regularisation is applicable. At the same time, quantum gravity is expected to resolve those singularities and thus to extend the range of applicability of AdS/CFT also in classically singular regimes. This paper exemplifies such a computation. We use an effective quantum corrected Kasner-AdS metric inspired by results from non-perturbative canonical quantum gravity to compute the 2-point correlator in the geodesic approximation for a negative Kasner exponent. The correlator derived in the classical gravity approximation has previously been shown to contain a pole at finite distance as a signature of the singularity. Using the quantum corrected metric, we show explicitly how the pole is resolved and that a new subdominant long-distance contribution to the correlator emerges, caused by geodesics passing arbitrarily close to the resolved classical singularity. In order to compute analytically in this paper, two key simplifications in the quantum corrected metric are necessary. They are lifted in a companion paper using numerical techniques, leading to the same qualitative results.

We obtain an action for a generalized conformal mechanics (GCM) coupled to Jackiw-Teitelboim (JT) gravity from a double scaling limit of the motion of a charged massive particle in the near-horizon geometry of a spherical black hole. When JT gravity is treated in the classical approximation, the backreaction of the particle’s wave function on the time-reparametrization mode (and therefore the bulk metric) vanishes while the conformal symmetry in GCM is reparametrized in a state-dependent way. We also construct the semi-classical Hilbert space of the full theory by explicitly solving the general time-dependent normalizable solutions of the Schrödinger equation for GCM, and show that the time-reparametrization mode can be inferred from the measurement of suitable observables. Since the full theory of the GCM coupled to JT gravity is amenable to quantization, it can lead to a solvable model for a detector coupled to quantum gravity. Published by the American Physical Society 2025

<p>In this thesis, the connections between thermodynamics and general relativity are explored. We introduce some of the history of the interaction between these two theories and take some time to individually study important concepts of both of them. Then, we move on to explore the concept of gravitationally induced temperature gradients in equilibrium states, first introduced by Richard Tolman. We explore these Tolman-like temperature gradients, understanding their physical origin and whether they can be generated by other forces or not. We then generalize this concept for fluids following generic four-velocities, which are not necessarily generated by Killing vectors, in general stationary space-times. Some examples are given. Driven by the interest of understanding and possibly extending the concept of equilibrium for fluids following trajectories which are not generated by Killing vectors, we dedicate ourselves to a more fundamental question: can we still define thermal equilibrium for non-Killing flows? To answer this question we review two of the main theories of relativistic non-perfect fluids: Classical Irreversible Thermodynamics and Extended Irreversible Thermodynamics. We also take a tour through the interesting concept of Born-rigid motion, showing some explicit examples of non-Killing rigid flows for Bianchi Type I space-times. These results are important since they show that the Herglotz–Noether theorem cannot be extended for general curved space-times. We then connect the Born-rigid concept with the results obtained by the relativistic fluid’s equilibrium conditions and show that the exact thermodynamic equilibrium can only be achieved along a Killing flow. We do, however, introduce some interesting possibilities which are allowed for non-Killing flows. We then launch into black hole thermodynamics, specifically studying the trans-Planckian problem for Hawking radiation. We construct a kinematical model consisting of matching two Vaidya spacetimes along a thin shell and show that, as long as the Hawking radiation is emitted only a few Planck lengths (in proper distance) away from the horizon, the trans-Plackian problem can be avoided. We conclude with a brief discussion about what was presented and what can be done in the future.</p>

<p>In this thesis, the connections between thermodynamics and general relativity are explored. We introduce some of the history of the interaction between these two theories and take some time to individually study important concepts of both of them. Then, we move on to explore the concept of gravitationally induced temperature gradients in equilibrium states, first introduced by Richard Tolman. We explore these Tolman-like temperature gradients, understanding their physical origin and whether they can be generated by other forces or not. We then generalize this concept for fluids following generic four-velocities, which are not necessarily generated by Killing vectors, in general stationary space-times. Some examples are given. Driven by the interest of understanding and possibly extending the concept of equilibrium for fluids following trajectories which are not generated by Killing vectors, we dedicate ourselves to a more fundamental question: can we still define thermal equilibrium for non-Killing flows? To answer this question we review two of the main theories of relativistic non-perfect fluids: Classical Irreversible Thermodynamics and Extended Irreversible Thermodynamics. We also take a tour through the interesting concept of Born-rigid motion, showing some explicit examples of non-Killing rigid flows for Bianchi Type I space-times. These results are important since they show that the Herglotz–Noether theorem cannot be extended for general curved space-times. We then connect the Born-rigid concept with the results obtained by the relativistic fluid’s equilibrium conditions and show that the exact thermodynamic equilibrium can only be achieved along a Killing flow. We do, however, introduce some interesting possibilities which are allowed for non-Killing flows. We then launch into black hole thermodynamics, specifically studying the trans-Planckian problem for Hawking radiation. We construct a kinematical model consisting of matching two Vaidya spacetimes along a thin shell and show that, as long as the Hawking radiation is emitted only a few Planck lengths (in proper distance) away from the horizon, the trans-Plackian problem can be avoided. We conclude with a brief discussion about what was presented and what can be done in the future.</p>

A few years ago, in my review of Sean Carroll's book in Classical and Quantum Gravity [1], I wrote that while the 1970s was the decade of Weinberg [2] and Misner, Thorne and Wheeler [3], and while the eighties was the decade of Schutz [4] and Wald [5], the 2000s was clearly the decade of Hartle [6] and Carroll [7]. In my opinion, these books continue to stand out in the surprisingly dense crowd of introductory textbooks on general relativity. At the dawn of this new decade I look forward to see what fresh pedagogical insights will be produced next, and who will be revealed as the winners of the 2010s. It is, of course, much too early to tell, but Schutz is back, and he will set the standard just as he did back in 1985.

The Clausius relation between entropy change and heat flux has previously been used to derive Einstein's field equations as an equation of state. In that derivation the entropy is proportional to the area of a local causal horizon, and the heat is the energy flux across the horizon, defined relative to an approximate boost Killing vector. We examine here whether a similar derivation can be given for extensions beyond Einstein gravity to include higher-derivative and higher-curvature terms. We review previous proposals which, in our opinion, are problematic or incomplete. Refining one of these, we assume that the horizon entropy depends on an approximate local Killing vector in a way that mimics the diffeomorphism Noether charge that yields the entropy of a stationary black hole. We show how this can be made to work if various restrictions are imposed on the nature of the horizon slices and the approximate Killing vector. Also, an integrability condition on the assumed horizon entropy density must hold. This can yield field equations of a Lagrangian constructed algebraically from the metric and Riemann tensor, but appears unlikely to allow for derivatives of curvature in the Lagrangian.

It is shown that the classical derivation of the diffusion equation uses two incompatible assumptions: (1) the diffusion coefficient, , is assumed to be finite ( is a characteristic velocity and λ is the mean-free path); but (2) the flux is approximated in the limit of λ → 0. The second assumption results in unphysical instantaneous propagation which disappears as this assumption is relaxed and the flux term is represented exactly (in the framework of the classical model). With this correction, the diffusion equation becomes (∂n/∂t) = (∂/∂x) where n is the density as a function of coordinate, x, and time, t. Keeping D finite and letting λ → 0 gives the classical equation (∂n/∂t) = (∂/∂x)[D(∂n/∂x)] and classical solution . However, this solution is valid only in the limits where |∂n/∂x| ≫ (λ2/3!)|∂3 n/∂x 3|; otherwise the classical approximation is not correct. Here, we have shown that this inequality translates into 24D 2 t 2/|6Dt − x 2| ≫ λ2 and gives limits for x 2: at any fixed t, x 2 cannot exceed . Relaxing the requirement of λ → 0 gives a new (finite-difference) type of diffusion equation for fluids . This equation and its more general (2D/3D) analogues reveal a qualitatively new, more complex diffusion paradigm than the classical ‘propagation of chaos’ concept.

Certain classical solutions of the field equations, hereafter sectons, were shown to dominate inclusive and semi-inclusive cross sections which are proportional to the absolute square of the Fourier transform of these solutions in zeroth order of a model independent approximation scheme. Higher order approximations are obtained by the loop expansion method, or by a perturbation expansion around the classical solutions. Sectons were shown to be associated with nonvanishing values of two conserved topological quantum numbers (that vanish for solitons). An application to phi/sup 4/ field theory gave an inclusive cross section with Feynman scaling, a total cross section with a power dependence on the primary energy and a Poisson-like multiplicity distribution. An SU(2) invariant Reggeon field theory containing three reggeon multiplets, one with negative mass, was studied in the classical approximation with zero transverse dimensions. A solution was found, qualitatively similar to a solution of supercritical Pomeron theory, giving a constant local cross section and decreasing cross sections for quantum number exchanges. A method, that allows the calculation of critical parameters of a lattice based on the strong coupling expansion of the finite renormalization group equations, was worked out and shown to give excellent results for a two dimensional lattice.

We propose an explicit form for the long-range limit of the SU(N) Yang-Mills Lagrangian expressed as a function of the dual (color-electric) vector potentials. While we cannot rigorously derive the Lagrangian, it can be made plausible that it follows from conventional Yang-Mills theory. This dual long-distance QCD Lagrangian has many of the properties of a magnetic superconductor. It has classical solutions corresponding to confined tubes of quantized electric color flux which result from a dual Meissner effect. However, the confining pressure is not produced by a scalar Higgs field, as in ordinary superconductivity, but by a magnetic condensate field which arises naturally from the nonlocal form of the dual Lagrangian. Within the classical approximation, we find the explicit distribution of color fields surrounding a flux tube. Semiclassical quantization around this solution can be expected to yield the QCD string, and the semiclassical expansion parameter is 1/N, where N is the number of colors.

Building on a recent proposal for a quantum reduction to spherical symmetry from full loop quantum gravity, we investigate the relation between a quantisation of spherically symmetric general relativity and a reduction at the quantum level. To this end, we generalise the previously proposed quantum reduction by dropping the gauge fixing condition on the radial diffeomorphisms, thus allowing us to make direct contact with previous work on reduced quantisation. A dictionary between spherically symmetric variables and observables with respect to the reduction constraints in the full theory is discussed, as well as an embedding of reduced quantum states to a subsector of the quantum symmetry reduced full theory states. On this full theory subsector, the quantum algebra of the mentioned observables is computed and shown to qualitatively reproduce the quantum algebra of the reduced variables in the large quantum number limit for a specific choice of regularisation. Insufficiencies in recovering the reduced algebra quantitatively from the full theory are attributed to the oversimplified full theory quantum states we use.

In this work we concentrate and further develop the theory of isolated horizons in the context of loop quantum gravity. Recently, we have proposed a new computation of BH entropy in loop quantum gravity (LQG) that avoids the internal gauge-fixing used in prior works and makes the underlying structure more transparent. We show, in particular, that the degrees of freedom of Type I isolated horizons can be encoded (along the lines of the standard treatment) in an SU(2) boundary connection. The results of this work clarify the relationship between the theory of isolated horizons and SU(2) Chern-Simons theory first explored in Ref. 3, and vindicates Krasnov’s original intuition that SU(2) Chern-Simons should be used to describe the degress of freedom of gravity at the horizon. Moreover, our work makes the relationship with the usual treatment of degrees of freedom in loop quantum gravity clear-cut. In the present work, we provide a full detail derivation of the result of our resent work and discuss several important issues that were only briefly mentioned then. An important point should be enphasized concerning the logarithmic corrections mentioned above. The logarithmic corrections to the Bekenstein-Hawking area formula for black hole entropy in the loop quantum gravity literature were thought to be of the (universal) form ∆S = −1/2 log(aH/lp). In Ref. 6 Kaul and Majumdar pointed out that, due to the necessary SU(2) gauge symmetry of the isolated horizon system, the counting should be modified leading to corrections of the form ∆S = −3/2 log(aH/lp). This suggestion is particularly interesting because it would eliminate the apparent tension with other approaches to entropy calculation. In particular their result is in complete agreement with the seemenly very general treatment (which includes the string theory calculations) proposed by Carlip. Our analysis confirms Kaul and Majumdar’s proposal and eliminates in this way the apparent discrepancy between different approaches.

Uranium (VI) sorption on colloidal magnetite under anoxic environment: experimental study and surface complexation modelling

Wald's formula for black hole entropy, applied to extremal black holes, leads to the entropy function formalism. We manipulate the entropy computed this way to express it as the logarithm of the ground state degeneracy of a dual quantum mechanical system. This provides a natural definition of the extremal black hole entropy in the full quantum theory. Our analysis also clarifies the relationship between the entropy function formalism and the Euclidean action formalism.

We discuss a no-boundary proposal for a subregion of the universe. In the classical approximation, this density matrix involves finding a specific classical solution of the equations of motion with no boundary. Beyond the usual no boundary condition at early times, we also have another no boundary condition in the region we trace out. We can find the prescription by starting from the usual Hartle-Hawking proposal for the wavefunction on a full slice and tracing out the unobserved region in the classical approximation. We discuss some specific subregions and compute the corresponding solutions. These geometries lead to phenomenologically unacceptable probabilities, as expected.We also discuss how the usual Coleman de Luccia bubble solutions can be interpreted as a possible no boundary contribution to the density matrix of the universe. These geometries lead to local (but not global) maxima of the probability that are phenomenologically acceptable.