Universality and a generalized C-function in CFTs with AdS duals

We argue that the thermodynamics of conformal field theories with AdS duals exhibits a remarkable universality. At strong coupling, a Cardy–Verlinde entropy formula holds even when R-charges or bulk supergravity scalars are turned on. In such a setting, the Casimir entropy can be identified with a generalized C-function that changes monotonically with temperature as well as when non-trivial bulk scalar fields are introduced. We generalize the Cardy–Verlinde formula to cases where no subextensive part of the energy is present and further observe that such a formula is valid for the super Yang–Mills theory in D=4 even at weak coupling. Finally we show that a generalized Cardy–Verlinde formula holds for asymptotically flat black holes in any dimension.     

Cardy–Verlinde formula in FRW Universe with inhomogeneous generalized fluid and dynamical entropy bounds near the future singularity

We derive a formula for the entropy for a multicomponent coupled fluid, which under special conditions reduces to the Cardy–Verlinde form relating the entropy of a closed FRW universe to its energy together with its Casimir energy. The generalized fluid obeys an inhomogeneous equation of state. A viscous dark fluid is included, and also modified gravity is included in terms of its fluid representation. It is demonstrated how such an expression reduces to the standard Cardy–Verlinde formula corresponding to the 2d CFT entropy in some special cases. The dynamical entropy bound for a closed FRW universe with dark components is obtained. The universality of the dynamical entropy bound near a future singularity (of all known four types), as well as near the Big Bang singularity, is investigated. It is demonstrated that, except from some special cases of Type II and Type IV singularities, the dynamical entropy bound is violated near the singularity even if quantum effects are taken into account. The dynamical entropy bound seems to be universal for the case of a regular universe, including the asymptotic de Sitter universe.     

Boundary conditions in rational conformal field theories

We develop further the theory of Rational Conformal Field Theories (RCFTs) on a cylinder with specified boundary conditions emphasizing the role of a triplet of algebras: the Verlinde, graph fusion and Pasquier algebras. We show that solving Cardy's equation, expressing consistency of a RCFT on a cylinder, is equivalent to finding integer valued matrix representations of the Verlinde algebra. These matrices allow us to naturally associate a graph G to each RCFT such that the conformal boundary conditions are labelled by the nodes of G. This approach is carried to completion for sl(2) theories leading to complete sets of conformal boundary conditions, their associated cylinder partition functions and the A-D-E classification. We also review the current status for WZW sl(3) theories. Finally, a systematic generalisation of the formalism of Cardy–Lewellen is developed to allow for multiplicities arising from more general representations of the Verlinde algebra. We obtain information on the bulk–boundary coefficients and reproduce the relevant algebraic structures from the sewing constraints.     

Implications,Consequences and Interpretations of Generalized Entropy in the Cosmological Setups

Recently, it was argued (Tsallis and Cirto, Eur. Phys. J. C 73, 2487 2013) that the total entropy of a gravitational system should be related to the volume of system instead of the system surface. Here, we show that this new proposal cannot satisfy the unified first law of thermodynamics and the Friedmans equation simultaneously, unless the effects of dark energy candidate on the horizon entropy are considered. In fact, our study shows that some types of dark energy candidate may admit this proposal. Some general properties of required dark energy are also addressed. Moreover, our investigation shows that this new proposal for entropy, while combined with the second law of thermodynamics (as the backbone of Verlinde’s proposal), helps us in provideing a thermodynamic interpretation for the difference between the surface and bulk degrees of freedom which, according to Padmanabhan’s proposal, leads to the emergence of spacetime and thus the universe expansion. In fact, our investigation shows that the entropy changes of system may be equal to the difference between the surface and bulk degrees of freedom falling from surface into the system volume. Briefly, our results signal us that this new proposal for entropy may be in agreement with the thermodynamics laws, the Friedmann equation, Padmanabhan’s holographic proposal for the emergence of spacetime and therefore the universe expansion. In fact, this new definition of entropy may be used to make a bridge between Verlinde’s and Padmanabhan’s proposals.     

Cosmological Consequences of the Holographic Scenario

In this paper we discuss some consequences of Verlinde’s holographic gravity model. Among other things, it yields the observed acceleration of the universe and the inflationary period at early universe obviating the Dark energy. From the Verlinde’s theory of gravity the first phenomenological Modified Newtonian dynamics obviating the Dark matter can be deduced. Moreover through the connection with the Modification of inertia resulting from a Hubble-scale Casimir effect (MiHsC) of McCulloch the model gives a promising possible explanation to the Pioneer anomaly, the flyby anomalies, the Tajmar effect and the minimum mass observed in the disc galaxies.     

Modified Friedmann equations from Debye entropic gravity

A remarkable new idea on the origin of gravity was recently proposed by Verlinde who claimed that the laws of gravitation are no longer fundamental, but rather emerge naturally as an entropic force. In Verlinde derivation, the equipartition law of energy on the holographic screen plays a crucial role. However, the equipartition law of energy fails at the very low temperature. Therefore, the formalism of the entropic force should be modified while the temperature of the holographic screen is very low. Considering the Debye entropic gravity and following the strategy of Verlinde, we derive the modified Newton’s law of gravitation and the corresponding Friedmann equations which are valid in all range of temperature. In the limit of strong gravitational field, i.e. high temperature compared to Debye temperature, T » T _D , one recovers the standard Newton’s law and Friedmann equations. We also generalize our study to the entropy corrected area law and derive the dynamical cosmological equations for all range of temperature. Some limits of the obtained results are also studied.     

Some notes on the big trip

The big trip is a cosmological process thought to occur in the future by which the entire universe would be engulfed inside a gigantic wormhole and might travel through it along space and time. In this Letter we discuss different arguments that have been raised against the viability of that process, reaching the conclusions that the process can actually occur by accretion of phantom energy onto the wormholes and that it is stable and might occur in the global context of a multiverse model. We finally argue that the big trip does not contradict any holographic bounds on entropy and information.     

The dark-baryonic matter mass relation for observational verification in Verlinde’s emergent gravity

Recently, a new interesting idea of origin of gravity has been developed by Verlinde. In this scheme of emergent gravity, where horizon entropy, microscopic de Sitter states and relevant contribution to gravity are involved, an entropy displacement resulting from matter behaves as a memory effect and can be exhibited at sub-Hubble scales, namely, the entropy displacement and its “elastic” response would lead to emergent gravity, which gives rise to an extra gravitational force. Then galactic dark matter effects may origin from such extra emergent gravity. We discuss some concepts in Verlinde’s theory of emergent gravity and point out some possible problems or issues, e.g., the gravitational potential caused by Verlinde’s emergent apparent dark matter may no longer be continuous in spatial distribution at ordinary matter boundary (such as a massive sphere surface). In order to avoid the unnatural discontinuity of the extra emergent gravity of Verlinde’s apparent dark matter, we suggest a modified dark-baryonic mass relation (a formula relating Verlinde’s apparent dark matter mass to ordinary baryonic matter mass) within this framework of emergent gravity. The modified mass relation is consistent with Verlinde’s result at relatively small scales (e.g., \(R<3h_{70}^{-1}\) Mpc). However, it seems that, compared with Verlinde’s relation, at large scales (e.g., gravitating systems with \(R>3h_{70}^{-1}\) Mpc), the modified dark-baryonic mass relation presented here might be in better agreement with the experimental curves of weak lensing analysis in the recent work of Brouwer et al. Galactic rotation curves are compared between Verlinde’s emergent gravity and McGaugh’s recent model of MOND (Modified Newtonian Dynamics established based on recent galaxy observations). It can be found that Verlinde rotational curves deviate far from those of McGaugh MOND model when the MOND effect (or emergent dark matter) dominates. Some applications of the modified dark-baryonic mass relation inspired by Verlinde’s emergent gravity will be addressed for galactic and solar scales. Potential possibilities to test this dark-baryonic mass relation as well as apparent dark matter effects, e.g., planetary perihelion precession at Solar System scale, will be considered. This may enable to place some constraints on the magnitudes of the MOND characteristic acceleration at the small solar scale.     

Entropic force, holography and thermodynamics for static space-times

Recently Verlinde has suggested a new approach to gravity which interprets gravitational interaction as a kind of entropic force. The new approach uses the holographic principle by stating that the information is kept on the holographic screens which coincide with equipotential surfaces. Motivated by this new interpretation of gravity (but not being limited by it) we study equipotential surfaces, the Unruh–Verlinde temperature, energy and acceleration for various static space-times: generic spherically symmetric solutions, axially symmetric black holes immersed in a magnetic field, traversable spherically symmetric wormholes of an arbitrary shape function, system of two and more extremely charged black holes in equilibrium. In particular, we have shown that the Unruh–Verlinde temperature of the holographic screen reaches absolute zero on the wormhole throat independently of the particular form of the wormhole solution.     

Black holes and information theory

During the past three decades investigators have unveiled a number of deep connections between physical information and black holes whose consequences for ordinary systems go beyond what has been deduced purely from the axioms of information theory. After a self-contained introduction to black hole thermodynamics, we review from its vantage point topics such as the information conundrum that emerges from the ability of incipient black holes to radiate, the various entropy bounds for non-black hole systems (holographic bound, universal entropy bound, etc.) which are most easily derived from black hole thermodynamics, Bousso's covariant entropy bound, the holographic principle of particle physics, and the subject of channel capacity of quantum communication channels.     

Null Wave Front and Ryu–Takayanagi Surface

The Ryu–Takayanagi formula provides the entanglement entropy of quantum field theory as an area of the minimal surface (Ryu–Takayanagi surface) in a corresponding gravity theory. There are some attempts to understand the formula as a flow rather than as a surface. In this paper, we consider null rays emitted from the AdS boundary and construct a flow representing the causal holographic information. We present a sufficient and necessary condition that the causal information surface coincides with Ryu–Takayanagi surface. In particular, we show that, in spherical symmetric static spacetimes with a negative cosmological constant, wave fronts of null geodesics from a point on the AdS boundary become extremal surfaces and therefore they can be regarded as the Ryu–Takayanagi surfaces. In addition, from the viewpoint of flow, we propose a wave optical formula to calculate the causal holographic information.     

Hořava–Lifshitz cosmology

The cosmological equations suggested by the non-relativistic renormalizable gravitational theory proposed by Hořava are considered. It is pointed out that the early universe cosmology has features that may give an alternative to inflation and the theory may be able to escape singularities.     

Trigonometric Solutions of the WDVV Equations from Root Systems

By introduction of an additional variable and addition of a Weyl invariant correction term to the perturbative prepotential in five-dimensional Seiberg-Witten theory we construct solutions of the Witten–Dijkgraaf–Verlinde–Verlinde equations of trigonometric type for all crystallographic root systems.     

Thermodynamics of charged Brans–Dicke AdS black holes

It is well known that in four dimensions, black hole solution of the Brans–Dicke–Maxwell equations is just the Reissner–Nordstrom solution with a constant scalar field. However, in n4 dimensions, the solution is not yet the (n+1)-dimensional Reissner–Nordstrom solution and the scalar field is not a constant in general. In this Letter, by applying a conformal transformation to the dilaton gravity theory, we derive a class of black hole solutions in (n+1)-dimensional (n4) Brans–Dicke–Maxwell theory in the background of anti-de Sitter universe. We obtain the conserved and thermodynamic quantities through the use of the Euclidean action method. We find a Smarr-type formula and perform a stability analysis in the canonical ensemble. We find that the solution is thermally stable for small α, while for large α the system has an unstable phase, where α is a coupling constant between the scalar and matter field.     

Holographic dark energy in a cyclic universe

In this paper we study the cosmological evolution of the holographic dark energy in a cyclic universe, generalizing the model of holographic dark energy proposed by Li. The holographic dark energy with c<1 can realize a quintom behavior; namely, it evolves from a quintessence-like component to a phantom-like one. The holographic phantom energy density grows rapidly and dominates the late-time expanding phase, helping to realize a cyclic universe scenario in which the high energy regime is modified by the effects of quantum gravity, causing a turn-around (and a bounce) of the universe. The dynamical evolution of holographic dark energy in the regimes of low energy and high energy is governed by two differential equations, respectively. It is of importance to link the two regimes for this scenario. We propose a link condition giving rise to a complete picture of holographic evolution of a cyclic universe.     

Scalar and neutrino fields in the Gödel universe

Some properties of the Gödel universe are demonstrated, such as closed timelike lines, and new coordinates are found. The scalar and neutrino field equations are solved and the eigenvalue spectra are calculated. The scalar field has a discrete spectrum, but the neutrino field has, in addition, a continuous spectrum due to the coupling of neutrino spin and rotation in the Gōdel universe. The mode solutions do not form a complete set for either the scalar or neutrino fields; therefore, a quantum field theory cannot be constructed in the usual manner.     

Spectral/hp least-squares finite element formulation for the Navier–Stokes equations

We consider the application of least-squares finite element models combined with spectral/hp methods for the numerical solution of viscous flow problems. The paper presents the formulation, validation, and application of a spectral/hp algorithm to the numerical solution of the Navier–Stokes equations governing two- and three-dimensional stationary incompressible and low-speed compressible flows. The Navier–Stokes equations are expressed as an equivalent set of first-order equations by introducing vorticity or velocity gradients as additional independent variables and the least-squares method is used to develop the finite element model. High-order element expansions are used to construct the discrete model. The discrete model thus obtained is linearized by Newton’s method, resulting in a linear system of equations with a symmetric positive definite coefficient matrix that is solved in a fully coupled manner by a preconditioned conjugate gradient method. Spectral convergence of the L₂ least-squares functional and L₂ error norms is verified using smooth solutions to the two-dimensional stationary Poisson and incompressible Navier–Stokes equations. Numerical results for flow over a backward-facing step, steady flow past a circular cylinder, three-dimensional lid-driven cavity flow, and compressible buoyant flow inside a square enclosure are presented to demonstrate the predictive capability and robustness of the proposed formulation.