Weyl Curvature Conjecture Research Articles

Gravitational Entropy and the Second Law of Thermodynamics

The spontaneous violation of Lorentz and diffeomorphism invariance in a phase near the big bang lowers the entropy, allowing for an arrow of time and the second law of thermodynamics. The spontaneous symmetry breaking leads to O(3,1) → O(3) × R , where O(3) is the rotational symmetry of the Friedmann–Lemaître–Robertson–Walker spacetime. The Weyl curvature tensor Cμνρσ vanishes in the FLRW spacetime satisfying the Penrose zero Weyl curvature conjecture. The requirement of a measure of gravitational entropy is discussed. The vacuum expectation value 〈0|ψμ|0〉 ≠ 0 for a vector field ψμ acts as an order parameter and at the critical temperature Tc a phase transition occurs breaking the Lorentz symmetry spontaneously. During the ordered O(3) symmetry phase the entropy is vanishingly small and for T < Tc as the universe expands the anti-restored O(3,1) Lorentz symmetry leads to a disordered phase and a large increase in entropy creating the arrow of time.

An Analysis of a Regular Black Hole Interior Model

We analyze the thermodynamical properties of the regular static and spherically symmetric black hole interior model presented by Mbonye and Kazanas. Equations for the thermodynamical quantities valid for an arbitrary density profile are deduced, and from them we show that the model is thermodynamically unstable. Evidence is also presented pointing to its dynamical instability. The gravitational entropy of this solution based on the Weyl curvature conjecture is calculated, following the recipe given by Rudjord, Grøn and Sigbjørn, and it is shown to have the expected behavior.

Gravitational Entropy and Inflation

The main topic of this paper is a description of the generation of entropy at the end of the inflationary era. As a generalization of the present standard model of the Universe dominated by pressureless dust and a Lorentz invariant vacuum energy (LIVE), we first present a flat Friedmann universe model, where the dust is replaced with an ideal gas. It is shown that the pressure of the gas is inversely proportional to the fifth power of the scale factor and that the entropy in a comoving volume does not change during the expansion. We then review different measures of gravitational entropy related to the Weyl curvature conjecture and calculate the time evolution of two proposed measures of gravitational entropy in a LIVE-dominated Bianchi type I universe, and a Lemaitre-Bondi-Tolman universe with LIVE. Finally, we elaborate upon a model of energy transition from vacuum energy to radiation energy, that of Bonanno and Reuter, and calculate the time evolution of the entropies of vacuum energy and radiation energy. We also calculate the evolution of the maximal entropy according to some recipes and demonstrate how a gap between the maximal entropy and the actual entropy opens up at the end of the inflationary era.

Gravitational Entropy of Black Holes and Wormholes

Pure thermodynamical considerations to describe the entropic evolution of the universe seem to violate the Second Law of Thermodynamics. This suggests that the gravitational field itself has entropy. In this paper we expand recent work done by Rudjord, Gr{\O}n and Sigbj{\O}rn where they suggested a method to calculate the gravitational entropy in black holes based on the so-called `Weyl curvature conjecture'. We study the formulation of an estimator for the gravitational entropy of Reissner-Nordstr\"om, Kerr, Kerr-Newman black holes, and a simple case of wormhole. We calculate in each case the entropy for both horizons and the interior entropy density. Then, we analyse whether the functions obtained have the expected behaviour for an appropriate description of the gravitational entropy density.

The Weyl curvature conjecture and black hole entropy

The universe today, containing stars, galaxies and black holes, seems to have evolved from a very homogeneous initial state. From this it appears as if the entropy of the universe is decreasing, in violation of the second law of thermodynamics. It has been suggested by Roger Penrose that this inconsistency can be solved if one assigns an entropy to the spacetime geometry. He also pointed out that the Weyl tensor has the properties one would expect to find in a description of a gravitational entropy. In this paper, we make an attempt to use this so-called Weyl curvature conjecture to describe the Hawking–Bekenstein entropy of black holes and the entropy of horizons due to a cosmological constant. Our analysis indicates that in the static spherically symmetric case this is not possible.

Braneworld singularities

We study the behaviour of spatially homogeneous braneworlds close to the initial singularity in the presence of both local and nonlocal stresses. It is found that the singularity in these braneworlds can be locally either isotropic or anisotropic. We then investigate the Weyl curvature conjecture, according to which some measure of the Weyl curvature is related to a gravitational entropy. In particular, we study the Weyl curvature conjecture on the brane with respect to the dimensionless ratio of the Weyl invariant and the Ricci square and the measure proposed by Grøn and Hervik. We also argue that the Weyl curvature conjecture should be formulated on brane (i.e., in the four-dimensional context).

Gravitational entropy and quantum cosmology

We investigate the evolution of different measures of `gravitationalentropy'in Bianchi type I and Lemaître-Tolman universe models.A new quantity behaving in accordance with the second law ofthermodynamics is introduced. We then go on and investigate whether aquantum calculation of initial conditions for the universe based upon theWheeler-DeWitt equation supports Penrose's Weyl curvatureconjecture, according to which the Ricci part of the curvature dominatesover the Weyl part at the initial singularity of the universe. The theory isapplied to the Bianchi type I universe models with dust and acosmological constant and to the Lemaître-Tolman universe models. Weinvestigate two different versions of the conjecture. First, we investigatea local version which fails to support the conjecture. Thereafter weconstruct a non-local entity which shows more promising behaviourconcerning the conjecture.