On the Noether charge form of the first law of black hole mechanics

The first law of black hole mechanics was derived by Wald in a general covariant theory of gravity for stationary variations around a stationary black hole. It is formulated in terms of Noether charges, and has many advantages. In this paper several issues are discussed to strengthen the validity of the Noether charge form of the first law. In particular, a gauge condition used in the derivation is justified. After that, we justify the generalization to non-stationary variations done by Iyer-Wald.Comment: Latex, 16 pages, arguments on gauge conditions and near-stationary entropy are added, accepted for publication in Physical Review

Lagrangian perfect fluids and black hole mechanics

The first law of black hole mechanics (in the form derived by Wald), is expressed in terms of integrals over surfaces, at the horizon and spatial infinity, of a stationary, axisymmetric black hole, in a diffeomorphism invariant Lagrangian theory of gravity. The original statement of the first law given by Bardeen, Carter and Hawking for an Einstein-perfect fluid system contained, in addition, volume integrals of the fluid fields, over a spacelike slice stretching between these two surfaces. When applied to the Einstein-perfect fluid system, however, Wald's methods yield restricted results. The reason is that the fluid fields in the Lagrangian of a gravitating perfect fluid are typically nonstationary. We therefore first derive a first law-like relation for an arbitrary Lagrangian metric theory of gravity coupled to arbitrary Lagrangian matter fields, requiring only that the metric field be stationary. This relation includes a volume integral of matter fields over a spacelike slice between the black hole horizon and spatial infinity, and reduces to the first law originally derived by Bardeen, Carter and Hawking when the theory is general relativity coupled to a perfect fluid. We also consider a specific Lagrangian formulation for an isentropic perfect fluid given by Carter, and directly apply Wald's analysis. The resulting first law contains only surface integrals at the black hole horizon and spatial infinity, but this relation is much more restrictive in its allowed fluid configurations and perturbations than that given by Bardeen, Carter and Hawking. In the Appendix, we use the symplectic structure of the Einstein-perfect fluid system to derive a conserved current for perturbations of this system: this current reduces to one derived ab initio for this system by Chandrasekhar and Ferrari.Comment: 26 pages LaTeX-2

Entropy of Constant Curvature Black Holes in General Relativity

We consider the thermodynamic properties of the constant curvature black hole solution recently found by Banados. We show that it is possible to compute the entropy and the quasilocal thermodynamics of the spacetime using the Einstein-Hilbert action of General Relativity. The constant curvature black hole has some unusual properties which have not been seen in other black hole spacetimes. The entropy of the black hole is not associated with the event horizon; rather it is associated with the region between the event horizon and the observer. Further, surfaces of constant internal energy are not isotherms so the first law of thermodynamics exists only in an integral form. These properties arise from the unusual topology of the Euclidean black hole instanton.Comment: 4 pages LaTeX2e (RevTeX), 2 PostScript figures. Small corrections in the text and the reference

A stochastic-Lagrangian particle system for the Navier-Stokes equations

This paper is based on a formulation of the Navier-Stokes equations developed by P. Constantin and the first author (\texttt{arxiv:math.PR/0511067}, to appear), where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. In this paper, we take $N$ copies of the above process (each based on independent Wiener processes), and replace the expected value with $\frac{1}{N}$ times the sum over these $N$ copies. (We remark that our formulation requires one to keep track of $N$ stochastic flows of diffeomorphisms, and not just the motion of $N$ particles.) We prove that in two dimensions, this system of interacting diffeomorphisms has (time) global solutions with initial data in the space \holderspace{1}{\alpha} which consists of differentiable functions whose first derivative is $\alpha$ H\"older continuous (see Section \ref{sGexist} for the precise definition). Further, we show that as $N \to \infty$ the system converges to the solution of Navier-Stokes equations on any finite interval $[0,T]$. However for fixed $N$, we prove that this system retains roughly $O(\frac{1}{N})$ times its original energy as $t \to \infty$. Hence the limit $N \to \infty$ and $T\to \infty$ do not commute. For general flows, we only provide a lower bound to this effect. In the special case of shear flows, we compute the behaviour as $t \to \infty$ explicitly.Comment: v3: Typo fixes, and a few stylistic changes. 17 pages, 2 figure

String Theory and Water Waves

We uncover a remarkable role that an infinite hierarchy of non-linear differential equations plays in organizing and connecting certain {hat c}<1 string theories non-perturbatively. We are able to embed the type 0A and 0B (A,A) minimal string theories into this single framework. The string theories arise as special limits of a rich system of equations underpinned by an integrable system known as the dispersive water wave hierarchy. We observe that there are several other string-like limits of the system, and conjecture that some of them are type IIA and IIB (A,D) minimal string backgrounds. We explain how these and several string-like special points arise and are connected. In some cases, the framework endows the theories with a non-perturbative definition for the first time. Notably, we discover that the Painleve IV equation plays a key role in organizing the string theory physics, joining its siblings, Painleve I and II, whose roles have previously been identified in this minimal string context.Comment: 49 pages, 4 figure

Statistical Mechanics of Charged Black Holes in Induced Einstein-Maxwell Gravity

The statistical origin of the entropy of charged black holes in models of induced Einstein-Maxwell gravity is investigated. The constituents inducing the Einstein-Maxwell action are charged and interact with an external gauge potential. This new feature, however, does not change divergences of the statistical-mechanical entropy of the constituents near the horizon. It is demonstrated that the mechanism of generation of the Bekenstein-Hawking entropy in induced gravity is universal and it is basically the same for charged and neutral black holes. The concrete computations are carried out for induced Einstein-Maxwell gravity with a negative cosmological constant in three space-time dimensions.Comment: 16 pages, latex, no figure

Morphological and taxonomic studies of Gracilaria and Gracilariopsis species (Gracilariales, Rhodophyta) from South Africa

Southern Africa has an extremely rich and diverse seaweed flora with a wide variety of marine habitats. Increased commercial interest in these seaweed resources has been the stimulus for biodiversity studies. The Gracilariaceae (Rhodophyta) has emerged as one of the families that possess economic potential as a source of agar and as a potential feed for abalone. A lack of knowledge concerning the taxonomic status of many members of this family is a concern. Gross morphological characters have been the main means of identification and incorrect applications have led to a number of misidentifications. Consequently, a comprehensive reappraisal and revision of these species was carried out. The species count for the South African Gracilariaceae is now two Gracilariopsis species, and nine Gracilaria species. Gracilaria crassa has been reduced to a synonym of G. canaliculata. It is believed that G. foliifera was erroneously identified and specimens in South Africa referred to as G. millardetii and G. protea are assigned to G. corticata. South African Gracilariopsis, previously referred to as Gs. lemaneiformis, is confirmed to be conspecific with European Gs. longissima. This species occurs along the west and south coasts of South Africa, co-existing in a few habitats with G. gracilis. The taxonomic identity of G. vieillardii specimens from South Africa and the differentiation of G. canaliculata and G. salicornia has been confirmed based on morphology. Three species of Gracilaria (G. aculeata, G. beckeri and G. capensis) are endemic or near endemic to the South African coast, and a fourth species, G. denticulata is localised in southeast Africa

Energy, Hamiltonian, Noether Charge, and Black Holes

It is shown that in general the energy ${\cal E}$ and the Hamiltonian ${\cal H}$ of matter fields on the black hole exterior play different roles. ${\cal H}$ is a generator of the time evolution along the Killing time while ${\cal E}$ enters the first law of black hole thermodynamics. For non-minimally coupled fields the difference ${\cal H}-{\cal E}$ is not zero and is a Noether charge $Q$ analogous to that introduced by Wald to define the black hole entropy. If fields vanish at the spatial boundary, $Q$ is reduced to an integral over the horizon. The analysis is carried out and an explicit expression for $Q$ is found for general diffeomorphism invariant theories. As an extension of the results by Wald et al, the first law of black hole thermodynamics is derived for arbitrary weak matter fields.Comment: 20 pages, latex, no figure

Statistical Entropy of a Stationary Dilaton Black Hole from Cardy Formula

With Carlip's boundary conditions, a standard Virasoro subalgebra with corresponding central charge for stationary dilaton black hole obtained in the low-energy effective field theory describing string is constructed at a Killing horizon. The statistical entropy of stationary dilaton black hole yielded by standard Cardy formula agree with its Bekenstein-Hawking entropy only if we take period $T$ of function $v$ as the periodicity of the Euclidean black hole. On the other hand, if we consider first-order quantum correction then the entropy contains a logarithmic term with a factor $-{1/2}$, which is different from Kaul and Majumdar's one, $-{3/2}$. We also show that the discrepancy is not just for the dilaton black hole, but for any one whose corresponding central change takes the form $\frac{c}{12}= \frac{A_H}{8\pi G}\frac{2\pi}{\kappa T}$.Comment: 11 pages, no figure, RevTex. Accepted for publication in Phys. Rev.