Geometry of Higher-Dimensional Black Hole Thermodynamics

We investigate thermodynamic curvatures of the Kerr and Reissner-Nordstr\"om (RN) black holes in spacetime dimensions higher than four. These black holes possess thermodynamic geometries similar to those in four dimensional spacetime. The thermodynamic geometries are the Ruppeiner geometry and the conformally related Weinhold geometry. The Ruppeiner geometry for $d=5$ Kerr black hole is curved and divergent in the extremal limit. For $d \geq 6$ Kerr black hole there is no extremality but the Ruppeiner curvature diverges where one suspects that the black hole becomes unstable. The Weinhold geometry of the Kerr black hole in arbitrary dimension is a flat geometry. For RN black hole the Ruppeiner geometry is flat in all spacetime dimensions, whereas its Weinhold geometry is curved. In $d \geq 5$ the Kerr black hole can possess more than one angular momentum. Finally we discuss the Ruppeiner geometry for the Kerr black hole in $d=5$ with double angular momenta.Comment: 8 pages, 2 figures, RevTex, References adde

All conformally flat pure radiation metrics

The complete class of conformally flat, pure radiation metrics is given, generalising the metric recently given by Wils.Comment: 7 pages, plain Te

Riemann-Cartan Space-times of G\"odel Type

A class of Riemann-Cartan G\"odel-type space-times are examined in the light of the equivalence problem techniques. The conditions for local space-time homogeneity are derived, generalizing previous works on Riemannian G\"odel-type space-times. The equivalence of Riemann-Cartan G\"odel-type space-times of this class is studied. It is shown that they admit a five-dimensional group of affine-isometries and are characterized by three essential parameters $\ell, m^2, \omega$: identical triads ($\ell, m^2, \omega$) correspond to locally equivalent manifolds. The algebraic types of the irreducible parts of the curvature and torsion tensors are also presented.Comment: 24 pages, LaTeX fil

Whole grain foods and health - A Scandinavian perspective

The food-based dietary guidelines in the Scandinavian countries that recommend an intake of minimum 75 g whole grain per 10 MJ (2,388 kcal) per day are mainly derived from prospective cohort studies where quantitative but little qualitative details are available on whole grain products. The objective of the current paper is to clarify possible differences in nutritional and health effects of the types of whole grain grown and consumed in the Scandinavian countries. A further objective is to substantiate how processing may influence the nutritional value and potential health effects of different whole grains and whole grain foods. The most commonly consumed whole grain cereals in the Scandinavian countries are wheat, rye, and oats with a considerable inter-country variation in the consumption patterns and with barley constituting only a minor role. The chemical composition of these different whole grains and thus the whole grain products consumed vary considerably with regard to the content of macro- and micronutrients and bioactive components. A considerable amount of scientific substantiation shows that processing methods of the whole grains are important for the physiological and health effects of the final whole grain products. Future research should consider the specific properties of each cereal and its processing methods to further identify the uniqueness and health potentials of whole grain products. This would enable the authorities to provide more specific food-based dietary guidelines in relation to whole grain to the benefit of both the food industry and the consumer

The type N Karlhede bound is sharp

We present a family of four-dimensional Lorentzian manifolds whose invariant classification requires the seventh covariant derivative of the curvature tensor. The spacetimes in questions are null radiation, type N solutions on an anti-de Sitter background. The large order of the bound is due to the fact that these spacetimes are properly $CH_2$, i.e., curvature homogeneous of order 2 but non-homogeneous. This means that tetrad components of $R, \nabla R, \nabla^{(2)}R$ are constant, and that essential coordinates first appear as components of $\nabla^{(3)}R$. Covariant derivatives of orders 4,5,6 yield one additional invariant each, and $\nabla^{(7)}R$ is needed for invariant classification. Thus, our class proves that the bound of 7 on the order of the covariant derivative, first established by Karlhede, is sharp. Our finding corrects an outstanding assertion that invariant classification of four-dimensional Lorentzian manifolds requires at most $\nabla^{(6)}R$.Comment: 7 pages, typos corrected, added citation and acknowledgemen

Where are the trapped surfaces?

We discuss the boundary of the spacetime region through each point of which a trapped surface passes, first in some simple soluble examples, and then in the self-similar Vaidya solution. For the latter the boundary must lie strictly inside the event horizon. We present a class of closed trapped surfaces extending strictly outside the apparent horizon.Comment: 6 pages, 1 figure; talk at the Spanish Relativity Meeting ERE09 in Bilba

On limits of spacetimes -- a coordinate-free approach

A coordinate-free approach to limits of spacetimes is developed. The limits of the Schwarzschild metric as the mass parameter tends to 0 or $\infty$ are studied, extending previous results. Besides the known Petrov type D and 0 limits, three vacuum plane-wave solutions of Petrov type N are found to be limits of the Schwarzschild spacetime.Comment: 19 p

On the Symmetries of the Edgar-Ludwig Metric

The conformal Killing equations for the most general (non-plane wave) conformally flat pure radiation field are solved to find the conformal Killing vectors. As expected fifteen independent conformal Killing vectors exist, but in general the metric admits no Killing or homothetic vectors. However for certain special cases a one-dimensional group of homotheties or motions may exist and in one very special case, overlooked by previous investigators, a two-dimensional homethety group exists. No higher dimensional groups of motions or homotheties are admitted by these metrics.Comment: Plain TeX, 7 pages, No figure

Equivalence of three-dimensional spacetimes

A solution to the equivalence problem in three-dimensional gravity is given and a practically useful method to obtain a coordinate invariant description of local geometry is presented. The method is a nontrivial adaptation of Karlhede invariant classification of spacetimes of general relativity. The local geometry is completely determined by the curvature tensor and a finite number of its covariant derivatives in a frame where the components of the metric are constants. The results are presented in the framework of real two-component spinors in three-dimensional spacetimes, where the algebraic classifications of the Ricci and Cotton-York spinors are given and their isotropy groups and canonical forms are determined. As an application we discuss Goedel-type spacetimes in three-dimensional General Relativity. The conditions for local space and time homogeneity are derived and the equivalence of three-dimensional Goedel-type spacetimes is studied and the results are compared with previous works on four-dimensional Goedel-type spacetimes.Comment: 13 pages - content changes and corrected typo

Geometro-thermodynamics of tidal charged black holes

Tidal charged spherically symmetric vacuum brane black holes are characterized by their mass m and tidal charge q, an imprint of the 5-dimensional Weyl curvature. For q>0 they are formally identical to the Reissner-Nordstr\"om black hole of general relativity. We study the thermodynamics and thermodynamic geometries of tidal charged black holes and discuss similarities and differences as compared to the Reissner-Nordstr\"om black hole. As a similarity, we show that (for q>0) the heat capacity of the tidal charged black hole diverges on a set of measure zero of the parameter space, nevertheless both the regularity of the Ruppeiner metric and a Poincar\'e stability analysis shows no phase transition at those points. The thermodynamic state spaces being different indicates that the underlying statistical models could be different. We find that the q<0 parameter range, which enhances the localization of gravity on the brane, is thermodynamically preferred. Finally we constrain for the first time the possible range of the tidal charge from the thermodynamic limit on gravitational radiation efficiency at black hole mergers.Comment: v3: 23 pages, 8 figures, 1 table, published versio