Lagrangians which transform homogeneously under a global transformation of the fields (a global rescaling, for instance) can be written on-shell as a total derivative which has a universal, solution-independent expression, using a functional version of the Euler theorem for homogeneous functions. We study the uniqueness of this expression and how this result can be used in the construction of generalized Komar charges.

Spain

José Cerdeira

Tomás Ortín

On-shell Lagrangians as total derivatives and the generalized Komar charge

The Cano-Ruipérez 4-dimensional string effective action is the simplest consistent truncation of the first-order in $\textit{α}$ heterotic string effective action compactified on T . This theory contains, on top of the metric, a dilaton and an axion that couple to the Gauss-Bonnet term and to the Pontrjagin density which suggests a very strong relation between the physical and geometrical properties of its solutions. In this paper we study the thermodynamics of the string black-hole solutions of this theory using Wald’s formalism. We construct the on-shell closed generalized Komar, dilaton and axion 2-form charges, and we used them to find the Smarr formula (that we test on the Schwarzschild and Kerr solutions) and to prove no-(primary)-hair theorems for the scalar. We find that $\textit{α}$ plays the role of a thermodynamical variable with an associated chemical potential. We also notice the occurrence of non-standard gravitational charges in the Wald entropy and scalar charges.

Germany

Matteo Zatti

On the thermodynamics of the black holes of the Cano-Ruipérez 4-dimensional string effective action

We find the defining equations for a Killing vector and its superpartner (called the generalized Killing spinor) and use them to construct the generalized Komar superform of minimal $$ \mathcal{N} $$ = 2 $\textit{d}$ = 4 supergravity using the superspace formulation. The superspace procedure presented here can be used for the construction of generalized Komar forms in more general supergravity theories. We also present the (more cumbersome) calculation of the generalized Komar 2-form and of the on-shell closed 2-form used to prove the first law in the component formalism, as an independent confirmation of our main result.

Igor Bandos

Patrick Meessen

Komar charge of $$ \mathcal{N} $$ = 2 supergravity and its superspace generalization

We consider spacetime metrics with a given (but quite generic) dependence on a dimensionful parameter such that in the 0 and ∞ limits of that parameter the metric becomes singular. We study the isometry groups of the original spacetime metrics and of the singular metrics that arise in the limits and the corresponding symmetries of the motion of p-branes evolving in them, showing how the Killing vectors and their Lie algebras can be found in general. We illustrate our general results with several examples which include limits of anti-de Sitter spacetime in which the holographic screen is one of the singular metrics and of pp-waves.

Netherlands

Eric Bergshoeff

Javier Matulich

On the symmetries of singular limits of spacetimes

We study the invariance of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1, d = 4 supergravity solutions under diffeormophisms and show that, in order to obtain consistent conditions (“Killing equations”) invariant under local supersymmetry transformations, one has to perform supersymmetry transformations generated by the superpartner of the vector that generates standard diffeomorphisms, just as a superspace analysis indicates. Using these transformations, we construct a Noether-Wald charge of   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 1, d = 4 supergravity with fermionic contributions which is diff- Lorentz- and supersymmetry-invariant (up to a total derivative).

Noether-Wald charge in supergravity: the fermionic contribution

Motivated by the current research of generalized symmetries and the construction of conserved charges in pure Einstein gravity linearized over Minkowski spacetime in Cartesian coordinates, we investigate, from a purely classical point of view, the construction of these charges in a coordinate- and frame-independent language in order to generalize them further. We show that all the charges constructed in that context are associated to the conformal Killing-Yano 2-forms of Minkowski spacetime. Furthermore, we prove that those associated to closed conformal Killing-Yano 2-forms are identical to the charges constructed by Kastor and Traschen for their dual Killing-Yano (d − 2)-forms. We discuss the number of independent and non-trivial gravitational charges that can be constructed in this way.

Carmen Gómez-Fayrén

Covariant generalized conserved charges of General Relativity

We study the thermodynamics of the 4-dimensional electrically charged black-hole solutions of the simplest 5-dimensional Kaluza-Klein theory using Wald’s formalism. We show how the electric work term present in the 4-dimensional first law of black-hole thermodynamics arises in the purely gravitational 5-dimensional framework. In particular, we find an interesting geometric interpretation of the 4-dimensional electrostatic potential similar to the angular velocity in rotating black holes. Furthermore, we show how the momentum map equation arises from demanding compatibility between the timelike Killing vector of the black-hole solution and the spatial Killing vector of the 5-dimensional background.

Wald entropy in Kaluza-Klein black holes

We revisit the first law of black hole thermodynamics in 4-dimensional theories containing scalar and Abelian vector fields coupled to gravity using Wald’s formalism and a new definition of scalar charge as an integral over a 2-surface which satisfies a Gauss law in the background of stationary black-hole spacetimes. We focus on ungauged supergravity-inspired theories with symmetric sigma models whose symmetries generate electric-magnetic dualities leaving invariant their equations of motion. Our manifestly duality-invariant form of the first law is compatible with the one obtained by of Gibbons, Kallosh and Kol. We also obtain the general expression for the scalar charges of a stationary black hole in terms of the other physical parameters of the solution and the position of the horizon, generalizing the expression obtained by Pacilio for dilaton black holes.

Chile

Romina Ballesteros

On scalar charges and black hole thermodynamics

We compute the first-order α′ corrections of the non-extremal Strominger-Vafa black hole and its non-supersymmetric counterparts in the framework of the Bergshoeff-de Roo formulation of the heterotic superstring effective action. The solution passes several tests: its extremal limit is the one found in an earlier publication and the effect of a T duality transformation on it is another solution of the same form with T dual charges. We compute the Hawking temperature and Wald entropy showing that they are related by the first law and Smarr formula. On the other hand, these two contain additional terms in which the dimensionful parameter α′ plays the role of thermodynamical variable.

Belgium

Pablo Cano

Italy

Alejandro Ruipérez

Non-extremal, α′-corrected black holes in 5-dimensional heterotic superstring theory

We study black-hole thermodynamics in theories that contain dimensionful constants such as the cosmological constant or coupling constants in Wald’s formalism. The most natural way to deal with these constants is to promote them to scalar fields introducing a (d − 1)-form Lagrange multiplier that forces them to be constant on-shell. These (d − 1)-form potentials provide a dual description of them and, in the context of superstring/supergravity theories, a higher-dimensional origin/explanation. In the context of gauged supergravity theories, all these constants can be collected in the embedding tensor. We show in an explicit 4-dimensional example that the embedding tensor can also be understood as a thermodynamical variable that occurs in the Smarr formula in a duality-invariant fashion. This establishes an interesting link between black-hole thermodynamics, gaugings and compactifications in the context of superstring/supergravity theories.