Programa Fondecyt Informe Final Etapa 2010 Comisión

PROGRAMA FONDECYT

INFORME FINAL

ETAPA 2010

COMISIÓN NACIONAL DE INVESTIGACION CIENTÍFICA Y TECNOLÓGICA

VERSION OFICIAL

FECHA: 02/11/2010

Nº PROYECTO : 3095018 DURACIÓN : 2 años AÑO ETAPA : 2010 TÍTULO PROYECTO : BLACK HOLES IN HIGHER DIMENSIONS.

DISCIPLINA PRINCIPAL : RELATIVIDAD GENERAL Y COSMOLOGIA GRUPO DE ESTUDIO : ASTRON.,COSMOL.Y PAR INVESTIGADOR(A) RESPONSABLE : SOURYA RAY CHAKRABORTY DIRECCIÓN : Edificio Emilio Pugin, Instituto de Ciencias Físicas y Matemáticas COMUNA : CIUDAD : Valdivia REGIÓN : XIV REGION

FONDO NACIONAL DE DESARROLLO CIENTIFICO Y TECNOLOGICO (FONDECYT) Moneda 1375, Santiago de Chile - casilla 297-V, Santiago 21 Telefono: 2435 4350 FAX 2365 4435 Email: [email protected] INFORME FINAL PROYECTO FONDECYT POSTDOCTORADO

OBJETIVOS

Cumplimiento de los Objetivos planteados en la etapa final, o pendientes de cumplir. Recuerde que en esta sección debe referirse a objetivos desarrollados, NO listar actividades desarrolladas. Nº OBJETIVOS CUMPLIMIENTO FUNDAMENTO 1 To study the mechanics and thermodynamics of TOTAL black hole spacetimes. 2 Investigation of symmetries, construction of TOTAL conserved charges and to derive thermodynamical first law type statements using Hamiltonian perturbation techniques. 3 To study the properties of black holes in higher TOTAL dimensions in higher derivative gravity theories.

Otro(s) aspecto(s) que Ud. considere importante(s) en la evaluación del cumplimiento de objetivos planteados en la propuesta original o en las modificaciones autorizadas por los Consejos. RESULTS OBTAINED: For each specific goal, describe or summarize the results obtained. Relate each one to work already published and/or manuscripts submitted. In the Annex section include additional information deemed pertinent and relevant to the evaluation process. The maximum length for this section is 5 pages. (Arial or Verdana, font size 10).

One of the main objective of the project was to investigate the role of symmetries in black hole spacetimes in higher dimensions. It is well-known that spacetime symmetries manifest themselves in conservation laws, which in turn governs the mechanics of the black holes. Eventhough the mechanics of stationary black holes in Einstein's theory of General Relativity is well understood by now, surprisingly there has not been much progress in understanding the black hole mechanics in presence of a cosmological constant. Moreover, in higher dimensions, there exists a simple generalization of Einstein's General Relativity, known as Lovelock gravity, which allows to explore the effect of higher curvature terms in the black hole thermodynamics. However, in order to fully understand the mechanics of black holes in general Lovelock gravity theories, it is essential to understand the role of the cosmological constant in black hole spacetimes.

Another objective of the project was to study black hole spacetimes in higher derivative gravities. Higher derivative terms appear in the low energy effective action of string theory. Generically it is much more difficult to integrate the field equations resulting from these terms. However, one can construct interesting carvature invariants such that for symmetric spacetimes the field equations are integrable. This allows to obtain non-trivial black hole configurations and then study the effect of higher derivatives in the laws of black hole mechanics. These objectives were fulfilled in the following projects.

Enthalpy and the Mechanics of AdS Black Holes.

One of the objectives of the project was to study the mechanics of black hole spacetimes in presence of a cosmological constant. Black hole solutions with a non-vanishing cosmological constant have received considerable recent attention. This is both due to the role they play in the phenomenology of the AdS/CFT correspondence and to the observational data suggesting that the universe has a small, positive value of Λ.

Along with David Kastor and Jennie Traschen, I have studied the mechanics of AdS black holes to bring our understanding more closely in parallel with well known results in the asymptotically flat case. Specifically, we presented a geometric derivation of the Smarr formula for AdS black holes. In asymptotically flat spacetimes, the Smarr formula gives an exact relation between the mass, angular momentum and the entropy of stationary black holes [5]. This formula can be obtained by integrating the Komar integral over all the boundaries of a spatial slice of a stationary black hole spacetime [4]. The Komar integral has recently been generalized for spacetimes with non- vanishing cosmological constant [2]. This is done by adding a Killing potential to the usual Komar integrand. This Killing potential then substracts of the infinite contribution from the stress energy of the cosmological constant within the boundary integral itself, without requiring any regularization procedure via background subtraction. This construction does not rely on the background and gives a sensible answer for relaxed asymptotic falloff conditions. Using this construction we have shown that in asymptotically AdS spacetimes, the Smarr relation gets modified by a new “work” term where the cosmological constant plays the role of pressure and the difference of the boundary integral of the Killing potential evaluated at spatial infinity and at the black hole horizon plays the role of effective volume.

Another objective of the project was to investigate the role of symmetries in the construction of conserved charges and to derive first law type statements using Hamiltonian perturbative techniques [3]. We have presented an extended version of the first law of black hole mechanics by including the variations of the cosmological constant. In close analogy to the Smarr relation, the coefficient of the variation of Λ in the first law is determined by the surface integrals of the Killing potential. This coefficient is proportional to a finite, effective volume of the region outside the AdS black hole horizon, which can also be interpreted as minus the volume excluded from a spatial slice by the black hole horizon. Thus, the new term in the first law has the form of effective volume times change in pressure that arises in the variation of the enthalpy in classical thermodynamics. This and related arguments suggest that the mass parameter m of an AdS black hole should be interpreted as the enthalpy of the spacetime. We have further related the expanded first law and the modified Smarr relation by a scaling argument based on Euler's theorem. This work has been published [1].

Smarr Formula and an Extended First Law for Lovelock Gravity.

Another objective of the project was to study black holes in alternative theories of gravity. In dimensions greater than four, the simplest generalization of Einstein's theory of General Relativity is Lovelock gravity. In Lovelock gravity, the Lagrangian density is a sum of dimensionally extended Euler densities. Lovelock gravity is the most general second order gravity theory in higher dimensional spacetimes.

Along with David Kastor and Jennie Traschen, I have investigated the mechanics of AdS black holes in general Lovelock theories of gravity. In [2], Kastor has generalized the Komar integral for general Lovelock theories. The key ingredient in the construction is a sequence of Killing- Lovelock potentials that are in one-to-one correspondence with the higher curvature terms in the Lagrangian of a Lovelock gravity theory. Using the Hamiltonian pertubation techniques of [3], we have derived an extended first law which includes variations of each of the dimensionful coupling constants bk of Lovelock theory. This introduces new thermodynamic potentials Ψk conjugate to the individual Lovelock couplings bk. We have given a general expression for these potentials. It has been shown that, in general, each of these potentials receives three different contributions. The first is given in terms of the boundary integrals of the Killing-Lovelock potentials. It can be thought of as coming from the effective stress-energy introduced by the higher curvature Lovelock interactions. The second and third contribution come from the explicit dependence of the mass and entropy on the Lovelock couplings. In the simple case of Einstein gravity with a non-zero cosmological constant Λ studied in [1], only first of these contributions arose, because the mass and entropy have no explicit dependence on Λ.

Using the extended first law, we have derived the Smarr forumla for static AdS black holes in Lovelock gravity via the scaling argument mentioned previously. The Smarr formula gives the mass of static asymptotically AdS black holes in terms of certain thermodynamic properties, the surface gravity, the entropy, and the potentials Ψk. The Smarr formula becomes particularly useful when such solutions are not known explicitly, as is the case in higher order Lovelock theories. We have also indecated how the Smarr formula may be used in analyzing the Hwaking-Page phase transition in general Lovelock gravity theories. This work has been accepted for publication [6].

A Classification of Six Derivative Lagrangians of Gravity and Static Spherically Symmetric Solutions.

In this work we continue to investigate exact black hole solutions in higher derivative theories of gravity. Higher derivative theories emerge as correction terms in the effective action in the low energy limit of string theory. However, due to the presence of higher derivatives, they often offer very limited calculational control over the theory. Specifically, it is substantially more difficult to obtain exact black hole solutions even in presence of exact symmetries. Recently, an interesting higher derivative theory of gravity was constructed in three dimensions. This theory, known as the BHT new massive gravity, supplements to the usual Einstein-Hilbert term, a particular combination of quadratic curvature invariants in the action. This quadratic curvature invariant has the unique feature that in three dimensions, it is the only invariant, quadratic in curvature, such that the trace of the field equations, obtained by its metric variations, is of second order. Consequently, the theory has a linearized conformal invariance. One naturally wonders, if there are higher dimensional counterparts of this theory. It turns out that the higher-dimensional counterpart of the quadratic invariant is a particular linear combination of the Gauss-Bonnet density ( 4) and a scalar invariant obtained by contracting the corresponding indices of two E 2 2 conformal tensors with each other (C ). In fact, in higher dimensions, 4 and C are the only two E linearly independent invariant which gives field equations with second order trace. This is known as the BHT new massive gravity theory. Apart from many other features, one aspect of this theory is that even though the field equations are of fourth order, the trace of all the field equations is of order two. The theory also permits to integrate the field equations to obtain exact stationary black hole solutions. One naturally wonders, if there are higher dimensional counterparts of this theory. Along with Julio Oliva, I have classified all the cubic invariants whose traced field equations are of second order. We have found that, in dimensions higher than five, such invariants can be expressed as a linear combination of the six-dimensional Euler density and two independent contractions of three conformal tensors (local conformal invariants). We then obtained the general static spherically symmetric solutions of the theories described by the local conformal invariants in arbitrary dimensions. Most interestingly however, we have found that in five dimensions, there exists a special cubic invariant which shares this feature but cannot be expressed in terms of Euler density and the local conformal invariants. We have shown that this invariant is the generalization of the qudratic invariant in the new massive gravity Lagrangian and subsequently presented a recipe to obtain all the higher order generalizations. This work has been submitted to Phys. Rev. D and is expected to be published soon.

A new cubic theory of gravity in five dimensions: Black hole, Birkhoff's theorem and C- function:

In this work, we continue with the objective of investigating black holes in higher derivative theories of gravity. Along with Julio Oliva, I have studied the static spherically symmetric solution of an interesting higher derivative theory in five dimensions. The Lagrangian for this theory was obtained as a result of classifying all the six-derivative invariants whose field equations, obtained by the metric variations, have second order trace. It was further found that even though the field equations are generically of fourth order, for spacetimes with spherical, hyperbolic or planar symmetry, all the field equations reduce to second order. This allows one to integrate the field equations and prove a Birkhoff's theorem. Further, we show the existence of a C-function when the theory is coupled to matter fields satisfying null-energy condition for static black hole with planar or hyperbolic symmetry, which is a monotonically increasing function of the outward radial coordinate. The C-function is obtained by extending the Wald's entropy formula following along the same lines of argument in [9]. We have also calculated the temperature and the entropy of the topological black hole and the mass assuming the validity of the first law of black hole mechanics. From this, we have show that the specific heat of the black hole is positive which implies local thermodynamic stability. We then supplement this theory with Einstein, Gauss- Bonnet and the cosmological terms and find a new asymptotically AdS black hole when the coupling constants are tuned in such a way that the three different vacuua degenerate into one. The black hole is shown to possess a Cauchy horizon in addition to the event horizon. Finally, we generalize the cubic theory to arbitrary higher order k and discuss the generic aspects of the static solution in dimensions 2k-1. This work has been published in [7].

References:

[1] D. Kastor, S. Ray and J. Traschen, “Enthalpy and the Mechanics of AdS Black Holes,” Class. Quant. Grav. 26, 195011 (2009) arXiv:0904.2765 [hep-th].

[2] D. Kastor, “Komar Integrals in Higher (and Lower) Derivative Gravity,” Class. Quant. Grav. 25, 175007 (2008) [arXiv:0804.1832 [hep-th]].

[3] V. Iyer and R. M. Wald, “Some properties of Noether charge and a proposal for dynamical black hole entropy,” Phys. Rev. D 50, 846 (1994) [arXiv:gr-qc/9403028].

[4] A. Komar, “Covariant conservation laws in general relativity,” Phys. Rev. 113, 934 (1959).

[5] L. Smarr, “Mass formula for Kerr black holes,” Phys. Rev. Lett. 30, 71-73, (1973). [6] D. Kastor, S. Ray and J. Traschen, “Smarr Formula and an Extended First Law for Lovelock Gravity,” accepted in Class. Quant. Grav. (2010) arXiv:1005.5053 [hep-th].

[7] J. Oliva and S. Ray, “A new cubic theory of gravity in five dimensions: Black hole, Birkhoff's theorem and C- function,” Class. Quant. Grav. 27, 225002 (2010) arXiv:1003.4773 [hep-th].

[8] J. Oliva and S. Ray, “A Classification of Six Derivative Lagrangians of Gravity and Static Spherically Symmetric Solutions,” submitted to Phys. Rev. D (2010) arXiv:1004.0737 [hep-th].

[9] M. M. Anber and D. Kastor, “C-functions in Lovelock Gravity,” JHEP 0805, 061(2008) arXiv:0802.1290 [hep-th].

OTHER ACHIEVEMENTS OF THE PROJECT: Research visit(s) to other institution(s). Outreach activities related to the project’s main topic. Any other contribution, not addressed elsewhere, that you consider important. The maximum length for this section is 1 page. (Arial or Verdana, font size 10).

• Talk given at Twelfth Marcel Grossmann Meeting on General relativity and Gravitation in Paris, France, from 12th - 18th July 2009. Title: Enthalpy and Mechanics of AdS Black Holes. Session: BHT4-Black Hole Thermodynamics. Date: 15th July. • Talk given in the Physics department at Indian Association for the Cultivation of Science, Kolkata, India. Title: Mechanics of AdS Black Holes. Date: 12th August. • Talk given at Theoretical Physics Seminar at Centro De Estudios Cientificos, Valdivia, Chile. Title: Enthalpy of AdS Black Holes. Date: 9th June. • Research visit to Max-Planck-Institut-fur-Physik, Munich, Germany to meet and discuss certain aspects of the project with collaborator Dr. Koushik Dutta. From 19th July-24th July. • Attended 19th International Conference on General Relativity and Gravitation (GR19), Mexico City, Mexico, from 5th - 9th July 2010. • Attended International School in Quantum Gravity at La Plata, Buenos Aires, Argentina, 19th - 27th July 2010. • Attended the conference Quantum Gravity in the Southern Cone in Buenos Aires, Argentina, 28th - 30th July 2010. • Talk given at Theoretical Physics Seminar at Centro De Estudios Cientificos, Valdivia, Chile. Title: Extending the BHT new massive gravity Lagrangian to arbitrary order. Date: 13th May. PROJECT SUMMARY: Using non-specialist language, provide a precise and brief description of the project goals, objectives and results obtained. This summary may be published in CONICYT's web page. The maximum length for this section is 1 page. (Arial or Verdana, font size 10).

Einstein's General Relativity is a theory of gravitation. It demonstrates how the gravitational force that we experience in our everyday life is actually a manifestation of curvature in the spacetime. One of the surprising predictions of General Relativity is the existence of highly compact bodies whose gravitational forces are so strong that even light cannot escape. Such objects are known as black holes. Black holes are one of the most fascinating objects in nature. They are widely believed to exist as a final state of gravitational collapse.

Even though General Relativity is the most successful theory of gravitation, it doesn't explain some of the observational facts about nature. One of them is why does the universe appear to have three space and one time dimensions. Indeed, string theory, which has emerged into a promising candidate for unifying all the forces of nature into a single framework, predicts that we may live in a higher dimensional world. So, it is natural to wonder how the laws of gravitation would manifest in a higher dimensional world. Since, black holes are also the simplest objects to study, it is natural to investigate the properties of black holes in higher dimensions.

In the last few decades, higher dimensional black holes have been a subject of intense research in the field of gravitational physics. It is found that they have a much richer structure than their four dimensional counterpart. Furthermore, in higher dimensions, there exists other alternate theories of gravitation which share many of the nice features of General Relativity. Some of them can be thought of as modifications of General Relativity which become dominant at very small scales. Others are thought to explain the accelerated expansion of the universe.

In this project, I have investigated the properties of black holes in various alternate theories of gravitation. Specifically, I have found how the laws of black hole mechanics are modified in some of these theories.

EVALUATION REPORT FROM POSTDOCTORAL RESEARCHER SPONSOR

SPONSOR NAME: CRISTIAN MARTINEZ

Dr. Sourya Ray has successfully addressed this (two years) research project with two published articles and other one accepted in a good journal. Moreover, he actively participated in the scientific activities organized by our group. Therefore, I give a very good evaluation to Dr. Ray by his work and the results of this project.

Sponsor signature

Date: 02.11.2010

1 PRODUCTOS

ARTÍCULOS Para trabajos en Prensa/ Aceptados/Enviados adjunte copia de carta de aceptación o de recepción.

Nº : 1 Autor (a)(es/as) : Kastor, D.; Ray. S.; Traschen, J. Nombre Completo de la Revista : Classical and Quantum Gravity Título (Idioma original) : Enthalpy and the Mechanics of AdS Black Holes Indexación : ISI ISSN : Año : 2009 Vol. : 26 Nº : 195011 Páginas : 16 Estado de la publicación a la fecha : Publicada Otras Fuentes de financiamiento, si las hay :

Envía documento en papel : no Archivo(s) Asociado(s) al artículo : Enthalpy_and_the_mechanics_of_AdS_black_holes.pdf http://sial.fondecyt.cl/index.php/investigador/f4_articulos/descarga/22833788/3095018/2010/9982/1/

Nº : 2 Autor (a)(es/as) : Kastor, D.; Ray. S.; Traschen, J. Nombre Completo de la Revista : Classical and Quantum Gravity Título (Idioma original) : Smarr Formula and an Extended First Law for Lovelock Gravity Indexación : ISI ISSN : Año : 2010 Vol. : Nº : Páginas : Estado de la publicación a la fecha : Aceptada Otras Fuentes de financiamiento, si las hay :

Envía documento en papel : no Archivo(s) Asociado(s) al artículo : Smarr_Formula_and_an_Extended_First_Law_for_Lovelock_Gravity.pdf http://sial.fondecyt.cl/index.php/investigador/f4_articulos/descarga/22833788/3095018/2010/9983/1/ Nº : 3 Autor (a)(es/as) : Oliva, J;Ray, S. Nombre Completo de la Revista : Classical and Quantum Gravity Título (Idioma original) : A new cubic theory of gravity in five dimensions: Black hole, Birkhoff's theorem and C-function Indexación : ISI ISSN : Año : 2010 Vol. : 27 Nº : 225002 Páginas : 16 Estado de la publicación a la fecha : Publicada Otras Fuentes de financiamiento, si las hay :

Envía documento en papel : no Archivo(s) Asociado(s) al artículo : A_new_cubic_theory_of_gravity_in_five_dimensions:_black_hole,_Birkhoffs_theorem_and_C-function.pdf http://sial.fondecyt.cl/index.php/investigador/f4_articulos/descarga/22833788/3095018/2010/9984/1/

Nº : 4 Autor (a)(es/as) : Oliva, J;Ray, S. Nombre Completo de la Revista : Physical Review D Título (Idioma original) : A Classification of Six Derivative Lagrangians of Gravity and Static Spherically Symmetric Solutions Indexación : ISI ISSN : Año : 2010 Vol. : Nº : Páginas : Estado de la publicación a la fecha : Enviada Otras Fuentes de financiamiento, si las hay :

Envía documento en papel : no Archivo(s) Asociado(s) al artículo : ClassifPRD_V2.pdf http://sial.fondecyt.cl/index.php/investigador/f4_articulos/descarga/22833788/3095018/2010/10004/1/

OTRAS PUBLICACIONES / PRODUCTOS Sin información ingresada.

CONGRESOS

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ANEXOS

Nº : 1 Archivo Asociado : output.pdf http://sial.fondecyt.cl/index.php/investigador/f5_anexos/descarga/22833788/3095018/2010/14826/

Nº : 2 Archivo Asociado : prdreceitlettersixorder.pdf http://sial.fondecyt.cl/index.php/investigador/f5_anexos/descarga/22833788/3095018/2010/14869/

A continuación se detallan los anexos físicos/papel que no se incluyen en el informe en formato PDF.