Thin accretion disk signatures in dynamical ChernSimons-modified gravity Tiberiu Harko, Zoltán Kovács, Francisco S N Lobo To cite this version: Tiberiu Harko, Zoltán Kovács, Francisco S N Lobo. Thin accretion disk signatures in dynamical ChernSimons-modified gravity. Classical and Quantum Gravity, IOP Publishing, 2010, 27(10), pp.105010. 10.1088/0264-9381/27/10/105010. hal-00597854 HAL Id: hal-00597854 https://hal.archives-ouvertes.fr/hal-00597854 Submitted on 2 Jun 2011 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Confidential: not for distribution. Submitted to IOP Publishing for peer review 8 March 2010 Thin accretion disk signatures in dynamical Chern-Simons modified gravity Tiberiu Harko∗ and Zolt´an Kov´acs† Department of Physics and Center for Theoretical and Computational Physics, The University of Hong Kong, Pok Fu Lam Road, Hong Kong Francisco S. N. Lobo‡ Centro de F´ısica Te´orica e Computacional, Faculdade de Ciˆencias da Universidade de Lisboa, Campo Grande, Ed. C8 1749-016 Lisboa, Portugal (Dated: March 8, 2010) ApromisingextensionofgeneralrelativityisChern-Simons(CS)modifiedgravity,inwhichthe Einstein-Hilbert action is modified by adding a parity-violating CS term, which couples to gravity via a scalar field. In this work, we consider the interesting, yet relatively unexplored, dynamical formulation of CS modified gravity, where the CS coupling field is treated as a dynamical field, endowed with its own stress-energy tensor and evolution equation. We consider the possibility of observationally testing dynamical CS modified gravity by using the accretion disk properties around slowly-rotating black holes. The energy flux, temperature distribution, the emission spectrum as well as the energy conversion efficiency are obtained, and compared to the standard general rela- tivistic Kerr solution. It is shown that the Kerr black hole provide a more efficient engine for the transformation of the energy of the accreting mass into radiation than their slowly-rotating coun- terparts in CS modified gravity. Specific signatures appear in the electromagnetic spectrum, thus leading to the possibility of directly testing CS modified gravity by using astrophysical observations of the emission spectra from accretion disks. PACS numbers: 04.50.Kd, 04.70.Bw, 97.10.Gz I. INTRODUCTION date, has considered the nondynamical formulation [7– 10], whereas the dynamical formulation remains mostly unexplored territory. Recently, modified theories of gravity have received a Relative to rotating black hole spacetimes, several solu- considerable amount of attention mainly motivated by tions in the nondynamical formulation were found in CS the problems of dark energy (see [1] for reviews) and modified gravity [8–10]. The first solutions were found dark matter [2], and from quantum gravity. A promis- by Alexander and Yunes [8, 9], using a far-field approxi- ing extension of general relativity is Chern-Simons (CS) mation (where the field point distance is considered to be modified gravity [3–5], in which the Einstein-Hilbert ac- much larger than the black hole mass). The second ro- tion is modified by adding a parity-violating CS term, tating black hole solution was found by Konno et al [10], which couples to gravity via a scalar field. It is interest- using a small slow-rotation approximation, where the ing to note that the CS correction introduces a means to spin angular momentum is assumed to be much smaller enhance parity violation through a pure curvature term, than the black hole mass. However, it is interesting to as opposed to through the matter term, as is usually con- note that recently, using the dynamical formulation of sidered in general relativity. In fact, CS modified grav- CS modified gravity, spinning black hole solutions in the ity can be obtained explicitly from superstring theory, slow-rotation approximation have been obtained [11, 12]. where the CS term in the Lagrangian density is essen- tial due to the Green-Schwarz anomaly-canceling mech- An interesting feature of CS modified gravity is that it anism, upon four-dimensional compactification [6]. Two has a characteristic observational signature, which could formulations of CS modified gravity exist as independent allow to discriminate an effect of this theory from other theories, namely, the nondynamical formulation and the phenomena. However, most of the tests of CS modified dynamical formulation (see [5] for an excellent recent re- gravity to date have been performed with astrophysical view). In the former, the CS scalar is an aprioripre- observations and concern the non-dynamical framework. scribed function, where its effective evolution equation In particular, it was found that the CS modified theory reduces to a differential constraint on the space of allowed predicts an anomalous precession effect [14], which was solutions; in the latter, the CS is treated as a dynami- tested [15] with LAGEOS [16]. Another constraint on the cal field, possessing an effective stress-energy tensor and non-dynamical theory was proposed in [17], where it was an evolution equation. The majority of the work, up to considered that the CS correction could be used to ex- plain the flat rotation curves of galaxies. However, in [18] aboundwasplacedonthenon-dynamicalmodelwitha canonical CS scalar that is eleven orders of magnitude ∗Electronic address: [email protected] stronger than the Solar System one, using double binary †Electronic address: [email protected] pulsar data. Recently, using the dynamical formulation ‡Electronic address: [email protected] of CS modified gravity, a stringent constraint was placed 2 on the coupling parameter associated to the dynamical The first term is the standard Einstein-Hilbert action coupling of the scalar field [19]. √ Z 4 In this work, we further extend the constraints placed SEH = κ d x −gR , (2) on the dynamical formulation of CS gravity by using the observational signatures of thin disk properties around where κ−1 =16πG and R is the Ricci scalar. The second rotating black holes. In the context of stationary axisym- term defined as metric spacetimes, the mass accretion around rotating α √ black holes was studied in general relativity for the first S = Z d4x −gϑ ∗RR, (3) time in [20], by extending the theory of non-relativistic CS 4 accretion [21]. The radiation emitted by the disk sur- face was also studied under the assumption that black is the Chern-Simons correction; the third term body radiation would emerge from the disk in thermo- √ β Z 4 µν dynamical equilibrium [22, 23]. More recently, the emis- Sϑ = − d x −g [g (∇µϑ)(∇ν ϑ)+2V (ϑ)] , (4) 2 sivity properties of the accretion disks were investigated for exotic central objects, such as wormholes [24], and is the scalar field term. The matter action is given by non-rotating or rotating quark, boson or fermion stars, √ brane-world black holes or gravastars [25–34]. Z 4 Smat = d x −gLmat , (5) Thus, it is the purpose of the present paper to study the thin accretion disk models for slowly-rotating black where L the matter Lagrangian. holes in the dynamical formulation of CS modified the- mat The parameters α and β are dimensional coupling con- ories of gravity, and carry out an analysis of the prop- stants; the CS coupling field, ϑ,isafunctionofspace- erties of the radiation emerging from the surface of the time that parameterizes deformations from GR [12]; ∇ disk. As compared to the standard general relativistic µ is the covariant derivative associated with the metric ten- case, significant differences appear in the energy flux and sor g ;andthequantity∗RR is the Pontryagin density electromagnetic spectrum for CS slowly-rotating black µν defined as holes, thus leading to the possibility of directly testing CS modified gravity by using astrophysical observations ∗ ∗ τ µν σ RR = R σ R τµν , (6) of the emission spectra from accretion disks. ∗ τ µν The present paper is organized as follows. In Sec. II, we where the dual Riemann tensor is given by R σ = 1 µναβ τ µναβ review the dynamical formulation of CS modified grav- 2 R σαβ,with the 4-dimensional Levi-Civita ity, and present the Yunes-Pretorius (YP) slowly-rotating tensor. solution found in [12]. In Sec. III, we review the formal- Varying the action S with respect to the metric gµν ism and the physical properties of the thin disk accretion one obtains the gravitational field equation given by onto compact objects, for stationary axisymmetric space- times. In Sec. IV, we analyze the basic properties of mat- α 1 mat ϑ Gµν + Cµν = Tµν + Tµν , (7) ter forming a thin accretion disk in slowly-rotating black κ 2κ hole spacetimes in CS modified gravity. We discuss and where Gµν is the Einstein tensor, and Cµν is the cotton conclude our results in Sec. V. Throughout this work, tensor defined as we use a system of units so that c = G =¯h = kB =1, µν σαβ(µ ν) ∗ α(µν)σ where kB is Boltzmann’s constant. C = ∇σϑ ∇β R α + ∇σ∇αϑ R . (8) The total stress-energy tensor is split into the matter µν µν term Tmat,andthescalarfieldcontributionTϑ ,which II. DYNAMICAL CHERN-SIMONS MODIFIED is provided by the following relationship GRAVITY ϑ 1 ν Tµν = β (∇µϑ)(∇ν ϑ) − gµν (∇µϑ)(∇ ϑ) − gµν V (ϑ) . In this Section, we write down the field equations of the 2 Chern-Simons gravity, and present the Yunes-Pretorius (9) (YP) slowly-rotating solution found in [12].