Does a black hole rotate in Chern-Simons modified gravity?

Transcription

1 PHYSICAL REVIEW D 76, (2007) Does a black hole otate in Chen-Simons modified gavity? Kohkichi Konno,, * Toyoki Matsuyama, 2 and Satoshi Tanda Depatment of Applied Physics, Hokkaido Univesity, Sappoo , Japan 2 Depatment of Physics, Naa Univesity of Education, Naa , Japan (Received 20 Apil 2007; published 20 July 2007) Rotating black hole solutions in the 3 -dimensional Chen-Simons modified gavity theoy ae discussed by taking account of petubation aound the Schwazschild solution. The zenith-angle dependence of a metic function elated to the fame-dagging effect is detemined fom a constaint equation independently of a choice of the embedding coodinate. We find that at least within the famewok of the fist-ode petubation method, the black hole cannot otate fo finite black hole mass if the embedding coodinate is taken to be a timelike vecto. Howeve, the otation can be pemitted in the limit of M=! 0 (whee M is the black hole mass and is the adius). Fo a spacelike vecto, the otation can also be pemitted fo any value of the black hole mass. DOI: 0.03/PhysRevD PACS numbes: Bw, h I. INTRODUCTION The latest obsevational esults of the cosmic micowave backgound (CMB) anisotopy fom the Wilkinson Micowave Anisotopy Pobe (WMAP) [] ae successfully explained by the cold dak matte standad model. Howeve, two big issues still emain: what is dak matte, and what is dak enegy? Accoding to the WMAP esults, unfotunately about 96% of the contents of the Univese is given by the dak components that we still do not know. Theefoe, popeties of dak matte and dak enegy have eagely been investigated fom obsevations [2,3]. In contast with the odinay appoaches [2,3], in which the existence of the dak components is assumed, it is of geat inteest to investigate altenative gavity theoies [4 8] to solve the dak matte and dak enegy poblem. In this pape, we focus ou attention on the Chen-Simons modified gavity theoy [9]. This gavity theoy was constucted by Dese et al. [9] in(2) spacetime dimensions fo the fist time by analogy with the topologically massive U() and SU(2) gauge theoies. The Chen-Simons modified gavity theoy was elatively ecently extended by Jackiw and Pi [0] to(3) spacetime dimensions. In the extended theoy, the Schwazschild solution holds without any modification [0]. Theefoe, the theoy passes the classical tests of geneal elativity []. In this gavity theoy, howeve, the Ke solution does not hold. Thus, the solution fo a otating black hole should have a diffeent fom fom the Ke solution. In 2 -dimensional Chen-Simons modified gavity, a family of otating black hole solutions was found by Moussa et al. [2]. The solutions have a fascinating featue that obseves in this spacetime behave like ones inside the egosphee of the Ke spacetime. This featue is simila to that of the otation of galaxies [3]. Theefoe, it is vey inteesting to investigate otating black hole solutions in the 3 -dimensional Chen-Simons modified gavity theoy. In *konno@topology.coe.hokudai.ac.jp this pape, we discuss otating black hole solutions taking account of petubation aound the Schwazschild solution. This pape is oganized as follows. In Sec. II, we biefly eview the 3 -dimensional Chen-Simons modified gavity theoy. In Sec. III, we conside the petubation aound the Schwazschild solution to discuss slow otation of the black hole. Fist we investigate a constaint equation independently of a choice of the embedding coodinate. In Sec. III A, fom the fist-ode equations of the field equation, we obtain the metic solution taking the embedding coodinate to be timelike. In Sec. III B, we investigate the metic solution fo the case in which the embedding coodinate is spacelike. Finally, we povide a summay in Sec. IV. In this pape, we use a unit in which c G. II. BRIEF REVIEW OF CHERN-SIMONS MODIFIED GRAVITY THEORY We biefly eview the Chen-Simons modification of geneal elativity developed by Jackiw and Pi [0]. The Chen-Simons modified gavity theoy is povided by the action I Z dx 4 L Z dx 4 p g R 6 4 # RR Z dx 4 p g R 6 2 v K ; () whee the fist tem in the integand is the Einstein-Hilbet action, and # is an extenal 4-vecto, which is called the embedding coodinate. The Pontyagin density RR is defined by RR R R, using the dual Riemann tenso R 2 " R. The Chen- Simons topological cuent K is given by K 2 3 ; (2) which is elated to the Pontyagin density K 2 RR =2007=76(2)=024009(5) The Ameican Physical Society

2 KOHKICHI KONNO, TOYOKI MATSUYAMA, AND SATOSHI TANDA PHYSICAL REVIEW D 76, (2007) Fom the vaiation of the Lagange density L with espect to the metic g, it tuns out that the field equation has the fom G C 8T ; (3) whee G R 2 g R is the Einstein tenso, T is the enegy-momentum tenso, and C is the Cotton tenso defined as C p 2 v g " R " R v R R : (4) Hee # is a symmetic tenso. Coesponding to the Bianchi identity G 0 and the equation of motion T, the following condition should be imposed: 0 C p 8 v RR: (5) g This constaint equation implies that diffeomophism symmety beaking is suppessed [0]. III. PERTURBATIVE APPROACH TO ROTATING BLACK HOLE SOLUTIONS In the Chen-Simons modified gavity theoy, the Schwazschild solution as a nonotating black hole solution holds without any modification as mentioned above. The Schwazschild metic is given by ds 2 2M dt 2 2M d 2 2 d 2 sin 2 d 2 ; (6) whee M is the black hole mass. This solution gives C 0 and C v p RR=8 g 0 tivially. In ode to take account of otation of the black hole, let us conside petubation aound the Schwazschild solution. In the petubation, the expansion paamete is elated to the angula momentum J of the black hole, i.e., J O. Unde the assumption of stationay, axisymmetic spacetime, we can wite the fom of the petubed metic as [4 6] ds 2 2M h; dt 2 2M m; d 2 2 k; d 2 sin 2 d!; dt 2 : (7) The functions h;, m;, k;, and!; ae of the fist ode in. Heeafte, we take account of equations up to the fist ode in. Using this petubed metic, fom the condition (5), we obtain 0 C v 3M 3 sin! ; 2 cot! ; ; (8) whee a subscipt comma denotes the patial diffeentiation with espect to the coodinates. In this expession, the function!; only appeas. Theefoe, we find that solutions fo!; should have the functional fom!; $ sin 2 ; (9) whee $ is a function of only. While!; is singula on the otation axis ( and ), the metic is egula at least up to the fist ode, because g t 2 $ O 2. Note that g t does not vanish on the otation axis unless $ is identically zeo. This means that the shift vecto N i g ti i ; ; defined in the (3 ) fomalism [7] is singula on the otation axis. It should also be noted that this esult is independent of a choice of the embedding coodinate v. A. Linea petubation equations and the metic solution fo timelike v We adopt a timelike vecto fo v, i.e., v = ; 0; 0; 0, which is deived fom # t=. In ou univese, thee exists the fame of efeence in which the CMB adiation can be seen as an isothemal distibution except fo small fluctuations. The fame of efeence is specified by a timelike vecto. Such a timelike vecto is a candidate fo the timelike vecto v. Fom the (tt), (), (), (), (), (), and ()- components of Eq. (3), we can obtain homogeneous diffeential equations fo the functions h, m, and k. Hence, the homogeneous diffeential equations have a simple solution of h; m; k; 0. These diffeential equations ae completely decoupled fom the function!. Since we ae now inteested in the otation of the black hole, i.e., the function!, we do not seek any othe solutions fo the functions h, m, and k. Fom the (t), (t), and t-components of Eq. (3), we obtain the equations fo! 0! ; 2M! ; 2 2M cot! ; 5 cot! ; 22 5M! ; 42 5M cot! ; 3cot 2! ; ; (0) 0 2 2M! ;! ; 3 cot! ; 6 2M! ; 4 3M! ; ; () 0 2M! ; 4 2M! ;! ; 3 cot! ; : (2) Hee, Eqs. (0) and () ae obtained, espectively, only fom the nonvanishing (t) and (t)-components of the

3 DOES A BLACK HOLE ROTATE IN CHERN-SIMONS... PHYSICAL REVIEW D 76, (2007) Cotton tenso, and Eq. (2) is obtained only fom the (t)- component of the Einstein tenso. Note that these equations do not include. This is due to a shotcoming of the fistode petubation method. Fom the esult of Eq. (9), Eq. (0) is automatically satisfied. Fom Eqs. () and (2), we obtain the diffeential equations fo $, espectively, 2 $ 000 6$ 00 6$ 0 ; (3) 2M$ M$ 0 2$ ; (4) whee a pime denotes the diffeentiation with espect to the coodinate. The solution of Eq. (3) is given by the black hole cannot otate in the Chen-Simons modified gavity fo the timelike vecto. Howeve, in the limit of M=! 0, the deivative of Eq. (4) coincides with Eq. (3). Hence, the solution i.e., $ C C 2 2 ; (7) g t C C 2 ; (8) is pemitted in this limit. Since the metic component g t is popotional to at infinity, the fame-dagging effect of this solution woks in the whole space. $ C 0 C C 2 2 ; (5) whee C 0, C, and C 2 ae constants of integation. On the othe hand, the solution of Eq. (4) is given by $ D 2M 3 D M 4M 2 4M 2M ln 2M; (6) whee D and D 2 ae constants of integation. Thus, the solution that satisfies both diffeential equations (3) and (4) is given only by $ 0. Theefoe, we conclude that within the famewok of the fist-ode petubation, B. Linea petubation equations and the metic solution fo spacelike v Next we take anothe choice of # cos=. This povides a spacelike vecto v 0; cos= ; sin= ; 0, which becomes a unit vecto paallel to the otation axis at infinity. The discepancy between the obsevational esult and the theoetical pediction in the quadupole moment of the CMB anisotopy may imply the existence of such a spacelike vecto [8]. Fom the (tt), (t), (), () (), and ()- components of the field equation, we obtain the nonvanishing equations, espectively, 2 2Mk ; 2 2Mm ; 23 5Mk ; k ; cotk ; m ; cotm ; 2k 2m 2M 2 2M 2 $ 000 2M6 M$ M3 2M$ 0 2M$; (9) sin 2 2M$00 4 2M$ 0 2$ 2 3 2M2 h ; coth ; 2M 2 k ; cotk ; 2Mh ; k ; 2Mcoth ; k ; 2Mf2 Mh ; 3Mm ; 32Mk ; g 2Mcotf3 Mh ; 32Mk ; 3Mm ; g25mh ; 3Mm ; 2Mk ; 2Mcot 2 h ; k ; 2 2 9M M 2 h ; 9M 2Mm ; 2 2 9M 4M 2 k ; 3 3M coth ; m ; 2Mcot 3 h ; k ; 22Mm k; (20) 2Mh ; coth ; k ; cotk ; 2 2Mh ; k ; 2km 3M 2M$ M$ 0 2$; (2) 2Mh ; k ; 3Mh ; Mm ; cot 2M$ M$ 0 2$; (22)

4 KOHKICHI KONNO, TOYOKI MATSUYAMA, AND SATOSHI TANDA PHYSICAL REVIEW D 76, (2007) 2Mh ; k ; Mh ; 2 Mk ; Mm ; coth ; k ; cot2 2M$ M$ 0 2$; (23) 2Mh ; k ; h ; m ; Mh ; 2 Mk ; Mm ; sin 2 2M$00 4 2M$ 0 2$ f 2M$ M$ 00 6$ 0 gsin 2 : (24) While the ight-hand side of Eq. (22) has the zenith-angle dependence of cot, the left-hand side is composed of the fist-ode deivatives of h, m, and k with espect to. Thus the solution has the fom of h; m; k /lnsin. Howeve, these functions become singula along the otation axis. Hence, the zenith-angle dependence of the functions h, m, and k should vanish, and theefoe these functions depend on only. On the othe hand, this esult, i.e., h h, m m, and k k, conflicts with Eq. (23), since the lefthand side becomes a function of only, and the ight-hand side has the dependence of cot 2. Theefoe, the functions h, m, and k should vanish. Then, we deive the diffeential equations 0 2 2M 2 $ 000 2M6 M$ M3 2M$ 0 2M$; (25) 0 2M$ M$ 0 2$; (26) 0 2M$ M$ 00 6$ 0 : (27) Equations (25) and (27) can be deived consistently fom Eq. (26). Thus the equation that we have to solve is Eq. (26). In the same way as the case fo the timelike v, the diffeential equation does not include the paamete. The solution of Eq. (26) is given by the same expession as Eq. (6), which leads to 2M D g t ~D ~ 2 2 2M 4M 2 4M 2M ln 2M; (28) whee ~D and ~D 2 ae constants. Theefoe, fo the spacelike vecto v, the spacetime otation is pemitted fo any value of the black hole mass. Howeve, if ~D 2, then the fame-dagging effect extends to infinity, because the second tem in Eq. (28) diveges as inceases. Futhemoe, the esult of Eq. (28) means that the above-mentioned sting singulaity of the shift vecto N i extends to infinity even if ~D 2. IV. SUMMARY We have investigated the otation of a black hole in the Chen-Simons modified gavity theoy. In paticula, we have consideed slow otation of a black hole using the petubation method, in which the Schwazschild solution was taken to be the backgound. Fom the constaint equation, we obtained the zenith-angle dependence of the metic function!; elated to the fame-dagging effect, independently of a choice of the embedding coodinate v. Futhemoe, by solving the field equation, we found that the black hole cannot otate fo the timelike vecto v at least within the famewok of the fist-ode petubation method. Howeve, in the limit of M=! 0, the spacetime otation is pemitted, whose fame-dagging effect extends to infinity. In contast, fo the spacelike vecto v, the spacetime otation is pemitted fo any value of the black hole mass. Its fame-dagging effect also extends to infinity. Theefoe, it is still an open question which fom of the metic coesponds to the Ke solution, which educes to the Minkowski metic at infinity. Deivation of exact solutions fo stationay, axisymmetic spacetimes in the Chen-Simons modified gavity theoy may solve this poblem. Then, we could also undestand effects of the paamete o, which appeas in the Chen-Simons tem, on the black hole physics. The deivation of exact solutions will be a futue wok. Futhemoe, it should be noted that the above-mentioned esults might be modified by the extension of the theoy in which # in the Chen-Simons tem is taken to be a dynamical vaiable. This will also be discussed elsewhee. ACKNOWLEDGMENTS This wok was suppoted in pat by a Gant-in-Aid fo Scientific Reseach fom The 2st Centuy COE Pogam Topological Science and Technology. Analytical calculations wee pefomed in pat by Mathematica (Wolfam Reseach, Inc.) on computes at YITP in Kyoto Univesity

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