Taub-NUT Black Holes in Third order Lovelock Gravity
aa r X i v : . [ h e p - t h ] A ug Taub-NUT Black Holes in Third order Lovelock Gravity
S. H. Hendi , ∗ and M. H. Dehghani , † Physics Department, College of Sciences,Yasouj University, Yasouj 75914, Iran Physics Department and Biruni Observatory,College of Sciences, Shiraz University, Shiraz 71454, Iran Research Institute for Astrophysics and Astronomyof Maragha (RIAAM), Maragha 55134, Iran
Abstract
We consider the existence of Taub-NUT solutions in third order Lovelock gravity with cosmologi-cal constant, and obtain the general form of these solutions in eight dimensions. We find that, as inthe case of Gauss-Bonnet gravity and in contrast with the Taub-NUT solutions of Einstein gravity,the metric function depends on the specific form of the base factors on which one constructs thecircle fibration. Thus, one may say that the independence of the NUT solutions on the geometryof the base space is not a robust feature of all generally covariant theories of gravity and is peculiarto Einstein gravity. We find that when Einstein gravity admits non-extremal NUT solutions withno curvature singularity at r = N , then there exists a non-extremal NUT solution in third orderLovelock gravity. In 8-dimensional spacetime, this happens when the metric of the base space ischosen to be CP . Indeed, third order Lovelock gravity does not admit non-extreme NUT solutionswith any other base space. This is another property which is peculiar to Einstein gravity. We alsofind that the third order Lovelock gravity admits extremal NUT solution when the base space is T × T × T or S × T × T . We have extended these observations to two conjectures about theexistence of NUT solutions in Lovelock gravity in any even-dimensional spacetime. ∗ email address: [email protected] † email address: [email protected] . INTRODUCTION The question as to why the Planck and electroweak scales differ by so many ordersof magnitude remains mysterious. In recent years, attempts have been made to addressthis hierarchy issue within the context of theories with extra spatial dimensions. In higher-dimensional spacetimes even with the assumption of Einstein – that the left-hand side of thefield equations is the most general symmetric conserved tensor containing no more than twoderivatives of the metric – the field equations need to be generalized. This generalizationhas been done by Lovelock [1], and he found a second rank symmetric conserved tensorin d dimensions which contains upto second order derivative of the metric. Other highercurvature gravities which have higher derivative terms of the metric, e.g., terms with quarticderivatives, have serious problems with the presence of tachyons and ghosts as well as withperturbative unitarity, while the Lovelock gravity is free of these problems [2].Many authors have considered the possibility of higher curvature terms in the field equa-tions and how their existence would modify the predictions about the gravitating system.Here, we are interested in the properties of the black holes, and we want to know which prop-erties of the black holes are peculiar to Einstein gravity, and which are robust features of allgenerally covariant theories of gravity. This fact provide a strong motivation for consideringnew exact solutions of Lovelock gravity. We show that some properties of NUT solutionsare peculiar to Einstein gravity and not robust feature of all generally covariant theories ofgravity. Although the nonlinearity of the field equations causes to have a few exact blackhole solutions in Lovelock gravity, there are many papers on this subject [3, 4, 5]. In thisletter we introduce Taub-NUT metrics in third order Lovelock gravity, and investigate whichproperties of these kinds of solutions will be modified by considering higher curvature termsin the field equations.The original four-dimensional solution [6] is only locally asymptotic flat. The spacetimehas as a boundary at infinity a twisted S bundle over S , instead of simply being S × S .There are known extensions of the Taub-NUT solutions to the case when a cosmologicalconstant is present. In this case the asymptotic structure is only locally de Sitter (for pos-itive cosmological constant) or anti-de Sitter (for negative cosmological constant) and thesolutions are referred to as Taub-NUT-(A)dS metrics. In general, the Killing vector thatcorresponds to the coordinate that parameterizes the fibre S can have a zero-dimensional2xed point set (called a NUT solution) or a two-dimensional fixed point set (referred to asa ‘bolt’ solution). Generalizations to higher dimensions follow closely the four-dimensionalcase [7, 8, 9]. Also, these kinds of solutions have been generalized in the presence of elec-tromagnetic field and their thermodynamics have been investigated [10, 11]. It is thereforenatural to suppose that the generalization of these solutions to the case of Lovelock gravity,which is the low energy limit of supergravity, might provide us with a window on someinteresting new corners of M-theory moduli space.The outline of this letter is as follows. We give a brief review of the field equations ofthird order Lovelock gravity in Sec. II. In Sec. III, we obtain Taub-NUT solutions of thirdorder Lovelock gravity in eight dimensions and then we check the conjectures given in Ref.[4]. We finish this letter with some concluding remarks. II. FIELD EQUATIONS
The vacuum gravitational field equations of third order Lovelock gravity may be writtenas: α g µν + α G (1) µν + α G (2) µν + α G (3) µν = 0 , (1)where α i ’s are Lovelock coefficients, G (1) µν is just the Einstein tensor, and G (2) µν and G (3) µν arethe second and third order Lovelock tensors given as G (2) µν = 2( R µσκτ R σκτν − R µρνσ R ρσ − R µσ R σν + RR µν ) − L g µν , (2) G (3) µν = − R τρσκ R σκλρ R λντµ − R τρλσ R σκτµ R λνρκ + 2 R τσκν R σκλρ R λρτµ − R τρσκ R σκτρ R νµ + 8 R τνσρ R σκτµ R ρκ + 8 R σντκ R τρσµ R κρ +4 R τσκν R σκµρ R ρτ − R τσκν R σκτρ R ρµ + 4 R τρσκ R σκτµ R νρ + 2 RR κτρν R τρκµ +8 R τνµρ R ρσ R στ − R σντρ R τσ R ρµ − R τρσµ R στ R νρ − RR τνµρ R ρτ +4 R τρ R ρτ R νµ − R τν R τρ R ρµ + 4 RR νρ R ρµ − R R νµ ) − L g µν . (3)In Eqs. (2) and (3) L = R µνγδ R µνγδ − R µν R µν + R is the Gauss-Bonnet Lagrangian and L = 2 R µνσκ R σκρτ R ρτµν + 8 R µνσρ R σκντ R ρτµκ + 24 R µνσκ R σκνρ R ρµ +3 RR µνσκ R σκµν + 24 R µνσκ R σµ R κν + 16 R µν R νσ R σµ − RR µν R µν + R (4)3s the third order Lovelock Lagrangian. Equation (1) does not contain the derivative of thecurvatures, and therefore the derivatives of the metric higher than two do not appear. Inorder to have the contribution of all the above terms in the field equation, the dimension ofthe spacetime should be equal or larger than seven. Here, for simplicity, we consider the NUTsolutions of the dimensionally continued gravity in eight dimensions. The dimensionallycontinued gravity in D dimensions is a special class of the Lovelock gravity, in which theLovelock coefficients are reduced to two by embedding the Lorentz group SO ( D − , SO ( D − ,
2) [12]. By choosing suitable unit, the remaining twofundamental constants can be reduced to one fundamental constant l . Thus, the Lovelockcoefficients α i ’s can be written as α = − l , α = 3 , α = 3 l , α = l . III. EIGHT-DIMENSIONAL SOLUTIONS
In this section we study the eight-dimensional Taub-NUT solutions of third order Lovelockgravity. In constructing these metrics the idea is to regard the Taub-NUT spacetime asa U (1) fibration over a 6-dimensional base space endowed with an Einstein-K¨ahler metric d Ω B . Then the Euclidean section of the 8-dimensional Taub-NUT spacetime can be writtenas: ds = F ( r )( dτ + N A ) + F − ( r ) dr + ( r − N ) d Ω B , (5)where τ is the coordinate on the fibre S and A has a curvature F = dA , which is proportionalto some covariantly constant 2-form. Here N is the NUT charge and F ( r ) is a function of r . The solution will describe a ‘NUT’ if the fixed point set of the U (1) isometry ∂/∂τ (i.e.the points where F ( r ) = 0) is less than 6-dimensional and a ‘bolt’ if the fixed point set is6-dimensional. Here, we consider only the cases where all the factor spaces of B have zero orpositive curvature. Thus, the base space B can be the 6-dimensional space CP , a productof three 2-dimensional spaces ( T or S ), or the product of a 4-dimensional space CP with4 2-dimensional one. The 1-forms and metrics of S , T , CP and CP are [13] A S = 2 cos θ i dφ i , d Ω S = dθ i + sin θ i dφ i , (6) A T = 2 η i dζ i , d Ω T = dη i + dζ i , (7) A CP = 6 sin θ ( dφ + sin θ dφ ) , (8) d Ω CP = 6 { dθ + sin θ cos θ ( dφ + sin θ dφ ) + sin θ ( dθ + sin θ cos θ dφ ) } , (9) A CP = 12 (cid:18)
12 (cos θ − sin θ ) dφ − cos θ cos θ dφ − sin θ cos θ dφ (cid:19) ,d Ω CP = 8 { dθ + 14 sin θ cos θ ( dφ − cos θ dφ + cos θ dφ ) + 14 cos θ ( dθ + sin θ dφ ) + 14 sin θ ( dθ + sin θ dφ ) } , (10)respectively. To find the metric function F ( r ), one may use any components of Eq. (1).After some calculation, we find that the metric function F ( r ) for any base space B can bewritten as F ( r ) = 1Γ l Ψ / β − r − N ) B + ( r − N ) l E / + Ω ! , (11)where Ψ = (cid:0) r − N (cid:1) C + s C + 8 β ( r − N ) (cid:20) ( r − N ) B + l E (cid:21)! , Ω = p (cid:0) r − N (cid:1) l + (cid:0) r − N (cid:1) (cid:0) r + 3 N (cid:1) , Γ = 5 r + 6 N r + 5 N , and p is the dimension of the curved factor spaces of B . The constant β and the functions B , E and C depend on the choice of the base space as:5 β E B CP N − l ) S × CP r − r N + 13 N ) 72 N (4 N − l ) S × S × S r + N ) 24 N (4 N − l ) T × CP − r + 6 r N + N ) 48 N (6 N − l ) T × S × S (12 N − l ) T × T × S − r + N ) N (12 N − l ) T × T × T N and B C CP ( l − N ) ( r + N ) Υ + mr Γ S × CP
324 ( r + N ) { N (3 l − N ) Υ + l [7 ( r + N ) + 38 r N ( N + r N + r )] − l N (93 N + 652 r N + 558 r N + 652 r N + 93 r ) } + 81 mr Γ S × S × S
72 ( r + N ) { N (3 l − N ) Υ + l (2 N + 9 r N + 10 r N + 9 r N + 2 r ) − l N (3 N + 2 r N + 3 r ) ( N + 6 r N + 3 r ) } + 9 mr Γ T × CP
108 ( r + N ) { N ( l − N ) Υ − l ( r − N ) − l N (9 N + 92 r N +54 r N + 92 r N + 9 r ) } + 81 mr Γ T × S × S − r + N ) { N ( l − l N + 48 N ) Υ + l ( r + N ) ( r − N ) (7 N +2 r N + 7 r ) } + 9 mr Γ T × T × S r + N ) h N ( l − N ) Υ + l ( r − N ) + 18 l N ( r − N ) i + 27 mr Γ T × T × T − N ( r + N ) Υ + mr Γ where Υ = (11 r + 84 N r + 66 N r + 84 N r + 11 N ) . Although we have written the metric function F ( r ) for specific values of Lovelock coefficientsbelonging to dimensionally continued Lovelock gravity in 8 dimensions, the form of F ( r ) forarbitrary values of α i ’s is the same as Eq. (11) with more complicated E , B and C .6 . Taub-NUT Solutions: The solutions given in Eq. (11) describe NUT solutions, if (i) F ( r = N ) = 0 and (ii) F ′ ( r = N ) = 1 / (4 N ). The first condition comes from the fact that all the extra dimensionsshould collapse to zero at the fixed point set of ∂/∂τ , and the second one is to avoid conicalsingularity with a smoothly closed fiber at r = N . Using these conditions, one finds thatthe third order Lovelock gravity, in eight dimensions admits NUT solutions only with CP base space when the mass parameter is fixed to be m n = 2 N (cid:0) N − l (cid:1) . (12)Computation of the Kretschmann scalar at r = N for the solutions in eight dimensionsshows that the spacetimes with base spaces S × S × T , S × S × S , T × CP or S × CP have a curvature singularity at r = N in Einstein gravity, while the spacetime with B = CP has no curvature singularity at r = N . Thus, the conjecture given in [4] is confirmed forthird order Lovelock gravity too. Here we generalize this conjecture for Lovelock gravity as“ For the non-extremal NUT solutions of Einstein gravity which have no curvature singularityat r = N , the Lovelock gravity admits NUT solutions, while the Lovelock gravity does notadmit non-extremal NUT when the spacetime has curvature singularity at r = N .” Indeed,we have non-extreme NUT solutions in 2 + 2 k dimensions with non-trivial fibration when the2 k -dimensional base space is chosen to be CP k . Although we have not written the solutionsin 2 + 2 k dimensions and with arbitrary α i ’s, but calculations confirm the above conjecture. B. Extreme Taub-NUT Solution
The solutions (11) with the base space B = T × T × T and B = S × T × T satisfythe extremal NUT solutions provided the mass parameter is fixed to be m B n = 128 N , (13) m B n = 16 N (cid:0) N − l (cid:1) . (14)Indeed for these two cases F ′ ( r = N ) = 0, and therefore the NUT solutions should beregarded as extremal solutions. Computing the Kretschmann scalar, we find that there isa curvature singularity at r = N for the spacetime with B = S × T × T , while the7pacetime with B = T × T × T has no curvature singularity at r = N . This leads usto the generalization of second conjecture of Ref. [4]: “ Lovelock gravity has extremal NUTsolutions whenever the base space is a product of 2-torii with at most one -dimensionalspace of positive curvature ”. Indeed, calculations in other dimensions show that when thebase space has at most one two dimensional curved space as one of its factor spaces, thenLovelock gravity admits an extreme NUT solution even though there exists a curvaturesingularity at r = N . IV. CONCLUDING REMARKS
We considered the existence of Taub-NUT solutions in 8-dimensional third order Lovelockgravity with cosmological constant. Although one can do the calculations for any arbitraryvalues of Lovelock coefficients, we chose them as those of dimensionally continued Lovelockgravity in eight dimensions to have more compact form for the solutions. It is worthwhileto mention that this choice of Lovelock coefficients has no effect on the properties if thesolutions. These solutions are constructed as circle fibrations over even dimensional spacesthat in general are products of Einstein-K¨ahler spaces. We found that as in the case ofGauss-Bonnet gravity, the function F ( r ) of the metric depends on the specific form of thebase factors on which one constructs the circle fibration. In other words we found that thesolutions are sensitive to the geometry of the base space, in contrast to Einstein gravity wherethe metric in any dimension is independent of the specific form of the base factors. Thus,one may say that the sensitivity of the NUT solutions on the geometry of the base spaceis a common feature of higher order Lovelock gravity, which does not happen in Einsteingravity.We have found that when Einstein gravity admits non-extremal NUT solutions with nocurvature singularity at r = N , then there exists a non-extremal NUT solution in thirdorder Lovelock gravity. In 8-dimensional spacetime, this happens when the metric of thebase space is chosen to be CP . Indeed, third order Lovelock gravity does not admit non-extreme NUT solutions with any other base spaces. Although we have not written the NUTsolutions in other dimensions, we found that in any dimension 2 k + 2, we have only one non-extremal NUT solution with CP k as the base space. That is, the Lovelock gravity singlesout the preferred non-singular base space CP k in 2 + 2 k dimensions. We also found that8nly when the base space is T × T × T or S × T × T , eight-dimensional third orderLovelock gravity admits extremal NUT solution. Calculations show that the extremal NUTblack holes exist for the base spaces T × T × T ..... × T and S × T × T ..... × T . Wehave extended these observations to two conjectures about the existence of NUT solutionsin Lovelock gravity. The study of thermodynamic properties of these solutions remains tobe carried out in future. Acknowledgments
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