Charged Randall-Sundrum black holes in Higher Dimensions
CCharged Randall-Sundrum black holes in Higher Dimensions
M. Meiers , L. Bovard and R.B. Mann Department of Physics & Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
Abstract
We extend some solutions for black holes in the Randall-Sundrum theory witha single brane. We consider a generalised version of the extremal black hole on thebrane in n + 1 dimensions and determine an asymptotic value of the geometry forlarge black holes. [email protected] [email protected] [email protected] a r X i v : . [ g r- q c ] D ec Introduction
Some time ago Randall and Sundrum proposed [1] a novel approach to add an extra un-experienced dimension to use in quantum gravity by considering a (4 + 1) dimensionalworld where non-gravitational physics is constrained to a (3 + 1) dimensional hypersurface(or brane), while gravitational effects are allowed to propagate through a bulk dimen-sional AdS spacetime. At low energies, the theory reduces to 4D general relativity at largedistances compared with the AdS length (cid:96) .In understanding whether and how the RS model is capable of recovering the strongfield predictions of general relativity for bulk dimensions N > [2], localized black holesolutions provide a useful tool. While a considerable amount of numerical work suggestssuch solutions exist (at least for small black holes) [3][4], there is still no analytic solution,except in (2 + 1) dimensions [5]. However it is known that there are no asymptoticallyflat black hole solutions to general relativity in (2 + 1) dimensional spacetime, leading tothe conclusion that the existence of this solution is due to quantum corrections from thedual Conformal Field Theory. These corrections turn what would be a conical singularityclassically into a regular horizon [6]. While inducing a negative cosmological constanton the brane yields braneworld black hole solutions that are similar to those of (2 + 1) dimensional AdS general relativity [7], large black holes are not localized on the brane.In an attempt to make progress on this issue, Kaus and Reall (KR) [8], considered anextreme black hole in (4+1) dimensions charged with respect to an electromagnetic fieldon the brane. By examining the extremal solution one can take advantage of symmetriesof the near horizon geometry in the bulk solution. This approach has the bulk equationsreduce to ordinary differential equations that integrate to yield a 1-parameter family ofsolutions. Solving the Israel junction conditions to obtain the gravitational effect on thebrane yields a relationship between this parameter and the charge on the brane, whichthen serves to label this family of solutions.Here we seek to find extremal black hole solutions in N = ( n + 1) bulk dimensionsthat contains an electromagnetic field on an n -dimensional brane. We intend to deter-mine whether the structure exhibited by the (4 + 1) dimensional extremal black hole isunique to that dimension of the brane and whether pathologies exist in higher dimensionalbraneworld theories. There is hope that there is some robustness in the properties foundin models in the style of Randall and Sundrum, as seen in other higher dimensional ex-tensions of other models [9] [10]. For properties that are robust, one could hope to betterunderstand them in the large N limit where general relativity reduces to a simpler model[11]. There may also be advantages of allowing N to take continuous values and look atlimits where the deviation of N from an integer value becomes small.We proceed by employing a warped product ansatz for the near-horizon metric whosespace transverse to the horizon is a sphere of dimension D (as opposed to 2 in KR) andsolve the resulting field equations in this context. By solving the Israel junction conditionson the brane we are able to determine the gravitational affect of the brane on the overallgeometry. For D = 2 the structure of the Israel junction conditions determine the specific1eometry of the spacetime, k = − [8]. These arguments can then be continued to higherdimensions to again restrict to branes of the form AdS × S D .Using our ansatz we find equations of motion that can be solved with a single parameterfamily of solutions. The action on an electromagnetically charged brane is then used to putjunction conditions on the extrinsic curvature in the bulk. We then look at the possiblesolutions in greater depth and delve into uncovering the structural relations between thecharge and radius of the black hole. We observe how the entropy measured in the braneand bulk differ, and construct an argument to find that for dimensions greater than thescaling of the entropy of small black holes is fundamentally different.We begin with generalizing the derivations [8] of the equations of motion and junctionconditions for higher dimension. Following our equations we look at the analytic solutionspossible before we consider numeric solutions. We conclude by discussing the propertiesof these solutions and examining the entropy scaling observed. The near horizon geometry of a static extreme black hole can be written in the warpedproduct form [12] ds = A ( z ) d Σ + dz + R ( z ) d Ω D (1)with d Σ = − dt + S ( k ) dr , where S (0) = 1 , S ( −
1) = sin( t ) and S (1) = sinh( t ) . Both t and r have been made unitless by using either the (A)dS radius (cid:96) for k = ± or somearbitrary length scale for k = 0 . The co-ordinate z corresponds to the transverse distancefrom the brane and d Ω D is the line element on S D , with N − D + 2 the dimension ofthe brane, and N = D + 3 the overall dimension of the bulk.The corresponding bulk Einstein field equations are given by R µν = − ( D + 2) (cid:96) g µν (2)and inserting the black hole ansatz (1) into the field equations yields kA − A (cid:18) dAdz (cid:19) − DAR dAdz dRdz − A d Adz = − ( D + 2) (cid:96) A d Adz + DR d Rdz = ( D + 2) (cid:96) (3) ( D − R − ( D − R (cid:18) dRdz (cid:19) − AR dAdz dRdz − R d Rdz = − ( D + 2) (cid:96) which reduce to the 5-dimensional version of [8] with D = 2 . By denoting R = (cid:96) ρ ( z/(cid:96) ) A = (cid:96) α ( z/(cid:96) ) we can re-write these equations as k − α (cid:48) α − Dα (cid:48) ρ (cid:48) αρ − α (cid:48)(cid:48) α = − ( D + 2)2 α (cid:48)(cid:48) α + Dρ (cid:48)(cid:48) ρ = ( D + 2) (4) ( D − − ρ (cid:48) ) ρ − α (cid:48) ρ (cid:48) αρ − ρ (cid:48)(cid:48) ρ = − ( D + 2) where the prime denotes the deriviative with respect to x = z/(cid:96) . The Hamiltonian con-straint is given by combining these three equations to eliminate the second order deriva-tives. k − α (cid:48) ) α − Dα (cid:48) ρ (cid:48) αρ + D ( D − − ρ (cid:48) ) ρ = − ( D + 2)( D + 1) (5)Since the horizon is compact in the bulk, the D- sphere of the geometry must contractto a point. As a result ρ ( z/l ) must vanish somewhere; this location can be chosen to be z = 0 without loss of generality. Requiring that the equations of motions are smooth at z = 0 then implies α ( z/(cid:96) ) = (cid:114) βD + 2 + β + kD + 1 (cid:18) D + 2 β (cid:19) / (cid:16) z(cid:96) (cid:17) + β + kD + 1 (cid:18) ( D − β − (3 D + 5) k ( D + 1)( D + 3) (cid:19) (cid:18) D + 2 β (cid:19) / (cid:16) z(cid:96) (cid:17) . . . (6) ρ ( z/(cid:96) ) = z(cid:96) + ( D − β − kD ( D + 1) (cid:18) D + 2 β (cid:19) (cid:16) z(cid:96) (cid:17) + ( D + D + 19 D + 3) β + 4 (5 D + 16 D + 3) kβ + 4 (6 D + 13 D + 3) k D ( D + 1) ( D + 3) (cid:18) D + 2 β (cid:19) (cid:16) z(cid:96) (cid:17) . . . (7)where α (0) = (cid:113) βD +2 for some positive β . The n = N − dimensional brane action is given by S brane = (cid:90) d n z √− h (cid:18) − σ − πG n F ij F ij (cid:19) (8)where h ij is the induced metric on the brane, σ is the brane tension, G n is Newton’sconstant on the brane, and F is the electromagnetic field on the brane. The tension of the3rane is given by σ = ( N − / πG N (cid:96) and G n = ( N − G N / (cid:96) [7]. The action results inan energy-momentum tensor T ab = 14 πG n F ai F bi − h ab (cid:18) σ + 116 πG n F ij F ij (cid:19) (9) T = − N − πG n F ij F ij − ( N − σ (10)localized on the brane. Employing the junction conditions (see (30) as in the appendix)and assuming that the brane is located at z = z is Z symmetric about z and thus K ab ( z +0 ) = − K ab ( z − ) we find K ab ( z ) = 8 πG N (cid:18) N − (cid:18) − N − πG n F ij F ij − ( N − σ (cid:19) h ab + − πG n F ai F bi + h ab (cid:18) σ + 116 πG n F ij F ij (cid:19)(cid:19) = − πG N σN − h ab + G N G n (cid:18) N − h ab F ij F ij − F ai F bi (cid:19) = − (cid:96) h ab + 2 (cid:96)N − (cid:18) N − F ij F ij h ab − F ai F bi (cid:19) (11)which up to a sign convention reduces down to the N = 5 case of [8]. We assume theelectromagnetic field to be spherically symmetric and purely electric, yielding (cid:63) n F = Q D dΩ D (12)where (cid:63) n is the Hodge dual in n dimensions, F is the Faraday differential form, and d Ω D isthe volume form on a D sphere. Given that for a − form ω on a Lorentzian manifold theHodge dual satisfies (cid:63) n (cid:63) n ω = − ω the sign is ( − k ( n − k ) s (where s is the sign of our metricso (-1) and k is the rank of the tensor) we find that the electromagnetic field strength isgiven by F = − (cid:63) n (cid:63) n F = − Q D (cid:63) n (dΩ D ) = − Q D S ( k ) A ( z ) R D ( z ) d t ∧ d r (13)We can use the definition of extrinsic curvature on our chosen metric and normal vectorto write K ab = − ∂ z ( g ab ) (cid:12)(cid:12) z = z . The junction conditions (11) then become ∂ z ( g ab ) (cid:12)(cid:12)(cid:12)(cid:12) z = z = 1 (cid:96) h ab + 2 (cid:96)N − (cid:18) N − Q D R D ( z ) h ab + Q D A ( z ) R D ( z ) (cid:0) δ ta δ tb − S ( k ) δ ra δ rb (cid:1)(cid:19) (14)which reduce to two independent constraints α (cid:48) ( z /(cid:96) ) α ( z /(cid:96) ) = 1 − N − N − N − q D ρ D ( z /(cid:96) ) ρ (cid:48) ( z /(cid:96) ) ρ ( z /(cid:96) ) = 1 + 3( N − N − q D ρ D ( z /(cid:96) ) (15)4n terms of α and ρ , where q D = Q/(cid:96) D − . These conditions can be combined and rear-ranged to the form α (cid:48) ( z /(cid:96) ) α ( z /(cid:96) ) + (2 D − ρ (cid:48) ( z /(cid:96) ) ρ ( z /(cid:96) ) = 2( D + 1) q D = ρ D ( z ) (cid:115) D (cid:18) ρ (cid:48) ( z /(cid:96) ) ρ ( z /(cid:96) ) − α (cid:48) ( z /(cid:96) ) α ( z /(cid:96) ) (cid:19) (16)The Hamiltonian constraint can also be evaluated at z = z to give D − D ( D + 1) q D ρ D ( z /(cid:96) ) + 2( D − D q D ρ D ( z /(cid:96) ) + D ( D − ρ ( z /(cid:96) ) = − kα ( z /(cid:96) ) (17)and, for D ≥ each term on the left hand side is positive; hence k = − , eliminatingthe other choices. There may be interest in the D=1 case where k does not have theserestrictions from the Hamiltonian constraint. We can now restrict ourselves to k = − which affords us two exact solutions. We willfirst look at the properties of the two exact solutions and then explore the remainder ofthe parameter space.The two analytic solutions are a generalization of those found in [8]. For the first casewe set β = D + 2 or α (0) = 1 and find α ( z/(cid:96) ) = cosh( z/(cid:96) ) ρ ( z/(cid:96) ) = sinh( z/(cid:96) ) (18)However the first condition in (16) requires either z /(cid:96) = arctanh (cid:0) (2 D − (cid:1) or z /(cid:96) =arctanh(1) = ∞ . The former solution is not real for D ≥ . For the latter solution both α = ∞ and ρ = ∞ , forcing q D = 0 . Under this (17) becomes D ( D −
1) = 2 for D ≥ ,and so this analytic solution requires D = 2 . As we will see in the following sections, for D > there are solutions with z /(cid:96) = ∞ , which result from α (0) < ; however they arenot analytic.The second exact solution sets β = 1 which results in α being constant α ( z/(cid:96) ) = 1 √ D + 2 ρ ( z/(cid:96) ) = (cid:114) DD + 2 sinh (cid:32)(cid:114) D + 2 D z(cid:96) (cid:33) . (19)Using the Israel junction conditions, we find that we can express q D , ρ exactly as z = (cid:96) (cid:114) DD + 2 arccoth (cid:32) D + 22 D − (cid:114) DD + 2 (cid:33) (20) q D = (cid:114) D ( D + 1)2 D − (cid:32) (2 D − (cid:114) D D + 11 D − (cid:33) D (21)5 α , ρ D = 2 D = 30 1 2 3 z/‘ α , ρ D = 4 1 2 3 z/‘ D = 5 Figure 1: Plots of the behaviour of ρ ( z/(cid:96) ) (blue) and α ( z/(cid:96) ) (orange). The dimension ofthe hypersphere D runs from to starting in the top left then proceeding across andthen down. β takes the values of / (dot dashed), (solid) and (dashed). For β < we place a vertical line at the location of the brane to demonstrate that z occurs before α = 0 . 6 α / α , ρ / ρ z/‘ α / α , ρ / ρ Figure 2: ρ (cid:48) /ρ (blue) and α (cid:48) /α (orange) for various D = 3 , , (solid, dashed, dash dot)with fixed β (top), and various β = 1 . , . , (solid, dashed, dash dot) with fixed D (bottom). This behaviour persists for all β > tested. We conclude α ( z/(cid:96) ) , ρ ( z/(cid:96) ) ∝ e z/(cid:96) for large z/(cid:96) .For general values of β , we will rely on numeric techniques. As our equations of motionare singular at , we must rely on the expansions (6) and (7) taken to O ( (cid:0) z(cid:96) (cid:1) ) whichcan be evaluated at z = 10 − (cid:96) to generate initial conditions. In general, we find twokinds of behaviour surrounding the case of constant α which can be seen in figure (1). For β > we must have α (0) > (cid:112) / ( D + 2) , and we find that α and ρ tend towards beingproportional to e z/(cid:96) for large z , as illustrated in figure (2). Conversely, for β < we seenew behaviour where α monotonically decreases to , for some finite z , at which point ρ also diverges. A calculation of the Kretschmann scalar indicates that there is a curvaturesingularity at z . This singularity would be naked if the brane, and therefore the Z flipacross it was not placed before it. In all our simulations we find z < z meaning the areaof the solution is not reached in the full RS model.The equations of motion always have solutions and the junction conditions eliminateadditional regions parameter space present in these solutions. For D = 2 the limitingcases are the first analytic solutions (18) and (19) described above [8], and for all α (0) < solutions can be found. However for D > this is no longer the case. We find that the7 . . . . . . α (0)0 . . . . . . . z / ‘ D = 2 D = 3 D = 4 D = 5 D = 50 D = 100 Figure 3: The location of the brane z /(cid:96) of differing initial radii of the AdS space forvarious spherical dimensions.value of the upper bound of α (0) is not easily characterized. In figure (3) we plot z /(cid:96) forvarious values of α (0) and we can see that z seems to diverge for some value of α (0) thatdecreases with D , see table (1) for the value of the bounding α (0) . Below this bound, wefind that solutions exist for all positive α (0) with z ∝ α (0) for small initial values.8pherical Dimension ( D = N − ) Bound on α (0) ≈ . ≈ . ≈ . ≈ . ≈ . ≈ . ≈ . ≈ . Table 1: The bounding value of α (0) of higher bulk dimensionWe now switch to a parametrization in terms of the more physically natural parameterof charge, q D to find the behaviour of α ( z ) , ρ ( z ) for large z . We can use the scaling of q D with (cid:96) to guess that α ( z ) = γ ( α,D ) q /D − D ρ ( z ) = γ ( ρ,D ) q /D − D for q D (cid:29) (22)where γ ( α,D ) and γ ( ρ,D ) are constants dependent on dimension. Figure (4) demonstratesthis behaviour for the first few dimensions and figure (5) shows the behaviour of thenumerically found constants. We note that for extremely small q D both α ( z ) /q / ( D − and ρ ( z ) /q / ( D − diverge, but the scale is incredibly small. Figure (3) suggests that for small z , z ∝ α (0) . As such, by their expansions (6) and (7) tofirst order, α ( z ) and ρ ( z ) all scale as α (0) . From the second relation of (16), we find that ρ ( z ) ∝ q / (2 D − D in the regime of small z where ρ (cid:48) ( z ) ≈ and α (cid:48) ( z ) = 0 . Consequently,by transitivity z , α ( z ) and, α (0) ∝ q / (2 D − D . Thus α ( z ) /q / ( D − D ∝ q − / (2 D − D − D ,which demonstrates the divergence of that ratio for q D → . Identical scaling applies to ρ ( z ) and accounts for the similar behaviour of its ratio for small q D . We then find S D +3 ∝ (cid:90) z d zρ D ( z/(cid:96) ) ≈ (cid:96)D + 1 ρ D +1 ( z ) ∝ (cid:96)q D +1) / (2 D − D unlike the expected scaling ∼ q D/D − D from (22). It is worth noting that unlike the specificcase of D = 2 , for general D we have DD − (cid:54) = D +1)2 D − and as a result we find for small z that our entropy scaling changes for D > . 9 q ( α , ρ ) / q / D = 2 0 2 4 6 q ( α , ρ ) / q / D = 30 2 4 6 q ( α , ρ ) / q / D = 4 0 2 4 6 q ( α , ρ ) / q / D = 5 Figure 4: The ratios between α ( z /(cid:96) ) (orange) and ρ ( z /(cid:96) ) (blue) to q / ( D − D . The dimen-sion of the hypersphere D runs from to starting in the top left then proceeding acrossand then down. 10
20 40 60 80 100 120 140 D − − γ γ ρ γ α Figure 5: Numerical ratio, γ , between α ( z /(cid:96) ) (orange) and ρ ( z /(cid:96) ) (blue) and q / ( D − D for increasing dimension D . 11 Conclusion
We have obtained extremally charged black hole solutions for a higher dimensional RandallSundrum models. We find a dimensional robustness to brane world black holes that iscommensurate with previous D = 2 results [8]. While the equations of motion for thenear horizon geometry allow for the space to have a subspace that is any 2D Lorentzianmanifold, only the choice of AdS allow for satisfaction of the junction conditions. Underthese constraints we find solutions in any dimension.However we find that some properties of black holes for the D = 2 case are unique.For black holes that are large compared to the AdS radius we find an identical scalingdependence of entropy with respect to charge. However in contrast to this, small blackholes break the scaling behaviour except for the specific case D = 2 . This change in entropyscaling may be expected as the RS model can only recover perturbative Newtonian gravityat which the scales are large compared to (cid:96) . For D = 2 [8] it may just be a coincidencethat such matching holds for smaller black holes.In the large q D regime we also found that the constants of proportionality for thescaling relations are no longer equal to one another, with both becoming less than unity.As seen in Figure (5) it appears the scaling for the AdS’ subspace size to q D falls off inthe large dimension limit. The spherical subspace’s scaling distinquishes itself by initiallydecreasing from unity of the initial N = 5 case but slowly recovering becoming the largercontribution to the size of the space.An exploration into how to better obtain the numerically found constant of propor-tionality for entropy for given dimension or a formulation of the dependence would alsobe advantageous. The divergence from the standard entropy scaling relations leaves roomfor inquiry concerning on if and how these new relations extend to non-static geometrysystems. Acknowledgements
This work was supported in part by the Natural Sciences and Engineering Research Councilof Canada.
Junction Condition
We generalize the junction conditions made for N=5 bulk in [2] to bulk spacetimes ofarbitrary dimension. We make use of Israel’s technique and break up our N dimensionalbulk into a family of N − dimensional sub-manifolds described by the coordinates x a ,and a normal distance from a particular surface z . To distinguish between tensors whichlie in the full space or only on the hypersurfaces, we will use Greek for the former andLatin for the latter. Let the metric in the bulk take the form d s = h ab ( x, z )d x a d x b + d z h ab lies in the tangent space of the sub-manifolds which encapsulates the informationin the intrinsic metric. We can also bring h into the full space to act as a projection metricto find the surface parallel components of tensors in the tangent space of the bulk. Tobring h up, let us define n µ = δ µz which describes the direction normal to each surface.The projection tensor thus takes the form ˜ h µν = g µν − n µ n ν . (23)We will use the tilde to bring an element of the tangent space of the N − dimensionalstructure into the tangent space of the N dimensional space via an inclusion map. Thedata encoded in h could be employed to find the intrinsic curvature on the surface, butwe have more interest in connecting the bulk’s curvature to the h . In order to do thisconnection, we need a means to measure the bending of the surface in the larger space.This bending is measured using the extrinsic curvature K µν = ˜ h αµ ˜ h βν ∇ α n β which clearly is tangential to the hypersurface, and although subtle in this form, it can beshown to be symmetric. There are a few results which also follow from our choice of theGauss Normal gauge. The first uses that n µ n µ = 1 for all coordinates which implies that ∇ α ( n µ n µ ) = n µ ∇ α ( n µ ) . Consequentially, K µν = ˜ h αµ ˜ h βν ∇ α n β = ˜ h αµ ( δ βν − n β n ν ) ∇ α n β = ˜ h αµ ( ∇ α n ν − n ν n β ∇ α n β ) = ˜ h αµ ∇ α n ν . (24)which allows us to not need the second projection. In fact, because n µ = δ µz implying ∂ µ n ν = 0 and g zµ = δ zµ one can conclude n µ ∇ µ n ν = n µ (cid:18) ∂ z g µν (cid:19) = (cid:18) ∂ z g zν (cid:19) = 0 . As a result of this property, we can further simplify the extrinsic curvature to require noprojections K µν = ˜ h αµ ∇ α n ν = ( δ αµ − n µ n α ) ∇ α n ν = ∇ µ n ν . (25)Finally, we make note of two identities ∇ µ ˜ h βα = − K µα n β − K µβ n α (26)which follow from (25) and the Leibniz rule, and another for the Lie derivative of theextrinsic curvature L n K µν = n λ ∇ λ K µν + K λν ∇ µ n λ + K µλ ∇ ν n λ = n λ ∇ λ K µν + 2 K µλ K ν λ . (27)13sing these tools, a well-known exercise yields the Gauss-Codazzi relation ˜ R ( N − µλαβ = (cid:16) K αµ ˜ h ρβ − K βµ ˜ h ρα (cid:17) K ρλ + ˜ h µµ (cid:48) ˜ h λ (cid:48) λ ˜ h α (cid:48) α ˜ h β (cid:48) β R µ (cid:48) λ (cid:48) α (cid:48) β (cid:48) . (28)and the relation L n K αβ = ˜ R ( N − αβ + 2 K αλ K βλ − πG N ˜ h α (cid:48) α ˜ h β (cid:48) β T α (cid:48) β (cid:48) + (cid:18) N − l + 8 πG N N − T (cid:19) ˜ h αβ . (29)Thus, if we posit that there exists a infinitesimal surface of non zero energy momentum, wecan treat T αβ as the distribution δ ( z ) T αβ . Integrating z from ( − (cid:15), (cid:15) ) , under the reasonableassumption of a finite discontinuity for all other terms, we find in the limit as (cid:15) tends to 0 K αβ ( z = 0 + ) − K αβ ( z = 0 − ) = 8 πG N (cid:18) N − T ˜ h αβ − ˜ h α (cid:48) α ˜ h β (cid:48) β T α (cid:48) β (cid:48) (cid:19) (30)which constitute the Israel junction conditions. References [1] L. Randall and R. Sundrum, “An alternative to compactification,”
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