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3 Basic Concepts and Solutions

In this section we present the basic framework for general relativity in higher dimensions, beginning with the definition of conserved charges in vacuum, i.e., mass and angular momentum, and the introduction of a set of dimensionless variables that are convenient for describing the phase space and phase diagrams of higher-dimensional rotating black holes. Then we introduce the Tangherlini solutions that generalize the four-dimensional Schwarzschild solution. The analysis that proves their classical stability is then reviewed. Black strings and black p-branes, and their Gregory–Laflamme instability, are briefly discussed for their relevance to novel kinds of rotating black holes.

3.1 Conserved charges

The Einstein–Hilbert action is generalized to higher dimensions in the form

1 ∫ √ --- I = ------ ddx − gR + Imatter. (5 ) 16πG
This is a straightforward generalization, and the only aspect that deserves some attention is the implicit definition of Newton’s constant G in d dimensions. It enters the Einstein equations in the conventional form
1 Rμν − --gμνR = 8 πGT μν , (6 ) 2
where Tμν = 2(− g)−1βˆ•2(δImatterβˆ•δgμν). This definition of the gravitational coupling constant, without any additional dimension-dependent factors, has the notable advantage that the Bekenstein-Hawking entropy formula takes the same form
π’œH S = ---- (7 ) 4G
in every dimension. This follows, e.g., from the standard Euclidean quantum gravity calculation of the entropy.

Mass, angular momenta, and other conserved charges of isolated systems are defined through comparison to the field created near asymptotic infinity by a weakly gravitating system ([154] gives a careful Hamiltonian analysis of conserved charges in higher-dimensional asymptotically-flat spacetimes). The Einstein equations for a small perturbation around flat Minkowski space

g μν = ημν + h μν (8 )
in linearized approximation take the conventional form
¯ β–‘ hμν = − 16πGT μν, (9 )
where ¯hμν = hμν − 1h ημν 2 and we have imposed the transverse gauge condition ∇ μ¯hμν = 0.

Since the sources are localized and we work at linearized perturbation order, the fields in the asymptotic region are the same as those created by point-like sources of mass M and angular momentum with antisymmetric matrix Jij, at the origin xk = 0 of flat space in Cartesian coordinates,

(d−1) k Ttt = M δ (x ), (10 ) 1- (d− 1) k Tti = − 2Jij∇j δ (x ). (11 )
The equations are easily integrated, assuming stationarity, to find
¯ ---16πG------M--- htt = (d − 3)Ω rd−3, (12 ) dk−2 ¯h = − 8πG--x-Jki, (13 ) ti Ωd−2 rd−1
where √ -i-i r = x x, and (d−1)βˆ•2 (d−-1) Ωd −2 = 2π βˆ• Γ 2 is the area of a unit (d − 2 )-sphere. From here we recover the metric perturbation hμν = ¯hμν − d1−2 ¯hημν as
h = ---16-πG-----M---, (14 ) tt (d − 2)Ωd −2rd−3 16πG M hij = --------------------d−3δij, (15 ) (d − 2)(d − 3)Ωd− 2r 8πG xkJki hti = −-------d−1-. (16 ) Ωd −2 r
It is often convenient to have the off-diagonal rotation components of the metric in a different form. By making a suitable coordinate rotation the angular momentum matrix Jij can be put into block-diagonal form, each block being a 2 × 2 antisymmetric matrix with parameter
Ja ≡ J2a−1,2a. (17 )
Here a = 1,...,N labels the different independent rotation planes. If we introduce polar coordinates on each of the planes
(x2a−1,x2a) = (racosφa, rasin φa) (18 )
then (no sum over a)
8 πGJa r2 8πGJa μ2 htφa = − --------da−1-= − --------d−a3. (19 ) Ωd −2 r Ωd−2 r
In the last expression we have introduced the ‘direction cosines’
ra μa = r . (20 )

Given the abundance of black-hole solutions in higher dimensions, one is interested in comparing properties, such as the horizon area π’œ H, of different solutions characterized by the same set of parameters (M, Ja ). A meaningful comparison between dimensionful magnitudes requires the introduction of a common scale, so the comparison is made between dimensionless magnitudes obtained by factoring out this scale. Since classical general relativity in vacuum is scale invariant, the common scale must be one of the physical parameters of the solutions, and a natural choice is the mass. Thus we introduce dimensionless quantities for the spins ja and the area aH,

d−3 J da−3 d−3 π’œdH−3 ja = cJ GM--d−2-, aH = cπ’œ (GM--)d−2 , (21 )
where the numerical constants are
d−2 ( ) d−23 c = Ωd−-3(d-−-2)--- , c = --Ωd-−3--(d − 2)d−2 d-−-4- (22 ) J 2d+1 (d − 3)d−23 π’œ 2(16π)d−3 d − 3
(these definitions follow the choices in [80Jump To The Next Citation Point]). Studying the entropy, or the area π’œH, as a function of Ja for fixed mass is equivalent to finding the function aH (ja).

Note that, with our definition of the gravitational constant G, both the Newtonian gravitational potential energy,

1 Φ = − -htt, (23 ) 2
and the force law (per unit mass)
(d − 3)8πG M F = − ∇ Φ = ----------------ˆr (24 ) (d − 2)Ωd− 2rd−2
acquire d-dependent numerical prefactors. Had we chosen to define Newton’s constant so as to absorb these factors in the expressions for Φ or F, Equation (7View Equation) would have been more complicated.

To warm up before dealing with black holes, we follow John Michell and Simon de Laplace and compute, using Newtonian mechanics, the radius at which the escape velocity of a test particle in this field reaches the speed of light. The kinetic energy of a particle of unit mass with velocity v = c = 1 is K = 1βˆ•2, so the equation K + Φ = 0 that determines the Michell–Laplace ‘horizon’ radius is

( 16πGM ) 1d−3- htt(r = rML ) = 1 ⇒ rML = ------------ . (25 ) (d − 2)Ωd−2
We will see in the next section 3.2 that, just like in four dimensions, this is precisely equal to the horizon radius for a static black hole in higher dimensions.

3.2 The Schwarzschild–Tangherlini solution and black p-branes

Consider the linearized solution above for a static source (14View Equation) in spherical coordinates, and pass to a gauge where r is the area radius,

r → r − -------8πG---------M---. (26 ) (d − 2 )(d − 3)Ωd −2rd−3
The linearized approximation to the field of a static source is then
( ) ( ) 2 --μ-- 2 --μ-- 2 2 2 ds(lin) = − 1 − rd−3 dt + 1 + rd− 3 dr + r dΩ d−2, (27 )
where, to lighten the notation, we have introduced the ‘mass parameter’
μ = --16πGM-----. (28 ) (d − 2)Ωd− 2
This suggests that the Schwarzschild solution generalizes to higher dimensions in the form
( μ ) dr2 ds2 = − 1 − -d−3- dt2 + -----μ-- + r2dΩ2d− 2. (29 ) r 1 − rd−3
In essence, all we have done is change the radial falloff 1βˆ•r of the Newtonian potential to the d-dimensional one, d−3 1βˆ•r. As Tangherlini found in 1963 [232], this turns out to give the correct solution: it is straightforward to check that this metric is indeed Ricci flat. It is apparent that there is an event horizon at r0 = μ1βˆ•(d−3), which coincides with the Michell–Laplace result (25View Equation).

Having this elementary class of black-hole solutions, it is easy to construct other vacuum solutions with event horizons in d ≥ 5. The direct product of two Ricci-flat manifolds is itself a Ricci-flat manifold. So, given any vacuum black-hole solution ℬ of the Einstein equations in d dimensions, the metric

∑p ds2d+p = ds2d(ℬ) + dxidxi (30 ) i=1
describes a black p-brane, in which the black-hole horizon β„‹ ⊂ ℬ is extended to a horizon p β„‹ × β„, or p β„‹ × π•‹ if we identify periodically i i x ∼ x + Li. A simple way of obtaining another kind of vacuum solution is the following: unwrap one of the directions xi, perform a boost t → coshαt + sinh αxi, xi → sinh αt + cosh αxi, and re-identify points periodically along the new coordinate xi. Although locally equivalent to the static black brane, the new boosted black-brane solution is globally different from it.

These black brane spacetimes are not (globally) asymptotically flat, so we only introduce them insofar as they are relevant for understanding the physics of asymptotically-flat black holes.

3.3 Stability of the static black hole

The stability of the d > 4 Schwarzschild solution against linearized gravitational perturbations can be analyzed by decomposing such perturbations into scalar, vector and tensor types according to how they transform under the rotational-symmetry group SO (d − 1)  [105Jump To The Next Citation Point163Jump To The Next Citation Point151Jump To The Next Citation Point]. Assuming a time dependence e− iωt and expanding in spherical harmonics on Sd −2, the equations governing each type of perturbation reduce to a single ODE governing the radial dependence. This equation can be written in the form of a time-independent Schrödinger equation with “energy” eigenvalue 2 ω.

In investigating stability, we consider perturbations that are regular on the future horizon and outgoing at infinity. An instability would correspond to a mode with Im ω > 0. For such modes, the boundary conditions at the horizon and infinity imply that the left-hand side (LHS) of the Schrödinger equation is self-adjoint, and hence 2 ω is real. Therefore, an unstable mode must have negative imaginary ω. For tensor modes, the potential in the Schrödinger equation is manifestly positive, hence ω2 > 0 and there is no instability [105]. For vectors and scalars, the potential is not everywhere positive. Nevertheless, it can be shown that the operator appearing on the LHS of the Schrödinger equation is positive, hence ω2 > 0 and there is no instability [151Jump To The Next Citation Point]. In conclusion, the d > 4 Schwarzschild solution is stable against linearized gravitational perturbations.

3.4 Gregory–Laflamme instability

The instabilities of black strings and black branes [118119] have been reviewed in [167128Jump To The Next Citation Point], so we shall be brief in this section and only mention the features that are most relevant to our subject. We shall only discuss neutral black holes and black branes; when charges are present, the problem becomes more complex.

This instability is the prototype for situations in which the size of the horizon is much larger in some directions than in others. Consider, as a simple, extreme case of this, the black string obtained by adding a flat direction z to the Schwarzschild solution. One can decompose linearized gravitational perturbations into scalar, vector and tensor types according to how they transform with respect to transformations of the Schwarzschild coordinates. Scalar and vector perturbations of this solution are stable [117Jump To The Next Citation Point]. Tensor perturbations that are homogeneous along the z-direction are also stable, since they are the same as tensor perturbations of the Schwarzschild black hole. However, there appears to be an instability for long-wavelength tensor perturbations with nontrivial dependence on z; the frequency ω of perturbations ∼ e−i(ωt−kz) acquires a positive imaginary part when k < kGL ∼ 1βˆ•r0, where r0 is the Schwarzschild horizon radius. Thus, if the string is compactified on a circle of length L > 2πβˆ•kGL ∼ r0, it becomes unstable. Of the unstable modes, the fastest one (with the largest imaginary frequency) occurs for k roughly one half of kGL. The instability creates inhomogeneities along the direction of the string. Their evolution beyond the linear approximation has been followed numerically in [42]. It is unclear yet what the endpoint is; the inhomogeneities may well grow until a sphere pinches down to a singularity.3 In this case, the Planck scale will be reached along the evolution, and fragmentation of the black string into black holes, with a consistent increase in the total horizon entropy, may occur.

Another important feature of this phenomenon is the appearance of a zero-mode (i.e., static) perturbation with k = kGL. Perturbing the black string with this mode yields a new static solution with inhomogeneities along the string direction [117Jump To The Next Citation Point120Jump To The Next Citation Point]. Following numerically these static perturbations beyond the linear approximation has given a new class of inhomogeneous black strings [248Jump To The Next Citation Point].

These results easily generalize to black p-branes; for a wavevector k along the p directions tangent to the brane, the perturbations ∼ exp (− iωt + ik ⋅ z)) with |k | ≤ kGL are unstable. The value of kGL depends on the codimension of the black brane, but not on p.

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