Topological black holes in Einstein-Yang-Mills theory
with a negative cosmological constant
We investigate the phase space of topological black hole solutions of Einstein-Yang-Mills theory in anti-de Sitter space with a purely magnetic gauge potential. The gauge field is described by magnetic gauge field functions , . For gauge group, the function has no zeros. This is no longer the case when we consider a larger gauge group. The phase space of topological black holes is considerably simpler than for the corresponding spherically symmetric black holes, but for and a flat event horizon, there exist solutions where at least one of the functions has one or more zeros. For most of the solutions, all the functions have no zeros, and at least some of these are linearly stable.
keywords:Topological black holes, Einstein-Yang-Mills theory
Pacs:04.20Jb, 04.40Nr, 04.70Bw
Many properties of black holes in asymptotically anti-de Sitter (adS) space-time are rather different from those of black holes in asymptotically flat space-time. For example, stationary vacuum black holes in four-dimensional asymptotically flat space-time must have spherical event horizon topology Hawking:1971vc (); Hawking:1973uf () but in four-dimensional adS space-time, vacuum black holes with other event horizon topologies exist (see, for example, Birmingham:1998nr (); Brill:1997mf (); Lemos:1994fn (); Lemos:1994xp (); Lemos:1995cm (); Vanzo:1997gw (); Cai:1996eg (); Mann:1996gj (); Smith:1997wx (); Mann:1997zn ()).
Many properties of black holes with nontrivial matter field hair in asymptotically adS space-time also differ from the properties of their asymptotically flat counterparts. In Einstein-Yang-Mills (EYM) theory, spherically symmetric, asymptotically flat “coloured” black holes Bizon:1990sr () are unstable under linear, spherically symmetric, perturbations Straumann:1990as (). In contrast, there exist stable, spherically symmetric, asymptotically adS, black hole solutions of EYM with a negative cosmological constant Winstanley:1998sn (); Bjoraker:1999yd (); Bjoraker:2000qd (). Analogues of these spherically symmetric, asymptotically adS black holes with nonspherical event horizon topology also exist when the gauge group is VanderBij:2001ia (). Furthermore, these topological EYM black holes are linearly stable VanderBij:2001ia ().
Since the discovery of stable black holes in EYM in adS, black hole solutions with the larger gauge group have been extensively studied (see Winstanley:2008ac (); Winstanley:2015loa () for reviews). For all , there exist stable, spherically symmetric, purely magnetic, asymptotically adS black holes Baxter:2007at (); Baxter:2008pi (); Baxter:2015gfa (). For gauge group, the purely magnetic gauge field is described by independent functions, so these stable black holes have an unlimited amount of gauge field “hair”.
A natural question is whether there exist analogues of the purely magnetic, spherically symmetric black holes in EYM in adS with nonspherical event horizon topology. Recently this question was answered in the affirmative Baxter:2014nka (), in two regimes: (a) for a negative cosmological constant with sufficiently large magnitude and (b) in a neighbourhood of an embedded solution. Furthermore, the proof of the existence of stable spherically symmetric EYM black holes in adS Baxter:2015gfa () can be extended to these topological EYM black holes, showing that at least some purely magnetic topological black holes, in the intersection of the regimes (a) and (b) above, are linearly stable Baxter:2015xfa ().
The analytic work in Baxter:2014nka (); Baxter:2015xfa () proves the existence of stable topological black holes in EYM in adS, but does not tell us about the nature of the phase space of topological black hole solutions. In the spherically symmetric case, the phase space of purely magnetic black hole solutions of EYM in adS has a very rich structure Baxter:2007au (). In this letter we explore the phase space of numerical topological black hole solutions of EYM in adS, drawing comparisons with the spherically symmetric solution space studied in Baxter:2007au (). We begin in section 2 with a review of the salient features of EYM in adS before presenting our numerical results in section 3 and our conclusions in section 4.
2 Einstein-Yang-Mills theory in adS
We consider Einstein-Yang-Mills (EYM) theory in four space-time dimensions, having the action:
where is the Ricci scalar and the cosmological constant. Here and throughout this paper we use units in which . We also set the gauge coupling constant equal to unity. The nonabelian gauge field strength is given in terms of the gauge field potential as follows:
Varying the action (1) gives the EYM field equations:
where the stress-energy tensor of the Yang-Mills field is:
with Tr denoting a Lie algebra trace.
In this letter we are interested in asymptotically anti-de Sitter (adS) topological black holes. The metric ansatz we employ takes the form:
where the metric functions and depend on the radial co-ordinate only. The metric function can be written in an alternative form in terms of a new function :
When , the metric (5) is spherically symmetric, with the , hypersurfaces being two-spheres. For , the , hypersurfaces are two-dimensional Euclidean spaces, while for these hypersurfaces have constant negative curvature (see Birmingham:1998nr (); Brill:1997mf (); Lemos:1994fn (); Lemos:1994xp (); Lemos:1995cm (); Vanzo:1997gw (); Cai:1996eg (); Mann:1996gj (); Smith:1997wx (); Mann:1997zn () for further details).
where and are matrices. The matrix has hermitian conjugate and is upper-triangular, with nonzero entries only immediately above the diagonal. These nonzero entries can be written in terms of functions of only:
The matrix is constant and diagonal:
The ansatz (8) reduces in the case to that in Kunzle:1991 () for a purely magnetic spherically-symmetric gauge field potential. For EYM, it is the ansatz used in VanderBij:2001ia () for topological black holes. The general form (8) was derived in Baxter:2014nka (). The gauge potential (8) is therefore described by the magnetic field functions , .
and coupled Yang-Mills equations
As in the case, the equation (11) for the metric variable decouples from the remaining equations. Furthermore, the field equations (11, 13) are invariant under the transformation for each independently, and also under for all .
We are interested in topological black holes with a regular event horizon at , where and . We assume that the field variables , and have regular Taylor series expansions in a neighbourhood of , of the form
The condition fixes to be
The parameters and are a priori arbitrary; the value of will be fixed by the boundary conditions at infinity. From the field equations (11, 13) the values of , and are fixed in terms of , and the values of the magnetic gauge field functions on the horizon Baxter:2014nka (). For a regular event horizon at , we require and this implies that
which places a (weak) constraint on the values of the magnetic gauge field functions on the horizon . For the case, the relation (17) implies the existence of a minimum value of for fixed :
As , the field variables have the following expansions Baxter:2014nka ():
where and are arbitrary constants. The value of the metric function on the horizon is fixed by the requirement that as . The local existence of solutions of the field equations (11, 13) satisfying the boundary conditions (15, 19) is proven in Baxter:2014nka ().
where is an arbitrary constant and the magnetic charge is given by
For , we have and the solution is the embedded Schwarzschild-adS black hole with planar event horizon topology. For , the magnetic charge and we have an embedded (magnetically-charged) Reissner-Nordström-adS black hole with constant negative curvature horizon.
3 Topological black hole solutions
We now integrate the field equations (11, 13) numerically. We begin our integration close to the event horizon , using the expansion of the field variables (15) as initial conditions. We then integrate for increasing until either the solution approaches the boundary conditions (19) to within a suitable tolerance or the solution becomes singular. The field equations (11, 13) are invariant under the transformation for each independently, so without loss of generality we may restrict our attention to values of the parameters such that for all .
In this section we plot the phase spaces of topological black hole solutions of the field equations, fixing , so that all length scales are in units of the event horizon radius, and considering various . The plots show the spaces of initial parameters which determine the solutions, and we explore the values of for which there is a regular event horizon, so that (17) holds. In each plot, we show the region of the parameter space for which there are nontrivial topological black hole solutions and label these solutions by the quantities , with being the number of zeros of the magnetic gauge field function . We review the numerical solutions for gauge group VanderBij:2001ia () before discussing our new solutions for gauge group.
3.1 topological black holes
Topological black hole solutions of the EYM field equations (11, 13) were first found in VanderBij:2001ia (). In this case the YM field is described by a single gauge field function . In figure 1 we plot the phase space of solutions for and in plots (a) and (b) respectively, with and varying cosmological constant . The black holes are parameterized by and , the value of the single magnetic gauge field function on the horizon. In figure 1 we show the space of the parameters satisfying the condition (17) for a regular black hole event horizon at . This is the region underneath the dotted curves. For some values of the parameters such that (17) is satisfied, the numerical solution of the field equations (11, 13) becomes singular at some . Such points are contained in the region labelled “no solution”, bounded by the dotted and solid curves.
Nontrivial topological black holes exist in the regions below the solid curves in the plots in figure 1. The line corresponds to the trivial embedded Reissner-Nordström-adS black hole with magnetic charge given by (21). As shown in VanderBij:2001ia (), for all the solutions for both and the single magnetic gauge field function has no zeros, so .
For , we find nontrivial solutions for all values of with , but for smaller than the range of values of for which there are solutions is very small, as in this case the condition (17) reduces to . In accordance with this inequality, as increases, we find an increasingly large range of for which nontrivial black hole solutions exist.
For and , the minimum value of for which there is a regular event horizon, given by (18), reduces to . As in the case, as increases for , we find an increasing range of values of for which there are nontrivial black hole solutions.
The phase spaces in figure 1 for and are much simpler than the corresponding phase space of spherically symmetric () black holes (see, for example, Baxter:2007au ()). In the case, for fixed and sufficiently large , all the black holes are such that has no zeros. However, as decreases, increases (for example, with , there are black holes with having up to four zeros Baxter:2007au ()). As , the phase space of black holes fragments and reduces to the asymptotically flat “coloured” black holes Bizon:1990sr (). If , there is no asymptotically flat limit. For the phase space of topological black holes simply ends when , while for the phase space shrinks to zero size as .
3.2 topological black holes
and defining rescaled quantities , , as follows:
it is found that and , considered as functions of , satisfy the EYM field equations (11, 13) with cosmological constant Baxter:2014nka (). In Baxter:2014nka () it is proven that there exist nontrivial (that is, nonembedded) topological black holes in a neighbourhood of these embedded black holes.
In figures 2 and 3 we plot the phase space of topological black holes for and respectively. In each case we fix . The black holes are now parameterized by , and . We fix a few selected values of for both and and plot the phase space in in each case. As in the case, we scan over those values of for which the condition (17) is satisfied.
In each plot in figures 2 and 3, the condition (17) for a regular event horizon is satisfied by parameter values in the region enclosed by the axes and the dotted curve. It can be seen how the size of this region expands as increases, in accordance with (17). For some values of the parameters such that (17) holds, we are unable to find a suitable numerical solution, and these values of the parameters are in the region labelled “no solution” between the dotted and solid curves. We find nontrivial topological black hole solutions for values of the parameters in the regions bounded by solid curves and the axes. The regions with solutions are labelled according to the number of zeros of the magnetic gauge field functions , respectively.
From figures 2, 3, the region of parameter space where we have nontrivial topological black hole solutions expands as increases, both in absolute size and in its size relative to the size of the region of parameter space for which (17) holds. For both and , embedded topological black hole solutions lie on the line . From (22) and the discussion in section 3.1, for these embedded solutions the two magnetic gauge field functions are equal and have no zeros so . Since the field equations (11, 13) are invariant under the transformation , the phase space of black holes is symmetric about the line . The point where gives the trivial embedded Reissner-Nordström-adS black hole with magnetic charge (21).
For , the condition (17) makes no constraints on the magnitude of the cosmological constant and we find nontrivial topological black holes for all values of examined. For each we find a region of nontrivial topological black hole solutions for which both gauge field functions and have no zeros, whose existence was proven in Baxter:2014nka () for sufficiently large . This region contains the embedded solutions. For larger values of , all nontrivial solutions found are such that both and have no zeros.
However, for smaller values of we find nontrivial topological black hole solutions for which at least one of , has zeros. The corresponding regions of parameter space lie between the nodeless regions (which contain all the embedded solutions) and the “no solution” region. When we find solutions with , and , in very small regions exterior to the region. The size of the regions of parameter space containing , or , solutions increases as decreases, as a proportion of the total size of the region of parameter space for which there are nontrivial solutions. For we also find numerical solutions with , and , , but the relevant regions of parameter space are too small to be seen in figure 2(a). They lie just outside the regions where , or , .
The overall shape of the region of parameter space for which there are nontrivial solutions with also changes as increases: for small we see a cusp-like shape at the maximum values of and in this region, but this smooths out for larger values of . This qualitative feature is also seen the spherically symmetric black holes discussed in Baxter:2007au ().
The phase space shown in figure 2 is much simpler than that for spherically symmetric solutions with Baxter:2007au (). When there are nodeless solutions for any value of , no matter how small (in a neighbourhood of the embedded solutions), whereas when and there are no nodeless solutions Baxter:2007au (). For a fixed , we also find a smaller range of values of and when compared with . For example, with , when we find the following combinations of : , and , but with there are also the combinations , , , , and Baxter:2007au (). As decreases towards zero, in the case presented here we do not see the fragmentation of the phase space seen when Baxter:2007au (). This is because for there is no solution in the limit; instead the phase space simply shrinks in size as decreases.
For , with , the minimum value of for which there is a regular event horizon is . Unlike the situation for , for all the topological black hole solutions that we find numerically are such that both magnetic gauge field functions have no zeros, so . Furthermore, the shape of the region of parameter space for which there are nontrivial solutions does not vary much as increases.
In this letter we have constructed numerical solutions of the Einstein-Yang-Mills (EYM) equations in anti-de Sitter (adS) space-time representing purely magnetic topological black holes. These solutions generalize the numerical spherically-symmetric black holes and solitons discussed in detail in Baxter:2007au (). We have explored the phase space of solutions for and gauge groups, for both (planar event horizon topology) and (when the event horizon is a surface of constant negative curvature).
In the case, the (single) magnetic gauge field function has no zeros for all topological black holes with both and VanderBij:2001ia (). For , the existence of topological black holes for which all the magnetic gauge field functions have no zeros is proven in Baxter:2014nka (). These solutions are of particular interest because at least some of them have been proven to be stable under linear perturbations of the metric and gauge field Baxter:2015xfa (). In the case, all the numerical solutions we find for have nodeless magnetic gauge field functions, but this is not the case for . When , as well as the nodeless solutions whose existence was proven in Baxter:2014nka (), we also find topological black holes for which at least one of the gauge field functions has one or more zeros.
Topological black holes with Ricci-flat () event horizons have attracted a great deal of attention recently as models of holographic superconductors (see, for example, Cai:2015cya () for a recent review). In particular, EYM black holes in adS with have been studied as gravitational analogues of -wave superconductors Gubser:2008wv (). Unlike the situation in this paper (where we consider only purely magnetic gauge fields), in holographic superconductor models the gauge field is dyonic, with nonzero electric and magnetic parts. The magnetic part of the gauge field forms a nontrivial condensate outside the event horizon and vanishes as Gubser:2008wv (); Gubser:2008zu (). A natural question is whether enlarging the gauge group to yields solutions which also model holographic superconductors. Recently the existence of dyonic topological black holes in EYM in adS was proven Baxter:2015tda (), but in that paper the solutions shown to exist are such that all the gauge field functions have no zeros (including at infinity), which are not relevant for holographic superconductors. A numerical study of the solutions of the field equations for a dyonic gauge field, extending the work in this letter, is therefore needed. We leave an investigation of this for future work.
The work of E.W. is supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant ST/L000520/1.
- (1) S.W. Hawking, Commun. Math. Phys. 25 (1972) 152–166.
- (2) S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time, Cambridge University Press, Cambridge, 1973.
- (3) D. Birmingham, Class. Quant. Grav. 16 (1999) 1197–1205.
- (4) D.R. Brill, J. Louko and P. Peldan, Phys. Rev. D 56 (1997) 3600–3610.
- (5) J.P.S. Lemos, Class. Quant. Grav. 12 (1995) 1081–1086.
- (6) J.P.S. Lemos, Phys. Lett. B 353 (1995) 46–51.
- (7) J.P.S. Lemos and V.T. Zanchin, Phys. Rev. D 54 (1996) 3840–3853.
- (8) L. Vanzo, Phys. Rev. D 56 (1997) 6475–6483.
- (9) R.G. Cai and Y.Z. Zhang, Phys. Rev. D 54 (1996) 4891–4898.
- (10) R.B. Mann, Class. Quant. Grav. 14 (1997) L109–L114.
- (11) W.L. Smith and R.B. Mann, Phys. Rev. D 56 (1997) 4942–4947.
- (12) R.B. Mann, Nucl. Phys. B 516 (1998) 357–381.
- (13) P. Bizon, Phys. Rev. Lett. 64 (1990) 2844–2847.
- (14) N. Straumann and Z. H. Zhou, Phys. Lett. B 243 (1990) 33–35.
- (15) E. Winstanley, Class. Quant. Grav. 16 (1999) 1963–1978.
- (16) J. Bjoraker, Y. Hosotani, Phys. Rev. Lett. 84 (2000) 1853–1856.
- (17) J. Bjoraker, Y. Hosotani, Phys. Rev. D 62 (2000) 043513.
- (18) J.J. Van der Bij, E. Radu, Phys. Lett. B 536 (2002) 107–113.
- (19) E. Winstanley, Lect. Notes Phys. 769 (2009) 49–87.
- (20) E. Winstanley, arXiv:1510.01669 [gr-qc].
- (21) J.E. Baxter, M. Helbling, E. Winstanley, Phys. Rev. Lett. 100 (2008) 011301.
- (22) J.E. Baxter, E. Winstanley, Class. Quant. Grav. 25 (2008) 245014.
- (23) J.E. Baxter, E. Winstanley, arXiv:1501.07541 [gr-qc].
- (24) J.E. Baxter, Gen. Rel. Grav. 47 (2015) 1829.
- (25) J.E. Baxter, arXiv:1507.03127 [gr-qc].
- (26) J.E. Baxter, M. Helbling, E. Winstanley, Phys. Rev. D 76 (2007) 104017.
- (27) H.P. Kunzle, Class. Quant. Grav. 8 (1991) 2283â-2297.
- (28) R.G. Cai, L. Li, L.F. Li, R.Q. Yang, Sci. China Phys. Mech. Astron. 58 (2015) 060401.
- (29) S.S. Gubser and S.S. Pufu, JHEP 0811 (2008) 033.
- (30) S.S. Gubser, Phys. Rev. Lett. 101 (2008) 191601.
- (31) J.E. Baxter, arXiv:1507.05314 [gr-qc].