Dyonic black holes in Kaluza-Klein theory with a Gauss-Bonnet action

We consider a five-dimensional Einstein-Gauss-Bonnet model, which gives rise after dimensional reduction to Einstein gravity nonminimally coupled to nonlinear electrodynamics. The black hole solutions of the four-dimensional model modify the Reissner-Nordstrom solutions of general relativity. The gravitational field presents the standard singularity at $r=0$, while the electric field can be regular everywhere if the magnetic charge vanishes


Introduction
In his career, Richard Kerner has given important contributions to the Kaluza-Klein framework, in particular showing that also nonabelian gauge theories can be obtained from the process of dimensional reduction by increasing the number of internal dimensions [1].He was also one of the first to notice the relevance of Gauss-Bonnet (GB) terms in the action of higher-dimensional theories, showing that they give rise to nonlinear contributions to the electrodynamics in the reduced theory [2]. 1onlinear electrodynamics was first proposed by Heisenberg and Euler [4] to give an effective classical description of quantum electrodynamics in a suitable limit.A general formulation, which includes also the model studied in [2.3] as a special case, was given by Plebanski [5].However, the action obtained from dimensional reduction of the GB action enjoys peculiar algebraic properties.Some solutions of this model in flat space, hence neglecting gravity, have been discussed in [2,6,7].
In the present paper, we are interested in the full theory containing both gravity and Maxwell fields.We have recently shown that a five-dimensional Kaluza-Klein theory containing GB contributions admits exact solutions that modify the Reissner-Nordström (RN) metric of general relativity [7].In particular, when the nonminimal coupling between gravity and Maxwell fields arising from the dimensional reduction is neglected, its dyonic solutions display an everywhere regular electric field [7].These modifications could in principle give experimental evidence of the existence of extra dimensions.
Here, we give a more complete discussion of the dyonic solutions of the model, that takes into account also the nonminimal interaction terms.It turns out that, contrary to the case where these terms are neglected, the electric field can be regular everywhere only for vanishing magnetic field. 2 This may be considered as a favourable feature of the model, since magnetic monopoles are not observed in nature.

The model
As is well known, the Einstein-Hilbert action can be generalized in dimensions higher than four, by the introduction of the Lovelock terms [9].These give the most general extensions of the Einstein-Hilbert action that give rise to second order field equations in arbitrary dimensions.One of their most notable properties is that they do not introduce new degrees of freedom in the spectrum in addition to the graviton, and therefore avoid the presence of ghosts or tachyons, in contrast with most higher-derivative actions [10].In lower dimensions they are total derivatives and do not contribute to the equations of motion.In particular, in five dimension, the only term of this type is the so-called Gauss-Bonnet term, The dimensional reduction of these generalized actions gives rise to models of gravity coupled to nonlinear electrodynamics, which as a consequence of the properties of the higher-dimensional theory, contain only graviton and photon excitations, and are therefore relevant from a phenomenological perspective.
Therefore, we consider a five-dimensional Einstein-Gauss-Bonnet theory, with action [1,2,7] where α is a coupling constant and R the Ricci scalar.The coupling constant α has dimension [L] 2 , and is usually assumed to be positive for stability reasons.Arguments based on quantum gravity or string theory fix it to be of Planck scale, but in any case observations set a very small upper limit on its value [8].We make the simple ansatz for the five-dimensional metric3 [9-10] where A i is the Maxwell potential and we have chosen the normalization in order to simplify the dimensionally reduced action.
Discarding total derivatives, the action reduces to [2,3,13] where If one neglects L int , the action describes a model of gravity minimally coupled to a specific form of nonlinear electrodynamics.Exact asymptotically flat spherically symmetric solutions of (3) in the absence of L int have been investigated in [7], where it was shown that the Reissner-Nordström solution of general relativity is modified in the dyonic case.In particular, for α > 0, the solutions read with P the magnetic monopole charge, and radial electric field E ≡ F 01 , while the metric function is where we have set α = √ 3αP 2 .The asymptotic behavior of the solution is given by and has therefore the same form as for RN up to order 1/r 5 .It follows that the integration constants M and Q can be identified with the mass and the electric charge.The electric field is regular everywhere, and the properties of the metric are analogous to those of the RN solution: a curvature singularity is present at the origin and for M greater than its extremal value is shielded by two horizons.If α < 0, instead, a singularity of the electric field appears at r 0 = √ α, where now α = √ −3αP 2 .The metric takes a slightly different form, A spherical curvature singularity occurs at r = r 0 and the solution can have one or two horizons depending on the specific values of the parameters, while the asymptotic behavior is still given by (9).
It is evident that the effects of nonlinear electrodynamics are more relevant at small r, and tend to vanish at infinity.

The solution
We now consider the asymptotically flat spherically symmetric solution of the field equations stemming from the action (3), when also the term L int is included.In this case it is not possible to find an exact solution, and we must proceed perturbatively.
We look for spherically symmetric solutions with electromagnetic field of the dyonic form (5) and where ν, λ and ρ are functions of r.
After integration by parts, the action becomes where ′ = d/dr.
Varying with respect to the fields, and then choosing the gauge e ρ = r, the independent field equations can be put in the form r 2 e −λ 1 + 12α An important effect of the nonminimal gravity-Maxwell coupling L int is that the metric field λ no longer vanishes.This is a common feature in presence of nonminimally coupled Maxwell fields.Also the radial electric field E ≡ F 01 is modified with respect to the solution (7), since (15) gives where Q is an integration constant that can be identified with the electric charge.The equations ( 13)-( 15) do not admit a solution in analytical form, so we perturb in the small parameter α around the Reissner-Nordström background with The perturbative expansion will be valid for large values of r, namely r ≫ √ α.As is well known, the horizons of the RN metric are located at In order to simplify some expressions, in the following we shall write the RN mass M in terms of the outer horizon and of the charges as We now define the perturbations σ(r), γ(r) and φ(r) through an expansion at order α around a RN background, where of course, λ = 0 for RN.Integrating (13), we obtain at order Substitution in (15) gives Finally, substituting the previous results in ( 13) and integrating, one obtains In all the solutions, we have chosen boundary conditions such that the corrections vanish at infinity.Hence, at order α, Our approximation works well for r → ∞.At leading orders in 1/r the asymptotic behavior is the same as in RN.Therefore, we can still identify M with the mass of the black hole and Q and P with its electric and magnetic charge.Notice that the corrections to the RN solutions are much larger than in the case where the L int term is neglected, since they are now o(1/r 4 ).Also, they are no longer symmetric in Q and P at leading order in α.
The horizons are displaced with respect to the RN solutions, where they are located at r * ± .At first order in α, one has r ± = r * ± + α∆r ± , where It follows that where to simplify the expression we have chosen as independent parameters r * + (r * − ), Q and P .The actual values of these displacements strongly depend on the charges.
A calculation shows that the condition of extremality is, at first order in α, Depending on the values of Q and P the correction with respect to the RN case can be both positive or negative.We recall however that, while the value of r + obtained in this way is in general well approximated by (29), the value of r − is reliable only for very small values of α.
In this approximation, the metric function e 2ν does not differ much from that of RN and the causal structure should therefore be analog.Hence, for M greater than extremality, one has two horizon, while a naked singularity is present for M less than its extremal value.However, this is not necessarily true for greater values of α, where the approximation fails.
Using standard definitions it is possible to derive the thermodynamical quantities associated to the black hole from the behaviour of the metric functions near the outer horizon,.The temperature can be calculated from the formula Hence, The entropy S is usually identified with the area of the horizon, namely, It follows that the thermodynamical quantities display a complicate dependence on the charges.It is also interesting to investigate the behavior of the solutions near the singularity.This can be calculated by an expansion in powers of r near r = 0. Setting and substituting in the field equations ( 13)-( 15), one obtains h = −2, l = −3 and k = −1.It follows that the metric functions and the electric field diverge for r = 0, thus destroying the nice property of the solution of sect. 2 to have a finite electric field at the origin.The only exception is for P = 0.In this case, the electric field vanishes at the origin.In fact now h = −1, l = 0 and k = 1.The existence of electric solutions regular at the origin in absence of magnetic field has also been noticed in [8].However, for small values of Q, numerical solutions show that the solutions become singular at a point r 0 > 0, presenting a spherical singularity, similarly with the α < 0 solutions of sect.2.
Remarkably, the behaviour of the solution is therefore opposite to those with L int = 0, since regular solution can now exist only if P = 0.

Numerical calculations
The solutions of ( 13)-( 15) can also be obtained numerically.This is especially interesting for r ≪ α, where the perturbative calculation of the previous section fails.In fig. 1 are reported the metric functions and the electric field for α = 0.01, M = 1, and several values of Q and P , such that Q 2 + P 2 = 1/2, so that r + ∼ 0.7.The metric functions e ν and e λ do not change much for different values of the charges.In particular, ǫ 2ν is similar to the RN solution, while e −2λ = 1 for r ≫ √ α and then fades to 0 for r → 0. In general, a curvature singularity is present at r = 0 and the causal structure is essentially the same as that of the RN solution.Also the electric field is singular at the origin as in the RN solution.
As mentioned before, an interesting special case is given by P = 0.In fig. 2 are depicted the metric functions e 2ν , e −2λ and the electric field F for M = 1 and different values of the electric charge.For our choice of the parameters, if Q > 0.44 the electric field is regular at the origin.The possibility of such behaviour had been noticed in [8] using different methods.However, for smaller Q a singularity occurs for a finite value of r and the metric functions and the electric field diverge there.The metric function e 2ν and e −2λ and the electric field F for black holes with mass M = 1, P = 0, and Q = 0.1 (in cyano), Q = 0.3 (in blue), Q = 0.5 (in green), Q = 0.7 (in red), Q = 0.9 (in magenta).As it is evident from the graphs, the curves with Q = 0.1, Q = 0.3 display a singularity at r0 = 0.08, r0 = 0.06 respectively.