Internal structure of Einstein-Yang-Mills black holes

Abstract

It is shown that a generic black hole solution of the SU (2) Einstein-Yang-Mills (EYM) equations develops a new type of an infinitely oscillating behavior near the singularity. Only for certain discrete values of the event horizon radius exceptional solutions exist possessing an inner structure of the Schwarzschild or Reissner-Nordström type. 04.20.Jb, 97.60.Lf, 11.15.Kc Discovered soon after the regular Bartnik-McKinnon (BK) solutions [1], EYM black holes (BH) [2][3][4] provided new insights into the BH physics related to the no-hair and uniqueness theorems [5]. They share sphaleronic properties of the BK particle-like solutions [6] and also exhibit an unusual discreteness ('quantization' of the YM field on the event horizon) due to a singular non-linear boundary value problem in the domain between the horizon and the asymptotically flat (AF) infinity. Still, the existing knowledge of the EYM BH's (unlike the BK objects) is incomplete since only external solutions have been constructed so far (though a qualitative discussion of inner solutions is available [3]). Here we present brief results of our investigation of the interior structure of the EYM BH's which reveal new surprising features due to coupling of non-linear fields to gravity.Assume the static spherically symmetric magnetic ansatz for the YM potential(T ϕ,θ are spherical projections of the SU (2) generators) and the following parametrization of the metricwhere dΩ 2 = dθ 2 + sin 2 θdϕ 2 , and ∆, σ depend on r. The field equations include a coupled system for W , ∆where V = (W 2 − 1), F = 1 − V 2 /r 2 , and a decoupled equation for σ:These equations admit BH solutions in the domain r ≥ r h for any radius of the event horizon r h . The solutions are specified by the number n ∈ N of nodes of W thus forming a discrete set for each r h . Although it is not guaranteed a priori that the chart (1) is extendible to the full region r < r h , for AF solutions we did not meet any singularity in the interior region unless the genuine one r = 0 is reached. In terms of coordinates (1) one can find three distinct local power series solutions. The first one is Schwarzschild-like (S), it corresponds to the vacuum value of the YM field |W (0)| = 1. Using the mass function m(r), ∆ = r 2 − 2mr, one gets [3]where m 0 , b are (the only) free parameters. The second is the Reissner-Nordström (RN) type of solution which can be found assuming the leading term of ∆ to be a positive constant. This requires W (0) = W 0 = ±1, 0 and gives [3]what corresponds to the RN metric of the mass m 0 and the (magnetic) charge. The expansion contains three free parameters W 0 , m 0 , c.We have also found the third local power series solution assuming a negative value for ∆(0) (i.e. imaginary P ):Here there is only one free parameter (W 0 ) for W , ∆. The corresponding space-time near the singularity is conformal to the cylinder (after a time rescaling):However, one may suspect that such asymptotics can not correspond to a generic BH. Imposing 'boundary conditions' in the singularity we obta...

Internal structure of the Skyrme black hole

“…This method was adopted in Refs. [4,5] and it was found that the point Q is the exceptional case (i). In Fig.…”

“…3. Three types of behaviors are considered assuming that variables are expressed by power series near the center [4], i.e., Schwarzschild-like, Reissner-Nortström (RN)-like and imaginary charged RN-like behaviors. By expanding the field variables…”

“…To calculate this behavior correctly, it is necessary to use suitable variables as introduced in the Appendix of Ref. [4]. As the colored black hole, infinitely oscillating behavior appears in the Skyrme black hole case.…”

“…They are important as counterexamples of the black hole no hair conjecture [3]. Moreover, recent investigations show that the internal structure of a black hole in the SU(2) EYM system exhibits interesting behaviors which are not found in the Kerr-Newman black holes [4,5]. A generic solution in the EYM system exhibits an infinitely oscillating behavior near the sigularity, which can be understood by reducing the equations to a simplified dynamical system.…”

Universal area spectrum in single-horizon black holes

“…Unfortunately, since black holes with nonAbelian fields, which have one horizon in general, show complicated behavior near the origin [31,32,33,34,35], we need further analysis to include these cases.…”