Meissner effect for axially symmetric charged black holes

Kurzfassung

I. INTRODUCTION The Meissner effect of black holes describes their property to expel any exterior magnetic field in case the black holes become extremal. This was discovered in Refs. [1–3] and has since then been discussed for electromagnetic test fields [4,5] and particular classes of black holes surrounded by a strong magnetic field [6–10]. In the recent paper [11], we proved that the Meissner effect holds even for generic uncharged black holes in equilibrium that are distorted by exterior matter and electromagnetic fields. This implies that the expulsion of the electromagnetic field is not due to the specific geometries like the Kerr geometry investigated so far. Still, its physical origin is not entirely understood. Therefore, the current paper serves two purposes. First, we will show that charged black holes exhibit the Meissner effect for strong electromagnetic fields as well, which will include all previous results. Second, we will relax as many assumptions as possible to still be able to prove the result to identify the physically necessary ones. In order to understand the conceptual problems related to the first task, it is instructive to discuss first the classical example of superconductors in external magnetic fields, which lends its name to this property of black holes. Let a neutral superconducting sphere that is above its critical temperature Tc be embedded in a magnetic field. When cooling it belowTc, the magnetic field is expelled. The behavior of uncharged black holes is analogous to this standard case. Now, consider a superconducting sphere at a temperature above Tc that carries some net charge, say, because it is doted with ions. If that sphere rotates, it generates itself already a magnetic field that penetrates its surface. Ifwe apply now an external field and cool the superconductor below its critical temperature, the external field will still be expelled. However, the magnetic field produced by the moving interior charges still causes a nonvanishingmagnetic flux across its surface. Hence, the Meissner effect does not predict that the total magnetic flux vanishes but only that the flux caused by external fields vanishes. Similarly, consider the Kerr-Newman metric, which has an electric charge QE. Due to its rotation with respect to inertial observers at infinity, it also exhibits a magnetic field penetrating the horizon. This would be additional to any other magnetic test fields generated by external sources. Clearly, also here, the Meissner effect discussed in Ref. [4] can only make a statement about the flux caused by the external matter. In the test field approach, both contributions to the magnetic flux, the one from the black hole and the one from the test field, can be trivially disentangled, and the Meissner effect can be formulated for the test field alone; see, e.g., Ref. [4]. However, for strong magnetic fields, this disentanglement is more involved because of the nonlinearity of the theory. Thus, it is not obvious which part of the flux needs to vanish. Technically, this means that in our proof in Ref. [11], the argument that the vanishing charge of the horizon implies that the integration constants and consequently the fields vanish, does not apply anymore. But as described above, it should not be expected that the total magnetic flux is vanishing. It should rather be understood that the magnetic field penetrating an extremal horizon does solely depend on the properties of the horizon like its electric and magnetic charge and not on the configuration of the external matter. This can be made more precise using the initial value problem of the underlying partial differential equations. To solve them, initial data need to be provided at the horizon and a null surface N intersecting it. The former describes horizon properties, whereas the latter describes the external matter and fields. The flux is now supposed to be independent of any initial data given on N. This implies then that the matter and electromagnetic fields in the exterior can be distributed arbitrarily without changing the magnetic flux across the horizon. In contrast to our previous work [11], we will allow for more general situations, in addition to allowing for a charge of the black hole. For example, we will allow strings piercing the horizon as it is the case for the C-metric [12], where the Meissner effect was already observed for test fields [13]. Additionally, we will relax the equilibrium condition further generalizing the notion of isolated horizons to almost isolated horizons. Lastly, we will not assume that the full Einstein equations hold but rather that only one of its projections does to allow for modified theories of gravity. The paper is structured as follows. In Sec. II, we introduce the notion of charged almost isolated horizons as well as the constraint equations, which need to be solved at the horizon. In Sec. III, we assume additionally axial symmetry to solve the constraint equation explicitly. This shows that the electromagnetic flux across the horizon depends only on horizon properties, which we parametrize in terms of the electric and magnetic charge and dipole and quadrupole moment of the almost isolated horizon, for illustration. The results are summarized and discussed in Sec. IV, where we also give the main Theorem 1. Throughout the paper, we use geometric units, in which c ¼ G ¼ 1, and the metric signature diagðþ1;−1;−1;−1Þ. Moreover, we use the abstract index notation, cf. Ref. [14]. By some abuse of notation, we use the same alphabet for the abstract indices of quantities regardless of the manifolds on which they are defined.