Hairy Black Holes and Null Circular Geodesics

📝 Abstract

Einstein-matter theories in which hairy black-hole configurations have been found are studied. We prove that the nontrivial behavior of the hair must extend beyond the null circular orbit (the photonsphere) of the corresponding spacetime. We further conjecture that the region above the photonsphere contains at least 50% of the total hair’s mass. We support this conjecture with analytical and numerical results.

💡 Analysis

Einstein-matter theories in which hairy black-hole configurations have been found are studied. We prove that the nontrivial behavior of the hair must extend beyond the null circular orbit (the photonsphere) of the corresponding spacetime. We further conjecture that the region above the photonsphere contains at least 50% of the total hair’s mass. We support this conjecture with analytical and numerical results.

📄 Content

arXiv:1112.3286v1 [gr-qc] 14 Dec 2011 Hairy Black Holes and Null Circular Geodesics Shahar Hod The Ruppin Academic Center, Emeq Hefer 40250, Israel and The Hadassah Institute, Jerusalem 91010, Israel (Dated: February 6, 2018) Einstein-matter theories in which hairy black-hole configurations have been found are studied. We prove that the non-trivial behavior of the hair must extend beyond the null circular orbit (the “photonsphere”) of the corresponding spacetime. We further conjecture that the region above the photonsphere contains at least 50% of the total hair’s mass. We support this conjecture with analytical and numerical results. The influential ‘no-hair conjecture’ of Wheeler [1] has played a key role in the development of black-hole physics [2, 3]. This conjecture suggests that black holes are fun- damental objects in general relativity, Einstein’s theory of gravity – they should be described by only a few pa- rameters, very much like atoms in quantum mechanics. The no-hair conjecture was motivated by earlier uniqueness theorems on black-hole solutions of the Ein- stein vacuum theory and the Einstein-Maxwell theory [4–8]. According to these uniqueness theorems, all sta- tionary solutions of the Einstein-Maxwell equations are uniquely described by only three conserved parameters which are associated with a Gauss-like law: mass, charge, and angular momentum. The belief in the no-hair conjecture was based on a sim- ple physical picture according to which all matter fields left in the exterior of a newly born black hole would even- tually be radiated away to infinity or be swallowed by the black hole itself (except when those fields were asso- ciated with conserved charges). In accord with this logic, early no-hair theorems indeed excluded scalar [9], mas- sive vector [10], and spinor [11] fields from the exterior of stationary black holes. However, the interplay between particle physics and general relativity in the following years has led to the somewhat surprising discovery of various types of “hairy” black holes, the first of which were the “colored black holes” [12]. These are black-hole solutions of the Einstein-Yang-Mills (EYM) theory that require for their complete specification not only the value of the mass but also an additional integer, n, which counts the number of nodes of the Yang-Mills field outside the horizon. Re- markably, this integer is not associated with any con- served charge. Soon after this discovery, a variety of hairy black- hole solutions equipped with different types of exterior fields have been discovered [13–24]. These include the Einstein-Skyrme, Einstein-non Abelian-Proca, Einstein- Yang-Mills-Higgs, and Einstein-Yang-Mills-Dilaton hairy black holes. It has become clear [3] that the non-linear character of the matter fields mentioned above plays a key role in the construction of these hairy black-hole configurations. N´u˜nez et. al. [3] have presented a nice heuristic pic- ture according to which it is the self-interaction between the part of the field in a region near the black-hole hori- zon (a loosely defined region from which the hair tends to be swallowed by the black hole) and the part of the field in a region relatively far from the black hole (a re- gion from which the hair tends to be radiated away to infinity) which is responsible, together with gravity, for the existence of stationary black-hole solutions with exte- rior matter fields (hair). The non-linear (self-interaction) character of the fields thus plays an essential role in bind- ing together the hair in these two regions in such a way that the “near-horizon” hair does not collapse into the black hole while the “far-region” hair does not escape to infinity. Thus, according to the heuristic picture of [3], the non- trivial (non-linear) behavior of the matter fields which constitute the hair is expected to extend into some loosely defined “far region” well above the black-hole horizon. But is it possible to provide a more explicit characteri- zation of the hair’s length? Here we turn our attention to another important characteristic of black-hole spacetimes: null geodesics. Geodesic motions provide important information on the structure of the spacetime geometry. Among the different kinds of geodesic motion, circular geodesics are especially important [25, 26]. In particular, the null circular orbit (also known as the “photon orbit” or “photonsphere”) is the boundary between two qualitatively different regions in the exterior of a black hole: No stationary spherically- symmetric configurations made of test particles (with no self-interactions) can exists below this orbit [27]. Grav- ity is simply too strong there. Relating this property of the null circular geodesic to our former discussion on hairy black holes, we conjecture that the “near region” (the region from which the hair tends to be sucked into the black hole) extends at least up to the height of the photon orbit. The aim of this Letter is to prove a theorem which supp

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