A Tale of Two Horizons

I revisit the fate of coinciding horizons and the volume between them in the extremal limit of spherically symmetric black holes in four spacetime dimensions, focusing on the Schwarzschild de Sitter black hole for concreteness. The two Killing horizons in the limit spacetime that are traditionally identified with the limiting event horizons of the non-extremal black hole are shown to instead be generated by an enhanced symmetry of the near horizon geometry (NHG). This dismantles the interpretation of the 4-volume between the horizons remaining finite in the extremal limit. The NHG is reinterpreted as a tangent spacetime to the degenerate black hole horizon, and geometrical objects, such as Killing vectors and Killing horizons, are carefully mapped between the bulk and the NHG. The implications for extremal black hole entropy are then discussed.Comment: 5 pages, 4 figures, submitted to Can. J. Phys. as part of Theory Canada 9 conference proceeding

Supergravity on an Atiyah-Hitchin Base

We construct solutions to five dimensional minimal supergravity using an Atiyah-Hitchin base space. In examining the structure of solutions we show that they generically contain a singularity either on the Atiyah-Hitchin bolt or at larger radius where there is a singular solitonic boundary. However for most points in parameter space the solution exhibits a velocity of light surface (analogous to what appears in a Goedel space-time) that shields the singularity. For these solutions, all closed time-like curves are causally disconnected from the rest of the space-time in that they exist within the velocity of light surface, which null geodesics are unable to cross. The singularities in these solutions are thus found to be hidden behind the velocity of light surface and so are not naked despite the lack of an event horizon. Outside of this surface the space-time is geodesically complete, asymptotically flat and can be arranged so as not to contain closed time-like curves at infinity. The rest of parameter space simply yields solutions with naked singularities.Comment: 29 pages, 5 figures, citations added, analytic solution added, figures changed, main results unaltere

Another Mass Gap in the BTZ Geometry?

We attempt the construction of perturbative rotating hairy black holes and boson stars, invariant under a single helical Killing field, in 2+1-dimensions to complete the perturbative analysis in arbitrary odd dimension recently put forth in \cite{Stotyn:2011ns}. Unlike the higher dimensional cases, we find evidence for the non-existence of hairy black holes in 2+1-dimensions in the perturbative regime, which is interpreted as another mass gap, within which the black holes cannot have hair. The boson star solutions face a similar impediment in the background of a conical singularity with a sufficiently high angular deficit, most notably in the zero-mass BTZ background where boson stars cannot exist at all. We construct such boson stars in the AdS_3 background as well as in the background of conical singularities of periodicities \pi,2\pi/3,\pi/2.Comment: 13 pages, 2 appendices, Invited Contribution to an IOP special volume of Journal of Physics A in honor of Stuart Dowker's 75th birthday, v2: discussion in section 4 expande

Quasilocal Smarr relation for an asymptotically flat spacetime

A quasilocal Smarr relation is obtained from Euler's theorem for Einstein-Maxwell(-Dilaton) theory for an asymptotically flat spacetime, and its associated first law is studied. To check both, we calculate quasilocal variables by employing Brown-York quasilocal method along with Mann-Marolf counterterms, which are consistent with Tolman temperature. We also derive entropy by constructing a quasilocal thermodynamic potential via Euclidean method. Here we found that the Euclidean action value in a quasilocal frame just yields a usual thermodynamic potential form, which do not include a $PA$ term, and entropy just becomes the Bekenstein-Hawking one. Through the examples, we confirmed that our quasilocal Smarr relation is satisfied with all cases, and its first law is also exactly satisfied except the dyonic black hole with the dilaton coupling constant $a=\sqrt{3}$. In that case when making a large $R$ expansion, the first law is satisfied up to $1/R$ order but it does not hold for higher sub-leading order of $R$. This issue should be resolved in future.Comment: 24 page

Exact Solutions and Black Hole Stability in Higher Dimensional Supergravity Theories

This thesis examines exact solutions to gauged and ungauged supergravity theories in space-time dimensions D⩾5 as well as various instabilities of such solutions. I begin by using two solution generating techniques for five dimensional minimal ungauged supergravity, the first of which exploits the existence of a Killing spinor to generate supersymmetric solutions, which are time-like fibrations over four dimensional hyper-Kähler base spaces. I use this technique to construct a supersymmetric solution with the Atiyah-Hitchin metric as the base space. This solution has three independent parameters and possesses mass, angular momentum, electric charge and magnetic charge. Via an analysis of a congruence of null geodesics, I determine that the solution contains a region with naked closed time-like curves. The centre of the space-time is a conically singular pseudo-horizon that repels geodesics, otherwise known as a repulson. The region exterior to the closed time-like curves is outwardly geodesically complete and possesses an asymptotic region free of pathologies provided the angular momentum is chosen appropriately. The second solution generating technique exploits a hidden G2 symmetry in five dimensional minimal supergravity. I use this hidden symmetry approach to construct the most general black string solution in five dimensions, which is endowed with mass, angular momentum, linear momentum, electric charge and magnetic charge. This general black string satisfies the first law of thermodynamics, with the Bekenstein-Hawking entropy being reproduced via a microstate counting in terms of free M-branes in the decoupling limit. Furthermore it reduces to all previously known black string solutions in its various limits. A phase diagram for extremal black strings is produced to draw conclusions about extremal black rings, in particular why supersymmetric black rings exhibit a lower bound on the electric charge. The same phase diagram further suggests the existence of a new class of supersymmetric black rings, which are completely disconnected from the previously known class. A particular limit of this general black string is the magnetically charged black string, whose thermodynamic phase behaviour and perturbative stability were previously studied but not very well understood. I construct magnetically charged topological solitons, which I then show play an important role in the phase structure of these black strings. Topological solitons in Einstein-Maxwell gravity, however, were previously believed to generically correspond to unstable "bubbles of nothing" which expand to destroy the space-time. I show that the addition of a topological magnetic charge changes the stability properties of these Kaluza-Klein bubbles and that there exist perturbatively stable, static, magnetically charged bubbles which are the local vacuum and the end-point of Hawking evaporation of magnetic black strings. In gauged supergravity theories, bubbles of nothing are stabilised by the positive energy theorem for asymptotically anti-de Sitter space-times. For orbifold anti-de Sitter space-times in odd dimensions, a local vacuum state of the theory is just such a bubble, known as the Eguchi-Hanson soliton. I study the phase behaviour of orbifold Schwarzschild-anti-de Sitter black holes, thermal orbifold anti-de Sitter space-times, and thermal Eguchi-Hanson solitons from a gravitational perspective; general agreement is found between this analysis and the previous analysis from the gauge theory perspective via the AdS/CFT correspondence. I show that the usual Hawking-Page phase structure is recovered and that the main effect of the soliton in the phase space is to widen the range of large black holes that are unstable to decay despite the positivity of their specific heat. Furthermore, using topological arguments I show that the soliton and orbifold AdS geometry correspond to a confinement phase in the boundary gauge theory while the black hole corresponds to a deconfinement phase. An important instability for rotating asymptotically anti-de Sitter black holes is the superradiant instability. Motivated by arguments that the physical end point of this instability should describe a clump of scalar field co-rotating with the black hole, I construct asymptotically anti-de Sitter black hole solutions with scalar hair. Perturbative results, i.e. low amplitude boson stars and small radius black holes with low amplitude scalar hair, are presented in odd dimensions relevant to gauged supergravity theories, namely D=5,7. These solutions are neither stationary nor axisymmetric, allowing them to evade the rigidity theorem; instead the space-time plus matter fields are invariant under only a single helical Killing vector. These hairy black holes are argued to be stable within their class of scalar field perturbations but are ultimately unstable to higher order perturbative modes

Black Holes and Boson Stars with One Killing Field in Arbitrary Odd Dimensions

We extend the recent D=5 results of Dias, Horowitz and Santos by finding asymptotically AdS rotating black hole and boson star solutions with scalar hair in arbitrary odd spacetime dimension. Both the black holes and the boson stars are invariant under a single Killing vector field which co-rotates with the scalar field and, in the black hole case, is tangent to the generator of the horizon. Furthermore, we explicitly construct boson star and small black hole ($r_+ \ll \ell$) solutions perturbatively assuming a small amplitude for the scalar field, resulting in solutions valid for low energies and angular momenta. We find that just as in D=5, the angular momentum is primarily carried by the scalar field in $D>5$, whereas unlike D=5 the energy is also primarily carried by the scalar field in $D>5$; the thermodynamics in D=5 are governed by both the black hole and scalar field whereas in $D>5$ they are governed primarily by the scalar field alone. We focus on cataloguing these solutions for the spacetime dimensions of interest in string theory, namely $D=5,7,9,11$.Comment: 28 pages, 1 table, 2 Appendices. v2: minor typos corrected, references added, small discussion added to section 4. v3: typos corrected, thermodynamic discussion expanded, accepted in PR