This short review surveys mass for two-dimensional asymptotically locally hyperbolic initial data sets. I explain the difficulties in defining mass in spatial dimension two, which are resolved via minimisation using a positive energy theorem, and review how gluing theorems can be used to construct novel initial data sets with controlled mass.

Light circling around an astrophysical black hole can spend a long time skirting its unstably bound photon orbits before escaping to infinity. To a distant observer, this orbiting light would appear as a bright ring encircling the image of the black hole. Though not yet resolved by radio-interferometric observations from the ground, this ``photon ring'' will be the target of future space-based black hole observations. Motivated by this experimental prospect, studies have sought to elucidate the theoretical connections between the photon ring -- an observable, classical effect -- and the putative holographic description of black holes in quantum gravity. General relativity predicts that the detailed structure of the photon ring encodes the high-frequency (eikonal) spectrum of quasinormal modes (QNMs) emitted by a perturbed black hole as it rings down, and also that the photon ring displays an emergent conformal symmetry that acts upon this spectrum. In holography, the classical QNM frequencies are expected to map to Ruelle resonances of the dual quantum theory. In this paper, we explore these connections in a lower-dimensional toy model based on Warped AdS$_3$ black holes that shares many features with the (3+1)-dimensional Kerr background -- including a photon ring at finite radius -- while still providing analytic control of the QNM frequencies.

We review notions of mass of asymptotically locally Anti-de Sitter three-dimensional spacetimes, and apply them to some known solutions. For two-dimensional general relativistic initial data sets the mass is not invariant under asymptotic symmetries, but a unique mass parameter can be obtained either by minimisation, or by a monodromy construction, or both. We give an elementary proof of positivity, and of a Penrose-type inequality, in a natural gauge. We carry-out a gluing construction at infinity to time-symmetric asymptotically locally hyperbolic vacuum initial data sets and derive mass/entropy formulae for the resulting manifolds. Finally, we show that all mass aspect functions can be realised by constant scalar curvature metrics on complete manifolds which are smooth except for at most one conical singularity.

We devise new boundary conditions for the near-horizon geometries of extremal BTZ and Kerr black holes, as well as for the ultra-cold limit of the Kerr-de Sitter black hole. These boundary conditions are obtained as the higher-dimensional uplift of recently proposed boundary conditions in two-dimensional gravity. Their asymptotic symmetries consist in the semi-direct product of a Virasoro and a current algebra, of which we determine the central extensions.

We derive a formula for the energy of asymptotically locally hyperbolic (ALH) manifolds obtained by a gluing at infinity of two ALH manifolds. As an application we show that there exist three-dimensional conformally compact ALH manifolds either without boundary or with toroidal black hole boundary, with connected conformal infinity of higher genus, with constant negative scalar curvature, and with negative mass.

A bstract We develop basic tools and matching conditions to interpolate between asymptotic and near horizon symmetries. We focus on black holes in three dimensions. In particular, we match Brown-Henneaux boundary conditions at infinity, which yields two Virasoro algebras, to Heisenberg boundary conditions at the horizon yielding two û (1) current algebras. Our construction allows to equip BTZ black holes with soft hair excitations at the horizon invisible to the asymptotic observer.

A bstract We study warped flat geometries in three-dimensional topologically massive gravity. They are quotients of global warped flat spacetime, whose isometries are given by the 2-dimensional centrally extended Poincaré algebra. The latter can be obtained as a certain scaling limit of Warped AdS 3 space with a positive cosmological constant. We discuss the causal structure of the resulting spacetimes using projection diagrams. We study their charges and thermodynamics, together with asymptotic Killing vectors preserving a consistent set of boundary conditions including them. The asymptotic symmetry group is given by a Warped CFT algebra, with a vanishing current level. A generalization of the derivation of the Warped CFT Cardy formula applies in this case, reproducing the entropy of the warped flat cosmological spacetimes.

We introduce a new entropy formula for Kerr black holes inspired by recent results for 3-dimensional black holes and cosmologies with soft Heisenberg hair. We show that also Kerr-Taub-NUT black holes obey the same formula.

We present and discuss near horizon boundary conditions for flat space higher-spin gravity in three dimensions. As in related work our boundary conditions ensure regularity of the solutions independently of the charges. The asymptotic symmetry algebra is given by a set of \( \widehat{\mathfrak{u}}(1) \) current algebras. The associated charges generate higher-spin soft hair. We derive the entropy for solutions that are continuously connected to flat space cosmologies and find the same result as in the spin-2 case: the entropy is linear in the spin-2 zero-mode charges and independent from the spin-3 charges. Using twisted Sugawara-like constructions of higher-spin currents we show that our simple result for entropy of higherspin flat space cosmologies coincides precisely with the complicated earlier results expressed in terms of higher-spin zero mode charges.

The holographic principle proposes a solution to one of the most prominent problems of our time – the search for a consistent, quantized theory of gravity. According to this principle a theory of gravity in (d + 1) dimensions is equivalent to a quantum field theory (without gravity) in d dimensions. An important realization of this conjecture is the Anti-de-Sitter/conformal field theory (AdS/CFT) correspondence. However, since this correspondence is a strong-weak correspondence, it is hard to explicitly check the holographic principle by calculating observables on the field theory and the gravitational theory side. Conversely, higher-spin theories lead to weak-weak dualites, which can provide useful insights into aspects of the holographic principle [1–3]. Furthermore, since calculations in three dimensions may be done in the Chern-Simons formulation and are technically less challenging than in higher dimensions, it is often useful to restrict onself to three dimensions to clear up conceptional issues and obtain a better understanding of the holographic principle. In this thesis, we construct a new set of boundary conditions for spin-3 gravity in three-dimensional flat space. This set of boundary conditions is inspired by the recent “Soft Heisenberg hair”-proposal for Einstein gravity in three-dimensional Anti-de-Sitter space [4], which has subsequently been extended to flat space [5] and higher-spin gravity in AdS space [6]. In chapter 2 we discuss the peculiarities of restricting oneself to three dimensions and review the ChernSimons formalism and the canonical analysis. In chapter 3 we discuss boundary conditions for gravity in three-dimensional AdS space and give a review of the Brown-Henneaux boundary conditions [7] and the near horizon boundary conditions proposed in [4]. In chapter 4 we motivate the respective near horizon boundary conditions for spin-3 gravity in threedimensional flat space and compute the canonical boundary charges and the asymptotic symmetry algebra. As in previous, related work [4–6] the boundary conditions ensure regularity of the solutions independently of the charges. The asymptotic symmetry algebra is again given by a set of û(1) current algebras. We find that the vacuum descendants generated by the charges all have the same energy as the vacuum, i.e. they are higher-spin “soft hair” in the sense of Hawking, Perry and Strominger [8]. Furthermore, we derive the entropy for solutions that are continuously connected to flat space cosmologies and find the same result as in the spin-2 case: the entropy is linear in the spin-2 zero-mode charges and independent from the spin-3 charges. Using twisted Sugawara-like constructions of the higher-spin currents we show that our simple result for entropy of higher-spin flat space cosmologies coincides precisely with the complicated earlier results expressed in terms of higher-spin zero-mode charges.