We investigate charged Dirac quasinormal spectra on Reissner–Nordström black holes in a mirror-like cavity. For this purpose, we first derive charged Dirac equations, and $\textit{two}$ sets of Robin boundary conditions following the vanishing energy flux principle. The Dirac spectra are then computed both analytically and numerically. Our results reveal a symmetry hidden in the Dirac spectra between two boundary conditions. Moreover, when the cavity is placed close to the event horizon $$r_+$$ , we identify that, in the neutral background the Dirac spectra asymptote to $$-(3/8+N/2)i$$ [ $$-(1/8+N/2)i$$ ] for the first [second] boundary condition; while in the charged background the real part of charged Dirac spectra asymptote to $$qQ/r_+$$ for both boundary conditions; where $\textit{N}$ is the overtone number, $\textit{q}$ and $\textit{Q}$ are charges for the field and for the background. In particular, we uncover a striking anomalous decay pattern, i.e. the excited modes decay $\textit{slower}$ than the fundamental mode, when the charge coupling $\textit{qQ}$ is large. Our results further illustrate the robustness of vanishing energy flux principle, which are applicable not only to anti-de Sitter black holes but also to black holes in a cavity.

China

Jia Liu

Mengjie Wang

Zishuo Wang

Haoyu Liu

Jinshan An

Jiliang Jing

Charged Dirac perturbations on Reissner–Nordström black holes in a cavity: quasinormal modes with Robin boundary conditions

We investigate the scalar perturbation around a rotating Einstein-bumblebee BTZ black hole under Robin boundary conditions. It is shown that the relationship curves between real and imaginary parts of scalar quasinormal modes in the complex plane depend on the black hole spin parameter <italic>j</italic> and the Lorentz symmetry breaking parameter <italic>s</italic>. The shapes of these curves are similar for different <italic>s</italic>, but heavily depend on the spin parameter <italic>j</italic>. With the increase of <italic>j</italic>, the symmetry of these curves with respect to the imaginary axis in the complex plane is gradually broken for different <italic>s</italic>. We also discuss the energy and angular momentum fluxes across the black hole horizon under Robin boundary conditions, and probe the changes of the threshold parameter related to the superradiance in Robin boundary conditions with the symmetry breaking parameter and the spin parameter. The combination of the Lorentz symmetry violation and the Robin boundary condition provides richer physics in the scalar perturbation around a rotating Einstein-bumblebee BTZ black hole.

Chengjia Chen

Fengkai Ge

Qiyuan Pan

Scalar perturbation around rotating Einstein-bumblebee BTZ black holes under Robin boundary conditions: Quasinormal modes and superradiance

In this paper, we delve into the gravitational path integral of Gauss-Bonnet gravity in four spacetime dimensions, in the mini-superspace approximation. Our primary focus lies in investigating the transition amplitude between distinct boundary configurations. Of particular interest is the case of Robin boundary conditions, known to lead to a stable Universe in Einstein-Hilbert gravity, alongside Neumann boundary conditions. To ensure a consistent variational problem, we supplement the bulk action with suitable surface terms. This study leads us to compute the necessary surface terms required for Gauss-Bonnet gravity with the Robin boundary condition, which wasn’t known earlier. Thereafter, we perform an exact computation of the transition amplitude. Through ħ → 0 analysis, we discover that the Gauss-Bonnet gravity inherently favors the initial configuration, aligning with the Hartle-Hawking no-boundary proposal. Remarkably, as the Universe expands, it undergoes a transition from the Euclidean (imaginary time) to the Lorentzian signature (real time). To further reinforce our findings, we employ a saddle point analysis utilizing the Picard-Lefschetz methods. The saddle point analysis allows us to find the initial configurations which lead to Hartle-Hawking no-boundary Universe that agrees with the exact computations. Our study concludes that for positive Gauss-Bonnet coupling, initial configurations corresponding to the Hartle-Hawking no-boundary Universe gives dominant contribution in the gravitational path-integral.

India

Manishankar Ailiga

Shubhashis Mallik

Gaurav Narain

Lorentzian Robin Universe

We study hairy black holes in Einstein-Maxwell-complex scalar theory in four-dimensional asymptotically global anti-de Sitter (AdS) spacetime when the Robin boundary conditions are imposed on the scalar field. This setup is dual to the double trace deformation of strongly interacting field theory on R × S 2 by charged scalar operators. We identify the instability of the Reissner-Nordström-AdS (RNAdS) black holes under the Robin boundary conditions and construct backreacted geometries branching at the onset of the instability. Also considering associated horizonless geometries called boson stars, we obtain phase diagrams with fairly rich structure in the grand canonical ensemble depending on the boundary condition parameter or the deformation parameter, where phase transition occurs between thermal AdS, RNAdS, charged boson stars, and hairy black holes.

Japan

Tomohiro Harada

Takaaki Ishii

Denmark

Takuya Katagiri

Norihiro Tanahashi

Hairy black holes in AdS with Robin boundary conditions

We perform a systematic study of the Maxwell quasinormal spectrum in a mirror-like cavity following the generic Robin type vanishing energy flux principle, by starting with the Schwarzschild black holes in this paper. It is shown that, for black holes in a cavity, the vanishing energy flux principle leads to two different sets of boundary conditions. By solving the Maxwell equations with these two boundary conditions both analytically and numerically, we observe two distinct sets of modes. This indicates that the vanishing energy flux principle may be applied not only to asymptotically anti-de Sitter (AdS) black holes but also to black holes in a cavity. In the analytic calculations, the imaginary part of the Maxwell quasinormal modes are derived analytically for both boundary conditions, which match well with the numeric results. While in the numeric calculations, we complete a thorough study on the two sets of the Maxwell spectrum by varying the mirror radius  $$r_m$$  <math> <msub> <mi>r</mi> <mi>m</mi> </msub> </math>  , the angular momentum quantum number  $$\ell $$  <math> <mi>ℓ</mi> </math>  , and the overtone number N. In particular, we proclaim that the Maxwell spectrum may bifurcate for both modes when the mirror is placed around the black hole event horizon, which is analogous to the spectrum bifurcation effects found for the Maxwell fields on asymptotically AdS black holes. This observation provides another example to exhibit the similarity between black holes in a cavity and the AdS black holes.

Yunhe Lei

Maxwell perturbations in a cavity with Robin boundary conditions: two branches of modes with spectrum bifurcation on Schwarzschild black holes

Within quantum field theory on a global monopole spacetime, we study thermal effects on a naked singularity and their relation to boundary conditions. We first obtain the two-point functions for the ground state and for thermal states of a massive, arbitrarily coupled, free scalar field compatible with Robin boundary conditions at the singularity. We then probe these states using a static Unruh-Dewitt particle detector. The transition rate is analyzed for the particular cases of massless minimally or conformally coupled fields at finite temperature. To interpret the detector’s behavior, we compute the thermal contribution to the ground-state fluctuations and to the energy density. We verify that the behavior of the transition rate, the fluctuations, and the energy density are closely intertwined. In addition, we find that these renormalized quantities remain finite at the singularity for, and only for, the Dirichlet boundary condition.

Italy

Lissa de Souza Campos

Brazil

João Paulo M. Pitelli

Thermal effects on a global monopole with Robin boundary conditions

We compute the transition rate of an Unruh-DeWitt detector coupled both to a ground state and to a KMS state of a massless, conformally coupled scalar field on a static BTZ black hole with Robin boundary conditions. We observe that, although the anti-Hawking effect is manifest for the ground state, this is not the case for the KMS state. In addition, we show that our analysis applies with minor modifications also to the anti-Unruh effect on Rindler-AdS$_{3}$ spacetime.

Claudio Dappiaggi

The anti-Hawking effect on a BTZ black hole with Robin boundary conditions

We generalize our previous studies on the Maxwell quasinormal modes around Schwarzschild-anti-de-Sitter black holes with Robin type vanishing energy flux boundary conditions, by adding a global monopole on the background. We first formulate the Maxwell equations both in the Regge–Wheeler–Zerilli and in the Teukolsky formalisms and derive, based on the vanishing energy flux principle, two boundary conditions in each formalism. The Maxwell equations are then solved analytically in pure anti-de Sitter spacetimes with a global monopole, and two different normal modes are obtained due to the existence of the monopole parameter. In the small black hole and low frequency approximations, the Maxwell quasinormal modes are solved perturbatively on top of normal modes by using an asymptotic matching method, while beyond the aforementioned approximation, the Maxwell quasinormal modes are obtained numerically. We analyze the Maxwell quasinormal spectrum by varying the angular momentum quantum number  $$\ell $$  <math> <mi>ℓ</mi> </math>  , the overtone number N, and in particular, the monopole parameter  $$8\pi \eta ^2$$  <math> <mrow> <mn>8</mn> <mi>π</mi> <msup> <mi>η</mi> <mn>2</mn> </msup> </mrow> </math>  . We show explicitly, through calculating quasinormal frequencies with both boundary conditions, that the global monopole produces the repulsive force.

Zhou Chen

Maxwell quasinormal modes on a global monopole Schwarzschild-anti-de Sitter black hole with Robin boundary conditions

We compute the renormalized vacuum polarization for a massless, conformally coupled scalar field on asymptotically anti–de Sitter black hole backgrounds. Mixed (Robin) boundary conditions are applied on the spacetime boundary. We consider black holes with nonspherical event horizon topology as well as spherical event horizons. The quantum scalar field is in the Hartle-Hawking state, and we employ Euclidean methods to calculate the renormalized expectation values. Far from the black hole, we find that the vacuum polarization approaches a finite limit, which is the same for all boundary conditions except Dirichlet boundary conditions.

United Kingdom

Thomas Morley

Elizabeth Winstanley

Ireland

Peter Taylor

Vacuum polarization on topological black holes with Robin boundary conditions

Realizing the no-boundary proposal of Hartle and Hawking as a consistent gravitational path integral has been a long-standing puzzle. In particular, it was demonstrated by Feldbrugge, Lehners, and Turok that the sum over all universes starting from a zero size results in an unstable saddle point geometry. Here we show that, in the context of gravity with a positive cosmological constant, path integrals with a specific family of Robin boundary conditions overcome this problem. These path integrals are manifestly convergent and are approximated by stable Hartle-Hawking saddle point geometries. The price to pay is that the off-shell geometries do not start at a zero size. The Robin boundary conditions may be interpreted as an initial state with Euclidean momentum, with the quantum uncertainty shared between the initial size and momentum.

Germany

Alice Di Tucci

Jean-Luc Lehners

No-Boundary Proposal as a Path Integral with Robin Boundary Conditions