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A violation of the usual conservation law is proposed in Rastall theory. We consider two generalizations of Rastall theory: Brans-Dicke-Rastall theory and the newly established Rastall-scalar-tensor theory, the latter being a further generalization of the former. The field equations in these two generalized theories are studied across different parameter space, and the polarization modes of gravitational waves, as a key focus, are subsequently investigated. The results show that the polarization modes of gravitational waves in Brans-Dicke-Rastall theory are the same as those in Brans-Dicke theory; specifically, both theories exhibit the plus, cross, and breathing modes. However, in Rastall-scalar-tensor theory, the polarization modes of gravitational waves depend on the parameter space of the theory. Particularly, over a broad range of the parameter space, regardless of some special values of the parameters, it allows only two tensor modes, just as in general relativity, without introducing any additional degrees of freedom. This indicates that Rastall theory offers a novel approach to constructing modified gravity theories that propagate only two tensor degrees of freedom. In the remaining regions of the parameter space, there is also one scalar mode in addition to the two tensor modes. The scalar mode can be either a mixture of the breathing and longitudinal modes or just a pure breathing mode, depending on the parameter space. These results will play a crucial role in constraining the theoretical parameters through future gravitational wave detection projects, such as LISA, Taiji, and TianQin.

The Isaacson picture provides a general method for describing the effects of gravitational waves, established on two sets of basic equations. The first set of equations describes the propagation of gravitational waves within a given background, while the second set elucidates the interaction between the effective energy-momentum tensor of gravitational waves and the background metric. These two sets of equations are typically derived by expanding the field equations to second-order perturbations. In addition to the method of perturbing field equations, there is an alternative method: obtaining the two sets of basic equations by expanding the action to the second-order of perturbations. In this paper, we elaborate on this method, establishing its foundations more rigorously. Especially, the second-order perturbation action, with the Minkowski metric as the background, contains all necessary information to construct the Isaacson picture far from the source. We outline the derivation of the two sets of basic equations to describe the gravitational wave effects in the Isaacson picture, based on the second-order perturbation action. This will enable us to analyze the following effects: polarizations of gravitational waves, the wave speed for each polarization mode, the effective energy-momentum tensor of gravitational waves, and the nonlinear memory effect of gravitational waves. We illustrate this method by applying it to the most general second-order vector-tensor theory including parity-breaking terms. We further analyze the polarization modes of gravitational waves in this theory. We highlight that this method facilitates model-independent research on various gravitational wave effects in general modified gravity theories. These effects are anticipated to be identified by the upcoming generation of gravitational wave detectors, aimed at testing potential modifications to gravity theory., Comment: 85 pages, 1 figure

In this work, we investigate the universal classifications of black hole states by considering them as topological defects within the thermodynamic parameter space. Through the asymptotic behaviors of the constructed vector, our results indicate the existence of four distinct topological classifications, denoted as $W^{1-}$, $W^{0+}$, $W^{0-}$, and $W^{1+}$. Within these classifications, the innermost small black hole states are characterized as unstable, stable, unstable, and stable, respectively, while the outermost large ones exhibit an unstable, unstable, stable, and stable behavior. These classifications also display contrasting thermodynamic properties in both low and high Hawking temperature limits. Furthermore, we establish a systematic ordering of the local thermodynamically stable and unstable black hole states as the horizon radius increases for a specific topological classification. These results reveal the universal topological classifications governing black hole thermodynamics, providing valuable insights into the fundamental nature of quantum gravity., Comment: 6 pages, 2 figures, accepted for publication as a Letter in PRD

In recent years, the holographic duality between $T\bar{T}$-deformed conformal field theory (CFT) and Anti-de Sitter (AdS) space with finite radial truncation has received significant attention. The study of $T\bar{T}$ deformation within the framework of de Sitter (dS)/CFT duality has also progressed. This paper shows that the trace flow equation in dS spacetime can be analytically extended from its AdS counterpart through double Wick rotations. Additionally, we generalize the replica sphere method in both AdS and dS holography to derive a general expression for the entanglement entropy of arbitrary single spatial intervals. For both finite size and finite temperature systems, we obtain the entanglement entropy corrections due to $T\bar{T}$ deformation. Finally, using strong subadditivity and boosted strong subadditivity, we show that the dual field theory on the timelike boundary is non-local in dS holography.

The path integral approach to a quantum theory of gravity is widely regarded as an indispensable strategy. However, determining what additional elements, beyond black hole or AdS spacetime, should be incorporated into the path integral remains crucial yet perplexing. We argue that the spacetime with a conical singularity in its Euclidean counterpart should be the most important ingredient to append to the path integral. Therefore, physical quantities should be ensemble-averaged over all geometries since they are described by the same Lorentzian metric. When the ensemble average is introduced, the Hawking-Page transition for the Schwarzschild-AdS black hole and the small-large black hole transition for the Reissner-Nordstrom-AdS black hole naturally arise as semi-classical approximations, when the size of the black hole system is much larger than the Planck length. Away from the semi-classical limit, the system is a superposition of different geometries, and the averaged quantities would deviate from the black hole thermodynamics. Expanding around the classical saddles, the subleading order of Newton's constant contributions can be derived, which are both half of the Hawking temperature for the Schwarzschild and Reissner-Nordstrom black holes. The result may imply a universal structure. The subsubleading terms and more intriguing physics that diverge from black hole thermodynamics are revealed. The ensemble-averaged theory provides a new way of studying subleading effects and extending the traditional AdS/CFT correspondence., Comment: 7 pages, 4 figures

Motivated by the recent progress on the holographic dual of the extended thermodynamics for black holes in anti-de-Sitter (AdS) space, we investiggate the hologrphic thermodynamics for the five-dimensional neutral Gauss-Bonnet AdS black hole in the context of the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. Through the extended bulk thermodynamics for the five-dimensional Gauss-Bonnet AdS black hole, we derive the first law of the CFT thermodynamics which is obtained by directly translating the arbitrary conformal factors in the dual CFT. In addition to the newly defined chemical potential $\mu$ conjugating to the central charge $C$, we obtain other pairs of thermodynamics for the CFT, such as the temperature $\tilde{T}$ and the entropy $S$, the Gauss-Bonnet coupling constant $\tilde{\alpha}$ and its conjugate variable $ \tilde{\mathcal{A}}$, the pressure $\mathcal{P}$ and its conjugate volume $\mathcal{V}$. In the fixed $C$, $\mathcal{V}$ and $\tilde{\alpha}$ canonical ensemble, we obtain the canonical description of the CFT thermodynamics and observe the standard swallowtail behavior in the Helmholtz free energy vs the temperature plot. The self-intersection point of the Helmholtz free energy indicates the phase transition between the high and low entropy states of the CFT. By using Maxwell's equal area law, we get the critical point and coexistence curve for the high and low entropy phases of the CFT. Besides, we get the critical exponents for the CFT, and find that the critical point and critical exponents associated with the $\tilde{T}-S$ criticality of the CFT are the same as those of the five-dimensional Gauss-Bonnet AdS black hole., Comment: 22 pages,5 figures

In this paper, we present charged spherically symmetric black hole solutions and slowly rotating charged solutions in bumblebee gravity with and without a cosmological constant. The static spherically symmetric solutions describe the Reissner-Nordstr\"om-like black hole and ReissnerNordstr\"om-(anti) de Sitter-like black hole, while the stationary and axially symmetric soltuions describe Kerr-Newman-like black hole and Kerr-Newman-(anti) de Sitter-like black hole. We utilize the Hamilton-Jacobi formalism to study the shadows of the black holes. Additionally, we investigate the effect of the electric charge and Lorentz-violating parameters on the radius of the shadow reference circle and the distortion parameter. We find that the radius of the reference circle decreases with the Lorentz-violating parameter and charge parameter, while the distortion parameter increases with the Lorentz-violating parameter and the charge parameter., Comment: 25 pages, 6 figures

Extreme mass-ratio inspirals are crucial sources for future space-based gravitational wave detections. Gravitational waveforms emitted by extreme mass-ratio inspirals are closely related to the orbital dynamics of small celestial objects, which vary with the underlying spacetime geometry. Despite the tremendous success of general relativity, there are unsolved issues such as singularities in both black holes and cosmology. Loop quantum gravity, a theory addressing these singularity problems, offers a framework for regular black holes. In this paper, we focus on periodic orbits of a small celestial object around a supermassive quantum-corrected black hole in loop quantum gravity and compute the corresponding gravitational waveforms. We view the small celestial object as a massive test particle and obtain its four-velocity and effective potential. Our results indicate that the quantum parameter $\hat{\alpha}$ influences the shape of the effective potential. We explore the effects of quantum corrections on marginally bound orbits, innermost stable circular orbits, and other periodic orbits. Using the numerical kludge scheme, we further explore the gravitational waveforms of the small celestial object along different periodic orbits. The waveforms exhibit distinct zoom and whirl phases in a complete orbital period, closely tied to the quantum parameter $\hat{\alpha}$. We also perform a spectral analysis of the gravitational waves from these periodic orbits and assess their detectability. With the steady progress of space-based gravitational wave detection programs, our findings will contribute to utilizing extreme mass-ratio inspirals to test and understand the properties of quantum-corrected black holes., Comment: 16 pages, 12 figures, and 2 tables

Gravitational echoes can be used to probe the structure of spacetime. In this paper, we investigate the gravitational echoes in different braneworld models in five-dimensional spacetime. We derive the gravitational perturbation equations of these models, and obtain the time-dependent evolution equations of the extra-dimensional and radial components. Using a Gaussian wave packet as initial data, we study the time evolution of the gravitational perturbation. By monitoring the evolution of the Gaussian wave packet, the gravitational echoes are observed whether the wave packet is generated from inside or outside the braneworld. Furthermore, we can restrict the parameters of the braneworld by calculating the strength of the first gravitational echo and using the current gravitational wave data., Comment: 29 pages, 19 figures, 1 table

As the fingerprints of black holes, quasinormal modes are closely associated with many properties of black holes. Especially, the ringdown phase of gravitational waveforms from the merger of compact binary components can be described by quasinormal modes. Serving as a model-independent approach, the framework of parameterized quasinormal frequencies offers a universal method for investigating quasinormal modes of diverse black holes. In this work, we first obtain the Schr\"{o}dinger-like master equation of the axial gravitational perturbation of a holonomy-corrected black hole. We calculate the corresponding quasinormal frequencies using the Wentzel-Kramers-Brillouin approximation and asymptotic iteration methods. We investigate the numerical evolution of an initial wave packet on the background spacetime. Then, we deduce the parameterized expression of the quasinormal frequencies and find that $r_0 \leq 10^{-2}$ is a necessary condition for the parameterized approximation to be valid. We also study the impact of the quantum parameter $r_0$ on the greybody factor and Hawking radiation. With more ringdown signals of gravitational waves detected in the future, our research will contribute to the study of the quantum properties of black holes., Comment: 13 pages,6 figures,and 6 tables

In this work, we investigate the quasinormal modes of the Poincar\'e thick brane with a finite extra dimension. Unlike the case with an infinite extra dimension, the gravitational effective potential exhibits three distinct shapes within different ranges of the parameter $n$ in the warp factor: harmonic oscillator potential, P\"oschl-Teller potential, and volcano-like potential. We then study various types of perturbations in this system. Utilizing a combination of analytical, semi-analytical, and numerical methods, we obtain the quasinormal modes of the perturbed fields. Our findings reveal a set of discrete quasinormal modes for the thick brane, similar to those of black holes. Interestingly, when $n=1$, the quasinormal modes exhibit purely imaginary behavior. This study may provide a new way to detect the existence of extra dimensions.

We study the polarization modes of gravitational waves in general Einstein-vector theory with an arbitrary constant background vector field under a Minkowski background. We compare these polarization modes with those of other vector-tensor theories and constrain the parameter spaces based on the gravitational-wave event GW170817 with its electromagnetic counterpart GRB170817A and observations of pulsar timing arrays. The presence of the background vector field leads to the anisotropy of space and a rich variety of gravitational waves contents. Our results reveal that the polarization modes of gravitational waves depend on the parameter spaces. There are at least two and at most five independent polarization modes in one parameter space. In different parameter spaces, some mixture modes are allowed, including tensor-vector, tensor-scalar, tensor-vector-scalar, vector-scalar, and scalar-scalar mixture modes, as well as five independent modes (excluding $P_l$): $P_+$, $P_{\times}$, $P_x$, $P_y$, and $P_b$. In all regions of the parameter spaces, there are always two tensor modes, which can be either independent of or mixed with other modes. The independent $P_b$ mode consistently exhibits the same speed as light. If the speed of tensorial gravitational waves strictly equals that of light, only the $P_+$, $P_{\times}$, and $P_b$ modes are permitted. Furthermore, through comparisons of some vector-tensor theories and based on the observations of gravitational waves, the $P_b$ mode is expected to be allowed.

Recently, Cui et al. [Nature \textbf{621}, 711 (2023)] reported that the jet nozzle of M87* exhibits a precession with a period of approximately 11 years. This finding strongly suggests that M87* is a spinning black hole with a tilted accretion disk. In this paper, our aim is to utilize these observations to constrain the parameters of the black hole. Firstly, we investigate the properties of the tilted circular orbits and the innermost stable circular orbits. The corresponding angular momentum, energy, and Carter constant for both prograde and retrograde orbits are calculated. We find that, compared to equatorial circular orbits, these quantities exhibit significant differences for fixed tilt angles. Moreover, the Carter constant takes positive values for nonvanishing tilt angles. Notably, the presence of misalignment of the orbit angular momentum and black hole spin leads to a precession effect in these tilted circular orbits. We then make use of these circular orbits to model the warp radius of the tilted accretion disk, which allows us to determine the corresponding precession period through the motion of massive particles. Further comparing with the observation of M87*, the relationship between the black hole spin and the warp radius is given, through which if one of them is tested, the other one will be effectively determined. Additionally, our study establishes an upper bound on the warp radius of the accretion disk. These findings demonstrate that the precession of the jet nozzle offers a promising approach for testing the physics of strong gravitational regions near a supermassive black holes., Comment: 11 pages, 8 figures. Typos are corrected

We study the lensing phenomena of the strong gravity regime of the five-dimensional charged, equally rotating black holes with a cosmological constant, familiarly known as the Cveti\v c-L\"u-Pope black holes. These black holes are characterized by three observable parameters, the mass $M$, the charge $Q$ and the angular momentum $J$, in addition to the cosmological constant. We investigate the strong gravitational lensing observables, mainly the photon sphere radius, the minimum impact parameter, the deflection angle, the angular size, and the magnification of the relativistic images. We model the $M87$ and $SgrA^*$ for these observables. We also focus on the relativistic time delay effect in the strong field regime of gravity and the impact of the observable on it. The analytical expressions for the observables of the relativistic images with vanishing angular momentum ($j=0$) are discussed in some detail., Comment: 29 pages, 16 figures

Recently, a static spherically symmetric black hole solution was found in gravity nonminimally coupled a background Kalb-Ramond field. The Lorentz symmetry is spontaneously broken when the Kalb-Ramond field has a nonvanishing vacuum expectation value. In this work, we focus on the quasinormal modes and greybody factor of this black hole. The master equations for the perturbed scalar field, electromagnetic field, and gravitational field can be written into a uniform form. We use three methods to solve the quasinormal frequencies in the frequency domain. The results agree well with each other. The time evolution of a Gaussian wave packet is studied. The quasinormal frequencies fitted from the time evolution data agree well with that of frequency domain. The greybody factor is calculated by Wentzel-Kramers-Brillouin (WKB) method. The effect of the Lorentz-violating parameter on the quasinormal modes and greybody factor are also studied., Comment: 10 pages, 4 figures, and 3 tables

In this paper, we establish a unified parameterized framework for analyzing the polarization modes of gravitational waves in the general metric theory (gravity is only described by the metric) and the general scalar-tensor theory (gravity is described by the metric and an additional scalar field). Specifically, we study the polarization modes of gravitational waves in the most general metric theory and general scalar-tensor theory that satisfy the following conditions: (1) Spacetime is four-dimensional; (2) The theory satisfies the principle of least action; (3) The theory is generally covariant; (4) The action describing a free particle is $\int ds$. We find that the polarization modes of gravitational waves in the theory satisfying the above conditions depends on the selection of parameters in the framework, and the theory allows for up to all six polarization modes. Once we have established our framework, the analysis of the polarization modes of gravitational waves in specific theories will depend on determining the corresponding parameters within our framework. In our analysis, we also find that the polarization modes of gravitational waves in the general metric theory and the general scalar-tensor theory that satisfy the conditions also have some interesting universal properties., Comment: v3: 38 pages, 1 figure, 2 tables, typos corrected

Boson stars are potential sources of gravitational radiation. To improve our comprehension of these exotic objects, we present the study of geodesics within the space-times of stable, collapsing, and migrating boson stars. We focus on timelike geodesics that are initially circular or reciprocating. We verify that orbits initially bound within a stable boson stars persist in their bound states. For a collapsing boson star, we show that orbits initially bound and reciprocating finally either become unbound or plunge into the newly formed black hole, depending on their initial maximal radii. For initially circular geodesics, orbits with small radii plunge into the black hole; orbits with intermediate radii become unbound; and large radii orbits linger around the black hole with nonvanishing eccentricities. For the migrating case, a black hole does not form. In this case, the reciprocating orbits span a wider radial range. For initially circular geodesics, orbits with small radii become unbound, and orbits with large radii remain bound with nonvanishing eccentricities. This geodesic study provides a novel approach to investigate the gravitational collapse and migration of boson stars., Comment: 6 pages, 5 figures, comments are welcome

We study static kink solutions in a generalized two-dimensional dilaton gravity model, where the kinetic term of the dilaton is generalized to be an arbitrary function of the canonical one $\mathcal X= -\frac12 (\nabla \varphi)^2$, say $\mathcal F(\mathcal X)$, and the kink is generated by a canonical scalar matter field $\phi$. It is found that for arbitrary $\mathcal F(\mathcal X)$, the background field equations have a simple first-order formalism, and the linear perturbation equation can always be written as a Schr\"odinger-like equation with factorizable Hamiltonian operator. After choosing appropriate $\mathcal F(\mathcal X)$ and superpotential, we obtain a sine-Gordon type kink solution with pure AdS$_2$ metric. The linear perturbation issue of this solution becomes an exactly solvable conformal quantum mechanics problem, if one of the model parameter takes a critical value., Comment: 7 pages, 2 figures

In recent years, there has been significant interest in the field of extended black hole thermodynamics, where the cosmological constant and/or other coupling parameters are treated as thermodynamic variables. Drawing inspiration from the Iyer-Wald formalism, which reveals the intrinsic and universal structure of conventional black hole thermodynamics, we illustrate that a proper extension of this formalism also unveils the underlying theoretical structure of extended black hole thermodynamics. As a remarkable consequence, for any gravitational theory described by a diffeomorphism invariant action, it is always possible to construct a consistent extended thermodynamics using this extended formalism., Comment: 8 pages, no figures

Studying the system state of coexistence regions has important applications in revealing detailed microscopic interactions between different phases. However, the thermodynamic nature in the coexistence black hole regions has been neglected in previous studies. In this work, we introduce two new ratio parameters to investigate these coexistence states. The first parameter measures the ratio of the horizon radii of the statured coexistence small and large black holes. Of particular interest, we demonstrate that it can serve as an order parameter to characterize the first-order phase transition. Furthermore, combining with the ratio of the ratio of the small black hole molecule number to the total molecule number, each black hole state in the coexistence region is uniquely determined via these two introduced parameters bounded between 0 and 1. These results are quite significant in analytical study of black hole phase transition and uncover the thermodynamical microscopic nature of black hole in the coexistence regions., Comment: 5 pages, 5 figures

The Aschenbach effect, the increasing behavior of the angular velocity of a timelike circular orbit with its radius coordinate, is found to extensively exist in rapidly spinning black holes to a zero-angular-momentum observer. It also has potential observation in the high-frequency quasi-periodic oscillations of X-ray flux. However, observing such effect remains to be a challenge in static and spherically symmetric black hole backgrounds. In this paper, we mainly focus on such issue. Starting with the geodesics, we analytically study the underlying properties of the timelike circular orbits, and show the conditions under which the Aschenbach effect survives. We then apply it to three characteristic black holes exhibiting different features. The results state that this effect is absent for both the Schwarzschild and Reissner-Nordstr\"{o}m black holes. While, for the dyonic black hole in quasi-topological electromagnetics, there indeed exists the Aschenbach effect. This provides a first example that such effect exists in a non-spinning black hole background. Moreover, it also uncovers an intriguing property for understanding the black holes in nonlinear electrodynamics., Comment: 9 pages, 6 figures

Double holography offers a profound understanding of the island formula by describing a gravitational system on AdS$_d$ coupled to a conformal field theory on $\mathbb{R}^{1,d-1}$, dual to an AdS$_{d+1}$ spacetime with an end-of-the-world (EOW) brane. In this work, we extend the proposal in [A. Almheiri et al. JHEP 03 (2020) 149] by considering that the dual bulk spacetime has two EOW branes: one with a gravitational system and the other with a thermal bath. We demonstrate an equivalence between this proposal and the wedge holographic theory. We examine it in both Anti-de Sitter gravity and de Sitter gravity by calculating the entanglement entropy of the Hawking radiation. Finally, we employ the doubly holographic model to verify the formula for the entanglement entropy in a subregion within conformally flat spacetime.

Recently, the complexity equals any gravitational observable conjecture has been proposed in [Phys. Rev. Lett. 128, 081602 (2022)], which is an extension of the complexity equals volume proposal. These gravitational observables are referred to as generalized volumes. In this paper, we investigate the generalized volume-complexity for black holes with one or two horizons respectively. We verify that the turning time is universal and independent of the Cauchy horizon. Not only does this phase transition occur once, but it may also occur two or more times depending on the number and height of the effective potential peaks. On the other hand, we confirm that the generalized volume-complexity can be divided based on the shape of the effective potential. We then discuss the non-smooth transition from the Reissner-Nordstr\"om-AdS black hole to the Schwarzschild-AdS black hole., Comment: 29 pages, 22 figures, updated version

Stimulated by the recent researches of black hole thermodynamics for black hole with Newman-Unti-Tamburino (NUT) parameter, we investigate the thermodynamics and weak cosmic censorship conjecture for a Kerr-Newman Taub-NUT black hole. By defining the electric charge as a Komar integral over the event horizon, we construct a consistent first law of black hole thermodynamics for a Kerr-Newman Taub-NUT black hole through Euclidean action. Having the first law of black hole thermodynamics, we investigate the weak cosmic censorship conjecture for the black hole with a charged test particle and a complex scalar field. We find that an extremal black hole cannot be destroyed by a charged test particle and a complex scalar field. For a near-extremal black hole with small NUT parameter, it can be destroyed by a charged test particle but cannot be destroyed by a complex scalar field., Comment: 12 pages,3 figures,published version

In this paper, we study polarization modes of gravitational waves in generalized Proca theory in the homogeneous and isotropic Minkowski background. The results show that the polarizations of gravitational waves depend on the parameter space of this gravity theory and can be divided into quite rich cases by parameters. In some parameter space, it only allows two tensor modes, i.e., the $+$ and $\times$ modes. In some parameter space, besides tensor modes, it also allows one scalar mode, or two vector (vector-$x$ and vector-$y$) modes, or both one scalar mode and two vector modes. The scalar mode is a mixture mode of a breathing mode and a longitudinal mode, or just a pure breathing mode. Interestingly, it is found that the amplitude of the vector modes is related to the speed of the tensor modes. This allows us to give the upper bound of the amplitude of the vector modes by detecting the speed of the tensor modes. Specifically, if the speed of tensor modes is strictly equal to the speed of light, then the amplitude of vector modes is zero., Comment: v2: 28 pages, 1 figure, 2 tables, improved version

In this work, we investigate the gravitational quasinormal modes (QNMs) and the gravitational resonances of a thick brane model. We use the asymptotic iteration and shooting methods to obtain the quasinormal frequencies (QNFs) of the brane. On the other hand, we investigate the resonances and their evolution numerically. The results show that the oscillations of the resonances equal (up to numerical error) to the real parts of the QNFs, while the damping rates of the resonances equal to the imaginary parts of the QNFs. The QNMs and resonances, both of them can be regarded as the characteristic modes of the thick brane, are closely related with each other. In addition, the lifetime of these QNMs could be very long, perhaps they might be detected in future accelerator or gravitational wave detector., Comment: 10 pages, 6 figures, improved version

We study the axial gravitational quasinormal modes of a self-dual black hole in loop quantum gravity. Considering the axial perturbation of the background spacetime, we obtain the Schr\"{o}dinger-like master equation. Then we calculate the quasinormal frequencies with the Wentzel-Kramers-Brillouin approximation and the asymptotic iteration method. We also investigate the numerical evolution of an initial wave packet on the self-dual black hole spacetime. We find the quantum correction parameter $P$ positively affects the absolute values of both the real and imaginary parts of quasinormal frequencies. We derive the relation between the parameters of the circular null geodesics and quasinormal frequencies in the eikonal limit for the self-dual black hole, and numerically verify this relation., Comment: 14 pages, 3 figures, 5 tables

The connection between quantum information and quantum gravity has captured the imagination of physicists. Recently, a broad new class of gravitational observables have been proposed to provide new possibilities for holographic complexity \cite{Belin:2021bga} , which is an extension of volume in the Complexity=Volume proposal. In this paper, we investigate generalized volume-complexity for the 4-dimensional Reissner-Nordstr\"{o}m-AdS black hole. These new observables satisfy the characteristic of the thermofield double state, i.e., they grow linearly in time on the late stage. We find that there are multiple extremal hypersurfaces anchored at a certain boundary time. In other words, for the same boundary time, more than one observable (generalized volume-complexity) can exist in the bulk. The size relationship of the observables on the two hypersurfaces changes over time. This will result in the substitution of the maximum extreme hypersurface which is dual to the complexity of the thermofield double state. We call the time when one hypersurface replaces another to become the largest extreme hypersurface the turning time $\tau _{turning}$. That is, a hypersurface dual to the complexity of the thermofield double state defined on the boundary jumps from one branch to another. This discontinuous jump is highly reminiscent of a phase transition, and the turning time denotes the moment at which this phase transition occurs. Our findings propose a discontinuous variation in bulk physics that is dual to the complexity of the thermofield double state defined on the boundary., Comment: 20 pages, 9 figures

In this work, we investigate numerical evolution of massive Kaluza-Klein (KK) modes of a Dirac field on a thick brane. We deduce the Dirac equation in five-dimensional spacetime, and obtain the time-dependent evolution equation and Schr\"odinger-like equation of the extra-dimensional component. We use the Dirac KK resonances as the initial data and study the corresponding dynamics. By monitoring the decay law of the left- and right-chiral KK resonances, we compute the corresponding lifetimes and find that there could exist long-lived KK modes on the brane. Especially, for the lightest KK resonance with a large coupling parameter and a large three momentum, it will have an extremely long lifetime., Comment: 26 pages, 12 figures

Unique features of particle orbits produce novel signatures of gravitational observable phenomena, and are quite useful in testing compact astrophysical objects in general relativity or modified theories of gravity. Here we observe a representative example that a static, spherically symmetric black hole solution with nonlinear electrodynamics admits static points at finite radial distance. Each static point thus produces a static sphere, on which a massive test particle can remain at rest at arbitrary latitudes with respect to an asymptotic static observer. As a result, the well-known static Dyson spheres can be implemented by such orbits. More interestingly, employing a topological argument, we disclose that stable and unstable static spheres (if they exist) always come in pairs in an asymptotically flat spacetime. In contrast to this, the counterpart naked singularity has one more stable static sphere than the unstable one. Our results have potential applications in testing black holes in standard Maxwell and nonlinear electrodynamics, as well as in uncovering the underlying astronomical observation effects in other gravitational theories beyond general relativity., Comment: 7 pages, 4 figures. Published version

This study investigates the localization of spinor fields in braneworld models in six or higher dimensions. We study the relationship between fermions in spacetimes of $2n+2$ and $2n$ dimensions. In conformally flat extra-dimensional spacetime, fermions cannot be localized through minimal coupling with gravity. To achieve the localization of spinor fields, we introduce a tensor coupling term given by $\bar{\Psi} \Gamma^{M}\Gamma^{N}\Gamma^{P}\cdots T_{MNP\cdots} \Psi$, which ensures $SO(n, 1)$ symmetry. For a tensor $T_{MNP\cdots}$ of odd order, the left and right chiralities of high-dimensional spinors are decoupled. We find that a special form of tensor coupling $\bar{\Psi}\Gamma^{M} \partial_M {F(\phi, R, R^{\mu\nu}R_{\mu\nu},\cdots)} \Psi$ may facilitate the localization of the spinor field when $F(\phi) = \phi^n$., Comment: v3: 26 pages, 2 figures

For the stochastic gravitational wave backgrounds (SGWBs) search centred at the milli-Hz band, the galactic foreground produced by white dwarf binaries (WDBs) within the Milky Way contaminates the extra-galactic signal severely. Because of the anisotropic distribution pattern of the WDBs and the motion of the spaceborne gravitational wave interferometer constellation, the time-domain data stream will show an annual modulation. This property is fundamentally different from those of the SGWBs. In this Letter, we propose a new filtering method for the data vector based on the annual modulation phenomenon. We apply the resulted inverse variance filter to the LISA data challenge. The result shows that for the weaker SGWB signal, such as energy density $\Omega_{\rm astro}=1\times10^{-12}$, the filtering method can enhance the posterior distribution peak prominently. For the stronger signal, such as $\Omega_{\rm astro}=3\times10^{-12}$, the method can improve the Bayesian evidence from `substantial' to `strong' against null hypotheses. This method is model-independent and self-contained. It does not ask for other types of information besides the gravitational wave data., Comment: 6 pages, 5 figures

From the quantum point of view, singularity should not exist. Recently, Bah and Heidmann constructed a five-dimensional singularity free topology star/black hole [Phys. Rev. Lett. 126, 151101 (2021)]. By integrating the extra dimension, a four-dimensional static spherical black hole with a magnetic charge and scalar hair can be obtained. In this paper, we study the quasinormal modes (QNMs) of the magnetic field and gravitational field on the background of this four-dimensional charged black hole with scalar hair. The odd parity of the gravitational perturbations couples with the even parity of the magnetic field perturbations. Two coupled second-order derivative equations are obtained. Using the matrix-valued direct integration method, we obtain the fundamental QNM frequencies numerically. The effect of the magnetic charge on the QNMs is studied. The differences of the frequencies of the fundamental QNMs between the charged black hole with scalar hair and the Reissner-Norstr\"{o}m black hole are very small for the angular number $l=2$. However, some new interesting results are found for higher angular number., Comment: 10 pages, 3 figures, 2 tables, some mistakes have been corrected

In this paper, we investigate the possible parameter space of Palatini-Horndeski theory with gravitational waves in a spatially flat Universe. We find that if the theory satisfies the following condition: in any spatially flat cosmological background, the tensor gravitational wave speed is the speed of light $c$, then only $S = \int d^4x \sqrt{-g} \big[K(\phi,X)-G_{3}(\phi,X){\tilde{\Box}}\phi+G_{4}(\phi)\tilde{R}\big]$ is left as the possible action in Palatini-Horndeski theory., Comment: 23 pages, no figure

We examine the multifold complexity and Loschmidt echo for an inverted harmonic oscillator. We give analytic expressions for any number of precursors, implementing multiple backward and forward time evolutions of the quantum state, at the leading order in the perturbation. We prove that complexity is dominated by the longest permutation of the given time combination in an alternating ``zig-zag'' order, the exact same result obtained with holography. We conjecture that the general structure for multifold complexity should hold true universally for generic quantum systems, in the limit of a large number of precursors., Comment: 24 pages, 9 figures

In extra dimensional theories, the four-dimensional field theory is reduced from a fundamental field theory in the bulk spacetime by integrating the extra dimensional part. In this paper we investigate the effective action of a self-interacting scalar field on a brane in the five-dimensional thick braneworld scenario. We consider two typical thick brane solutions and obtain the P\"{o}schl-Teller and harmonic potentials of the Kaluza-Klein (KK) modes, respectively. The analytical mass spectra and wave functions along extra dimension of the KK modes are obtained. Further, the effective coupling constant between different KK particles, cross section, and decay rate for some processes of the KK particles are related to the fundamental coupling in five dimensions and the new physics energy scale. Some interesting properties of these interactions are found with these calculations. The KK particles with higher mode have longer lifetime, and they almost do not interact with ordinary matter on the brane if their mode numbers are large enough. Thus, these KK particles with higher modes might be a candidate of dark matter., Comment: 10 pages, 6 figures

We study domain wall solutions of a real spinor field coupling with gravitation in five dimensions. We find that the nonlinear spinor field supports a class of soliton configurations which could be viewed as a wall embedded in five dimensions. We begin with an illuminating solution of the spinor field in the absence of gravitation. In a further investigation, we exhibit three sets of solutions of the spinor field with nonconstant curvature bulk spacetimes and three sets of solutions corresponding to three constant curvature bulk spacetimes. We demonstrate that some of these solutions in specific conditions have the energy density distributions of domain walls for the spinor field, where the scalar curvature is regular everywhere. Therefore, the configurations of these walls can be interpreted as spinor walls which are interesting spinor field realizations of domain walls. In order to investigate the stability of these spinor configurations, the linear perturbations are considered. The localization of the zero mode of tensor perturbation is also discussed., Comment: 31 pages,5 figures; a comment added in conclusion; matches the published version

In vacuum, the gravitational recoil of the final black hole from the merger of two black holes depends exclusively on the mass ratio and spins of the coalescing black holes, and on the eccentricity of the binary. If matter is present, accretion by the merging black holes may modify significantly their masses and spins, altering both the dynamics of the binary and the gravitational recoil of the remnant black hole. This paper considers such scenario. We investigate the effects on the kick of the final black hole from immersing the binary in a scalar field cloud. We consider two types of configurations: one with non-spinning and unequal-mass black holes, and a second with equal mass and spinning holes. For both types, we investigate how the gravitational recoil of the final black hole changes as we vary the energy density of the scalar field. We find that the accretion of scalar field by the merging black holes could have a profound effect. For the non-spinning, unequal-mass binary black holes, the kicks are in general larger than in the vacuum case, with speeds of approximately 1,200 km/s for binaries with mass ratio 2:1, one order of magnitude larger than in vacuum. For equal mass, binaries with black holes with spins aligned with the orbital angular momentum, kicks larger than in vacuum are also found. For systems with spins in the super-kick configuration, the scalar field triggers a similar dependence of the kicks with the entrance angle at merger as in the vacuum case but in this case depending on the strength of the scalar field., Comment: 13 pages, 9 figures

In this work, employing the generalized off-shell free energy, we treat black hole solutions as defects in the thermodynamic parameter space. The results show that the positive and negative winding numbers corresponding to the defects indicate the local thermodynamical stable and unstable black hole solutions, respectively. The topological number $W$ defined as the sum of the winding numbers for all the black hole branches at an arbitrary given temperature is found to be a universal number independent of the black hole parameters. Moreover, this topological number only depends on the thermodynamic asymptotic behaviors of the black hole temperature at small and large black hole limits. Different black hole systems are characterized by three classes via this topological number. This number could help us well understanding the black hole thermodynamics, and further shed new light on the fundamental nature of quantum gravity., Comment: 7 pages, 4 figures, supplemental material is added. Version published in Physical Review Letters, selected as Editors' Suggestion

A topological approach has been successfully used to study the properties of the light ring and the null circular orbit, in a generic black hole background. However, for the equatorial timelike circular orbit, quite different from the light ring case, its radius is closely dependent of the energy and angular momentum of a test particle. This fact seems to restrict the extension of the topological treatment to the timelike circular orbit. In this paper, we confirm that the angular momentum does not affect the asymptotic behavior of the constructed vector with its zero points denoting the equatorial timelike circular orbits. As a result, a well-behaved topology to characterize the equatorial timelike circular orbits can be constructed. Our study shows that the total topological number of the timelike circular orbits vanishes for a generic black hole, which is dependent of the energy of the particle. Significantly, it reveals that if there the timelike circular orbits exist, they always come in pairs for fixed angular momentum. Meanwhile, the stable and unstable timelike circular orbits have positive or negative winding number. Of particular interest is that the marginally stable circular orbit corresponds to the bifurcation point of the zero point of the constructed vector. Moreover, we also examine the case when the particle energy acts as the control parameter. It is shown that there will be topological phase transition when the value of the particle energy is one. Below this value, the timelike circular orbits always come in pairs for fixed energy. Otherwise, we will have one more unstable timelike circular orbit. We further apply the treatment to the Kerr black hole. All the results given in a generic black hole background are exactly reproduced. These strongly indicate that our topological approach can be generalized to the equatorial timelike circular orbits., Comment: 20 pages, 10 figures

In this work, thick branes in Rastall gravity are investigated. Three types of maximally symmetric thick brane models are constructed and the linear tensor perturbation is analyzed. In the flat brane model, the tensor modes of the perturbation are either unstable or nonlocalizable for a nonvanishing Rastall parameter. In the de Sitter brane model, only the ground state of the tensor mode is localized. In the anti-de Sitter brane model, the number of the bound tensor states is infinity. For both the de Sitter and anti-de Sitter brane models, the condition of stability for the Rastall parameter is obtained., Comment: 13 pages, 7 figures

The direct detection of gravitational waves opens the possibility to test general relativity and its alternatives in the strong field regime. Here we focus on the test of the existence of extra dimensions. The classification of gravitational waves in metric gravity theories according to their polarizations in higher-dimensional space-time and the possible observation of these polarizations in 3-dimensional subspace are discussed in this work. And we show that the difference in the response of gravitational waves in detectors with and without extra dimensions can serve as evidence for the existence of extra dimensions., Comment: 19 pages, 4 figures

In the letter titled ''Traversable Wormholes in General Relativity'' [Phys. Rev. Lett. 128, 091104 (2022)], R. A. Konoplya and A. Zhidenko have constructed an asymmetric wormhole solution, which is not symmetric about the throat and is compounded from smooth gravitational and charged Dirac fields. However, the authors have claimed that a physically relevant condition on the throat is imposed to lead to no gravitational force experienced by a stationary observer at the throat. In this comment, we point out that the above condition is unnecessary., Comment: 1 page, 1 figure, 1 table

Extracting gravitational wave (GW) signals from individual Galactic binaries (GBs) against their self-generated confusion noise is a key data analysis challenge for space-borne detectors operating in the $\approx 0.1$ mHz to $\approx 10$ mHz range. Given the likely prospect that there will be multiple such detectors, namely LISA, Taiji, and Tianqin, with overlapping operational periods in the next decade, it is important to examine the extent to which the joint analysis of their data can benefit GB resolution and parameter estimation. To investigate this, we use realistic simulated LISA and Taiji data containing the set of $30\times 10^6$ GBs used in the first LISA data challenge (Radler), and an iterative source extraction method called GBSIEVER introduced in an earlier work. We find that a coherent network analysis of LISA-Taiji data boosts the number of confirmed sources by $\approx 75\%$ over that from a single detector. The residual after subtracting out the reported sources from the data of any one of the detectors is much closer to the confusion noise expected from an ideal, but infeasible, multisource resolution method that perfectly removes all sources above a given signal-to-noise ratio threshold. While parameter estimation for sources common to both the single detector and network improves broadly in line with the enhanced signal to noise ratio of GW sources in the latter, deviation from the scaling of error variance predicted by Fisher information analysis is observed for a subset of the parameters., Comment: 13 pages, 5 figures