The calculation of conserved charges of black holes is a rich problem, for which many methods are known. Until recently, there was some controversy on the proper definition of conserved charges in asymptotically anti-de Sitter (AdS) spaces in arbitrary dimensions. This paper provides a systematic and explicit Hamiltonian derivation of the energy and the angular momenta of both asymptotically flat and asymptotically AdS spacetimes in any dimension D bigger or equal to 4. This requires as a first step a precise determination of the asymptotic conditions of the metric and of its conjugate momentum. These conditions happen to be achieved in ellipsoidal coordinates adapted to the rotating solutions.The asymptotic symmetry algebra is found to be isomorphic either to the Poincare algebra or to the so(D-1, 2) algebra, as expected. In the asymptotically flat case, the boundary conditions involve a generalization of the parity conditions, introduced by Regge and Teitelboim, which are necessary to make the angular momenta finite. The charges are explicitly computed for Kerr and Kerr-AdS black holes for arbitrary D and they are shown to be in agreement with thermodynamical arguments.
The geodesic motion on anti-de Sitter spacetimes is studied pointing out how the trajectories are determined by the ten independent conserved quantities associated to the specific SO(2,3) isometries of these manifolds. The new result is that there are two conserved SO(3) vectors which play the same role as the Runge-Lenz vector of the Kepler problem, determining the major and minor semiaxes of the ellipsoidal anti-de Sitter geodesics.
The analytic proof of mode stability of the Kerr black hole was provided by Whiting. In his proof, the construction of a conserved quantity for unstable mode was crucial. We extend the method of the analysis for the Kerr-de Sitter geometry. The perturbation equations of massless fields in the Kerr-de Sitter geometry can be transformed into Heun's equations which have four regular singularities. In this paper we investigate differential and integral transformations of solutions of the equations. Using those we construct a conserved quantity for unstable modes in the Kerr-de Sitter geometry, and…
Conformal rescaling of conformal Yano--Killing tensors and relations between Yano and CYK tensors are discussed. Pullback of these objects to a submanifold is used to construct all solutions of a CYK equation in anti-de Sitter and de Sitter spacetimes. Properties of asymptotic conformal Yano--Killing tensors are examined for asymptotic anti-de Sitter spacetimes. Explicit asymptotic forms of them are derived. The results are used to construct asymptotic charges in asymptotic AdS spacetime. Well known examples like Schwarzschild-AdS, Kerr-AdS and NUT-AdS are examined carefully in the constructio…
The geodesics on the $(1+3)$-dimensional de Sitter spacetime are considered studying how their parameters are determined by the conserved quantities in the conformal Euclidean, Friedmann-Lemaître-Robertson-Walker, de Sitter-Painlevé and static local charts with Cartesian space coordinates. Moreover, it is shown that there exist a special static chart in which the geodesics are genuine hyperbolas whose asymptotes are given by the conserved momentum and the associated dual momentum.
We prove that sphaleron black holes in ${\mathfrak {su}}(2)$ Einstein-Yang-Mills-Higgs theory with a Higgs doublet in four-dimensional, asymptotically anti-de Sitter space-time are unstable.
Within the general class of Asymptotically Anti-de Sitter spacetimes that are asymptotic to the A-de-S Schwarzschild metric, we give a simple positive mass theorem based on arguments from causal structure. A general result for all asymptotically A-de-S spacetimes will be given elsewhere.
We determine all spacetime flows which lead to a Hamiltonian dynamics in the space of general relativistic initial data sets with asymptotically Kerr-de Sitter ends. The corresponding Hamiltonians are calculated. Some implications for black-hole thermodynamics are pointed out.
We compute the conserved charges for Kerr anti-de Sitter spacetimes in various dimensions using the conformal and the counterterm prescriptions. We show that the conserved charge corresponding to the global timelike killing vector computed by the two methods differ by a constant dependent on the rotation parameter and cosmological constant in odd spacetime dimensions, whereas the charge corresponding to the rotational killing vector is the same in either approach. We comment on possible implications of our results to the AdS/CFT correspondence.
The unexpected discovery of hairy black hole solutions in theories with scalar fields simply by considering asymptotically Anti de-Sitter, rather than asymptotically flat, boundary conditions is analyzed in a way that exhibits in a clear manner the differences between the two situations. It is shown that the trivial Schwarzschild Anti de Sitter becomes unstable in some of these situations, and the possible relevance of this fact for the ADS/CFT conjecture is pointed out.
Recent observations of the luminosity-red shift relation of distant type Ia supernovae established the fact that the expansion of the universe is accelerated. This is interpreted by saying that there exists some kind of agent (called dark energy), which exerts an overall repulsive effect on ordinary matter. Dark energy contributes today in the amount of about 73 % to the total energy content of the universe, and its spatial distribution is compatible with perfect uniformity. The simplest possible explanation for dark energy is to assume that it is just a universal constant, the so called cosmo…
Previous works on black hole shadows have been primarily focused on studying shadows in asymptotically flat and anti-de Sitter space times. In the present work, we find general expressions for asymptotically de Sitter black hole shadow as seen by static and comoving observers, for any spherically symmetric black hole solution, in any space time dimension in generic theories of gravity. As test cases, we use the derived general expressions to study the shadow of three different black hole solutions: the four dimensional Reissner-Nordström de Sitter black hole, the recently proposed four dimensi…
Numerical arguments are presented for the existence of spherically symmetric regular and black hole solutions of the EYMH equations with a negative cosmological constant. These solutions approach asymptotically the anti-de Sitter spacetime. The main properties of the solutions and the differences with respect to the asymptotically flat case are discussed. The instability of the gravitating sphaleron solutions is also proven.
We consider $n$-dimensional asymptotically anti-de Sitter spacetimes in higher curvature gravitational theories with $n \geq 4$, by employing the conformal completion technique. We first argue that a condition on the Ricci tensor should be supplemented to define an asymptotically anti-de Sitter spacetime in higher curvature gravitational theories and propose an alternative definition of an asymptotically anti-de Sitter spacetime. Based on that definition, we then derive a conservation law of the gravitational field and construct conserved quantities in two classes of higher curvature gravitati…
It is shown that the Nachtmann boosting method of introducing coordinates on de Siter manifolds can be completed with suitable gauge transformations able to keep under control the transformation under isometries of the conserved quantities. With this method, the rest local charts (or natural frames) are defined pointing out the role of the conserved quantities in investigating the relative geodesic motion. The advantages of this approach can be seen from the applications presented here. For the first time, the simple kinematic effects, the electromagnetic field of a free falling charge and the…
We start a systematic investigation of possible isometries of the asymptotically de Sitter solutions to Einstein equations. We reformulate the Killing equation as conformal equations for the initial data at $\mathcal{I}^+$. This allows for partial classification of possible symmetry algebras. In particular, if they are not maximal, they may be at most $4$-dimensional. We provide several examples. As a simple collorary it is shown that the only spacetime in which the Killing horizon intersects $\mathcal{I}^+$ (after a conformal completion) is locally the de Sitter universe.
The regularized four-dimensional Einstein-Gauss-Bonnet model has been recently proposed in [D. Glavan and C. Lin, Phys. Rev. Lett. \textbf{124}, 081301 (2020)] whose formulation is different of the Einstein theory, allowing us to bypass the Lovelock theorem. The action is formulated in higher dimensions ($D>4$) by adding the Gauss-Bonnet correction to the conventional Einstein-Hilbert action with a cosmological constant. The four-dimensional spacetime is constructed through dimensional regularization by taking the limit $D\rightarrow 4$. We find explicitly the parametric regions of stability o…
Our recent results concerning the transformation under isometries of the conserved quantities on de Sitter manifolds, allow us to define the rest frame and study the relative geodesic motion in terms of conserved momentum, revealing thus various relativistic effects [I. I. Cotaescu, arXiv:1701.08499]. Here we discuss the time dilation (of the twin paradox) and the Lorentz contraction on the de Sitter expanding portions, pointing out the dependence of these effects on the position and moment in which the time and lengths are measured.
Fermion production by the electric field of a charge on de Sitter expanding universe is analyzed. The amplitude and probability of pair production are computed. We obtain from our calculations that the modulus of the momentum is no longer conserved and that there are probabilities for production processes where the helicity is no longer conserved. The rate of pair production in an electric field is found to be important in the early universe when the expansion factor was large comparatively with the particle mass.
The relative geodesic motion on $(1+3)$-dimensional anti-de Sitter spacetimes is studied in terms of conserved quantities by adapting the Nachtmann boosting method created initially for the de Sitter spacetimes. In this approach the Lorentzian isometriy is derived, relating the coordinates of the local chart of a fixed observer with the coordinates of a mobile chart considered as the rest frame of a massive mobile object moving on a timelike anti-de Sitter geodesic. The transformation of the conserved quantities is also investigated, constructing thus a complete theory of the relative geodesic…
Conformally invariant scalar waves in black hole spacetimes which are asymptotically anti-de Sitter are investigated. We consider both the $(2+1)$-dimensional black hole and $(3+1)$-dimensional Schwarzschild-anti-de Sitter spacetime as backgrounds. Analytical and numerical methods show that the waves decay exponentially in the $(2+1)$ dimensional black hole background. However the falloff pattern of the conformal scalar waves in the Schwarzschild-anti-de Sitter background is generally neither exponential nor an inverse power rate, although the approximate falloff of the maximal peak is weakly …
We give a Hamiltonian definition of mass for spacelike hypersurfaces in space-times with metrics which are asymptotic to the anti-de Sitter one, or to a class of generalizations thereof. We show that our definition provides a geometric invariant for a spacelike hypersurface embedded in a space-time.
In this article we investigate cylindrical thin-shell wormholes which are asymptotically anti-de Sitter. We analyze their stability under perturbations preserving the symmetry by using two different methods. We compare the results with those corresponding to the wormholes constructed from the Levi-Civita spacetime. We find that the configurations always require the presence of exotic matter at the throat, and, in the case of the linearized stability analysis, they can be stable for suitable values of the parameters
Penrose's quasi-local mass construction is carried through for two-surfaces at infinity in asymptotically anti-de Sitter space-times. A modification of the Witten argument is given to prove a positivity property of the resulting conserved quantities. [This work formed part of Ron Kelly's Oxford D.Phil. thesis, and the first person pronoun refers to him. It appeared in hand-written form as `Asymptotically anti-de Sitter space-times' in Twistor Newsletter 20 (1985) pp11-23 (available at http://people.maths.ox.ac.uk/lmason/Tn/TN1-25), but is appearing type-set for the first time here. Footnotes m…
Starting with a procedure for dealing with general asymptotic behaviors, we construct a quantum theory for asymptotically anti-de Sitter wormholes. We follow both the path integral formalism and the algebraic quantization program proposed by Ashtekar. By adding suitable surface terms, the Euclidean action of the asymptoically anti-de Sitter wormholes can be seen to be finite and gauge invariant. This action determines an appropriate variational problem for wormholes. We also obtain the wormhole wave functions of the gravitational model and show that all the physical states of the quantum theor…
We study the apparent horizon for two boosted black holes in the asymptotically de Sitter space-time by solving the initial data on a space with punctures. We show that the apparent horizon enclosing both black holes is not formed if the conserved mass of the system (Abbott-Deser mass) is larger than a critical mass. The black hole with too large AD mass therefore cannot be formed in the asymptotically de Sitter space-time even though each black hole has any inward momentum. We also discuss the dynamical meaning of AD mass by examining the electric part of the Weyl tensor (the tidal force) for…
Based on the recent paper hep-th/0503045, we derive a formula of calculating conserved charges in even dimensional asymptotically {\it locally} anti-de Sitter space-times by using the definition of Wald and Zoupas. This formula generalizes the one proposed by Ashtekar {\it et al}. Using the new formula we compute the masses of Taub-Bolt-AdS space-times by treating Taub-Nut-AdS space-times as the reference solution. Our result agrees with those resulting from "background subtraction" method or "boundary counterterm" method. We also calculate the conserved charges of Kerr-Taub-Nut-AdS solutions …
Recently, Bañados, Teitelboim and Zanelli obtained spherically symmetric black hole solutions in a particular class of Einstein--Lovelock gravity. We derive the propagator in an exact form for a conformal scalar field in the asymptotically anti-de Sitter black hole spacetime so as to study the quantum effects of the scalar fields. We treat the cases in odd dimensions in this paper. We calculate the vacuum expectation value of $\langleφ^2\rangle$ and show its dependence on the radial coordinate for the five-dimensional case as an example.
A new prescription to calculate the total energies and angular momenta of asymptotically $(d+1)$-dimensional anti-de Sitter spacetimes is proposed. The method is based on an extension of the field theoretical approach to General Relativity to the case where there is an effective cosmological constant. A $(d-1)$-form $Ω$ is exhibited which, when integrated on asymptotic $(d-1)$-dimensional boundary surfaces, yields the values of those conserved quantities. The calculations are gauge independent once asymptotic conditions are not violated . Total energies and angular momenta of some known soluti…
We discuss the definition of conserved quantities in asymptotically locally de Sitter spacetimes. One may define an analogue of holographic charges at future and past infinity and at other Cauchy surfaces $I_t$ as integrals over the intersection of timelike surfaces $C$ and the Cauchy surface $I_t$. In general, the charges $Q^t$ defined on the Cauchy surface $I_t$ depend on $C$, but if gravitational flux is absent the charges are independent of $C$. On the other hand, if there is a net gravitational flux entering or leaving the spacetime region bounded by $C_1, C_2$ and two Cauchy surfaces the…
We present a covariant phase space construction of hamiltonian generators of asymptotic symmetries with `Dirichlet' boundary conditions in de Sitter spacetime, extending a previous study of Jäger. We show that the de Sitter charges so defined are identical to those of Ashtekar, Bonga, and Kesavan (ABK). We then present a comparison of ABK charges with other notions of de Sitter charges. We compare ABK charges with counterterm charges, showing that they differ only by a constant offset, which is determined in terms of the boundary metric alone. We also compare ABK charges with charges defined b…
Lower dimensional gravity has the potential of providing non-trivial and valuable insight into some of the conceptual issues arising in four dimensional relativistic gravitational analysis. The asymptotically anti-de Sitter (2+1) dimensional spacetime described by Ba$\tilde{n}$ados, Teitelboim and Zanelli (BTZ) which admits a black hole solution, has become a source of fascination for theoretical physicists in recent years. By suitably choosing the form of the mass function $m(r)$, we derive a new class of solutions for the interior of an isotropic star corresponding to the exterior (2+1) asym…
We have investigated the cosmological scenarios with a four dimensional effective action which is connected with multidimensional, supergravity and string theories. The solution for the scale factor is such that initially universe undergoes a decelerated expansion but in late times it enters into the accelerated expansion phase. Infact, it asymptotically becomes a de-Sitter universe. The dilaton field in our model is a decreasing function of time and it becomes a constant in late time resulting the exit from the scalar tensor theory to the standard Einstein's gravity. Also the dilaton field re…
We present a technique for obtaining spherically symmetric, asymptotically (anti)-de Sitter, black hole solutions of dilaton gravity with generic coupling to a Maxwell field, starting from exact asymptotically flat solutions and adding a suitable dilaton potential to the action.
In de Sitter (dS) special relativity (SR), two kinds of conserved currents are derived. The first kind is a 5-dimensional (5d) dS-covariant angular momentum (AM) current, which unites the energy-momentum (EM) and 4d AM current in an inertial-type coordinate system. The second kind is a dS-invariant AM current, which can be generalized to a conserved current for the coupling system of the matter field and gravitational field in dS gravity. Moreover, an inherent EM tensor is predicted, which comes from the spin part of the dS-covariant current. All the above results are compared to the ordinary …
We investigate nature of asymptotically de Sitter space-times containing a black hole. We show that if the matter fields satisfy the dominant energy condition and the cosmic censorship holds in the considering space-time, the area of the cosmological event horizon for an observer approaching a future timelike infinity does not decrease, i.e. the second law is satisfied. We also show under the same conditions that the total area of the black hole and the cosmological event horizon, a quarter of which is the total Bekenstein-Hawking entropy, is less than $12π/Λ$, where $Λ$ is a cosmological cons…
We establish the positive energy theorem for weak asymptotically anti-de Sitter initial data sets with distributional curvature under the weak dominant energy condition.
We study the occurrence, visibility, and curvature strength of singularities in dust-containing Szekeres spacetimes (which possess no Killing vectors) with a positive cosmological constant. We find that such singularities can be locally naked, Tipler strong, and develop from a non-zero-measure set of regular initial data. When examined along timelike geodesics, the singularity's curvature strength is found to be independent of the initial data.
Asymptotically anti-de Sitter space-times are considered in a general dimension $d\ge 4$. As one might expect, the boundary conditions at infinity ensure that the asymptotic symmetry group is the anti-de Sitter group (although there is an interesting subtlety if d=4). Asymptotic field equations imply that, associated with each generator $ξ$ of this group, there is a quantity $Q_ξ$ which satisfies the expected `balance equation' if there is flux of physical matter fields across the boundary $\I$ at infinity and is absolutely conserved in absence of this flux. Irrespective of the dimension d, al…
We calculate exactly the quasinormal frequencies of Klein-Gordon and Dirac test fields propagating in two-dimensional uncharged Achucarro-Ortiz black hole. For both test fields we study whether the quasinormal frequencies are well defined in the massless limit. We use their values to discuss the classical stability of the quasinormal modes in uncharged Achucarro-Ortiz black hole and to check the recently proposed Time Times Temperature bound. Furthermore we extend some of these results to the charged Achucarro-Ortiz black hole.
It is a standard fact that trapped or marginally trapped surfaces are not visible from conformal infinity, under the usual set of conditions on matter fields and the conformal completion, provided that the cosmological constant is non-positive. In this note we show that the situation is more delicate in the presence of a positive cosmological constant: we present examples of visible marginally trapped surfaces, and we provide a set of natural conditions which guarantee non-visibility.
We define the total energy-momenta for (4+1)-dimensional asymptotically anti-de Sitter spacetimes, and prove the positive energy theorem for such spacetimes.
We analyze the propagation of gravitational waves (GWs) in an asymptotically de-Sitter space by expanding the perturbation around Minkowski and introducing the effects of the Cosmological Constant ($Λ$), first as an additional source (de-Donder gauge) and after as a gauge effect ($Λ$-gauge). In both cases the inclusion of the Cosmological Constant $Λ$ impedes the detection of a gravitational wave at a distance larger than $L_{crit}=(6\sqrt{2}πf \hat{h}/\sqrt{5})r_Λ^2$, where $r_Λ=\frac{1}{\sqrtΛ}$ and f and $\hat{h}$ are the frequency and strain of the wave respectively. We demonstrate that $L…
We find the low lying quasinormal mode frequencies of the recently proposed novel four dimensional Gauss-Bonnet de Sitter black holes for scalar, electromagnetic and Dirac field perturbations using the third order WKB approximation as well as Padé approximation, as an improvement over WKB. We figure out the effect of the Gauss-Bonnet coupling $α$ and the cosmological constant $Λ$ on the real and imaginary parts of the QNM frequencies. We also study the greybody factors and eikonal limits in the above background for all three different types of perturbations.
This article considers the quasi-local conserved quantities with respect to a reference spacetime with a cosmological constant. We follow the approach developed by the authors in [25,26,7] and define the quasi-local energy as differences of surface Hamiltonians. The ground state for the gravitational energy is taken to be a reference configuration in the de Sitter (dS) or Anti-de Sitter (AdS) spacetime. This defines the quasi-local energy with respect to the reference spacetime and generalizes our previous definition with respect to the Minkowski spacetime. Through an optimal isometric embeddi…
We discuss several aspects of the relation between asymptotically AdS and asymptotically dS spacetimes including: the continuation between these types of spaces, the global stability of asymptotically dS spaces and the structure of limits within this class, holographic renormalization, and the maximal mass conjecture of Balasubramanian-deBoer-Minic.
Vanishing of the Fefferman-Graham obstruction tensor was used by Andersson and Chru{ś}ciel to show stability of the asymptotically de Sitter spaces in even dimensions. However, existing proofs of hyperbolicity of this equation contain gaps. We show in this paper that it is indeed a well-posed hyperbolic system with unique up to diffeomorphism and conformal transformations smooth development for smooth Cauchy data. Our method applies also to equations defined by various versions Graham-Jenne-Mason-Sparling operators. In particular, we use one of these operators to propagate Gover's condition of…
The aim of this manuscript is to review the studies about de Sitter solution and the null infinity of asymptotically flat and de Sitter space-times. Thus, after introducing the de Sitter space-time, the symmetry group is described. Also precise definitions of asymptotically flat and de Sitter space-times are reviewed. Henceforth, the null infinities and the asymptotic symmetry groups of these two space-times are considered, which lead to the Fefferman-Graham approach and the Penrose-Newman formalism.
A regular solution of the system of coupled equations of the nonlinear electrodynamics and gravity describing static and spherically-symmetric black holes in an asymptotically de Sitter universe is constructed and analyzed. Special emphasis is put on the degenerate configurations (when at least two horizons coincide) and their near horizon geometry. It is explicitly demonstrated that approximating the metric potentials in the region between the horizons by simple functions and making use of a limiting procedure one obtains the solutions constructed from maximally symmetric subspaces with diffe…
The general structure of the conformal boundary $\mathscr{I}^+$ of asymptotically de Sitter spacetimes is investigated. First we show that Penrose's quasi-local mass, associated with a cut ${\cal S}$ of the conformal boundary, can be zero even in the presence of outgoing gravitational radiation. On the other hand, following a Witten--type spinorial proof, we show that an analogous expression based on the Nester--Witten form is finite only if the Witten spinor field solves the 2-surface twistor equation on ${\cal S}$, and it yields a positive functional on the 2-surface twistor space on ${\cal …
The article considers a new mechanism of charge accumulation in the early Universe in theories with compact extra dimensions. The relaxation processes in the extra space metric that take place during its formation lead to the establishment of symmetrical extra space configuration. As a result, the initial accumulation of the number associated with the symmetry occurs. We demonstrate this mechanism using a simple example of a two-dimensional apple-like extra space metric with $U(1)$-symmetry. The conceptual idea of the mechanism can be used to develop a model for the production of the baryon or…
Despite the recent evidence that anti-de Sitter spacetime is nonlinearly unstable, we argue that many asymptotically anti-de Sitter solutions are nonlinearly stable. This includes geons, boson stars, and black holes. As part of our argument, we calculate the frequencies of long-lived gravitational quasinormal modes of AdS black holes in various dimensions. We also discuss a new class of asymptotically anti-de Sitter solutions describing noncoalescing black hole binaries.
We use the formulation of asymptotically anti-de Sitter boundary conditions given by Ashtekar and Magnon to obtain a coordinate expression for the general asymptotically AdeS metric in a neighbourhood of infinity. From this, we are able to compute the time delay of null curves propagating near infinity. If the gravitational mass is negative, so will be the time delay (relative to null geodesics at infinity) for certain null geodesics in the spacetime. Following closely an argument given by Penrose, Sorkin, and Woolgar, who treated the asymptotically flat case, we are then able to argue that a …
Recently, a general method for calculating conserved charges for (black hole) solutions to generally covariant gravitational theories, in any dimensions and with arbitrary asymptotic behaviors has been introduced. Equipped with this method, which can be dubbed as "solution phase space method," we calculate mass and angular momentum for the Kerr-dS black holes. Furthermore, for any choice of horizons, associated entropy and the first law of thermodynamics are derived. Interestingly, according to insensitivity of the analysis to the chosen cosmological constant, the analysis unifies the thermody…
We establish the inequality for Henneaux-Teitelboim's total energy-momentum for asymptotically anti-de Sitter initial data sets which are asymptotic to arbitrary $t$-slice in anti-de Sitter spacetime. In particular, when $t=0$, it generalizes Chruściel-Maerten-Tod's inequality in the center of AdS mass coordinates. We also show that the determinant of energy-momentum endomorphism ${\bf Q}$ is the geometric invariant of asymptotically anti-de Sitter spacetimes.
A family of exact solutions is presented which represents a rigidly rotating cylinder of dust in a background with a negative cosmological constant. The interior of the infinite cylinder is described by the Godel solution. An exact solution for the exterior solution is found which depends both on the rotation of the interior and on its radius. For values of these parameters less than a certain limit, the exterior solution is shown to be locally isomorphic to the Linet-Tian solution. For values larger than another limit, it is shown that the exterior solution extends into a region which contain…
