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As part of a programme to classify quasi-Einstein metrics (M, g, X) on closed manifolds and near-horizon geometries of extreme black holes, we study such spaces when the vector field X is divergence-free but not identically zero. This condition is satisfied by left-invariant quasi-Einstein metrics on compact homogeneous spaces (including the near-horizon geometry of an extreme Myers–Perry black hole with equal angular momenta in two distinct planes) and on certain bundles over Kähler–Einstein manifolds. We find that these spaces exhibit a mild form of rigidity: they always admit a one-parameter group of isometries generated by X. Further geometrical and topological restrictions are also obtained. [ABSTRACT FROM AUTHOR]

Marginally outer trapped surfaces (also referred to as apparent horizons) that are stable in 3-dimensional initial data sets obeying the dominant energy condition strictly are known to satisfy an area bound. The main purpose of this note is to show (in several ways) that such surfaces also satisfy a diameter bound. Some comments about higher dimensions are also presented. [ABSTRACT FROM AUTHOR]

In this paper, approximate Noether and Lie symmetries of 2nd order for Gibbons–Maeda–Garfinkle–Horowitz–Strominger (GMGHS) charged black hole in the Einstein frame are analyzed comprehensively. To explore the approximate Noether symmetries of 2nd order, Noether symmetries of Minkowski spacetime are used which forms a 17 dimensional Lie algebra. It is observed that no new approximate Noether symmetry is obtained at 1st and 2nd order. To examine the 1st and 2nd order approximate Lie symmetries of the GMGHS black hole spacetime, 35 Lie symmetries (exact) of the Minkowski spacetime are used which forms an algebra sl(6, R). It is shown that no new approximate Lie symmetry exists at 1st and 2nd order and only exact 35 symmetries are recouped as trivial approximate Lie symmetries at both orders. Furthermore, no energy rescaling factor is seen in this spacetime. [ABSTRACT FROM AUTHOR]

In this note, we show that compact static near-horizon geometries with negative cosmological constant are either Einstein or the product of a circle and an Einstein metric. Chruściel, Reall, and Todd proved rigidity when the cosmological constant vanishes, in which case one get the stronger result that the space is Ricci flat (Chruściel et al. in Class Quantum Gravity 23:549–554, 2006). It has been previously asserted that a stronger rigidity statement also holds for negative cosmological constant, but Bahuaud, Gunasekaran, Kunduri, and Woolgar recently pointed out that this was not the case (Bahuaud et al. in Lett Math Phys 112(6):116, 2022). They showed, moreover, that for a compact static near-horizon geometry with negative cosmological constant, the potential vector field X is constant length and divergence-free. We give an argument using the Bochner formula to improve their conclusion to X being a parallel field, which implies the optimal rigidity result. The result also holds more generally for m-Quasi Einstein metrics with m > 0 . [ABSTRACT FROM AUTHOR]

We derive new lower bounds for the first eigenvalue of the Dirac operator of an oriented hypersurface Σ bounding a noncompact domain in a spin asymptotically flat manifold (M n , g) with nonnegative scalar curvature. These bounds involve the boundary capacity potential and, in some cases, the capacity of Σ in (M n , g) yielding several new geometric inequalities. The proof of our main result relies on an estimate for the first eigenvalue of the Dirac operator of boundaries of compact Riemannian spin manifolds endowed with a singular metric which may have independent interest. [ABSTRACT FROM AUTHOR]

Static vacuum near-horizon geometries are solutions (M, g, X) of a certain quasi-Einstein equation on a closed manifold M, where g is a Riemannian metric and X is a closed 1-form. It is known that when the cosmological constant vanishes, there is rigidity: X vanishes and consequently g is Ricci-flat. We study this form of rigidity for all signs of the cosmological constant. It has been asserted that this rigidity also holds when the cosmological constant is negative, but we exhibit a counter-example. We show that for negative cosmological constant if X does not vanish identically, it must be incompressible, have constant norm, and be nontrivial in cohomology, and (M, g) must have constant scalar curvature and zero Euler characteristic. If the cosmological constant is positive, X must be exact (and vanishing if dim M = 2 ). Our results apply more generally to a broad class of quasi-Einstein equations on closed manifolds. We extend some known results for quasi-Einstein metrics with exact 1-form X to the closed X case. We consider near-horizon geometries for which the vacuum condition is relaxed somewhat to allow for the presence of a limited class of matter fields. An “Appendix” contains a generalization of a result of Lucietti on the Yamabe type of quasi-Einstein compact metrics (with arbitrary X). [ABSTRACT FROM AUTHOR]

We define a notion of extrinsic black hole in pure Lovelock gravity of degree k which captures the essential features of the so-called Lovelock-Schwarzschild solutions, viewed as rotationally invariant hypersurfaces with null 2k-mean curvature in Euclidean space R n + 1 , 2 ≤ 2 k ≤ n - 1 . We then combine a regularity argument with a rigidity result by Araújo and Leite (Indiana University Mathematics Journal pp. 1667–1693, 2012) to prove, under a natural ellipticity condition, a global uniqueness theorem for this class of black holes. As a consequence we obtain, in the context of the corresponding Penrose inequality for graphs established by Ge et al. (Advances in Mathematics 266: 84–119, 2014), a local rigidity result for the Lovelock-Schwarzschild solutions. [ABSTRACT FROM AUTHOR]

We establish the construction of a conformal scattering theory of the spin-1/2 massless Dirac equation on the Kerr spacetime by using the conformal geometric method and a pointwise logarithmic decay estimate for the massless Dirac field. In particular, our construction is valid on the exterior domains of Kerr black hole spacetimes in the full sub-extremal range | a | < M . [ABSTRACT FROM AUTHOR]

In this paper, we show that there is a breakdown of scattering between the event horizon (or the Cauchy horizon) and an intermediate Cauchy hypersurface in the dynamic interior of a Reissner–Nordström-like black hole. More precisely, we show that the trace operators and their analytic counterparts, the inverse wave operators, do not have bounded inverses, even though these operators themselves are bounded. This result holds for the natural energy given by the energy–momentum tensor of the wave equation using the timelike vector field of the Regge–Wheeler variable, which asymptotically becomes normal to the horizons. The behaviour of solutions at low spatial frequencies and their behaviour at high angular momenta are the only obstructions causing this breakdown of scattering. The breakdown follows from an analysis of a 1 + 1 -dimensional wave equation with exponentially decaying potential which we treat for general potentials, and we show that the breakdown is generic. [ABSTRACT FROM AUTHOR]

The decay of solutions to the Klein–Gordon equation is studied in two expanding cosmological spacetimes, namely the de Sitter universe in flat Friedmann–Lemaître–Robertson–Walker (FLRW) form and the cosmological region of the Reissner–Nordström–de Sitter (RNdS) model. Using energy methods, for initial data with finite higher-order energies, decay rates for the solution are obtained. Also, a previously established decay rate of the time derivative of the solution to the wave equation, in an expanding de Sitter universe in flat FLRW form, is improved, proving Rendall's conjecture. A similar improvement is also given for the wave equation in the cosmological region of the RNdS spacetime. [ABSTRACT FROM AUTHOR]

Einstein's theory of General Relativity is one of the pillars of modern physics. For decades, the theory has been mainly tested in the weak field regime with experiments in the Solar System and observations of binary pulsars. Thanks to a new generation of observational facilities, the past 5 years have seen remarkable changes in this field and there are now numerous efforts for testing General Relativity in the strong field regime with black holes and neutron stars using different techniques. Here I will review the work of my group at Fudan University devoted to test General Relativity with black hole X-ray data. [ABSTRACT FROM AUTHOR]

In this short note, we show how one can use established results to prove various versions of the positive mass theorem for initial data sets with boundary, in dimensions less than 8. [ABSTRACT FROM AUTHOR]

We consider spherically symmetric spacetimes with an outer trapping horizon. Such spacetimes are generalizations of spherically symmetric black hole spacetimes where the central mass can vary with time, like in black hole collapse or black hole evaporation. While these spacetimes possess in general no timelike Killing vector field, they admit a Kodama vector field which in some ways provides a replacement. The Kodama vector field allows the definition of a surface gravity of the outer trapping horizon. Spherically symmetric spacelike cross sections of the outer trapping horizon define in- and outgoing lightlike congruences. We investigate a scaling limit of Hadamard 2-point functions of a quantum field on the spacetime onto the ingoing lightlike congruence. The scaling limit 2-point function has a universal form and a thermal spectrum with respect to the time parameter of the Kodama flow, where the inverse temperature β = 2 π / κ is related to the surface gravity κ of the horizon cross section in the same way as in the Hawking effect for an asymptotically static black hole. Similarly, the tunnelling probability that can be obtained in the scaling limit between in- and outgoing Fourier modes with respect to the time parameter of the Kodama flow shows a thermal distribution with the same inverse temperature, determined by the surface gravity. This can be seen as a local counterpart of the Hawking effect for a dynamical horizon in the scaling limit. Moreover, the scaling limit 2-point function allows it to define a scaling limit theory, a quantum field theory on the ingoing lightlike congruence emanating from a horizon cross section. The scaling limit 2-point function as well as the 2-point functions of coherent states of the scaling limit theory is correlation-free with respect to separation along the horizon cross section; therefore, their relative entropies behave proportional to the cross-sectional area. We thus obtain a proportionality of the relative entropy of coherent states of the scaling limit theory and the area of the horizon cross section with respect to which the scaling limit is defined. Thereby, we establish a local counterpart, and microscopic interpretation in the setting of quantum field theory on curved spacetimes, of the dynamical laws of outer trapping horizons, derived by Hayward and others in generalizing the laws of black hole dynamics originally shown for stationary black holes by Bardeen, Carter and Hawking. [ABSTRACT FROM AUTHOR]

We construct dynamical many-black-hole spacetimes with well-controlled asymptotic behavior as solutions of the Einstein vacuum equation with positive cosmological constant. We accomplish this by gluing Schwarzschild–de Sitter or Kerr–de Sitter black hole metrics into neighborhoods of points on the future conformal boundary of de Sitter space, under certain balance conditions on the black hole parameters. We give a self-contained treatment of solving the Einstein equation directly for the metric, given the scattering data we encounter at the future conformal boundary. The main step in the construction is the solution of a linear divergence equation for trace-free symmetric 2-tensors; this is closely related to Friedrich's analysis of scattering problems for the Einstein equation on asymptotically simple spacetimes. [ABSTRACT FROM AUTHOR]

In a previous work of ours, the most general family of Kerr deformations—admitting a Carter constant—has been presented. This time a simple, necessary and sufficient condition in order for the aforementioned family to have a separable Klein-Gordon equation is exhibited. In addition, we provide a solid theoretical foundation that the maximum number of free to be chosen radial functions is three. [ABSTRACT FROM AUTHOR]

We give an elementary treatment of the curvature of surfaces of revolution in the language of vector calculus, using differentials rather than an explicit parameterization. We illustrate some basic features of curvature using embedding diagrams, and then use such a diagram to analyze the geometry of the Schwarzschild black hole. [ABSTRACT FROM AUTHOR]

The global hyperbolicity assumption present in gravitational collapse singularity theorems is in tension with the quantum mechanical phenomenon of black hole evaporation. In this work, I show that the causality conditions in Penrose's theorem can be almost completely removed. As a result, it is possible to infer the formation of spacetime singularities even in the absence of predictability and hence compatibly with quantum field theory and black hole evaporation. [ABSTRACT FROM AUTHOR]

In this paper we use Morawetz and geometric energy estimates—the so-called vector field method—to prove decay results for the Maxwell field in the static exterior region of the Reissner–Nordström–de Sitter black hole. We prove two types of decay: the first is a uniform decay of the energy of the Maxwell field on achronal hypersurfaces as the hypersurfaces approach timelike infinities. The second decay result is a pointwise decay in time with a rate of t - 1 which follows from local energy decay by Sobolev estimates. Both results are consequences of bounds on the conformal energy defined by the Morawetz conformal vector field. These bounds are obtained through wave analysis on the middle spin component of the field. The results hold for a more general class of spherically symmetric spacetimes with the same arguments used in this paper. [ABSTRACT FROM AUTHOR]

We begin with a review of the statements of non-linear, linear and mode stability of autonomous dynamical systems in classical mechanics, using symplectic geometry. We then discuss what the Arnowitt–Deser–Misner (ADM) phase space and the ADM Hamiltonian of general relativity are, what constitutes a dynamical system, and subsequently present a nascent attempt to draw a formal analogy between the notions of stability in these two theories. We wish to note that we have not discussed here the construction of the reduced phase space by forming the quotient space of the constrained phase space with the gauge orbits. Our approach here is pedagogical and geometric, and the motivation is to unify and simplify a formal understanding of the statements regarding the stability of stationary solutions of general relativity. That is, typically the governing equations of motion of a Hamiltonian dynamical system are simply the flow equations of the associated symplectic Hamiltonian vector field, defined on phase space, and the non-linear stability analysis of its critical points have simply to do with the divergence of its flow there. Further, the linear stability of a critical point is related to the properties of the tangent flow of the Hamiltonian vector field. In the second half of this work, we posit that a study of the genericity of a particular black hole or naked singularity spacetime forming as an endstate of gravitational collapse is equivalent to an inquiry of how sensitive the orbits of the symplectic Hamiltonian vector field of general relativity are to changes in initial data. We demonstrate this by conducting a restricted non-linear stability analysis of the formation of a Schwarzschild black hole, working in the usual initial value formulation of general relativity. [ABSTRACT FROM AUTHOR]

We have investigated effect of geometric optics as the rotation of polarization vector of light in spacetime of gravitational compact object in the fourth-order theory of conformal Weyl gravity. The Pineault–Roeder method is applied to the rotating Weyl metric, and analytical results are obtained in the limit of weak field and or slow rotation. For the photon traveling parallel to the symmetry axes from the equatorial plane to infinity, the rotation of the polarization plane depends on the Weyl parameter γ on the contrary to the Kerr spacetime where there is no rotation of polarization plane for this case. [ABSTRACT FROM AUTHOR]

A massive star undergoes a continual gravitational collapse when the pressures inside the collapsing star become insufficient to balance the pull of gravity. The Physics of gravitational collapse of stars is well studied. Using general relativistic techniques, one can show that the final fate of such a catastrophic collapse can be a black hole or a naked singularity, depending on the initial conditions of gravitational collapse. While stars are made of baryonic matter whose collapse is well studied, there is good indirect evidence that another type of matter, known as dark matter, plays an important role in the formation of large-scale structures in the universe, such as galaxies. It is estimated that some 85% of the total matter in the universe is dark matter. Since the particle constituent of dark matter is not known yet, the gravitational collapse of dark matter is less explored. Here, we consider first some basic properties of baryonic matter and dark matter collapse. Then, we discuss the final fate of gravitational collapse for different types of matter fields and the nature of the singularity which can be formed as an endstate of gravitational collapse. We then present a general relativistic technique to form equilibrium configurations, and argue that this can be thought of as a general relativistic analog of the standard virialization process. We suggest a modification, where the top-hat collapse model of primordial dark-matter halo formation is modified using the general relativistic technique of equilibrium. We also explain why this type of collapse process is more likely to happen in the dark-matter fields. [ABSTRACT FROM AUTHOR]

We study the peeling on Kerr spacetime for fields satisfying conformally invariant linear and semi-linear scalar wave equations. We follow the approach initiated by Mason and Nicolas (J Inst Math Jussieu 8(1):179–208, 2009; J Geom Phys 62(4):867–889, 2012. arXiv:1101.4333) for the Schwarzschild metric, based on a Penrose compactification and energy estimates. This approach provides a definition of the peeling at all orders in terms of Sobolev regularity near I instead of C k regularity at I , allowing to characterise completely and without loss the classes of initial data ensuring a certain order of peeling at I . This paper extends the construction to the Kerr metric, confirms the validity and optimality of the flat spacetime model (in the sense that the same regularity and fall-off assumptions on the data guarantee the peeling behaviour in flat spacetime and on the Kerr metric) and does so for the first time for a nonlinear equation. Our results are local near spacelike infinity and are valid for all values of the angular momentum of the spacetime, including for fast Kerr metrics. [ABSTRACT FROM AUTHOR]

We provide bounds on the first Betti number and structure results for the fundamental group of horizon cross sections for extreme stationary vacuum black holes in arbitrary dimension, without additional symmetry hypotheses. This is achieved by exploiting a correspondence between the associated near-horizon geometries and the mathematical notion of m-quasi-Einstein metrics, in addition to generalizations of the classical splitting theorem from Riemannian geometry. Consequences are analyzed and refined classifications are given for the possible topologies of these black holes. [ABSTRACT FROM AUTHOR]

We study the long-time existence of solutions to nonlinear wave equations with power-type nonlinearity (of orderp) and small data, on a large class of (1+n)-dimensional nonstationary asymptotically flat backgrounds, which include the Schwarzschild and Kerr black hole space-times. Under the assumption that uniform energy bounds and a weak form of local energy estimates hold forward in time, we give lower bounds of the lifespan whenn = 3,4 andpis not bigger than the critical one. The lower bounds for three-dimensional subcritical and four-dimensional critical cases are sharp in general. For the most delicate three-dimensional critical case, we obtain the first existence result up to, for many space-times including the nontrapping exterior domain, nontrapping asymptotically Euclidean space and Schwarzschild space-time. [ABSTRACT FROM AUTHOR]

We apply the confluent Heun functions to study the resonant frequencies (quasispectrum), the Hawking radiation and the scattering process of scalar waves, in a class of spacetimes, namely, the ones generated by a Kerr–Newman–Kasuya spacetime (dyon black hole) and a Reissner–Nordström black hole surrounded by a magnetic field (Ernst spacetime). In both spacetimes, the solutions for the angular and radial parts of the corresponding Klein–Gordon equations are obtained exactly, for massive and massless fields, respectively. The special cases of Kerr and Schwarzschild black holes are analyzed and the solutions obtained, as well as in the case of a Schwarzschild black hole surrounded by a magnetic field. In all these special situations, the resonant frequencies, Hawking radiation and scattering are studied. [ABSTRACT FROM AUTHOR]

Charged massive scalar fields are considered in the gravitational and electromagnetic field produced by a dyonic black hole with a cosmic string along its axis of symmetry. Exact solutions of both angular and radial parts of the covariant Klein–Gordon equation in this background are obtained, and are given in terms of the confluent Heun functions. The role of the presence of the cosmic string in these solutions is showed up. From the radial solution, we obtain the exact wave solutions near the exterior horizon of the black hole, and discuss the Hawking radiation spectrum and the energy flux. [ABSTRACT FROM AUTHOR]

I review Lorentzian causality theory paying particular attention to the optimality and generality of the presented results. I include complete proofs of some foundational results that are otherwise difficult to find in the literature (e.g. equivalence of some Lorentzian length definitions, upper semi-continuity of the length functional, corner regularization, etc.). The paper is almost self-contained thanks to a systematic logical exposition of the many different topics that compose the theory. It contains new results on classical concepts such as maximizing curves, achronal sets, edges, horismos, domains of dependence, Lorentzian distance. The treatment of causally pathological spacetimes requires the development of some new versatile causality notions, among which I found particularly convenient to introduce: biviability, chronal equivalence, araying sets, and causal versions of horismos and trapped sets. Their usefulness becomes apparent in the treatment of the classical singularity theorems, which is here considerably expanded in the exploration of some variations and alternatives. [ABSTRACT FROM AUTHOR]