We consider general initial data for the Einstein scalar-field system on a closed $3$-manifold $(M,γ)$ which is close to data for a Friedman-Lemaître-Robertson-Walker solution with homogeneous scalar field matter and a negative Einstein metric $γ$ as spatial geometry. We prove that the maximal globally hyperbolic development of such initial data in the Einstein scalar-field system is past incomplete in the contracting direction and exhibits stable collapse into a Big Bang curvature singularity. Under an additional condition on the first positive eigenvalue of $-Δ_γ$ satisfied, for example, by closed hyperbolic 3-manifolds of small diameter, we prove that the data evolves to a future complete spacetime in the expanding direction which asymptotes to a vacuum Friedman solution with $(M,γ)$ as the expansion normalized spatial geometry. In particular, the Strong Cosmic Censorship conjecture holds for this class of solutions in the $C^{2}$-sense.
Conjugate points play an important role in the proofs of the singularity theorems of Hawking and Penrose. We examine the relation between singularities and conjugate points in FLRW spacetimes with a singularity. In particular we prove a theorem that when a non-comoving, non-spacelike geodesic in a singular FLRW spacetime obeys conditions (39) and (40), every point on that geodesic is part of a pair of conjugate points. The proof is based on the Raychaudhuri equation. We find that the theorem is applicable to all non-comoving, non-spacelike geodesics in FLRW spacetimes with non-negative spatial…
It was recently shown that Strong Cosmic Censorship may be violated in highly charged black-hole spacetimes living in a universe with a positive cosmological constant. Several follow-up works have since suggested that such result, while conceptually interesting, cannot be upheld in practice. We focus here on the claim that the presence of charged massive scalars suffices to save Strong Cosmic Censorship. To the contrary, we show that there still exists a finite region in parameter space where Strong Cosmic Censorship is expected to be violated.
The FLRW spacetimes can be realized as submanifolds of $\mathbb{R}^6$. In this paper we relate the Laplace-Beltrami operator for an homogeneous scalar field $ϕ$ of $\mathbb{R}^6$ to its explicit restriction on FLRW spacetimes. We then make the link between the homogeneous solutions of the equation $\square_6 ϕ= 0$ in $\mathbb{R}^6$ and those of the Klein-Gordon equation $(\square_{f} - ξR^f + m^2)ϕ^f=0 $ for the free field $ϕ^f$ in the FLRW spacetime. We obtain as a byproduct a formula for the Ricci scalar of the FRLW spacetime in terms of the function $f$ defining this spacetime in $\mathbb{R…
Geodesics are used in a wide array of applications in cosmology and astrophysics. However, it is not a trivial task to efficiently calculate exact geodesic distances in an arbitrary spacetime. We show that in spatially flat (3+1)-dimensional Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetimes, it is possible to integrate the second-order geodesic differential equations, and derive a general method for finding both timelike and spacelike distances given initial-value or boundary-value constraints. In flat spacetimes with either dark energy or matter, whether dust, radiation, or a stiff fluid,…
We address the issue of strong cosmic censorship for T^2-symmetric spacetimes with positive cosmological constant. In the case of collisionless matter, we complete the proof of the C^2 formulation of the conjecture for this class of spacetimes. In the vacuum case, we prove that the conjecture holds for the special cases where the area element of the group orbits does not vanish on the past boundary of the maximal Cauchy development.
We prove strong cosmic censorship for T^2-symmetric cosmological spacetimes (with spatial topology T^3 and vanishing cosmological constant Lambda) with collisionless matter. Gowdy symmetric spacetimes constitute a special case. The formulation of the conjecture is in terms of generic C^2-inextendibility of the metric. Our argument exploits a rigidity property of Cauchy horizons, inherited from Killing fields.
The shape of the orbit of a free particle is examined in FLRW cosmologies. The spatial projection of the orbit is time-independent and has a simple geometric description. We relate this description to the expression in terms of standard FLRW coordinates.
The Ernst spacetime is a solution of the Einstein-Maxwell equations describing two charged black holes accelerating apart in a uniform electric (or magnetic) field. As the field approaches a critical value, the black hole horizon appears to touch the acceleration horizon. We show that weak cosmic censorship cannot be violated by increasing the field past this critical value: The event horizon remains intact. On the other hand, strong cosmic censorship does appear to be violated in this spacetime: For a certain range of parameters, we find evidence that the inner horizon is classically stable.
In quantum theory of the free Dirac field on spatially flat FLRW spacetimes we introduce a new type of vacuum able to separate the positive and negative frequencies in the rest frames. This is refereed as the rest frame vacuum shoving that this differs from the adiabatic vacuum apart from the Minkowski spacetime where these two vacua coincide.
Misner spacetime is among the simplest solutions of Einstein's equation that exhibits a Cauchy horizon with a smooth extension beyond it. Besides violating strong cosmic censorship, this extension contains closed timelike curves. We analyze the stability of the Cauchy horizon, and prove that neighboring spacetimes in one parameter families of solutions through Misner's in pure gravity, gravity coupled to a scalar field, or Einstein-Maxwell theory, end at the Cauchy horizon developing a curvature singularity.
In this talk we review the appearance of new types of singularities (big rip, sudden singularities...) in FLRW cosmological models that have arisen on considering explanations for accelerated expansion of our universe.
We analyze here the final fate of complete gravitational collapse of a massless scalar field within general relativity. A class of dynamical solutions with initial data close to the Friedmann-Lemaitre-Robertson-Walker (FLRW) collapse model is explicitly given and the Einstein equations are integrated numerically in a neighborhood of the center. We show that the initial data space is evenly divided between the dynamical evolutions that terminate in a black hole final state and those that produce a locally naked singularity. We comment on the genericity aspects of the collapse end-states and the…
Currently available topological censorship theorems are meant for gravitationally isolated black hole spacetimes with cosmological constant $Λ=0$ or $Λ<0$. Here, we prove a topological censorship theorem that is compatible with $Λ>0$ and which can be applied to whole universes containing possibly multiple collections of black holes. The main assumption in the theorem is that distinct black hole collections eventually become isolated from one another at late times, and the conclusion is that the regions near the various black hole collections have trivial fundamental group, in spite of there po…
Strong cosmic censorship conjecture has been one of the most important leap of faith in the context of general relativity, providing assurance in the deterministic nature of the associated field equations. Though it holds well for asymptotically flat spacetimes, a potential failure of the strong cosmic censorship conjecture might arise for spacetimes inheriting Cauchy horizon along with a positive cosmological constant. We have explicitly demonstrated that violation of the censorship conjecture holds true in the presence of a Maxwell field even when higher spacetime dimensions are invoked. In …
We point out that past-incompleteness of geodesics in FLRW spacetimes does not necessarily imply that these spacetimes start from a singularity. Namely, if a test particle that follows such a trajectory has a non-vanishing velocity, its energy was super-Planckian at some time in the past if it kept following that geodesic. That indicates a breakdown of the particle's description, which is why we should not consider those trajectories for the definition of an initial singularity. When one only considers test particles that do not have this breakdown of their trajectory, it turns out that the on…
Presently, the inclusion of the vacuum energy in the energy momentum tensor, and the inclusion of the extra dimensions in the spacetime, can not be rule out of the research in gravitation. In this work we study the influence of the vacuum energy in the collapse process of a stellar fluid, and consequently for the cosmic censorship conjecture, considering a homogeneous and isotropic spacetime with arbitrary number of dimensions. We discuss the active gravitational mass of the black hole formed, where the vacuum energy and the number of dimensions has a crucial role in the process.
We present here a brief review and discussion on recent developments in the theory of spacetime singularities. After mentioning some key motivations on the main ideas and concepts involved, we take the approach that the singularities will be eventually resolved by the quantum gravity effects. Some consequences are indicated when such singularities are visible to far away observers in the universe.
We first calculate equations of motion for particles in the Kerr-Sen-de Sitter black hole spacetime. Then in the eikonal regime, we analytically obtain the quasi-normal resonant modes of massless neutral scalar field perturbation and find its imaginary part to be characterized by the surface gravity of the near-extremal Kerr-Sen-de Sitter black hole with Cauchy horizon approaching the event horizon. We further show that the Penrose strong cosmic censorship conjecture is thus respected in this spacetime with dilaton scalar field and axion pseudoscalar field.
Gravitational collapse of radiation in an anti-de Sitter background is studied. For the spherical case, the collapse proceeds in much the same way as in the Minkowski background, i.e., massless naked singularities may form for a highly inhomogeneous collapse, violating the cosmic censorship, but not the hoop conjecture. The toroidal, cylindrical and planar collapses can be treated together. In these cases no naked singularity ever forms, in accordance with the cosmic censorship. However, since the collapse proceeds to form toroidal, cylindrical or planar black holes, the hoop conjecture in an …
The quantum electrodynamics (QED) on a spatially flat $(1+3)$-dimensional Friedmann-Lema\^ itre-Robertson-Walker (FLRW) spacetime with a Milne-type scale factor is outlined focusing on the amplitudes of the allowed effects in the first order of perturbations. The definition of the transition rates is reconsidered obtaining an appropriate angular behavior of the probability of the pair creation from a photon.
We prove nonlinear Lyapunov stability of a family of `$n+1$'-dimensional cosmological models of general relativity locally isometric to the Friedman Lemaître Robertson Walker (FLRW) spacetimes including a positive cosmological constant. In particular, we show that the perturbed solutions to the Einstein-Euler field equations around a class of spatially compact FLRW metrics (for which the spatial slices are compact negative Einstein spaces in general and hyperbolic for the physically relevant $n=3$ case) arising from regular Cauchy data remain uniformly bounded and decay to a family of metrics …
We establish the future non-linear stability of Friedmann-Lema\^ıtre-Robertson-Walker (FLRW) solutions to the Einstein-Euler equations of the universe filled with a large class of perfect fluids (the equations of state are allowed to be certain nonlinear or linear types both). Several previous results as specific examples can be covered in the results of this article. We emphasize that the future stability of FLRW metric for polytropic fluids with positive cosmological constant has been a difficult problem and can not be directly generalized from the previous known results. Our result in this …
It was recently shown that Strong Cosmic Censorship might be violated for near-extremally-charged black holes in 4-dimensional de Sitter space under scalar perturbations. Here, we extend the study of neutral massless scalar perturbations in higher dimensions and discuss the dimensional influence on the validity of Strong Cosmic Censorship hypothesis. By giving an elaborate description of neutral massless scalar perturbations of Reissner-Nordström-de Sitter black holes in $d=4,5$ and $6$ dimensions we conclude that Strong Cosmic Censorship is violated near extremality.
Spacetimes admitting appropriate spatial homothetic Killing vectors are called spatially homothetic spacetimes. Such spacetimes conform to the fact that gravity has no length-scale for matter inhomogeneities. The matter density for such spacetimes is (spatially) arbitrary and the matter generating the spacetime admits {\it any} equation of state. Spatially homothetic spacetimes necessarily possess energy-momentum fluxes. We first discuss spherically symmetric and axially symmetric examples of such spacetimes that do not form naked singularities for regular initial data. We then show that the C…
Within the framework of adiabatic regularization, we present a simple formalism to calculate number density and renormalized energy-momentum density of spin 1/2 particles in spatially flat FLRW spacetimes using an appropriate WKB ansatz for the adiabatic expansion for the field modes. The conformal and axial anomalies thus found are in exact agreement with those obtained from other renormalization methods. This formalism can be considered as an appropriate extension of the techniques originally introduced for scalar fields, applicable to fermions in curved space.
This paper provides an analytical treatment of accelerated and geodesic motion within the framework of the Friedmann -Lemaitre-Robertson-Walker (FLRW) spacetime. By employing conformal time transformations we manage to convert second order differential equations of motion in FLRW spacetime to first order equations in the conformally transformed spacetime. This allows us to derive a general analytical solution in closed-form for accelerated motion in spatially curved FLRW spacetime. We provide few examples of this general solution for the spatially flat cases. The last part of our work focuses …
School of Astronomy, Institute for Research in Fundamental Sciences (IPM)（基础科学研究学院天文学系）；Department of Physics and Astronomy, University of Missouri（密苏里大学物理与天文系）
In general relativity, an IDEAL (Intrinsic, Deductive, Explicit, ALgorithmic) characterization of a reference spacetime metric $g_0$ consists of a set of tensorial equations $T[g]=0$, constructed covariantly out of the metric $g$, its Riemann curvature and their derivatives, that are satisfied if and only if $g$ is locally isometric to the reference spacetime metric $g_0$. The same notion can be extended to also include scalar or tensor fields, where the equations $T[g,ϕ]=0$ are allowed to also depend on the extra fields $ϕ$. We give the first IDEAL characterization of cosmological FLRW spacet…
The quasinormal resonant modes of massless neutral fields in near-extremal Kerr-Newman-de Sitter black-hole spacetimes are calculated in the eikonal regime. It is explicitly proved that, in the angular momentum regime ${\bar a}> \sqrt{{1-2{\barΛ}\over{4+{\barΛ}/3}}}$, the black-hole spacetimes are characterized by slowly decaying resonant modes which are described by the compact formula $\Imω(n)=κ_+\cdot(n+{1\over2})$ [here the physical parameters $\{{\bar a},κ_+,{\barΛ},n\}$ are respectively the dimensionless angular momentum of the black hole, its characteristic surface gravity, the dimensio…
We investigate the asymptotia of decelerating and spatially flat FLRW spacetimes at future null infinity. We find that the asymptotic algebra of diffeomorphisms can be enlarged to the recently discovered Weyl-BMS algebra for asymptotically flat spacetimes by relaxing the boundary conditions. This algebra remains undeformed in the cosmological setting contrary to previous extensions of the BMS algebra. We then study the equations of motion for asymptotically FLRW spacetimes with finite fluxes and show that the dynamics is fully constrained by the energy-momentum tensor of the source. Finally, w…
The geodesic deviation equation (`GDE') provides an elegant tool to investigate the timelike, null and spacelike structure of spacetime geometries. Here we employ the GDE to review these structures within the Friedmann--Lema\^ıtre--Robertson--Walker (`FLRW') models, where we assume the sources to be given by a non-interacting mixture of incoherent matter and radiation, and we also take a non-zero cosmological constant into account. For each causal case we present examples of solutions to the GDE and we discuss the interpretation of the related first integrals. The de Sitter spacetime geometry …
We consider an exponentially expanding, flat, Friedmann-Lemaitre-Robertson-Walker (FLRW) Universe filled with a free Schroedinger field. The probability fluid of the latter is used to mimic the cosmological fluid (baryonic plus dark matter), thus providing the matter density and pressure terms in the corresponding Friedmann-Lemaitre equations. We first obtain the eigenfunctions of the Laplacian operator on flat FLRW space. A quantum operator qualifying as a cosmological constant is defined to act on the Schroedinger field. We then compute the matrix representing the cosmological constant in th…
In the present article, we show that a simple modification to the Einstein-Hilbert action can explain the possibility of mutual interaction between the cosmic fluids. That is achieved considering the Weyl Integrable Spacetime in the background of a nonflat Friedmann-Lemnaître-Robertson-Walker geometry for the universe. We show that widely-known phenomenological interacting cosmological scenarios can naturally appear in this context. We then performed the dynamical system analysis of the underlying cosmological scenario and explored many possibilities extracted from this gravitational theory.
Strong cosmic censorship holds that given suitable initial data on a spacelike hypersurface, the laws of general relativity should determine, completely and uniquely, the future evolution of the spacetime. Here it is argued that while strong cosmic censorship is enforced for all black holes residing in asymptotically flat spacetime, it is violated (within the classical formulation of general relativity) for some black holes residing in non asymptotically flat spacetime. It is suggested that the semi-classical formulation of general relativity might enforce strong cosmic censorship.
It has recently been shown that the strong cosmic censorship conjecture can be violated by the massless neutral scalar field in the nearly extremal Reissner-Nordstrom-de Sitter black hole. However, the formation of such a black hole by gravitational collapse necessitates the presence of the charged sector on top of the Einstein-Maxwell system. Thus we numerically calculate the quasi-normal modes for a massless charged scalar field in the Reissner-Nordstrom-de Sitter spacetime by generalizing the characteristic formulation to the charged case. As a result, the strong cosmic censorship turns out…
The general formalism of the free Dirac fermions on spatially flat $(1+3)$-dimensional Friedmann-Lema\^ itre-Robertson-Walker (FLRW) spacetimes is developed in momentum representation. The mode expansions in terms of the fundamental spinors satisfying the charge conjugation and normalization conditions are used for deriving the structure of the anti-commutator matrix-functions and, implicitly, of the retarded, advanced, and Feynman fermion propagators. The principal result is that the new type of integral representation we proposed recently in the de Sitter case can be applied to the Dirac fer…
We present the Fermi story of strong cosmic censorship in the near-extremal Reissner-Nordstrom-de Sitter black hole. To this end, we first derive from scratch the criterion for the quasi-normal modes of Dirac field to violate strong cosmic censorship in such a background, which turns out to be exactly the same as those for Bose fields, although the involved energy momentum tensor is qualitatively different from that for Bose fields. Then to extract the low-lying quasi-normal modes by Prony method, we apply Crank-Nicolson method to evolve our Dirac field in the double null coordinates. As a res…
A number of near-extremal conditions are utilized to simplify the equation of motion of the neutral scalar perturbations in generalized spherically symmetric black hole background into a differential equation with the Pöschl-Teller potential. An analytic formula for quasinormal frequencies is obtained. The analytic formula is then used to investigate Strong Cosmic Censorship conjectures~(SCC) of the generalized black hole spacetime for the smooth initial data. The Christodoulou version of the SCC is found to be violated for certain regions of the black hole parameter space including the black …
We construct a gedanken experiment in which an extremal Kerr black hole interacts with a test electromagnetic field. Using Teukolsky's solutions for electromagnetic perturbations in Kerr spacetime, and the conservation laws imposed by the energy momentum tensor of the electromagnetic field and the Killing vectors of the spacetime, we prove that this interaction cannot convert the black hole into a naked singularity, thus cosmic censorship conjecture is not violated in this case.
It is proved that dynamically formed Reissner-Nordström-de Sitter (RNdS) black holes, which have recently been claimed to provide counter-examples to the Penrose strong cosmic censorship conjecture, are characterized by unstable (singular) inner Cauchy horizons. The proof is based on analytical techniques which explicitly reveal the fact that {\it charged} massive scalar fields in the charged RNdS black-hole spacetime are characterized, in the large-coupling regime, by quasinormal resonant frequencies with $\Imω^{\text{min}}<{1\over2}κ_+$, where $κ_+$ is the surface gravity of the black-hole e…
We show that the big bang is a coordinate singularity for a large class of $k = -1$ inflationary FLRW spacetimes which we have dubbed 'Milne-like.' By introducing a new set of coordinates, the big bang appears as a past boundary of the universe where the metric is no longer degenerate -- a result which has already been investigated in the context of vacuum decay [12]. We generalize their results and approach the problem from a more mathematical perspective. Similar to how investigating the geometrical properties of the $r = 2m$ event horizon in Schwarzschild led to a better understanding of bl…
Generalizing earlier results on dust collapse in higher dimensions, we show here that cosmic censorship can be restored in gravitational collapse with tangential pressure present if we take the spacetime dimension to be $N\ge6$. This is under conditions to be motivated physically, such as the smoothness of initial data from which the collapse develops. The models considered here incorporating a non-zero tangential pressure include the Einstein cluster spacetime.
We investigate toroidal Marginally Outer Trapped Surfaces (MOTS) and Marginally Outer Trapped Tubes (MOTT) in closed Friedmann-Lemaitre-Robertson-Walker (FLRW) geometries. They are constructed by embedding Constant Mean Curvature (CMC) Clifford tori in a FLRW spacetime. This construction is used to assess the quality of certain isoperimetric inequalities, recently proved in axial symmetry. Similarly to spherically symmetric MOTS existing in FLRW spacetimes, the toroidal ones are also unstable.
