arXiv subjects
Publications and source records attributed to Gustavo Dotti.
The conjecture that compact trapped submanifolds (CTMs) of any codimension greater than one cannot intersect the domain of outer communications of a black hole is tested in symmetrically collapsing spacetimes of $n+1$ dimensions, $n \geq 2$, and on the entire Kerr-Newman sub-extreme family. The results provide evidence to the idea that CTMs of lower dimension, such as trapped loops, should be ragarded as black hole signatures.
We study the curvature conditions introduced in [Class. Quant. Grav. 27, 152002] to predict focal points for trapped spacelike submanifolds in spacetimes of arbitrary dimensions, with the purpose of generalizing Penrose's singularity theorem to compact trapped submanifolds (CTMs) of codimension higher than two. We find that these conditions do not apply in general but may apply for specific CTMs. As a result, higher codimension CTMs may still work as singularity predictors, although the possibility that they intersect the domain of outer communications cannot be ruled out using standard arguments.
We introduce the concept of $k-$future convex spacelike/null hypersurface $\Sigma$ in an $n+1$ dimensional spacetime $M$ and prove that no $k-$dimensional closed trapped submanifold (k-CTM) can be tangent to $\Sigma$ from its future side. As a consequence, k-CTMs cannot be found in open spacetime regions foliated by such hypersurfaces. In gravitational collapse scenarios, specific hypersurfaces of this kind act as past barriers for trapped submanifolds. A number of examples are worked out in detail, two of them showing 3+1 spacetime regions containing trapped loops ($k=1$) but no closed trapped surfaces ($k=2$). The use of trapped loops as an early indicator of black hole formation is briefly discussed.
A simple criterion is given to rule out the existence of closed trapped surfaces in large open regions inside black holes.
We study a family of solutions of Einstein-non linear sigma models with $S^2$ and $SU(2) \sim S^3$ target manifolds. In the $S^2$ case, the solutions are smooth everywhere, free of conical singularities, and approach asymptotically the metric of a cosmic string, with a mass per length that is proportional to the absolute value of the winding number from topological spheres onto the target $S^2$. This gives an interesting example of a relation between a mass and a topological charge. The case with target $SU(2)$ generalizes the stationary solution found in Eur. Phys. J. C (2021) 81:55 to parallel waves with a non-planar wavefront $\mathcal{W}$. We prove that these $\mathcal{W}$-fronted parallel waves are sub-quadratic in the classification in Class. Quant. Grav. \textbf{20} (2003) 2275, and thus causally well behaved. These spacetimes have a non-vanishing baryon current and their geometry has many striking features.
These notes follow from a course delivered at the V José Pl\'ınio Baptista School of Cosmology, held at Guarapari (Esp\'ırito Santo) Brazil, from 30 September to 5 October 2021. A review of the current status of the linear stability of black holes and naked singularities is given. The standard modal approach, that takes advantage of the background symmetries and analyze separately the harmonic components of linear perturbations, is briefly introduced and used to prove that the naked singularities in the Kerr--Newman family, as well as the inner black hole regions beyond Cauchy horizons, are unstable and therefore unphysical. The proofs require a treatment of the boundary condition at the timelike boundary, which is given in detail. The nonmodal linear stability concept is then introduced, and used to prove that the domain of outer communications of a Schwarzschild black hole with a non-negative cosmological constant satisfies this stronger stability condition, which rules out transient growths of perturbations, and also to show that the perturbed black hole settles into a slowly rotating Kerr black hole. The encoding of the perturbation fields in gauge invariant curvature scalars and the effects of the perturbation on the geometry of the spacetime is discussed.
This paper is a companion of [Phys. Rev. D 95, 124041 (2017)] in which, following a program on black hole nonmodal linear stability initiated in Phys. Rev. Lett. 112 (2014) 191101, odd perturbations of the Einstein-Maxwell equations around a Reissner-Nordström (A)dS black hole were analyzed. Here we complete the proof of the nonmodal linear stability of this spacetime by analyzing the even sector of the linear perturbations. We show that all the gauge invariant information in the metric and Maxwell field even perturbations is encoded in two spacetime scalars: ${\mathcal S}$, which is a gauge invariant combination of $δ(C_{αβγε}C^{αβγε})$ and $δ(C_{αβγδ} F_{αβ} F^{γδ})$, and ${\mathcal T}$, a gauge invariant combination of $δ( \nabla _μF_{αβ} \nabla^μF^{αβ})$ and $δ( \nabla_μ C_{αβγδ} \nabla^μC^{αβγδ})$. Here $C_{αβγδ}$ is the Weyl tensor, $F_{αβ}$ the Maxwell field and $δ$ means first order variation. We prove that $\mathcal{S}$ and $\mathcal{T}$ are are in one-one correspondence with gauge classes of even linear perturbations, and that the linearized Einstein-Maxwell equations imply that these scalar fields are pointwise bounded on the outer static region.
The static region outside the event horizon of an asymptotically anti de Sitter black hole has a conformal timelike boundary $\mathscr{I}$ on which boundary conditions have to be imposed for the evolution of linear fields from initial data to be a well posed problem. Only homogeneous Dirichlet, Neumann or Robin conditions preserve the action of the background isometry group on the solution space. We study the case in which the modal decomposition of the linear field leads to potentials not diverging at the conformal timelike boundary. We prove that there is always an instability if Robin boundary conditions with large enough $γ$ (the quotient between the values of the derivative of the field and the field at the boundary) are allowed. We explain the origin of this instability, show that for modes with nonnegative potentials there is a single unstable state and prove a number of properties of this state. Although our results apply in general to 1+1 wave equations on a half infinite domain with a potential that is not singular at the boundary, our motivation is to analyze the gravitational stability of the four dimensional Schwarzschild anti de Sitter black holes (SAdS${}_4$) in the context of the black hole non modal linear stability program initiated in Phys.\ Rev.\ Lett.\ {\bf 112}, 191101 (2014), and the related supersymmetric type of duality exchanging odd and even modes. We prove that this symmetry is broken except when a combination of Dirichlet conditions in the even sector and a particular Robin condition in the odd sector is enforced, or viceversa, and that only the first of these two choices leads to a stable dynamics.
Following a program on black hole nonmodal linear stability initiated in Phys.\ Rev.\ Lett.\ {\bf 112} (2014) 191101, we study odd linear perturbations of the Einstein-Maxwell equations around a Reissner-Nordström (A)dS black hole. We show that all the gauge invariant information in the metric and Maxwell field perturbations is encoded in the spacetime scalars $\mathcal{F} =δ(F^*_{αβ} F^{αβ})$ and $\mathcal{Q} =δ(\tfrac{1}{48} C^*_{αβγδ} C^{αβγδ})$, where $C_{αβγδ}$ is the Weyl tensor, $F_{αβ}$ the Maxwell field, a star denotes Hodge dual and $δ$ means first order variation, and that the linearized Einstein-Maxwell equations are equivalent to a coupled system of wave equations for $\mathcal{F}$ and $\mathcal{Q}$. For nonnegative cosmological constant we prove that $\mathcal{F}$ and $\mathcal{Q}$ are pointwise bounded on the outer static region. The fields are shown to diverge as the Cauchy horizon is approached from the inner dynamical region, providing evidence supporting strong cosmic censorship. In the asymptotically AdS case the dynamics depends on the boundary condition at the conformal timelike boundary and there are instabilities if Robin boundary conditions are chosen.
The nonmodal linear stability of the Schwarzschild black hole established in Phys. Rev. Lett. 112 (2014) 191101 is generalized to the case of a nonnegative cosmological constant $Λ$. Two gauge invariant combinations $G_{\pm}$ of perturbed scalars made out of the Weyl tensor and its first covariant derivative are found such that the map $[h_{αβ}] \to \left( G_- \left([h_{αβ}] \right), G_+ \left([h_{αβ}] \right) \right)$ with domain the set of equivalent classes $[h_{αβ}]$ under gauge transformations of solutions of the linearized Einstein's equation, is invertible. The way to reconstruct a representative of $[h_{αβ}]$ in terms of $(G_-,G_+)$ is given. It is proved that, for an arbitrary perturbation consistent with the background asymptote, $G_+$ and $G_-$ are bounded in the the outer static region. At large times, the perturbation decays leaving a linearized Kerr black hole around the Schwarzschild or Schwarschild de Sitter background solution. For negative cosmological constant it is shown that there is a choice of boundary conditions at the time-like boundary under which the Schwarzschild anti de Sitter black hole is unstable. The root of Chandrasekhar's duality relating odd and even modes is exhibited, and some technicalities related to this duality and omitted in the original proof of the $Λ=0$ case are explained in detail.
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.
We show that a spacetime satisfying the linearized vacuum Einstein equations around a type D background is generically of type I, and that the splittings of the Principal Null Directions (PNDs) and of the degenerate eigenvalue of the Weyl tensor are non analytic functions of the perturbation parameter of the metric. This provides a gauge invariant characterization of the effect of the perturbation on the underlying geometry, without appealing to differential curvature invariants. This is of particular interest for the Schwarzschild solution, for which there are no signatures of the even perturbations on the algebraic curvature invariants. We also show that, unlike the general case, the unstable even modes of the Schwarzschild naked singularity deforms the Weyl tensor into a type II one.
A proof is given that the space $Ł$ of solutions of the linearized vacuum Einstein's equation around a Schwarzschild black hole is parameterized by two scalar fields which are gauge invariant combinations of perturbed algebraic and differential invariants of the Weyl tensor, and encode the information on the odd (-) and even (+) sectors $Ł_{\pm}$ These fields measure the distortion of the geometry caused by a generic perturbation, and are shown to be pointwise bounded on the outer region $r \geq 2M$.
We study the scalar wave equation on the open exterior region of an extreme Reissner-Nordström black hole and prove that, given compactly supported data on a Cauchy surface orthogonal to the timelike Killing vector field, the solution, together with its $(t,s,θ,ϕ)$ derivatives of arbitrary order, $s$ a tortoise radial coordinate, is bounded by a constant that depends only on the initial data. Our technique does not allow to study transverse derivatives at the horizon, which is outside the coordinate patch that we use. However, using previous results that show that second and higher transverse derivatives at the horizon of a generic solution grow unbounded along horizon generators, we show that any such a divergence, if present, would be milder for solutions with compact initial data.
We show that both the interior region $r M^2$ Kerr naked singularity admit unstable solutions of the Teukolsky equation for any value of the spin weight. For every harmonic number there is at least one axially symmetric mode that grows exponentially in time and decays properly in the radial directions. These can be used as Debye potentials to generate solutions for the scalar, Weyl spinor, Maxwell and linearized gravity field equations on these backgrounds, satisfying appropriate spatial boundary conditions and growing exponentially in time, as shown in detail for the Maxwell case. It is suggested that the existence of the unstable modes is related to the so called "time machine" region, where the axial Killing vector field is time-like, and the Teukolsky equation, restricted to axially symmetric fields, changes its character from hyperbolic to elliptic.
We present a generalization of previous results regarding the stability under gravitational perturbations of nakedly singular super extreme Kerr spacetime and Kerr black hole interior beyond the Cauchy horizon. To do so we study solutions to the radial and angular Teukolsky's equations with different spin weights, particulary $s=\pm 1$ representing electromagnetic perturbations, $s=\pm 1/2$ representing a perturbation by a Dirac field and $s=0$ representing perturbations by a scalar field. By analizing the properties of radial and angular eigenvalues we prove the existence of an infinite family of unstable modes.
Reissner--Nordström black holes have two static regions: $r > \ro$ and $0 < r < \ri$, where $\ri$ and $\ro$ are the inner and outer horizon radii. The stability of the exterior static region has been established long time ago. In this work we prove that the interior static region is unstable under linear gravitational perturbations, by showing that field perturbations compactly supported within this region will generically excite a mode that grows exponentially in time. This result gives an alternative reason to mass inflation to consider the space time extension beyond the Cauchy horizon as physically irrelevant, and thus provides support to the strong cosmic censorship conjecture, which is also backed by recent evidence of a linear gravitational instability in the interior region of Kerr black holes found by the authors. The use of intertwiners to solve for the evolution of initial data plays a key role, and adapts without change to the case of super-extremal \rn black holes, allowing to complete the proof of the linear instability of this naked singularity. A particular intertwiner is found such that the intertwined Zerilli field has a geometrical meaning -it is the first order variation of a particular Riemann tensor invariant-. Using this, calculations can be carried out explicitely for every harmonic number.
The classification of certain class of static solutions for the Einstein-Gauss-Bonnet theory in vacuum is performed in $d\geq5$ dimensions. The class of metrics under consideration is such that the spacelike section is a warped product of the real line and an arbitrary base manifold. It is shown that for a generic value of the Gauss-Bonnet coupling, the base manifold must be necessarily Einstein, with an additional restriction on its Weyl tensor for $d>5$. The boundary admits a wider class of geometries only in the special case when the Gauss-Bonnet coupling is such that the theory admits a unique maximally symmetric solution. The additional freedom in the boundary metric enlarges the class of allowed geometries in the bulk, which are classified within three main branches, containing new black holes and wormholes in vacuum.