We analyze the relationship between $n$-dimensional conformal metrics and a certain class of partial differential equations (PDEs) that are in duality with the eikonal equation. In particular, we extend the Null Surface Formulation of General Relativity to higher dimensions and give explicit expressions for the components of the metric and the generalized Wünschmann-like metricity conditions. We also compute the equation that the conformal factor must satisfy in order the metric be a solution of the Einstein equations.
In this article we describe the formulation of null geodesics as null conformal geodesics and their description in the tractor formalism. A conformal extension theorem through an isotropic singularity is proven by requiring the boundedness of the tractor curvature and its derivatives to sufficient order along a congruence of null conformal geodesic. This article extends earlier work by Tod and Luebbe.
We study the space-times admitting a null conformal Killing-Yano tensor whose divergence defines a Killing vector. We analyze the similitudes and differences with the recently studied non null case (Gen. Relativ. Grav. (2015) {\bf 47} 1911). The results by Barnes concerning the Birkhoff theorem for the case of null orbits are analyzed and generalized.
Space-times admitting a 3-dimensional Lie group of conformal motions $C_3$ acting on null orbits are studied. Coordinate expressions for the metric and the conformal Killing vectors (CKV) are provided (irrespectively of the matter content) and then all possible perfect fluid solutions are found, although none of these verify the weak and dominant energy conditions over the whole space-time manifold.
A new approach to space-time asymptotics is presented, refining Penrose's idea of conformal transformations with infinity represented by the conformal boundary of space-time. Generalizing examples such as flat and Schwarzschild space-times, it is proposed that the Penrose conformal factor be a product of advanced and retarded conformal factors, which asymptotically relate physical and conformal null (light-like) coordinates and vanish at future and past null infinity respectively, with both vanishing at spatial infinity. A correspondingly refined definition of asymptotic flatness at both spati…
We analyze the decomposition of the enveloping algebra of the conformal algebra in arbitrary dimension with respect to the mass-squared operator. It emerges that the subalgebra that commutes with the mass-squared is generated by its Poincare subalgebra together with a vector operator. The special cases of the conformal algebras of two and three dimensions are described in detail, including the construction of their Casimir operators.
Space-times admitting a 3-dimensional Lie group of conformal motions $C_3$ acting on null orbits are studied. Coordinate expressions for the metric and the conformal Killing vectors (CKV) are then provided (irrespectively of the matter content) and all possible perfect fluid solutions are found, although none of them verifies the weak and dominant energy conditions over the whole space-time manifold.
We show that for general relativity in odd spacetime dimensions greater than 4, all components of the unphysical Weyl tensor for arbitrary smooth, compact spatial support perturbations of Minkowski spacetime fail to be smooth at null infinity at leading nonvanishing order. This implies that for nearly flat radiating spacetimes, the non-smoothness of the unphysical metric at null infinity manifests itself at the same order as it describes deviations from flatness of the physical metric. Therefore, in odd spacetime dimensions, it does not appear that conformal null infinity can be in any way use…
This article uses the conformal Einstein equations and the conformal representation of spatial infinity introduced by Friedrich to analyse the behaviour of the gravitational field near null and spatial infinity for the development of initial data which are, in principle, non-conformally flat and time asymmetric. This article is the continuation of the investigation started in Class. Quantum Grav. 21 (2004) 5457-5492, where only conformally flat initial data sets were considered. For the purposes of this investigation, the conformal metric of the initial hypersurface is assumed to have a very p…
The conformal formulation provides a method for constructing and parametrizing solutions of the Einstein constraint equations by mapping freely chosen sets of conformal data to solutions, provided a certain set of coupled, elliptic determined PDEs (whose expression depends on the chosen conformal data) admit a unique solution. For constant mean curvature (CMC) data, it is known in almost all cases which sets of conformal data allow these PDEs to have solutions, and which do not. For non CMC data, much less is known. Here we exhibit the first class of non CMC data for which we can prove that no…
In this work we find analytical solutions to the null geodesics around a black hole in the conformal Weyl gravity. Exact expressions for the horizons are found, and they depend on the cosmological constant and the coupling constants of the conformal Weyl gravity. Then, we study the radial motion from the point of view of the proper and coordinate frames, and compare it with that found in spacetimes of general relativity. The angular motion is also examined qualitatively by means of an effective potential; quantitatively, the equation of motion is solved in terms of $\wp$-Weierstrass elliptic f…
The Conformal Einstein equations and the representation of spatial infinity as a cylinder introduced by Friedrich are used to analyse the behaviour of the gravitational field near null and spatial infinity for the development of data which are asymptotically Euclidean, conformally flat and time asymmetric. Our analysis allows for initial data whose second fundamental form is more general than the one given by the standard Bowen-York Ansatz. The Conformal Einstein equations imply upon evaluation on the cylinder at spatial infinity a hierarchy of transport equations which can be used to calculat…
We show that the extreme six-dimensional black string admits a conformal isometry under inversion. Duality relations between null geodesics of various brane geometries are demonstrated, some of which have a geometrical origin through an optical metric.
We show that the symmetry algebra of asymptotically flat four dimensional spacetimes at null infinity in the sense of Newman and Unti is isomorphic to the direct sum of the abelian algebra of infinitesimal conformal rescalings with bms4. We then work out the local conformal properties of the relevant Newman-Penrose coefficients, as well as the surface charges and their algebra.
This is the third paper in a series describing a numerical implementation of the conformal Einstein equation. This paper describes a scheme to calculate (three) dimensional data for the conformal field equations from a set of free functions. The actual implementation depends on the topology of the spacetime. We discuss the implementation and exemplary calculations for data leading to spacetimes with one spherical null infinity (asymptotically Minkowski) and for data leading to spacetimes with two toroidal null infinities (asymptotically A3). We also outline the (technical) modifications of the…
This thesis is concerned with the development and application of conformal techniques to numerical calculations of asymptotically flat spacetimes. The conformal compactification technique enables us to calculate spatially unbounded domains, thereby avoiding the introduction of an artificial timelike outer boundary. We construct in spherical symmetry an explicit scri-fixing gauge, i.e. a conformal and a coordinate gauge in which the spatial coordinate location of null infinity is independent of time so that no resolution loss in the physical part of the conformal extension appears. Going beyond…
Recently there has been developed a reformulation of General Relativity - referred to as {\it the null surface version of GR} - where instead of the metric field as the basic variable of the theory, families of three-surfaces in a four-manifold become basic. From these surfaces themselves, a conformal metric, conformal to an Einstein metric, can be constructed. A choice of conformal factor turns them into Einstein metrics. The surfaces are then automatically characteristic surfaces of this metric. In the present paper we explore the linearization of this {\it null surface theory} and compare i…
Among the coordinates used to construct a conformal compactification of the Schwarzschild spacetime, none of them simultaneously extend smoothly both through an event horizon and beyond null infinity.To construct such coordinates, instead of starting with the Kruskal-Szekeres coordinates we assume direct analytic transformation between Schwarzschild and compactified coordinates and determine their behavior on the event horizon and at null infinity. We then propose an example of such coordinates and illustrate the way they cover the conformally extended Schwarzschild spacetime as well as their …
In this paper we investigate whether the holographic principle proposed in string theory has a classical counterpart in general relativity theory. We show that there is a partial correspondence: at least in the case of vacuum Petrov type D spacetimes that admit a non-trivial Killing tensor, which encompass all the astrophysical black hole spacetimes, there exists a one to one correspondence between gravity in bulk and a two dimensional classical conformal scalar field on a null boundary.
We study the Lie point symmetries of a general class of partial differential equations (PDE) of second order. An equation from this class naturally defines a second-order symmetric tensor (metric). In the case the PDE is linear on the first derivatives we show that the Lie point symmetries are given by the conformal algebra of the metric modulo a constraint involving the linear part of the PDE. Important elements in this class are the Klein--Gordon equation and the Laplace equation. We apply the general results and determine the Lie point symmetries of these equations in various general classe…
We study the concept of Carrollian spacetime starting from its underlying fiber-bundle structure. The latter admits an Ehresmann connection, which enables a natural separation of time and space, preserved by the subset of Carrollian diffeomorphisms. These allow for the definition of Carrollian tensors and the structure at hand provides the designated tools for describing the geometry of null hypersurfaces embedded in Lorentzian manifolds. Using these tools, we investigate the conformal isometries of general Carrollian spacetimes and their relationship with the BMS group.
Using Cartan's equivalence method for point transformations we obtain from first principles the conformal geometry associated with third order ODEs and a special class of PDEs in two dimensions. We explicitly construct the null tetrads of a family of Lorentzian metrics, the conformal group in three and four dimensions and the so called normal metric connection. A special feature of this connection is that the non vanishing components of its torsion depend on one relative invariant, the (generalized) Wünschmann Invariant. We show that the above mentioned construction naturally contains the Null…
We show how the manifold $T^*SU(2, 2)$ arises as a symplectic reduction from eight copies of the twistor space. Some of the constraints in the twistor space correspond to an octahedral configuration of twelve complex light rays in the Minkowski space. We discuss a mechanism to break the conformal symmetry down to the twistorial parametrisation of $T^*SL(2, C)$ used in loop quantum gravity.
We present an algorithm for the construction of conformal coordinates in the interior of a spherically symmetric, collapsing matter cloud in general relativity. This algorithm is based on the numerical integration of the radial null geodesics and a local analysis of their behavior close to the singularity. As an application, we consider a collapsing spherical dust cloud, generate the corresponding conformal diagram and analyze the structure of the resulting singularity. A new bound on the initial data which guarantees that the singularity is visible from future null infinity is also obtained.
When Einstein's equations for an asymptotically flat, vacuum spacetime are reexpressed in terms of an appropriate conformal metric that is regular at (future) null infinity, they develop apparently singular terms in the associated conformal factor and thus appear to be ill-behaved at this (exterior) boundary. In this article however we show, through an enforcement of the Hamiltonian and momentum constraints to the needed order in a Taylor expansion, that these apparently singular terms are not only regular at the boundary but can in fact be explicitly evaluated there in terms of conformally re…
We study the scalar, conformally invariant wave equation on a four-dimensional Minkowski background in spherical symmetry, using a fully pseudospectral numerical scheme. Thereby, our main interest is in a suitable treatment of spatial infinity, which is represented as a cylinder. We consider both Cauchy problems, where we evolve data from a Cauchy surface to future null infinity, as well as characteristic initial value problems with data at past null infinity, and demonstrate that highly accurate numerical solutions can be obtained for a small number of grid points.
The null curvature condition (NCC) is the requirement that the Ricci curvature of a Lorentzian manifold be nonnegative along null directions, which ensures the focusing of null geodesic congruences. In this note, we show that the NCC together with the causal structure significantly constrain the metric. In particular, we prove that any conformal rescaling of a vacuum spacetime introduces either geodesic incompleteness or negative null curvature, provided the conformal factor is non-constant on at least one complete null geodesic. In the context of bulk reconstruction in AdS/CFT, our results co…
The Null Surface Formulation of General Relativity is developed for 2+1 dimensional gravity. The geometrical meaning of the metricity condition is analyzed and two approaches to the derivation of the field equations are presented. One method makes explicit use of the conformal factor while the other only uses conformal information. The resulting set of equations contain the same geometrical meaning as the 4-D formulation without the technical complexities of the higher dimensional analog. A canonical family of null surfaces in this formulation, the light cone cuts of null infinity, are constru…
This paper presents an efficient technique for finding Killing, homothetic, or even proper conformal Killing vectors in the Newman-Penrose (NP) formalism. Leaning on, and extending, results previously derived in the GHP formalism we show that the (conformal) Killing equations can be replaced by a set of equations involving the commutators of the Lie derivative with the four NP differential operators applied to the four coordinates. It is crucial that these operators refer to a preferred tetrad relative to the (conformal) Killing vectors, a notion to be defined. The equations can then be readil…
Linear perturbations on Minkowski space are used to probe numerically the remote region of an asymptotically flat space-time close to spatial infinity. The study is undertaken within the framework of Friedrich's conformal field equations and the corresponding conformal representation of spatial infinity as a cylinder. The system under consideration is the (linear) zero-rest-mass equation for a spin-2 field. The spherical symmetry of the underlying background is used to decompose the field into separate non-interacting multipoles. It is demonstrated that it is possible to reach null-infinity fr…
We use a new, conformally-invariant method of analysis to test incomplete null geodesics approaching the singularity in a model of an evaporating black hole for the possibility of extensions of the conformal metric. In general, a local conformal extension is possible from the future but not from the past.
A physical interpretation is presented of the general class of conformally flat pure radiation metrics that has recently been identified by Edgar and Ludwig. It is shown that, at least in the weak field limit, successive wave surfaces can be represented as null (half) hyperplanes rolled around a two-dimensional null cone. In the impulsive limit, the solution reduces to a pp-wave whose direction of propagation depends on retarded time. In the general case, there is a coordinate singularity which corresponds to an envelope of the wave surfaces. The global structure is discussed and a possible va…
We resume former discussions of the conformally invariant wave equation on a Schwarzschild background, with a particular focus on the behaviour of solutions near the 'cylinder', i.e. Friedrich's representation of spacelike infinity. This analysis can be considered a toy model for the behaviour of the full Einstein equations and the resulting logarithmic singularities that appear to be characteristic for massive spacetimes. The investigation of the Cauchy problem for the conformally invariant wave equation (Frauendiener and Hennig 2018, Class. Quantum Grav. 35, 065015) showed that solutions gen…
We describe a class of impulsive gravitational waves which propagate either in a de Sitter or an anti-de Sitter background. They are conformal to impulsive waves of Kundt's class. In a background with positive cosmological constant they are spherical (but non-expanding) waves generated by pairs of particles with arbitrary multipole structure propagating in opposite directions. When the cosmological constant is negative, they are hyperboloidal waves generated by a null particle of the same type. In this case, they are included in the impulsive limit of a class of solutions described by Siklos t…
In this article we propose a new geometrization of the radiative phase space of asymptotically flat space-times: we show that the geometry induced on null-infinity by the presence of gravitational waves can be understood to be a generalisation of the tractor calculus of conformal manifolds adapted to the case of degenerate conformal metrics. It follows that the whole formalism is, by construction, manifestly conformally invariant. We first show that a choice of asymptotic shear amounts to a choice of linear differential operator of order two on the bundle of scales of null-infinity. We refer t…
We show that the DBI conformal galileons, derived from the world-volume theory of a 3-brane moving in an AdS bulk, admit a background, stable under quantum corrections, which violates the Null Energy Condition (NEC). The perturbations around this background are stable and propagate subluminally. Unlike other known examples of NEC violation, such as ghost condensation and conformal galileons, this theory also admits a stable, Poincare-invariant vacuum, with a Lorentz-invariant S-matrix satisfying standard analyticity conditions. Like conformal galileons, perturbations around deformations of the…
Using twistor methods, we explicitly construct all local forms of four--dimensional real analytic neutral signature anti--self--dual conformal structures $(M,[g])$ with a null conformal Killing vector. We show that $M$ is foliated by anti-self-dual null surfaces, and the two-dimensional leaf space inherits a natural projective structure. The twistor space of this projective structure is the quotient of the twistor space of $(M,[g])$ by the group action induced by the conformal Killing vector. We obtain a local classification which branches according to whether or not the conformal Killing vect…
We construct a relation between the Aretakis charge of any extreme black hole and the Newman-Penrose charge. This is achieved by constructing a conformal correspondence between extreme black holes and what we call weakly asymptotically flat space-times. Under this correspondence the Newman-Penrose charge of the weakly asymptotically flat space-time maps on to the Aretakis charge of the extreme black hole. Furthermore, we generalise the conformal isometry displayed by the extreme Reissner-Nordström solution to a novel conformal symmetry that acts within a class of static STU supergravity black …
It is well-known that the $(1+1)$ dimensional Schwarzschild and spatially flat FLRW spacetimes are conformally flat. This work examines entanglement harvesting from the conformal field vacuums in these spacetimes between two Unruh-DeWitt detectors, moving along outgoing null trajectories. In $(1+1)$ dimensional Schwarzschild spacetime, we considered the Boulware and Unruh vacuums for our investigations. In this analysis, one observes that while entanglement harvesting is possible in $(1+1)$ dimensional Schwarzschild and $(1+3)$ dimensional de Sitter spacetimes, it is not possible in the $(1+1)…
We consider averaged null energy conditions (ANEC) for strongly coupled quantum field theories in even (two and four) dimensional curved spacetimes by applying the no-bulk-shortcut principle in the context of the AdS/CFT duality. In the same context but in odd-dimensions, the present authors previously derived a conformally invariant averaged null energy condition (CANEC), which is a version of the ANEC with a certain weight function for conformal invariance. In even-dimensions, however, one has to deal with gravitational conformal anomalies, which make relevant formulas much more complicated …
We develop a global twistor correspondence for pseudo-Riemannian conformal structures of signature (++--) with self-dual Weyl curvature. Near the conformal class of the standard indefinite product metric on S^2 x S^2, there is an infinite-dimensional moduli space of such conformal structures, and each of these has the surprising global property that its null geodesics are all periodic. Each such conformal structure arises from a family of holomorphic disks in CP_3 with boundary on some totally real embedding of RP^3 into CP_3. An interesting sub-class of these conformal structures are represen…
In the Cauchy problem for asymptotically flat vacuum data the solution-jets along the cylinder at space-like infinity develop in general logarithmic singularities at the critical sets at which the cylinder touches future/past null infinity. The tendency of these singularities to spread along the null generators of null infinity obstructs the development of a smooth conformal structure at null infinity. For the solution-jets arising from time reflection symmetric data to extend smoothly to the critical sets it is necessary that the Cotton tensor of the initial three-metric h satisfies a certain…
In [arXiv:0803.3259] the equations describing the parallel transport of orthonormal frames along timelike (spacelike) geodesics in a spacetime admitting a non-degenerate principal conformal Killing-Yano 2-form h were solved. The construction employed is based on studying the Darboux subspaces of the 2-form F obtained as a projection of h along the geodesic trajectory. In this paper we demonstrate that, although slightly modified, a similar construction is possible also in the case of null geodesics. In particular, we explicitly construct the parallel-transported frames along null geodesics in …
The Kastor-Traschen metric is a time-dependent solution of the Einstein-Maxwell equations with positive cosmological constant $Λ$ which can be used to describe an arbitrary number of charged dynamical black holes. In this paper, we consider the null geodesic structure of this solution, in particular, focusing on the projection to the space of orbits of the timelike conformal retraction. It is found that these projected light rays arise as integral curves of a system of third order ordinary differential equations. This system is not uniquely defined, however, and we use the inherent freedom to …
It is argued that the symmetry algebra of asymptotically flat spacetimes at null infinity in 4 dimensions should be taken as the semi-direct sum of supertranslations with infinitesimal local conformal transformations and not, as usually done, with the Lorentz algebra. As a consequence, two dimensional conformal field theory techniques will play as fundamental a role in this context of direct physical interest as they do in three dimensional anti-de Sitter gravity.
We study the null-boundaries of Wheeler-de Witt (WdW) patches in three dimensional Poincare-AdS, when the selected boundary timeslice is an arbitrary (non-constant) function, presenting some useful analytic statements about them. Special attention will be given to the piecewise smooth nature of the null-boundaries, due to the emergence of caustics and null-null joint curves. This is then applied, in the spirit of our previous paper arXiv:1806.08376, to the problem of how complexity of the CFT$_2$ groundstate changes under a small local conformal transformation according to the action (CA) prop…
We consider the question of whether solutions of Klein--Gordon equations on asymptotically Anti-de Sitter spacetimes can be uniquely continued from the conformal boundary. Positive answers were first given by the second author with G. Holzegel, under suitable assumptions on the boundary geometry and with boundary data imposed over a sufficiently long timespan. The key step was to establish Carleman estimates for Klein--Gordon operators near the conformal boundary. In this article, we further improve upon the above-mentioned results. First, we establish new Carleman estimates---and hence new un…
Carroll-Cotton Tensors and Gravitational Radiation at Null Infinity
École polytechnique, Centre National de la Recherche Scientifique – Unité Mixte de Recherche 7644, Institut Polytechnique de Paris（巴黎综合理工学院，法国国家科学研究中心联合研究单位7644，巴黎理工学院）；Université Libre de Bruxelles and International Solvay Institutes（布鲁塞尔自由大学和国际索尔维研究所）；International School for Advanced Studies (SISSA)（高级国际研究学院（SISSA））；Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Trieste（意大利国家核物理研究所的里雅斯特分部）；Université de Mons（蒙斯大学）
The generic null geodesic of the Schwarzschild--Kruskal--Szekeres geometry has a natural complexification, an elliptic curve with a cusp at the singularity. To realize that complexification as a Riemann surface without a cusp, and also to ensure conservation of energy at the singularity, requires a branched cover of the space-time over the singularity, with the geodesic being doubled as well to obtain a genus two hyperelliptic curve with an extra involution. Furthermore, the resulting space-time obtained from this branch cover has a Hamiltonian that is null geodesically complete. The full comp…
We extend earlier numerical and analytical considerations of the conformally invariant wave equation on a Schwarzschild background from the case of spherically symmetric solutions, discussed in Class. Quantum Grav. 34, 045005 (2017), to the case of general, nonsymmetric solutions. A key element of our approach is the modern standard representation of spacelike infinity as a cylinder. With a decomposition into spherical harmonics, we reduce the four-dimensional wave equation to a family of two-dimensional equations. These equations can be used to study the behaviour at the cylinder, where the s…
