Our understanding of observed Gravitational Waves (GWs) comes from matching data to known signal models describing General Relativity (GR). These models, expressed in the post-Newtonian formalism, contain the mathematical constant $π$. Allowing $π$ to vary thus enables a strong, universal and generalisable null test of GR. From a population of 22 GW observations, we make an astrophysical measurement of $π=3.115^{+0.048}_{-0.088}$, and prefer GR as the correct theory of gravity with a Bayes factor of 321. We find the variable $π$ test robust against simulated beyond-GR effects.
A hypersurface composed of two null sheets, or "light fronts", swept out by the two congruences of future null normal geodesics emerging from a spacelike 2-disk can serve as a Cauchy surface for a region of spacetime. Already in the 1960s free (unconstrained) initial data for vacuum general relativity were found for hypersurfaces of this type. Here the Poisson brackets of such free initial data are calculated from the Hilbert action. The brackets obtained can form the starting point for a constraint free canonical quantization of general relativity and may be relevant to holographic entropy bo…
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 present a convenient null gauge for the construction of the balanced equations of motion. This null gauge has the property that the asymptotic structure is intimately related to the interior one; in particular there is a strong connexion between the field equation and the balanced equations of motion. We present the balanced equations of motion in second order of the acceleration. We solve the required components of the field equation at their respective required orders, $G^2$ and $G^3$. We indicate how this approach can be extended to higher orders.
In this paper we obtain a 2+2 double null Hamiltonian description of General Relativity using only the (complex) SO(3) connection and the components of the complex densitised self-dual bivectors. We carry out the general canonical analysis of this system and obtain the first class constraint algebra entirely in terms of the self-dual variables. The first class algebra forms a Lie algebra and all the first class constraints have a simple geometrical interpretation.
The Hamiltonian formulation of general relativity on a null surface is established in the teleparallel geometry. No particular gauge conditons on the tetrads are imposed, such as the time gauge condition. By means of a 3+1 decomposition the resulting Hamiltonian arises as a completely constrained system. However, it is structurally different from the the standard Arnowitt-Deser-Misner (ADM) type formulation. In this geometrical framework the basic field quantities are tetrads that transform under the global SO(3,1) and the torsion tensor.
We describe neutrino radiation in general relativity by introducing the energy-momentum tensor of a null fluid into the Einstein equations. Investigating the geometry and analysis at null infinity, we prove that a component of the null fluid enlarges the Christodoulou memory effect of gravitational waves. The description of neutrinos in general relativity as a null fluid can be regarded as a limiting case of a more general description using the massless limit of the Einstein-Vlasov system. The present authors with co-authors have work in progress to generalize the results of this paper using t…
In this paper, we analysis the counter-term for the general relativity in the Palatini framework. The expression is valid for both the null boundary and non-null boundary. We show that final results coincide with that in Ref.\cite{pad1} which starts form the Einstein-Hilbert action.
A hypersurface formed of two null sheets, or "light fronts", swept out by the future null normal geodesics emerging from a common spacelike 2-disk can serve as a Cauchy surface for a region of spacetime. Already in the 1960s free (unconstrained) initial data for general relativity were found for such hypersurfaces. Here an expression is obtained for the symplectic 2-form of vacuum general relativity in terms of such free data. This can be done, even though variations of the geometry do not in general preserve the nullness of the initial hypersurface, because of the diffeomorphism gauge invaria…
Based on work of Derrick, Coll, and Morales, we define a `symmetric' null coframe with {\it four real null covectors}. We show that this coframe is closely related to the GPS type coordinates recently introduced by Rovelli.
The characteristic formalism of numerical relativity is based on a system of coordinates aligned with outgoing null cones. While these coordinates were designed for studying gravitational waves, they can also be easily adapted to model cosmological past null cones (PNCs). Similar to observational coordinates in the observational approach to cosmology, this then provides a model that only makes use of information causally connected to an observer. However, the diameter distance, which is used as a radial coordinate, limits the model's cosmological application to the region prior to the PNC refo…
According to the theory of unimodular relativity developed by Anderson and Finkelstein, the equations of general relativity with a cosmological constant are composed of two independent equations, one which determines the null-cone structure of space-time, another which determines the measure structure of space-time. The field equations that follow from the restricted variational principle of this version of general relativity only determine the null-cone structure and are globally scale-invariant and scale-free. We show that the electromagnetic field may be viewed as a compensating gauge field…
We investigate the use of asymptotically null slices combined with stretching or compactification of the radial coordinate for the numerical simulation of asymptotically flat spacetimes. We consider a 1-parameter family of coordinates characterised by the asymptotic relation $r\sim R^{1-n}$ between the physical radius $R$ and coordinate radius $r$, and the asymptotic relation $K\sim R^{n/2-1}$ for the extrinsic curvature of the slices. These slices are asymptotically null in the sense that their Lorentz factor relative to stationary observers diverges as $Γ\sim R^{n/2}$. While $1<n\le 2$ slice…
All strategies for the treatment of future null-infinity in numerical relativity involve some form of regularization of the field equations. In a recent proposal that relies on the dual foliation formalism this is achieved by the use of an asymptotically Minkowskian generalized harmonic tensor basis. For the scheme to work however, derivatives of certain coordinate light-speeds must decay fast enough. Presently, we generalize the method of asymptotic expansions for nonlinear wave equations to treat first order symmetric hyperbolic systems. We then use this heuristic tool to extract the expecte…
Null test of general relativity based on gravitational waves
University of Chinese Academy of Sciences（中国科学院大学）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；Beijing Normal University（北京师范大学）
This preliminary report proposes integrating the Maxwell equations in Minkowski spacetime using coordinates where the spacelike surfaces are hyperboloids asymptotic to null cones at spatial infinity. The space coordinates are chosen so that Scri+ occurs at a finite coordinate and a smooth extension beyond Scri+ is obtained. The question addressed is whether a Cauchy evolution numerical integration program can be easily modified to compute this evolution. In the spirit of the von Neumann and Richtmyer artificial viscosity which thickens a shock by many orders of magnitude to facilitate numerica…
A wave-front in a space-time $\cal M$ is a family of null geodesics orthogonal to a smooth spacelike two-surface in $\cal M$; it is of some interest to know how a wave-front can fail to be a smoothly immersed surface in $\cal M$. In this paper we see that the space of null geodesics $\cal N$ of $\cal M$, considered as a contact manifold, provides a natural setting for an efficient study of the stable singularities arising in the time evolution of wave-fronts.
We use the formalism developed by Wald and Zoupas to derive explicit covariant expressions for the charges and fluxes associated with the Bondi-Metzner-Sachs symmetries at null infinity in asymptotically flat spacetimes in vacuum general relativity. Our expressions hold in non-stationary regions of null infinity, are local and covariant, conformally-invariant, and are independent of the choice of foliation of null infinity and of the chosen extension of the symmetries away from null infinity. While similar expressions have appeared previously in the literature in Bondi-Sachs coordinates (to wh…
We define and discuss various quantum operators that describe the geometry of spacetime in quantum general relativity. These are obtained by combining the Null-Surface Formulation of general relativity, recently developed, with asymptotic quantization. One of the operators defined describes a ``fuzzy'' quantum light cone structure. Others, denoted ``spacetime-point operators'', characterize geometrically-defined physical points. We discuss the interpretation of these operators. This seems to suggest a picture of quantum spacetime as made of ``fuzzy'' physical points. We derive the commutation …
In the spirit of the Newtonian theory, we characterize spherically symmetric empty space in general relativity in terms of energy density measured by a static observer and convergence density experienced by null and timelike congruences. It turns out that space surrounding a static particle is entirely specified by vanishing of energy and null convergence density. The electrograv-dual$^{1}$ to this condition would be vanishing of timelike and null convergence density which gives the dual-vacuum solution representing a Schwarzschild black hole with global monopole charge$^{2}$ or with cloud of …
We show that the asymptotic charges associated with Lorentz symmetries on past and future null infinity match in the limit to spatial infinity in a class of asymptotically-flat spacetimes. These are spacetimes that obey the Ashtekar-Hansen definition of asymptotic flatness at null and spatial infinity and satisfy an additional set of conditions which we lay out explicitly. Combined with earlier results on the matching of supertranslation charges, this shows that all Bondi-Metzner-Sachs (BMS) charges on past and future null infinity match in the limit to spatial infinity in this class of spacet…
We revisit Winicour's affine-null metric initial value formulation of General Relativity, where the characteristic initial value formulation is set up with a null metric having two affine parameters. In comparison to past work, where the application of the formulation was aimed for the timelike-null initial value problem, we consider here a boundary surface that is a null hypersurface. All of the initial data are either metric functions or first derivatives of the metric. Given such a set of initial data, Einstein equations can be integrated in a hierarchical manner, where first a set of equat…
The oft-neglected issue of the causal structure in the flat spacetime approach to Einstein's theory of gravity is considered. Consistency requires that the flat metric's null cone be respected, but this does not happen automatically. After reviewing the history of this problem, we introduce a generalized eigenvector formalism to give a kinematic description of the relation between the two null cones, based on the Segre' classification of symmetric rank 2 tensors with respect to a Lorentzian metric. Then we propose a method to enforce special relativistic causality by using the naive gauge free…
We treat the calculation of gravitational radiation using the mixed timelike-null initial value formulation of general relativity. The determination of an exterior radiative solution is based on boundary values on a timelike worldtube $Γ$ and on characteristic data on an outgoing null cone emanating from an initial cross-section of $Γ$. We present the details of a 3-dimensional computational algorithm which evolves this initial data on a numerical grid, which is compactified to include future null infinity as finite grid points. A code implementing this algorithm is calibrated in the quasisphe…
General hypersurfaces of any causal character can be studied abstractly using the hypersurface data formalism. In the null case, we write down all tangential components of the ambient Ricci tensor in terms of the abstract data. Using this formalism, we formulate and solve in a completely abstract way the characteristic Cauchy problem of the Einstein vacuum field equations. The initial data is detached from any spacetime notion, and it is fully diffeomorphism and gauge covariant. The results of this paper put the characteristic problem on a similar footing as the standard Cauchy problem in Gene…
In general relativity, the double null foliation is one for which $d$-dimensional spacetime is foliated by two families of intersecting null hyper surfaces (i.e. surfaces whose normal vectors are null) of $(d-1)$ dimensions. Their intersection is at space like surfaces of dimension $(d-2)$. This means that the leaves of this foliation are the space like surfaces of two dimensions lower than that of spacetime which are located by looking at the bundles of light rays that are going into, and emanating from them. Using this foliation, we present a reformulation of the theory which makes explicit …
We construct a class of spacetimes (without symmetry assumptions) satisfying the vacuum Einstein equations with singular boundaries on two null hypersurfaces intersecting in the future on a 2-sphere. The metric of these spacetimes extends continuously beyond the singularities while the Christoffel symbols fail to be square integrable in a neighborhood of any point on the singular boundaries. The construction shows moreover that the singularities are stable in a suitable sense. These singularities are stronger than the impulsive gravitational spacetimes considered by Luk-Rodnianski and conjectu…
Gravitational waves with a space-translation Killing field are considered. In this case, the 4-dimensional Einstein vacuum equations are equivalent to the 3-dimensional Einstein equations with certain matter sources. This interplay between 4- and 3- dimensional general relativity can be exploited effectively to analyze issues pertaining to 4 dimensions in terms of the 3-dimensional structures. An example is provided by the asymptotic structure at null infinity: While these space-times fail to be asymptotically flat in 4 dimensions, they can admit a regular completion at null infinity in 3 dime…
We study geometric structures associated with shear-free null geodesic congruences in Minkowski space-time and asymptotically shear-free null geodesic congruences in asymptotically flat space-times. We show how in both the flat and asymptotically flat settings, complexified future null infinity acts as a "holographic screen," interpolating between two dual descriptions of the null geodesic congruence. One description constructs a complex null geodesic congruence in a complex space-time whose source is a complex world-line; a virtual source as viewed from the holographic screen. This complex nu…
Given a truncated incoming null cone and a truncated outgoing null cone intersecting at a two sphere $S$ with smooth characteristic initial data, a theorem of Rendall shows that the vacuum Einstein equations can be solved in a small neighborhood of $S$ in the future of $S$. We show that in fact the vacuum Einstein equations can be solved in a neighborhood in the future of the cones, as long as the constraint equations are initially satisfied on the null cones. The proof is based on energy type estimates and relies heavily on the null structure of the Einstein equations in the double null folia…
The main science driver for the coming generation of cosmological surveys is understanding dark energy which relies on testing General Relativity on the largest scales. Once we move beyond the simplest explanation for dark energy of a cosmological constant, the space of possible theories becomes both vast and extremely hard to compute realistic observables. A key discriminator of a cosmological constant, however, is that the growth of structure is scale-invariant on large scales. By carefully weighting observables derived from distributions of galaxies and a dipole pattern in their apparent si…
The simplest electromagnetic fields' (general- as well as special-relativistic) classification is formulated which is based on physically motivated ideas. According to this classification these fields can belong to three types (electric, magnetic and null), each of them being split in pure and impure subtypes. Only pure null type field propagates with the fundamental velocity $c$, all other fields have the propagation velocity less than that of light. The reference-frame-based methods of elimination of alternative three-fields (e.g., magnetic in the electric type case) are given for pure subty…
We show that the BMS-supertranslations and their associated supermomenta on past null infinity can be related to those on future null infinity, proving the conjecture of Strominger for a class of spacetimes which are asymptotically-flat in the sense of Ashtekar and Hansen. Using a cylindrical 3-manifold of both null and spatial directions of approach towards spatial infinity, we impose appropriate regularity conditions on the Weyl tensor near spatial infinity along null directions. The asymptotic Einstein equations on this 3-manifold and the regularity conditions imply that the relevant Weyl t…
This paper is aimed to elaborate the problem of energy-momentum in General Relativity. In this connection, we use the prescriptions of Einstein, Landau-Lifshitz, Papapetrou and Möller to compute the energy-momentum densities for two exact solutions of Einstein field equations. The spacetimes under consideration are the non-null Einstein-Maxwell solutions and the singularity-free cosmological model. The electromagnetic generalization of the Gödel solution and the Gödel metric become special cases of the non-null Einstein-Maxwell solutions. It turns out that these prescriptions do not provide co…
In general relativity, an idealized distant observer is situated at future null infinity where light rays emitted from the source approach. This article concerns conserved quantities such as mass, energy-momentum, angular momentum, and center of mass at future null infinity. The classical definitions of Bondi mass at future null infinity ascertains the mass radiated away in gravitational waves distinctively. However, the same question for other conserved quantities such as angular momentum has been a subtle issue since the discovery of "supertranslation ambiguity" in the 1960's. Recently, new …
The dual-frame formalism leads to an approach to extend numerical relativity simulations in generalized harmonic gauge (GHG) all the way to null infinity. A major setback is that without care, even simple choices of initial data give rise to logarithmically divergent terms that would result in irregular variables and equations on the compactified domain, which would in turn prevent accurate numerical approximation. It has been shown, however, that a suitable choice of gauge and constraint addition can be used to prevent their appearance. Presently we give a first order symmetric hyperbolic red…
It is possible that null paths in 5D appear as the timelike paths of massive particles in 4D, where there is an oscillation in the fifth dimension around the hypersurface we call spacetime. A particle in 5D may be regarded as multiply imaged in 4D, and the 4D weak equivalence principle may be regarded as a symmetry of the 5D metric.
Event and apparent horizons are key diagnostics for the presence and properties of black holes. In this article I review numerical algorithms and codes for finding event and apparent horizons in numerically-computed spacetimes, focusing on calculations done using the 3+1 ADM formalism. There are 3 basic algorithms for finding event horizons, based respectively on integrating null geodesics \emph{forwards} in time, integrating null geodesics \emph{backwards} in time, and integrating null \emph{surfaces} backwards in time. The last of these is generally the most efficient and accurate. There are…
A null path in 5D can appear as a timelike path in 4D, and for a certain gauge in 5D the motion of a massive particle in 4D obeys the usual quantization rule with an uncertainty-type relation. Generalizations of this result are discussed in regard to induced-matter and membrane theory.
The properties of multimomentum maps on null hypersurfaces, and their relation with the constraint analysis of General Relativity, are described. Unlike the case of spacelike hypersurfaces, some constraints which are second class in the Hamiltonian formalism turn out to contribute to the multimomentum map.
We use a trick similar to Weinberg's for adiabatic modes, in a Manton approximation for general relativity on manifolds with spatial boundary. This results in a description of the slow-time dependent solutions as null geodesics on the space of boundary diffeomorphisms, with respect to a metric we prove to be composed solely of the boundary data. We show how the solutions in the bulk space is determined with the constraints of general relativity. To give our description a larger perspective, we furthermore identify our resulting Lagrangian as a generalized version of the covariantized Lagrangia…
The peeling theorem of general relativity predicts that the Weyl curvature scalars Psi_n (n=0...4), when constructed from a suitable null tetrad in an asymptotically flat spacetime, fall off asymptotically as r^(n-5) along outgoing radial null geodesics. This leads to the interpretation of Psi_4 as outgoing gravitational radiation at large distances from the source. We have performed numerical simulations in full general relativity of a binary black hole inspiral and merger, and have computed the Weyl scalars in the standard tetrad used in numerical relativity. In contrast with previous result…
We provide a covariant derivation of plasma physics coupled to gravitation by utilizing the 3+1 formulation of general relativity, including a discussion of the Lorentz force law. We then reduce the system to the spherically symmetric case and show that all regions of the spacetime can be represented in a single coordinate system, thus revoking the need for junction conditions. We further show that the region exterior to the collapsing region is naturally described by the charged Vaidya spacetime in non-null coordinates.
The Goldberg-Sachs theorem is an exact result on shear-free null geodesics in a vacuum spacetime. It is compared and contrasted with an exact result for pressure-free matter: shear-free flows cannot both expand and rotate. In both cases, the shear-free condition restricts the way distant matter can influence the local gravitational field. This leads to intriguing discontinuities in the relation of the General Relativity solutions to Newtonian solutions in the timelike case, and of the full theory to the linearised theory in the null case. It is a pleasure to dedicate this paper to Josh Goldber…
We prove that for spacetimes solving the Einstein-Maxwell (EM) equations, the electromagnetic field contributes at highest order to the nonlinear memory effect of gravitational waves. In [5] D. Christodoulou showed that gravitational waves have a nonlinear memory. He discussed how this effect can be measured as a permanent displacement of test masses in a laser interferometer gravitational wave detector. Christodoulou derived a precise formula for this permanent displacement in the Einstein vacuum (EV) case. We prove in Theorem 6 that for the EM equations this permanent displacement exhibits a…
In general relativity, astrophysical black holes are simple objects, being described by just their mass and spin. These simple solutions are not exclusive to general relativity, as they also appear in theories that allow for an extra scalar degree of freedom. Recently, it was shown that some theories which couple a scalar field with the Gauss-Bonnet invariant can have the same classic general relativity black hole solutions as well as hairy black holes. These scalarized solutions can be stable, having an additional "charge" term that has an impact on the gravitational-wave emission by black ho…
Consider the characteristic initial value problem for the Einstein vacuum equations without any symmetry assumptions. Impose a sequence of data on two intersecting null hypersurfaces, each of which is foliated by spacelike $2$-spheres. Assume that the sequence of data is such that the derivatives of the metrics along null directions are only uniformly bounded in $L^2$ but the derivatives of the metrics along the directions tangential to the $2$-spheres obey higher regularity bounds uniformly. By the results in [J. Luk and I. Rodnianski, Nonlinear interaction of impulsive gravitational waves fo…
The accurate modeling of gravitational radiation is a key issue for gravitational wave astronomy. As simulation codes reach higher accuracy, systematic errors inherent in current numerical relativity wave-extraction methods become evident, and may lead to a wrong astrophysical interpretation of the data. In this paper, we give a detailed description of the Cauchy-characteristic extraction technique applied to binary black hole inspiral and merger evolutions to obtain gravitational waveforms that are defined unambiguously, that is, at future null infinity. By this method we remove finite-radius…
