Newman-Penrose Constants Research Articles

The non-linear perturbation of a black hole by gravitational waves. III. Newman–Penrose constants

In this paper we continue our study of the non-linear response of a Schwarzschild black hole to an ingoing gravitational wave by computing the Newman–Penrose (NP) constants. The NP constants are five complex, supertranslation-invariant quantities defined on null infinity I+ and although put forward in the 60’s, they have never been computed in a non-stationary setting. We accomplish this through a numerical implementation of Friedrich’s generalized conformal field equations whose semi-global evolution yields direct access to I+ . The integral expressions for the NP constants as given originally are expressed in a general gauge to enable their computation from numerically generated data. Canonical methods of fixing inherent degrees of freedom in their definitions are discussed. The NP expressions are then computed for a variety of different ingoing wave profiles in axisymmetry, and then with no symmetry assumptions in a general 3+1 setting for which all five complex quantities are non-zero. We discuss the consequence of the (numerical) constancy of the NP expressions for the (non)-smoothness of null-infinity.

On the conserved quantities in an Einstein-scalar system

The generalization of ten conserved quantities (Newman–Penrose constants) defined in asympotically flat vacuum spacetimes to asymptotically flat Einstein-scalar spacetimes is studied. It is found that these quantites, when suitably modified, are also conserved in a Einstein-scalar system.

Zero rest-mass fields and the Newman–Penrose constants on flat space

Zero rest-mass fields of spin 1 (the electromagnetic field) and spin 2 propagating on flat space and their corresponding Newman–Penrose (NP) constants are studied near spatial infinity. The aim of this analysis is to clarify the correspondence between data for these fields on a spacelike hypersurface and the value of their corresponding NP constants at future and past null infinity. To do so, Friedrich’s framework of the cylinder at spatial infinity is employed to show that, expanding the initial data in terms spherical harmonics and powers of the geodesic spatial distance ρ to spatial infinity, the NP constants correspond to the data for the second highest possible spherical harmonic at a fixed order in ρ. In addition, it is shown that for generic initial data within the class considered in this article, there is no natural correspondence between the NP constants at future and past null infinity—for both the Maxwell and spin-2 field. However, if the initial data are time-symmetric, then the NP constants at future and past null infinity have the same information.

On late time tails in an extreme Reissner–Nordström black hole: frequency domain analysis

In this paper, we revisit the study of the leading order late time decay tails of massless scalar perturbations outside an extreme Reissner–Nordström black hole. Previous authors have analysed this problem in the time domain; we analyse the problem in the frequency domain. We first consider initial perturbations with generic regular behaviour across the horizon on characteristic surfaces. For this set-up, we reproduce some of the previous results of Sela (2016 Phys. Rev. D 93 024054) using Fourier methods. Next we consider related initial data on hypersurfaces, and present decay results at timelike infinity, near future null infinity, and near the future horizon. Along the way, using the inversion symmetry of the extreme Reissner–Nordström spacetime, we relate the higher multipole Aretakis and Newman–Penrose constants for a massless scalar in this background.

The Characteristic Gluing Problem and Conservation Laws for the Wave Equation on Null Hypersurfaces

We obtain necessary and sufficient conditions for the existence of “conservation laws” on null hypersurfaces for the wave equation on general four-dimensional Lorentzian manifolds. Examples of null hypersurfaces exhibiting such conservation laws include the standard null cones of Minkowski spacetime and the degenerate horizons of extremal black holes. Another (limiting) example of such a conservation law is that which gives rise to the well-known Newman–Penrose constants along the null infinity of asymptotically flat spacetimes. The existence of such conservation laws can be viewed as an obstruction to a certain gluing construction for characteristic initial data for the wave equation. We initiate the general study of the latter gluing problem and show that the existence of conservation laws is in fact the only obstruction. Our method relies on a novel elliptic structure associated to a foliation with 2-spheres of a null hypersurface.

Relating the Newman-Penrose constants to the Geroch–Hansen multipole moments

In these proceedings we investigate the relation between the Newman-Penrose constants and the Geroch-Hansen multipole moments for stationary spacetimes. This is short description of the results of the paper [1] published in Classical and Quantum Gravity. This relation gives a simple characterization of which stationary spacetimes have vanishing Newman-Penrose constants.

Relating the Newman–Penrose constants to the Geroch–Hansen multipole moments

In this paper, we express the Newman–Penrose constants in terms of the Geroch–Hansen multipole moments for stationary spacetimes. These expressions are translation-invariant combinations of the multipole moments up to quadrupole order, which do not normally vanish.

Newman-Penrose constants of stationary electrovacuum space-times

A theorem related to the Newman-Penrose constants is proven. The theorem states that all the Newman-Penrose constants of asymptotically flat, stationary, asymptotically algebraically special electrovacuum space-times are zero. Straightforward application of this theorem shows that all the Newman-Penrose constants of the Kerr-Newman space-time must vanish.

Uniqueness of Kerr space-time near null infinity

We reexpress the Kerr metric in standard Bondi-Sachs coordinates near null infinity I(+). Using the uniqueness result of the characteristic initial value problem, we prove the Kerr metric is the only asymptotically flat, stationary, axially symmetric, type-D solution of the vacuum Einstein equation. The Taylor series of Kerr space-time is expressed in terms of Bondi-Sachs coordinates, and the Newman-Penrose constants have been calculated.

Newman-Penrose constants of the Kerr-Newman metric

The Newman-Unti formalism of the Kerr-Newman metric near future null infinity is developed, with which the Newman-Penrose constants for both the gravitational and electromagnetic fields of the Kerr-Newman metric are computed and shown to be zero. The multipole structure near future null infinity in the sense of Janis-Newman of the Kerr-Newman metric is then further studied. It is found that up to the ${2}^{4}$-pole, modulo a constant dependent upon the order of the pole, these multipole moments agree with those of Geroch-Hansen multipole moments defined at spatial infinity.

Asymptotic properties of the development of conformally flat data near spatial infinity

The analysis of the relation between Bondi-type systems (NP-gauge) and a gauge used in the analysis of the structure of spatial infinity (F-gauge) which was carried out by Friedrich and Kánnár (2000 J. Math Phys. 41 2195) is retaken and applied to the development of a suitable class of conformally flat initial data sets with non-vanishing second fundamental form. The calculations presented depend on a certain assumption about the existence and regularity of the solutions to the conformal Einstein field equations close to null and spatial infinity. As a result of the calculations the Newman–Penrose constants of both future and past null infinity are calculated in terms of initial data and are shown to be equal. It is also shown that the asymptotic shear goes to zero as one approaches spatial infinity along the generators of null infinity so that it is possible to select, in a canonical fashion, the Poincaré group out of the BMS group. An expansion—again in terms of initial data quantities—of the Bondi mass close to spatial infinity is calculated. This expansion shows that if the existence and regularity assumptions hold, the Bondi mass approaches the ADM mass. A discussion of possible conditions on the initial data which would render a peeling development is presented.

Light cone structure near null infinity of the Kerr metric

Motivated by our attempt to understand the question of angular momentum of a relativistic rotating source carried away by gravitational waves, in the asymptotic regime near future null infinity of the Kerr metric, a family of null hypersurfaces intersecting null infinity in shearfree (good) cuts are constructed by means of asymptotic expansion of the eikonal equation. The geometry of the null hypersurfaces as well as the asymptotic structure of the Kerr metric near null infinity are studied. To the lowest order in angular momentum, the Bondi-Sachs form of the Kerr metric is worked out. The Newman-Unti formalism is then further developed, with which the Newman-Penrose constants of the Kerr metric are computed and shown to be zero. Possible physical implications of the vanishing of the Newman-Penrose constants of the Kerr metric are also briefly discussed.

On Newman–Penrose constants of stationary spacetimes

We consider the general asymptotic expression of stationary spacetime. Using the Killing equation, we reduce the dynamical freedom of the Einstein equation to the in-going gravitational wave Ψ0. The general form of this function can be obtained. With the help of an asymptotically algebraic special condition, we prove that all Newman–Penrose constants vanish.

A New Class of Obstructions to the Smoothness of Null Infinity

Expansions of the gravitational field arising from the development of asymptotically Euclidean, time symmetric, conformally flat initial data are calculated in a neighbourhood of spatial and null infinities up to order 6. To this ends a certain representation of spatial infinity as a cylinder is used. This set up is based on the properties of conformal geodesics. It is found that these expansions suggest that null infinity has to be non-smooth unless the Newman-Penrose constants of the spacetime, and some other higher order quantities of the spacetime vanish. As a consequence of these results it is conjectured that similar conditions occur if one were to take the expansions to even higher orders. Furthermore, the smoothness conditions obtained suggest that if a time symmetric initial data which is conformally flat in a neighbourhood of spatial infinity yields a smooth null infinity, then the initial data must in fact be Schwarzschildean around spatial infinity.

Conserved quantities in a black hole collision

The Newman–Penrose constants of the spacetime corresponding to the development of the Brill–Lindquist initial data are calculated by making use of a particular representation of spatial infinity due to H Friedrich. The Brill–Lindquist initial data set represents the head-on collision of two non-rotating black holes. In this case one non-zero constant is obtained. Its value is given in terms of the product of the individual masses of the black holes and the square of a distance parameter separating the two black holes. This constant retains its value all along null infinity, and therefore it provides information about the late time evolution of the collision process. In particular, it is argued that the magnitude of the constants provides information about the amount of residual radiation contained in the spacetime after the collision of the black holes.

Boost-rotation symmetric typeDradiative metrics in Bondi coordinates

The asymptotic properties of the solutions to the Einstein-Maxwell equations with boost-rotation symmetry and Petrov type D are studied. We find series solutions to the pertinent set of equations which are suitable for a late time descriptions in coordinates which are well adapted for the description of the radiative properties of spacetimes (Bondi coordinates). By calculating the total charge, Bondi and NUT mass and the Newman-Penrose constants of the spacetimes we provide a physical interpretation of the free parameters of the solutions. Additional relevant aspects on the asymptotics and radiative properties of the spacetimes considered, such as the possible polarization states of the gravitational and electromagnetic field, are discussed through the way.

Bondi-type systems near spacelike infinity and the calculation of the Newman–Penrose constants

We relate Bondi systems near spacelike infinity to another type of gauge conditions. While the former are based on null infinity, the latter are defined in terms of Einstein propagation, the conformal structure, and data on some Cauchy hypersurface. For a certain class of time symmetric space–times we study an expansion which allows us to determine the behavior of various fields arising in Bondi systems in the region of space–time where null infinity touches spacelike infinity. The coefficients of these expansions can be read off from the initial data. We obtain, in particular, expressions for the constants discovered by Newman and Penrose in terms of the initial data. For this purpose we calculate a certain expansion introduced by Friedrich [J. Geom. Phys. 24, 83–163 (1998)] up to third order.

On Killing vector fields and Newman–Penrose constants

Asymptotically flat space–times with one Killing vector field are studied. The Killing equations are solved asymptotically using polyhomogeneous expansions (i.e., series in powers of 1/r and ln r), and solved order by order. The solution to the leading terms of these expansions yields the asymptotic form of the Killing vector field. The possible classes of Killing fields are discussed by analyzing their orbits on null infinity. The integrability conditions of the Killing equations are used to obtain constraints on the components of the Weyl tensor (Ψ0,Ψ1,Ψ2) and on the shear (σ). The behavior of the solutions to the constraint equations is studied. It is shown that for Killing fields that are non-supertranslational the characteristics of the constraint equations are the orbits of the restriction of the Killing field to null infinity. As an application, the particular case of boost-rotation symmetric space–times is considered. The constraints on Ψ0 are used to study the behavior of the coefficients that give rise to the Newman–Penrose constants, if the space–time is non-polyhomogeneous, or the logarithmic Newman–Penrose constants, if the space–time is polyhomogeneous.

Polyhomogeneity and zero-rest-mass fields with applications to Newman-Penrose constants

A discussion of polyhomogeneity (asymptotic expansions in terms of 1/r and lnr) for zero-rest-mass fields and gravity and its relation with the Newman-Penrose (NP) constants is given. It is shown that for spin-s zero-rest-mass fields propagating on Minkowski spacetime, the logarithmic terms in the asymptotic expansion appear naturally if the field does not obey the `peeling theorem'. The terms that give rise to the slower fall-off admit a natural interpretation in terms of an advanced field. The connection between such fields and the NP constants is also discussed. The case when the background spacetime is curved and polyhomogeneous (in general) is considered. The free fields have to be polyhomogeneous, but the logarithmic terms due to the connection appear at higher powers of 1/r. In the case of gravity, it is shown that it is possible to define a new auxiliary field, regular at null infinity, and containing some relevant information on the asymptotic behaviour of the spacetime. This auxiliary zero-rest-mass field `evaluated at future infinity (i+)' yields the logarithmic NP constants.

Logarithmic Newman-Penrose constants for arbitrarypolyhomogeneousspacetimes

A discussion of how to calculate asymptotic expansions for polyhomogeneous spacetimes using the Newman-Penrose formalism is made. The existence of logarithmic Newman-Penrose constants for a general polyhomogeneous spacetime (i.e. a polyhomogeneous spacetime such that 0 = O(r-3lnN3)) is addressed. It is found that these constants exist for the generic case.