Gravitational radiation near spatial and null infinity

1987 ◽  
Vol 410 (1838) ◽  
pp. 201-208 ◽  
Keyword(s):  
Radiation Field ◽  
Gravitational Radiation ◽  
Null Infinity ◽  
Adm Mass ◽  
Formal Expansion ◽  
Bondi Mass ◽  
Expansion Coefficients ◽  
Analytic Behaviour

Taking Bondi’s approach to an extreme by adding a 1/ u -expansion to the usual 1/ r -expansion, one can study the effects of the presence of mass at space-like infinity. Assuming convergence of the formal expansion one finds: (1) the limit of the Bondi mass agrees with the ADM-mass of certain space-like hypersurfaces; (2) relations between expansion coefficients on space-like hypersurfaces and the radiation field on J + can be given; (3) analytic properties in J + lead to non-analytic behaviour on space-like hypersurfaces.

scholarly journals ADM MASS, BONDI MASS, AND ENERGY CONSERVATION IN TWO-DIMENSIONAL DILATON GRAVITIES

1996 ◽  
Vol 11 (03) ◽  
pp. 553-561 ◽  
Author(s):  
WON T. KIM ◽  
JULIAN LEE
Keyword(s):  
Boundary Condition ◽  
Energy Conservation ◽  
Radiation Energy ◽  
Two Dimensional ◽  
Dilaton Gravity ◽  
Null Infinity ◽  
Adm Mass ◽  
Stress Energy ◽  
Bondi Mass ◽  
Definition Of

We show how a stress-energy pseudotensor can be constructed in two-dimensional dilaton gravity theories (classical, CGHS and RST) and derive from it the expression for the ADM mass in these theories. We compare this expression with the ones in the literature obtained by other methods. We define the Bondi mass for these theories by using the pseudotensor formalism. The resulting expression is the generalization of the expression for the ADM mass. The boundary condition needed for the energy conservation is also investigated. It is shown that under appropriate boundary conditions, our definition of the Bondi mass is exactly the ADM mass minus the matter radiation energy at null infinity.

Gravitational radiation field of an isolated system at null infinity

1990 ◽  
Vol 431 (1882) ◽  
pp. 337-344 ◽  
Keyword(s):  
Radiation Field ◽  
Gravitational Radiation ◽  
High Frequency ◽  
Radiation Flux ◽  
Flat Space ◽  
Background Field ◽  
Null Infinity ◽  
Present Treatment ◽  
New Formula ◽  
Space Result

The quadrupole and octupole contributions to the gravitational radiation flux at null infinity from an initially stationary isolated system are computed in terms of the asymptotic moments defined there. The present treatment incorporates the influence of the background field of the source while still neglecting the nonlinear self-interaction of the radiation. Compared with the flat space result, the new formula predicts a suppression of the contribution from the high-frequency modes for which the frequency ω satisfies GM 0 ω / c 3 ≫ 1, M 0 being the initial mass of the system.

Logarithmic corrections in the asymptotic expansion for the radiation field along null infinity

2019 ◽  
Vol 16 (01) ◽  
pp. 1-34 ◽  
Author(s):  
Yannis Angelopoulos ◽  
Stefanos Aretakis ◽  
Dejan Gajic
Keyword(s):  
Radiation Field ◽  
Late Time ◽  
Spherically Symmetric ◽  
Null Infinity ◽  
Asymptotically Flat ◽  
Black Hole Spacetimes ◽  
Logarithmic Corrections ◽  
Time Asymptotics ◽  
Physics Literature ◽  
Field Of Solutions

We obtain the second-order late-time asymptotics for the radiation field of solutions to the wave equation on spherically symmetric and asymptotically flat backgrounds including the Schwarzschild and sub-extremal Reissner–Nordström families of black hole spacetimes. These terms appear as logarithmic corrections to the leading-order asymptotic terms which were rigorously derived in our previous work. Such corrections have been heuristically and numerically derived in the physics literature in the case of a non-vanishing Newman–Penrose constant. In this case, our results provide a rigorous confirmation of the existence of these corrections. On the other hand, the precise logarithmic corrections for spherically symmetric compactly supported initial data (and hence, with a vanishing Newman–Penrose constant) explicitly obtained here appear to be new.

scholarly journals Taub numbers at future null infinity: III. The Bondi mass

1997 ◽  
Vol 14 (7) ◽  
pp. 1899-1909 ◽  
Author(s):  
E N Glass ◽  
M G Naber
Keyword(s):  
Null Infinity ◽  
Bondi Mass ◽  
Future Null Infinity

scholarly journals A new look at the Bondi-Sachs energy-momentum

2021 ◽  
Author(s):  
Jörg Frauendiener ◽  
Chris Stevens
Keyword(s):  
Unit Sphere ◽  
Gauss Curvature ◽  
Null Infinity ◽  
Conformally Invariant ◽  
Lorentzian Metric ◽  
Correct Definition ◽  
Ghp Formalism ◽  
Bondi Mass ◽  
Definition Of ◽  
Bondi Energy

Abstract How does one compute the Bondi mass on an arbitrary cut of null infinity I when it is not presented in a Bondi system? What then is the correct definition of the mass aspect? How does one normalise an asymptotic translation computed on a cut which is not equipped with the unit-sphere metric? These are questions which need to be answered if one wants to calculate the Bondi-Sachs energy-momentum for a space-time which has been determined numerically. Under such conditions there is not much control over the presentation of I so that most of the available formulations of the Bondi energy-momentum simply do not apply. The purpose of this article is to provide the necessary background for a manifestly conformally invariant and gauge independent formulation of the Bondi energy-momentum. To this end we introduce a conformally invariant version of the GHP formalism to rephrase all the well-known formulae. This leads us to natural definitions for the space of asymptotic translations with its Lorentzian metric, for the Bondi news and the mass-aspect. A major role in these developments is played by the “co-curvature”, a naturally appearing quantity closely related to the Gauß curvature on a cut of I.

scholarly journals The Case Against Smooth Null Infinity I: Heuristics and Counter-Examples

2021 ◽  
Author(s):  
Leonhard M. A. Kehrberger
Keyword(s):  
Asymptotic Expansion ◽  
Wave Equation ◽  
Gravitational Radiation ◽  
Scattering Problem ◽  
Scattering Data ◽  
Linear Wave ◽  
Spherically Symmetric ◽  
Field Equations ◽  
Null Infinity ◽  
World Scientific Publishing

AbstractThis paper initiates a series of works dedicated to the rigorous study of the precise structure of gravitational radiation near infinity. We begin with a brief review of an argument due to Christodoulou (in: The Ninth Marcel Grossmann Meeting, World Scientific Publishing Company, Singapore, 2002) stating that Penrose’s proposal of smooth conformal compactification of spacetime (or smooth null infinity) fails to accurately capture the structure of gravitational radiation emitted by N infalling masses coming from past timelike infinity $$i^-$$ i - . Modelling gravitational radiation by scalar radiation, we then take a first step towards a dynamical understanding of the non-smoothness of null infinity by constructing solutions to the spherically symmetric Einstein–Scalar field equations that arise from polynomially decaying boundary data, $$r\phi \sim t^{-1}$$ r ϕ ∼ t - 1 as $$t\rightarrow -\infty $$ t → - ∞ , on a timelike hypersurface (to be thought of as the surface of a star) and the no incoming radiation condition, $$r\partial _v\phi =0$$ r ∂ v ϕ = 0 , on past null infinity. We show that if the initial Hawking mass at $$i^-$$ i - is nonzero, then, in accordance with the non-smoothness of $${\mathcal {I}}^+$$ I + , the asymptotic expansion of $$\partial _v(r\phi )$$ ∂ v ( r ϕ ) near $${\mathcal {I}}^+$$ I + reads $$\partial _v(r\phi )=Cr^{-3}\log r+{\mathcal {O}}(r^{-3})$$ ∂ v ( r ϕ ) = C r - 3 log r + O ( r - 3 ) for some non-vanishing constant C. In fact, the same logarithmic terms appear already in the linear theory, i.e. when considering the spherically symmetric linear wave equation on a fixed Schwarzschild background. As a corollary, we can apply our results to the scattering problem on Schwarzschild: Putting compactly supported scattering data for the linear (or coupled) wave equation on $${\mathcal {I}}^-$$ I - and on $${\mathcal {H}}^-$$ H - , we find that the asymptotic expansion of $$\partial _v(r\phi )$$ ∂ v ( r ϕ ) near $${\mathcal {I}}^+$$ I + generically contains logarithmic terms at second order, i.e. at order $$r^{-4}\log r$$ r - 4 log r .

Gravitational waves and the radiative field

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
General Relativity ◽  
General Solution ◽  
Gravitational Waves ◽  
Radiation Field ◽  
Gravitational Radiation ◽  
Gravitational Force ◽  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Einstein Equations ◽  
Maxwell Theory

This chapter turns to the gravitational radiation produced by a system of massive objects. The discussion is confined to the linear approximation of general relativity, which is compared with the Maxwell theory of electromagnetism. In the first part of the chapter, the properties of gravitational waves, which are the general solution of the linearized vacuum Einstein equations, are studied. Next, it relates these waves to the energy–momentum tensor of the sources creating them. The chapter then turns to the ‘first quadrupole formula’, giving the gravitational radiation field of these sources when their motion is due to forces other than the gravitational force.

scholarly journals Asymptotically simple spacetimes and mass loss due to gravitational waves

2017 ◽  
Vol 27 (01) ◽  
pp. 1730027 ◽  
Author(s):  
Vee-Liem Saw
Keyword(s):  
Mass Loss ◽  
Gravitational Waves ◽  
Hamiltonian Formulation ◽  
Conformal Structure ◽  
De Sitter ◽  
Research Activity ◽  
Null Infinity ◽  
Open Problems ◽  
Timelike Hypersurface ◽  
Bondi Mass

The cosmological constant [Formula: see text] used to be a freedom in Einstein’s theory of general relativity (GR), where one had a proclivity to set it to zero purely for convenience. The signs of [Formula: see text] or [Formula: see text] being zero would describe universes with different properties. For instance, the conformal structure of spacetime directly depends on [Formula: see text]: null infinity [Formula: see text] is a spacelike, null, or timelike hypersurface, if [Formula: see text], [Formula: see text], or [Formula: see text], respectively. Recent observations of distant supernovae have taught us that our universe expands at an accelerated rate, and this can be accounted for by choosing [Formula: see text] in Einstein’s theory of GR. A quantity that depends on the conformal structure of spacetime, especially on the nature of [Formula: see text], is the Bondi mass which in turn dictates the mass loss of an isolated gravitating system due to energy carried away by gravitational waves. This problem of extending the Bondi mass to a universe with [Formula: see text] has spawned intense research activity over the past several years. Some aspects include a closer inspection on the conformal properties, working with linearization, attempts using a Hamiltonian formulation based on “linearized” asymptotic symmetries, as well as obtaining the general asymptotic solutions of de Sitter-like spacetimes. We consolidate on the progress thus far from the various approaches that have been undertaken, as well as discuss the current open problems and possible directions in this area.

scholarly journals Gravitational memory effects and Bondi-Metzner-Sachs symmetries in scalar-tensor theories

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shaoqi Hou ◽  
Zong-Hong Zhu
Keyword(s):  
General Relativity ◽  
Angular Momentum ◽  
Evolution Equations ◽  
Equations Of Motion ◽  
Memory Effects ◽  
Scalar Tensor Theory ◽  
Null Infinity ◽  
Infinite Dimensional ◽  
Tensor Theory ◽  
Bondi Mass

Abstract The relation between gravitational memory effects and Bondi-Metzner-Sachs symmetries of the asymptotically flat spacetimes is studied in the scalar-tensor theory. For this purpose, the solutions to the equations of motion near the future null infinity are obtained in the generalized Bondi-Sachs coordinates with a suitable determinant condition. It turns out that the Bondi-Metzner-Sachs group is also a semi-direct product of an infinite dimensional supertranslation group and the Lorentz group as in general relativity. There are also degenerate vacua in both the tensor and the scalar sectors in the scalar-tensor theory. The supertranslation relates the vacua in the tensor sector, while in the scalar sector, it is the Lorentz transformation that transforms the vacua to each other. So there are the tensor memory effects similar to the ones in general relativity, and the scalar memory effect, which is new. The evolution equations for the Bondi mass and angular momentum aspects suggest that the null energy fluxes and the angular momentum fluxes across the null infinity induce the transition among the vacua in the tensor and the scalar sectors, respectively.

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