We study the limit of quasilocal energy defined in [7] and [8] for a family of spacelike 2-surfaces approaching null infinity of an asymptotically flat spacetime. It is shown that Lorentzian symmetry is recovered and an energy-momentum 4-vector is obtained. In particular, the result is consistent with the Bondi-Sachs energy-momentum at a retarded time. The quasilocal mass in [7] and [8] is defined by minimizing quasilocal energy among admissible isometric embeddings and observers. The solvability of the Euler-Lagrange equation for this variational problem is also discussed in both the asymptotically flat and asymptotically null cases. Assuming analyticity, the equation can be solved and the solution is locally minimizing in all orders. In particular, this produces an optimal reference hypersurface in the Minkowski space for the spatial or null exterior region of an asymptotically flat spacetime.
We study the limit of quasilocal mass defined in [4] and [5] for a family of spacelike 2-surfaces in spacetime. In particular, we show the limit coincides with the ADM mass at spatial infinity. The limit for coordinate spheres of a boosted slice of the Schwarzchild solution is computed explicitly and shown to give the expected energy-momentum four vector.
We give a brief review of the definition of the Wang-Yau quasilocal mass and discuss the evaluation of which on surfaces of unit size at null infinity of an axi-symmetric spacetime in Bondi-van der Burg-Metzner coordinates.
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 …
An idealized observer of an astronomical event is situated at future null infinity, where light rays emitted from the source approach. Mathematically, null infinity corresponds to the portion of the spacetime boundary defined by equivalence classes of null geodesics. But what can we observe at future null infinity? In the note, we start by reviewing the description of null infinity in terms of the Bondi-Sachs coordinate system. In particular, we reformulate and extend the invariance of Bondi mass and the mass loss formula in terms of a modified mass aspect 2-form. At the end, we discuss new re…
I shall discuss the Chen-Wang-Yau quasilocal angular momentum, which is defined based on the theory of optimal isometric embedding and quasilocal mass of Wang-Yau, and the limits of which at spatial and null infinity of an isolated gravitating system. This is based on joint work with Po-Ning Chen, Jordan Keller, Ye-Kai Wang, and Shing-Tung Yau.
There are two important statements regarding the Trautman-Bondi mass at null infinity: one is the positivity, and the other is the Bondi mass loss formula, which are both global in nature. In this note, we compute the limit of the Wang-Yau quasi-local mass on unit spheres at null infinity of an asymptotically flat spacetime in the Bondi-Sachs coordinates. The quasi-local mass leads to a local description of the radiation that is purely gravitational at null infinity. In particular, the quasi-local mass is evaluated in terms of the news function of the Bondi-Sachs coordinates.
We give new definitions of null infinity and black hole in terms of causal boundaries, applicable to any strongly causal spacetime $(M,g)$. These are meant to extend the standard ones given in terms of conformal boundaries, and use the new definitions to prove a classic result in black hole theory for this more general context: if the null infinity is regular (i.e. well behaved in a suitable sense) and $(M,g)$ obeys the null convergence condition, then any closed trapped surface in $(M,g)$ has to be inside the black hole region. As an illustration of this general construction, we apply it to t…
In this paper, we extend the results of Klainerman and Rodnianski in \cite{KR:Trapped}, which were obtained for a finite region, by showing similar results from past null infinity. This allows us to recover and extend the results from past null infinity in the work of Christodoulou \cite{Chr:book}.
For a spacelike 2-surface in spacetime, we propose a new definition of quasi-local angular momentum and quasi-local center of mass, as an element in the dual space of the Lie algebra of the Lorentz group. Together with previous defined quasi-local energy-momentum, this completes the definition of conserved quantities in general relativity at the quasi-local level. We justify this definition by showing the consistency with the theory of special relativity and expectations on an axially symmetric spacetime. The limits at spatial infinity provide new definitions for total conserved quantities of …
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…
There are two important statements regarding the Trautman-Bondi mass [1,8,5] at null infinity: one is the positivity [7,6], and the other is the Bondi mass loss formula [1], which are both global in nature. The positivity of the quasi-local mass can potentially lead to a local description at null infinity. This is confirmed for the Vaidya spacetime in this note. We study the Wang-Yau quasi-local mass on surfaces of fixed size at the null infinity of the Vaidya spacetime. The optimal embedding equation is solved explicitly and the quasi-local mass is evaluated in terms of the mass aspect functi…
The supertranslation invariance of the Chen-Wang-Yau (CWY) angular momentum in the Bondi-Sachs formalism/gauge was ascertained by the authors in \cite{CKWWY_evol, CWWY_atmp}. In this article, we study the corresponding problem in the double null gauge. In particular, supertranslation ambiguity of this gauge is identified and the CWY angular momentum is proven to be free of this ambiguity. A similar result is obtained for the CWY center of mass integral.
In this short paper, we review recent progress on the positive mass theorem for spacelike hypersurfaces which approach to null infinity in asymptotically flat spacetimes. We use it to prove, if the functions $c(u, θ, ψ)$, $d(u, θ, ψ)$ vanish at certain retarded time in vacuum Bondi's radiating spacetimes, then the Bondi mass is nonnegative up to that time.
We calculate the limits of the quasi-local angular momentum and center-of-mass defined by Chen-Wang-Yau \cite{CWY} for a family of spacelike two-spheres approaching future null infinity in an asymptotically flat spacetime admitting a Bondi-Sachs expansion. Our result complements earlier work of Chen-Wang-Yau \cite{CWY2}, where the authors calculate the limits of the quasi-local energy and linear momentum at null infinity. Finiteness of the center-of-mass limit requires that the spacetime be in the so-called center-of-mass frame, a mild assumption on the mass aspect function amounting to vanish…
We study how conserved quantities such as angular momentum and center of mass evolve with respect to the retarded time at null infinity, which is described in terms of a Bondi-Sachs coordinate system. These evolution formulae complement the classical Bondi mass loss formula for gravitational radiation. They are further expressed in terms of the potentials of the shear and news tensors. The consequences that follow from these formulae are (1) Supertranslation invariance of the fluxes of the CWY conserved quantities. (2) A conservation law of angular momentum à la Christodoulou. (3) A duality pa…
We prove that an $(n+1)$-dimensional spin static vacuum with negative cosmological constant whose null infinity has a boundary admitting a non-trivial Killing spinor field is the AdS spacetime. As a consequence, we generalize previous uniqueness results by X. Wang \cite{Wa2} and by Chru{ś}ciel-Herzlich \cite{CH} and introduce, for this class of spin static vacua, some Lorentzian manifolds which are prohibited as null infinities.
We show that the spherically symmetric Einstein-scalar-field equations for wave-like decaying initial data at null infinity have unique local solutions and unique global solutions for small initial data. We also generalize Christodoulou's global generalized solutions to the wave-like decaying initial data. We emphasize that this decaying condition is sharp.
