Bondi Mass Research Articles

Calculation of radiated gravitational energy using the second-order Einstein tensor

In general relativity there is a well-defined prescription for defining a quantity that represents the radiated energy of an exact, asymptotically flat solution of Einstein’s equation. This quantity is called the Bondi energy flux. However, in linearized gravity off a stationary and asymptotically flat background, the second-order Einstein tensor has been used as a stress-energy tensor for the perturbed gravitational field, enabling one to calculate the energy radiated away in gravitational radiation. It is natural to ask how this method compares to the exact method for calculating the Bondi energy flux. In this paper, it is shown that if the metric perturbation satisfies certain falloff and gauge conditions, then the radiated energy calculated using the second-order Einstein tensor equals the second-order contribution to the Bondi energy flux associated with the perturbation. As an application, the second-order Einstein tensor is used to demonstrate gravitational superradiance from a Kerr black hole. Also, the Appendix contains a theorem that makes precise the notion that if ∇(aξb) and its derivative is ‘‘small,’’ then ξa is close to a Killing field.

Gravitational radiation reaction in quasistatic, axially symmetric systems with possibly strong fields

The leading radiation reaction in quasistatic, axially symmetric systems is calculated by matched asymptotic expansions. Earlier results on inductive transfer of near-zone “energy” are combined via matching with a wave-zone expansion in the Regge-Wheeler gauge. Two independent measures of the damping are computed: (1) the change of the 1/Rcoefficient in a Weyl-Levi-Civita type of multipole expansion in the near zone and (2) the change in Bondi energy at future null infinity. When an averaging process is justified, such as in (nearly) periodic systems, both quantities yield a generalized version of the usual quadrupole formula, provided the radiation is outgoing at future null infinity. It is also shown that terms nonanalytic in the slow-motion parameter ɛ make a time-even contribution to the near-zone monopole coefficient. The principal advantage of this calculation over previous work is that the sources are permitted to have strong gravitational fields. Moreover, no specific model for the sources is assumed.

Axially symmetric gravitational radiation from isolated sources

Asymptotically flat, dynamic, axially symmetric space-times are considered from the point of view of the characteristic initial value problem and the expansion of solutions in powers of r−1. Imposing conditions such that the system initially be exactly static and nonradiative finally, a news function is constructed which governs the dynamics of such a space-time and which yields a finite, nonvanishing total loss of the Bondi mass. Furthermore, the Riemann tensor (to order r−3) is known explicitly for all time since an expression for the mass aspect is also determined. It is also shown that the total change in the asymptotic shear is related to the total change in the Bondi mass. Finally the implications concerning transitions between two exactly stationary states are discussed.

On the positivity of the Bondi mass

A new way of showing local positivity of the Bondi mass is presented. The method used makes use only of the constraint equations. By looking at the constraint equations of general relativity on an asymptotically constant mean extrisic curvature hypersurface, a mass aspect is picked out, whose average over the 2-sphere coincides with the Bondi mass. This mass aspect differs from Bondi's mass aspect by terms linear in the news function. It is shown that the mass aspect so produced may be considered as a functional of the dependent data in the problem. This functional is shown to have Minkowski space as a critical point. The second variational of the functional about flat space is then shown to be positive. From this, one concludes that the functional is positive in a neighborhood of flat space. Hence the Bondi mass, too, is positive in a neighborhood of flat space.

Positivity of the Bondi energy

We give a new proof that the Bondi momentum is time-like and future directed. This is done by extending a theorem showing the existence of solutions of Witten’s equation on a complete, asymptotically flat surface, to the case where the surface is asyptotically hyperbolic and spans a cut Ŝ of ℐ+. By choosing appropriate boundary conditions the positivity of the Bondi energy is also shown for the case where the surface is no longer free of inner boundaries but is internally bounded by the union of a finite number of marginally trapped surfaces. This argument also applies to the case of asymptotically flat surfaces.

On the positivity of the Bondi mass: Sian M. Stumbles. Department of Mathematics, University of California, Berkeley, California

On the positivity of the Bondi mass: Sian M. Stumbles. Department of Mathematics, University of California, Berkeley, California

An inequality relating total mass and the area of a trapped surface in general relativity

Let M be a space-time which is asymptotically flat at past null infinity I-, satisfies the dominant energy condition and contains a trapped surface T. The authors show that if T can be connected to I- by means of a nonsingular null-hypersurface N, then m2>or=A/16 pi where m is the Bondi mass with respect to N and A in the area of T.

Positive mass theorems for black holes

We extend Witten's proof of the positive mass theorem at spacelike infinity to show that the mass is positive for initial data on an asymptotically flat spatial hypersurface Σ which is regular outside an apparent horizonH. In addition, we prove that if a black hole has electromagnetic charge, then the mass is greater than the modulus of the charge. These results are also valid for the Bondi mass at null infinity. Finally, in the case of the Einstein equation with a negative cosmological constant, we show that a suitably defined mass is positive for data on an asymptotically anti-de Sitter surface Σ which is regular outside an apparent horizon.

Momentum, angular momentum and their quasi-local null surface extensions

Let M be an asymptotically flat space-time which satisfies the dominant energy condition and let N be a null hypersurface of M which meets I+ in a space-like cross section S given by r= infinity , where r is an affine parameter along the generators of N. It is shown that there exists a quasi-local momentum Pa(r), defined by means of an integral over the r=constant cross sections of N, which tends to the Bondi momentum as r to infinity , and which satisfies the radial 'mass-gain condition' Pa(r2)ka>or=Pa(r1)ka when r2>or=r1, where ka is a constant future pointing null vector. This condition is then used to show that, in certain circumstances, the Bondi momentum is always future pointing or, in other words, that the Bondi mass is always positive. A quasi-local angular momentum is also defined, which tends to Bramson's (1975) angular momentum as r to infinity .

Trapped surfaces and the positivity of Bondi mass

It is shown that the Bondi mass of an asymptotically flat space-time which satisfies the dominant energy condition and which contains a number of trapped surfaces is positive.

Proof That the Bondi Mass is Positive

It is demonstrated that the total mass of a nontrivial isolated physical system is positive even after part of its mass has been lost because of gravitational radiation. This method is an extension of the one that the authors used in the proff of the classical positive-mass conjecture.

On the asymptotic flatness of theC metrics at spatial infinity

It is shown that a family of exact, radiating solutions of Einstein's field equations (theC metrics) is asymptotically flat at spatial infinity (in the sense of Ashtekar-Hansen). The ADM and Bondi masses are discussed.

A simple proof of the positivity of the Bondi mass

Using a nonsingular hypersurface that is not only asymptotically null, the authors prove the existence of spinor fields that yield a positive expression for the Bondi mass. Standard existence theorems based on compact hypersurfaces are applied.

The positivity of the Bondi mass

Two component spinor techniques, similar to those used by Witten, are used to express the Bondi momentum of an asymptotically flat space-time in the form of an integral over an asymptotically null, space-like hypersurface 2. It is then shown that the Bondi mass is positive, given the existence of a Green function for the 'Witten equation' on 2. In a recent paper Witten (1981) has given a simple proof of the positivity of the ADM mass of an asymptotically flat space-time by means of a simple argument using Dirac spinors. Witten's proof of the positive energy theorem is not, however, quite correct as it stands as it uses an invalid three-dimensional truncation of the four-dimensional Gauss divergence theorem. By giving a new four-dimensional covariant expression for the ADM four-momentum, Nester (1981) has been able to avoid this difficulty and give a simple proof of the positivity of the ADM mass. We show here how similar techniques may be used in the case of the Bondi mass. Let M be an asymptotically flat space-time with future null infinity $+. The Bondi-Sachs (Bondi et a1 1962, Sachs 1962) four-momentum Pa(S) of M is a four-vector function, defined on the space of all space-like cross sections (cuts) of 4+, which lies in the Minkowski space of BMS translations T. If we let T = YO 9 where Y is the space of two-spinors, then, on using the Penrose abstract index notation (Penrose 1968), we may write

The equilibrium of two charged masses in general relativity

The condition of equilibrium is obtained up to the ppN approximation by solving the Einstein-Maxwell equations. It is conjectured that for spherically symmetric particles the classical condition e 1 e 2 = Gm 1 m 2 applies.

Axially symmetric two-body problem in general relativity. III. Bondi mass loss and the failure of the quadrupole formula

Suggested difficulties and criticism regarding earlier work is addressed. It is demonstrated that the rate of gravitational energy loss from the authors' model free-fall system employing the widely accepted Bondi method agrees precisely with the results described in prior works. Origins of the breakdown of the quadrupole formalism for free fall, previously indicated, are now delineated in detail. The role of source structure in the energy loss rate re-emerges, bringing into question much of the earlier work of others. The iterative technique with flat-space wave operators is justified. A new approach to quasiperiodic systems such as binary stars is described. Ideally modeled upon the actual birth of such systems, it evolves from an initially stationary configuration, again avoiding the problems and ambiguities regarding incoming radiation.

Energy and momentum in general relativity

It is shown that in an asymptotically flat space-time (M ,g) a “flat metric”h μv can be naturally determined so that the energy and momenta of the space-time are covariantly defined by the Landau-Lifshitz complex. Specifically, it is shown that the asymptotic form ofg μv determines uniquely the asymptotic form ofh μv to such an extent that the Landau-Lifshitz energy and momenta (a) are independent of the remaining freedom inh μv, (b) are equal to the corresponding components of the Bondi energy momentum, and (c) transform as components of a free Lorentz vector under general BMS transformations. Thus the Landau-Lifshitz prescription is completely equivalent to the Bondi method.

The center of mass in general relativity

In Dixon's theory of the dynamics of extended bodies in metric theories of gravity, a definition of a center-of-mass line is proposed. We prove the existence and uniqueness of a zero-linear-momentum vector field. Using this vector field we show the existence of a center-of-mass line which is a smooth timelike curve contained in a convex hull of the world-tube of the body.

The uniqueness of the center of mass in general relativity

Using the theorem of Poincare-Hopf we prove the uniqueness of the center-of-mass line (its existence has been shown in a different paper, see [2]) in Dixon's theory of the dynamics of extended bodies in metric theories of gravity.

Rotating, charged, and uniformly accelerating mass in general relativity

A new general class of solutions of the Einstein-Maxwell equations is presented. It depends on seven arbitrary parameters that group in a natural way into three complex parameters m + in, a + ib, e + ig, and the cosmological constant λ. They correspond to mass, NUT parameter, angular momentum per unit mass, acceleration, and electric and magnetic charge. The metric is in general stationary and axially symmetric. These solutions are of type D and contain as special cases all known solutions of type D belonging to this class. The known solutions are recovered by performing limiting transitions. An appropriate limit of our solutions describes an electromagnetic field in flat spacetime. We investigate the properties of that field. Its singular region corresponds in general to two circles moving with uniform acceleration in the positive and negative directions along the axis of symmetry. One can easily extend our solutions to the complex domain. Then it turns out that the metric can be written in a double Kerr-Schild form.