Bondi Mass Research Articles

Mass gap in the critical gravitational collapse of a kink

We study the gravitational collapse of a kink within spherical symmetry and the characteristic formulation of General Relativity. We explore some expected but elusive gravitational collapse issues which have not been studied before in detail, finding new features. The numerical one-parametric solution and the structure of the spacetime are calculated using finite differences, Galerkin collocation techniques, and some scripting for automated grid coverage. We study the threshold of black hole formation and confirm a mass gap in the phase transition. In the supercritical case we find a mass scaling power law $M_{BH}={M^*_{BH}}+K[\lambda-\lambda^*]^{2\gamma}+f(K[\lambda-\lambda^*]^{2\gamma})$, with $\gamma\approx 0.37$ independent of the initial data for the cases considered, and $M^*_{BH}$, $K$ and $\lambda^*$ each depending on the initial datum. The spacetime has a self-similar structure with a period of $\Delta\approx 3.4$. In the subcritical case the Bondi mass at null infinity decays in cascade with $\Delta/2$ interval as expected.

Energy, momentum, and center of mass in general relativity

These notions in the title are of fundamental importance in any branch of physics. However, there have been great difficulties in finding physically acceptable definitions of them in general relativity since Einstein's time. I shall explain these difficulties and progresses that have been made. In particular, I shall introduce new definitions of center of mass and angular momentum at both the quasi-local and total levels, which are derived from first principles in general relativity and by the method of geometric analysis. With these new definitions, the classical formula p=mv is shown to be consistent with Einstein's field equation for the first time. This paper is based on joint work [14][15] with Po-Ning Chen and Shing-Tung Yau.

Quasilocal angular momentum and center of mass in general relativity

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 an isolated system, which do not depend on any asymptotically flat coordinate system or asymptotic Killing field. The new proposal is free of ambiguities found in existing definitions and presents the first definition that precisely describes the dynamics of the Einstein equation.

A lower bound of the Trautman–Bondi energy

We obtain an energy inequality on null surfaces u = const in the Bondi–Sachs formalism. We show that for a sufficiently regular event horizon H there is an affine radial coordinate which is constant on H. Then the energy inequality can be prolongated to the horizon giving an estimation which is closely related to the Penrose inequality. We test it for the Kerr solution written in Fletcher-Lun coordinates.

A positive Bondi-type mass in asymptotically de Sitter spacetimes

The general structure of the conformal boundary of asymptotically de Sitter spacetimes is investigated. First we show that Penrose’s quasi-local mass, associated with a cut of the conformal boundary, can be zero even in the presence of outgoing gravitational radiation. On the other hand, following a Witten-type spinorial proof, we show that an analogous expression based on the Nester–Witten form is finite only if the Witten spinor field solves the two-surface twistor equation on and it yields a positive functional on the two-surface twistor space on provided the matter fields satisfy the dominant energy condition. Moreover, this functional is vanishing if and only if the domain of dependence of the spacelike hypersurface which intersects in the cut is locally isometric to the de Sitter spacetime. For non-contorted cuts this functional yields an invariant analogous to the Bondi mass.

Lovelock black holes with a nonconstant curvature horizon

This paper studies a class of $D=n+2(\ge 6)$ dimensional solutions to Lovelock gravity that is described by the warped product of a two-dimensional Lorentzian metric and an $n$-dimensional Einstein space. Assuming that the angular part of the stress-energy tensor is proportional to the Einstein metric, it turns out that the Weyl curvature of an Einstein space must obey two kinds of algebraic conditions. We present some exact solutions satisfying these conditions. We further define the quasilocal mass corresponding to the Misner-Sharp mass in general relativity. It is found that the quasilocal mass is constructed out of the Kodama flux and satisfies the unified first law and the monotonicity property under the dominant energy condition. Making use of the quasilocal mass, we show Birkhoff's theorem and address various aspects of dynamical black holes characterized by trapping horizons.

The asymptotic behaviour of the Hawking energy along null asymptotically flat hypersurfaces

In this work we obtain the limit of the Hawking energy of a large class of foliations along general null hypersurfaces Ω satisfying a weak notion of asymptotic flatness. The foliations are not required to be either geodesic or approaching large spheres at infinity. The limit is obtained in terms of a reference background geodesic foliation approaching large spheres and a positive function, constant along the null generators on Ω, which describes the relation between the two foliations at infinity. The integrand in the limit expression has interesting covariance and invariance properties with respect to change of background foliation. The standard result that the Hawking energy tends to the Bondi energy under suitable circumstances is recovered in this framework.

On curves with nonnegative torsion

We provide new results and new proofs of results about the torsion of curves in \({\mathbb{R}^3}\). Let \({\gamma}\) be a smooth curve in \({\mathbb{R}^3}\) that is the graph over a simple closed curve in \({\mathbb{R}^2}\) with positive curvature. We give a new proof that if \({\gamma}\) has nonnegative (or nonpositive) torsion, then \({\gamma}\) has zero torsion and hence lies in a plane. Additionally, we prove the new result that a simple closed plane curve, without any assumption on its curvature, cannot be perturbed to a closed space curve of constant nonzero torsion. We also prove similar statements for curves in Lorentzian \({\mathbb{R}^{2,1}}\) which are related to important open questions about time flat surfaces in spacetimes and mass in general relativity.

New tensorial estimates in Besov spaces for time-dependent (2 + 1)-dimensional problems

In this paper, we consider various tensorial estimates in geometric Besov-type norms on a one-parameter foliation of surfaces with evolving geometries. Moreover, we wish to accomplish this with only very weak control on these geometries. Several of these estimates were proved in [S. Klainerman and I. Rodnianski, Causal geometry of Einstein-vacuum spacetimes with finite curvature flux, Invent. Math. 159 (2005) 437–529; S. Klainerman and I. Rodnianski, Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux, Geom. Funct. Anal. 16(3) (2006) 164–229], but in very specific settings. A primary objective of this paper is to significantly simplify and make more robust the proofs of the estimates. Another goal is to generalize these estimates to more abstract settings. In [S. Alexakis and A. Shao, On the geometry of null cones to infinity under curvature flux bounds, Class. Quantum Grav. 31 (2014) 195012], we will apply these estimates in order to consider a variant of the problem in [S. Klainerman and I. Rodnianski, Causal geometry of Einstein-vacuum spacetimes with finite curvature flux, Invent. Math. 159 (2005) 437–529], that of a truncated null cone in an Einstein-vacuum spacetime extending to infinity. This analysis will then be used in [S. Alexakis and A. Shao, Bounds on the Bondi energy by a flux of curvature, to appear in J. Eur. Math. Soc.] to study and to control the Bondi mass and the angular momentum under minimal conditions.

On the energy of a null cone

We derive a formula for the Bondi mass aspect in terms of the asymptotic data of the Bondi–Sachs metric in the affine gauge. We prove the positivity of the total energy of a regular null cone in agreement with a recent result of Chruściel and Paetz.

On the geometry of null cones to infinity under curvature flux bounds

The main objective of this paper is to control the geometry of a future outgoing truncated null cone extending smoothly toward infinity in an Einstein-vacuum spacetime. In particular, we wish to do this under minimal regularity assumptions, namely, at the (weighted) L2-curvature level. We show that if the curvature flux and the data on an initial sphere of the cone are sufficiently close to the corresponding values in a standard Minkowski or Schwarzschild null cone, then we can obtain quantitative bounds on the geometry of the entire infinite cone. The same bounds also imply the existence of limits at infinity, along the null cone, of naturally scaled geometric quantities. In Alexakis and Shao (2013 Bounds on the Bondi energy by a flux of curvature arXiv:1308.4170), we will apply these results in order to control various physical quantities—e.g., the Bondi energy and (linear and angular) momenta—associated with such infinite null cones in vacuum spacetimes.

Time Flat Surfaces and the Monotonicity of the Spacetime Hawking Mass

We identify a condition on spacelike 2-surfaces in a spacetime that is relevant to understanding the concept of mass in general relativity. We prove a formula for the variation of the spacetime Hawking mass under a uniformly area expanding flow and show that it is nonnegative for these so-called “time flat surfaces.” Such flows generalize inverse mean curvature flow, which was used by Huisken and Ilmanen to prove the Riemannian Penrose inequality for one black hole. A flow of time flat surfaces may have connections to the problem in general relativity of bounding the mass of a spacetime from below by the quasi-local mass of a spacelike 2-surface contained therein.

Initial data for perturbed Kerr black holes on hyperboloidal slices

We construct initial data corresponding to a single perturbed Kerr black hole in vacuum. These data are defined on specific hyperboloidal (ACMC) slices, on which the mean extrinsic curvature K asymptotically approaches a constant at future null infinity . More precisely, we require that K obeys the Taylor expansion , where is a constant and σ describes a compactified spatial coordinate such that is represented by . We excise the singular interior of the black hole and assume a marginally outer trapped surface as the inner boundary of the computational domain. The momentum and Hamiltonian constraints are solved by means of pseudo-spectral methods and we find exponential rates of convergence of our numerical solutions. Some of the physical properties of the initial data are studied with the calculation of the Bondi mass, together with a multipole decomposition of the horizon. We probe the standard picture of gravitational collapse by assessing a family of Penrose-like inequalities and discuss in particular their rigidity aspects. Dynamical evolutions are planned in a future project.

Bondi Mass Cannot Become Negative in Higher Dimensions

We prove that the Bondi mass of an asymptotically flat, vacuum spacetime cannot become negative in any even dimension \({d \geq 4}\) . The notion of Bondi mass is more subtle in d > 4 dimensions because radiating metrics have a slower decay than stationary ones, and those subtleties are reflected by a considerably more difficult proof of positivity. Our proof holds for the standard spherical infinities, but it also extends to infinities of more general type which are (d − 2)-dimensional manifolds admitting a real Killing spinor. Such manifolds typically have special holonomy and Sasakian structures. The main technical advance of the paper is an expansion technique based on “conformal Gaussian null coordinates”. This expansion helps us to understand the consequences imposed by Einstein’s equations on the asymptotic tail of the metric as well as auxiliary spinorial fields. As a by-product, we derive a coordinate expression for the geometrically invariant formula for the Bondi mass originally given by Hollands and Ishibashi.

ACTIVE GALACTIC NUCLEUS FEEDBACK IN THE HOT HALO OF NGC 4649

Using the deepest available Chandra observations of NGC 4649, we find strong evidences of cavities, ripples, and ring-like structures in the hot interstellar medium that appear to be morphologically related with the central radio emission. These structures show no significant temperature variations in correspondence with higher pressure regions (0.5 kpc < r < 3 kpc). On the same spatial scale, a discrepancy between the mass profiles obtained from stellar dynamic and Chandra data represents the telltale evidence of a significant nonthermal pressure component in this hot gas, which is related to the radio jet and lobes. On a larger scale we find agreement between the mass profile obtained from Chandra data and planetary nebulae and globular cluster dynamics. The nucleus of NGC 4649 appears to be extremely radiatively inefficient, with highly sub-Bondi accretion flow. Consistent with this finding, the jet power evaluated from the observed X-ray cavities implies that a small fraction of the accretion power calculated for the Bondi mass accretion rate emerges as kinetic energy. Comparing the jet power to radio and nuclear X-ray luminosity, the observed cavities show behavior similar to those of other giant elliptical galaxies.

Perturbations of Keplerian orbits in stationary spherically symmetric spacetimes

We study spherically symmetric perturbations determined by alternative theories of gravity to the gravitational field of a central mass in General Relativity (GR). In particular, we focus on perturbations in the form of power laws and calculate their effect on the secular variations of the orbital elements of a Keplerian orbit. We show that, to lowest approximation order, only the argument of pericenter and mean anomaly undergo secular variations; furthermore, we calculate the variation of the orbital period. We give analytic expressions for these variations which can be used to constrain the impact of alternative theories of gravity.

On the Bondi mass of Maxwell–Klein–Gordon spacetimes

In this paper we calculate the Bondi mass of asymptotically flat spacetimes with interacting electromagnetic and scalar fields. The system of coupled Einstein–Maxwell–Klein–Gordon equations is investigated and corresponding field equations are written in the spinor form and in the Newman–Penrose formalism. Asymptotically flat solution of the resulting system is found near null infinity. Finally we use the asymptotic twistor equation to find the Bondi mass of the spacetime and derive the Bondi mass-loss formula. We compare the results with our previous work (Bičák et al. in Class Quantum Gravity 27(17):175011, 2010) and show that, unlike the conformal scalar field, the (Maxwell–)Klein–Gordon field has negatively semi-definite mass-loss formula.

Rigidity and minimizing properties of quasi-local mass

In this article, we survey recent developments in defining the quasi-local mass in general relativity. We discuss various approaches and the properties and applications of the different definitions. Among the expected properties, we focus on the rigidity property: for a surface in the Minkowski spacetime, one expects that the mass should vanish. We describe the Wang-Yau quasi-local mass whose definition is motivated by this rigidity property and by the Hamilton-Jacobi analysis of the Einstein-Hilbert action. In addition, we survey recent results on the minimizing property the Wang-Yau quasi-local mass.

Radiation from aD-dimensional collision of shock waves: Higher-order setup and perturbation theory validity

The collision of two D-dimensional, ultra-relativistic particles, described in General Relativity as Aichelberg-Sexl shock waves, is inelastic. In first order perturbation theory, the fraction of the initial centre of mass energy radiated away was recently shown to be 1/2 - 1/D. Here, we extend the formalism to higher orders in perturbation theory, and derive a general expression to extract the inelasticity, valid non-perturbatively, based on the Bondi mass loss formula. Then, to clarify why perturbation theory captures relevant physics of a strong field process in this problem, we provide one variation of the problem where the perturbative framework breaks down: the collision of ultra-relativistic charged particles. The addition of charge, and the associated repulsive nature of the source, originates an extra radiation burst, which we argue to be an artifact of the perturbative framework, veiling the relevant physics.

Tail terms in gravitational radiation reaction via effective field theory

Gravitational radiation reaction affects the dynamics of gravitationally bound binary systems. Here we focus on the leading "tail" term which modifies binary dynamics at fourth post-Newtonian order, as first computed by Blanchet and Damour. We re-produce this result using effective field theory techniques in the framework of the Lagrangian formalism suitably extended to include dissipation effects. We recover the known logarithmic tail term, consistently with the recent interpretation of the logarithmic tail term in the mass parameter as a renormalization group effect of the Bondi mass of the system.