Emanuel Gallo, Luis Lehner, Osvaldo M. Moreschi
The purpose of this note is to point out ambiguities that appear in the calculation of angular momentum and its radiated counterpart when some simple formulae are used to compute them. We illustrate, in two simple different examples, how incorrect results can be obtained with them. Additionally, we discuss the magnitude of possible errors in well known situations.
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This is a write-up of a talk for non-specialists on the treatment of angular momentum for radiating systems resolving the supertranslation problem.
There are several definitions of the notion of angular momentum in general relativity. However non of them can be said to capture the physical notion of intrinsic angular momentum of the sources in the presence of gravitational radiation. We present a definition which is appropriate for the description of intrinsic angular momentum in radiative spacetimes. This notion is required in calculations involving radiation of angular momentum, as for example is expected in binary coalescence of black holes.
We present a simplified expression for the gravitational angular momentum in the framework of the teleparallel equivalent of general relativity (TEGR). The expression arises from the constraints equations of the Hamiltonian formulation of the TEGR. We apply this expression to the calculation of the angular momentum of plane-fronted gravitational waves in an arbitrary three-dimensional volume $V$ of space and compare the results with those obtained for linearised gravitational waves.
Often it is asserted that only by using of the symmetric Landau-Lifschitz energy-momentum complex one is able to formulate a conserved angular momentum complex in General Relativity ({\bf GR}). Obviously, it is an uncorrect statement. For example, years ago, Bergmann and Thomson have given other, very useful expression on angular momentum. This expression is closely joined to the non-symmetric, Einstein canonical energy-momentum complex. In the paper we review the Bergmann-Thomson angular momentum complex and compare it with that of given by Landau and Lifschitz.
In an extended, new form of general relativity, which is a teleparallel theory of gravity, we examine the energy-momentum and angular momentum carried by gravitational wave radiated from Newtonian point masses in a weak-field approximation. The resulting wave form is identical to the corresponding wave form in general relativity, which is consistent with previous results in teleparallel theory. The expression for the dynamical energy-momentum density is identical to that for the canonical energy-momentum density in general relativity up to leading order terms on the boundary of a large sphere …
In de Sitter (dS) special relativity (SR), two kinds of conserved currents are derived. The first kind is a 5-dimensional (5d) dS-covariant angular momentum (AM) current, which unites the energy-momentum (EM) and 4d AM current in an inertial-type coordinate system. The second kind is a dS-invariant AM current, which can be generalized to a conserved current for the coupling system of the matter field and gravitational field in dS gravity. Moreover, an inherent EM tensor is predicted, which comes from the spin part of the dS-covariant current. All the above results are compared to the ordinary …
The characteristic approach to numerical relativity is a useful tool in evolving gravitational systems. In the past this has been implemented using two patches of stereographic angular coordinates. In other applications, a six-patch angular coordinate system has proved effective. Here we investigate the use of a six-patch system in characteristic numerical relativity, by comparing an existing two-patch implementation (using second-order finite differencing throughout) with a new six-patch implementation (using either second- or fourth-order finite differencing for the angular derivatives). We …
We apply the energy-momentum tensor to calculate energy, momentum and angular-momentum of two different tetrad fields. This tensor is coordinate independent of the gravitational field established in the Hamiltonian structure of the teleparallel equivalent of general relativity (TEGR). The spacetime of these tetrad fields is the charged dilaton. Our results show that the energy associated with one of these tetrad fields is consistent, while the other one does not show this consistency. Therefore, we use the regularized expression of the gravitational energy-momentum tensor of the TEGR. We inves…
Local Continuity of Angular Momentum and Noether Charge for Matter in General Relativity
该研究拓展广义相对论物质连续性研究至角动量与规范对称诺特流范畴，推导相关表达式并以三维玻色星碰撞为例验证方法。
Generalized definitions for angular and linear momentum are given and shown to reduce to the ADM (at spatial infinity) definitions and the definitions at null infinity in the appropriate limit. These definitions are used to express angular momentum in terms of linear momentum. The formalism allows one to see the connection with the classical and special relativitistic notions of momenta. Further, the techniques elucidate, for the first time, the geometric nature of these conserved quantities. The boosted Schwarzschild solution is used to illustrate some aspects. The definitions are useful and …
Angular Momentum for Black Hole Binaries in Numerical Relativity
本文基于数值相对论准圆轨道结果，提出了适用于小黑洞间距的双星轨道角动量表达式，用于研究自旋到轨道角动量的转移及潮汐耦合，并与Kerr几何粒子微扰结果高度一致。
There exist at least a few different kind of averaging of the differences of the energy-momentum and angular momentum in normal coordinates {\bf NC(P)} which give tensorial quantities. The obtained averaged quantities are equivalent mathematically because they differ only by constant scalar dimensional factors. One of these averaging was used in our papers [1-8] giving the {\it canonical superenergy and angular supermomentum tensors}. In this paper we present one other averaging of the energy-momentum and angular momentum differences which gives tensorial quantities with proper dimensions of t…
The energy-momentum tensor, which is coordinate independent, is used to calculate energy, momentum and angular-momentum of two different tetrad fields. Although, the two tetrad fields reproduce the same space-time their energies are different. Therefore, a regularized expression of the gravitational energy-momentum tensor of the teleparallel equivalent of general relativity, (TEGR), is used to make the energies of the two tetrad fields equal. The definition of the gravitational energy-momentum is used to investigate the energy within the external event horizon. The components of angular-moment…
Many years ago we have introduced into general relativity, {\bf GR}, the canonical superenergy tensors, $S_i^{~k}$, and the canonical angular supermomentum tensors, $S^{ikl}=(-)S^{kil}$, matter and gravitation. We have obtained these tensors by special averaging of the differences of the canonical energy-momentum and canonical angular momentum. The averaging was performed in Riemann normal coordinates, {\bf RNC(P)}; {\bf P} is beginning of these coordinates. About four years ago we have observed that these tensors can also be obtained in other, simpler way, by using the canonical superenergy a…
Geometrical inequalities show how certain parameters of a physical system set restrictions on other parameters. For instance, a black hole of given mass can not rotate too fast, or an ordinary object of given size can not have too much electric charge. In this article we are interested in bounds on the angular momentum and electromagnetic charges, in terms of total mass and size. We are mainly concerned with inequalities for black holes and ordinary objects. The former are the most studied systems in this context in General Relativity, and where most results have been found. Ordinary objects, …
The flux of angular momentum in electromagnetism cannot be expressed entirely in terms of the field's radiative degrees of freedom. Its expression also involves Coulombic pieces of the field, in the form of a charge aspect $q(θ,ϕ)$, a function of polar angles whose integral gives the total charge of the system. Guided by the strong analogy between radiative processes in electromagnetism and gravitation, we ask whether the flux of angular momentum in general relativity might also involve Coulombic pieces of the gravitational field. Further, we ask whether such terms might have been missed in th…
We define a procedure by which observers can measure a type of special-relativistic linear and angular momentum $(P^a, J^{ab})$ at a point in a curved spacetime using only the spacetime geometry in a neighborhood of that point. The method is chosen to yield the conventional results in stationary spacetimes near future null infinity. We also explore the extent to which spatially separated observers can compare the values of angular momentum that they measure and find consistent results. We define a generalization of parallel transport along curves which gives a prescription for transporting val…
New numerical methods have been applied in relativity to obtain a numerical evolution of Einstein equations much more robust and stable. Starting from 3+1 formalism and with the evolution equations written as a FOFCH (first-order flux conservative hyperbolic) system, advanced numerical methods from CFD (Computational Fluid Dynamics) have been successfully applied. A flux limiter mechanism has been implemented in order to deal with steep gradients like the ones usually associated with black hole spacetimes. As a test bed, the method has been applied to 3D metrics describing propagation of nonli…
In the paper we study propagation of light in general relativity through a spacetime filled with cold plasma with infinite conductivity. We use the geometric optics based on Synge's approach. As the main result we provide a formula for calculation of spectral flux of radiation emitted from a pointlike source. The formula employs connecting vectors that are obtained by integrating the ray deviation equation along the reference ray connecting the source and observation event. As a byproduct we formulate Etherington's reciprocity theorem with the inclusion of plasma, interrelating angular size di…
There exist at least a few different kind of averaging of the differences of the energy-momentum and angular momentum in normal coordinates {\bf NC(P)} which give tensorial quantities. The obtained averaged quantities are equivalent mathematically because they differ only by constant scalar dimensional factors. One of these averaging was used in our papers [1-8] giving the {\it canonical superenergy and angular supermomentum tensors}. In this paper we present another averaging of the differences of the energy-momentum and angular momentum which gives tensorial quantities with proper dimensions…
We present an introduction to mass and angular momentum in General Relativity. After briefly reviewing energy-momentum for matter fields, first in the flat Minkowski case (Special Relativity) and then in curved spacetimes with or without symmetries, we focus on the discussion of energy-momentum for the gravitational field. We illustrate the difficulties rooted in the Equivalence Principle for defining a local energy-momentum density for the gravitational field. This leads to the understanding of gravitational energy-momentum and angular momentum as non-local observables that make sense, at bes…
Balance flux laws beyond general relativity
本文研究了超越广义相对论的理论中平衡流律的存在和结构，推导了Horndeski理论中的流表达式，并通过比较验证了方法的通用性。
We exam the validity of the definition of the ADM angular momentum without the parity assumption. Explicit examples of asymptotically flat hypersurfaces in the Minkowski spacetime with zero ADM energy-momentum vector and finite non-zero angular momentum vector are presented. We also discuss the Beig-Ó Murchadha-Regge-Teitelboim center of mass and study analogous examples in the Schwarzschild spacetime.
Recently there have been suggestions that the Lorentz force law is inconsistent with special relativity. This is difficult to understand, since Einstein invented relativity in order to reconcile electrodynamics with mechanics. Here we investigate the momentum of an electric charge and a magnetic dipole in the frame in which both are at rest, and in an infinitesimally boosted frame in which both have a common velocity. We show that for a dipole composed of a magnetic monopole-antimonopole pair the torque is zero in both frames, while if the dipole is a point dipole, the torque is not zero, but …
Angular momentum in the teleparallel equivalent of general relativity
该研究聚焦广义相对论远平行等价理论中的角动量问题，分析了时间规范下3-角动量为零的现象及相关角动量物理解释模糊的问题，推导得到了该理论下角动量的若干通用性质。
Mapping between black-hole perturbation theory and numerical relativity: gravitational-wave energy flux
本研究验证了Islam等人提出的α-β映射可推广至双黑洞引力波能流，找到对应缩放参数使BHPT能流与NR结果对齐，并探究了参数与BHPT次级黑洞有限尺寸修正的关联。
For observers in curved spacetimes, elements of the dual space of the set of linearized Poincaré transformations from an observer's tangent space to itself can be naturally interpreted as local linear and angular momenta. We present an operational procedure by which observers can measure such quantities using only information about the spacetime curvature at their location. When applied by observers near spacelike or null infinity in stationary, vacuum, asymptotically flat spacetimes, there is a sense in which the procedure yields the well-defined linear and angular momenta of the spacetime. W…
We present numerical results on the hydrodynamic stability of coalescing binary stars in the first post Newtonian(1PN) approximation of general relativity. We pay particular attention to the hydrodynamical instability of corotating binary stars in equilibrium states assuming the stiff polytropic equation of state with the adiabatic constant $Γ=3$. In previous 1PN numerical studies on corotating binary stars in equilibrium states, it was found that along the sequence of binary stars as a function of the orbital separation, they have the energy and/or angular momentum minima where the secular in…
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 …
We analytically calculate the equilibrium sequence of the corotating binary stars of incompressible fluid in the first post-Newtonian(PN) approximation of general relativity. By calculating the total energy and total angular momentum of the system as a function of the orbital separation, we investigate the innermost stable circular orbit for corotating binary(we call it ISCCO). It is found that by the first PN effect, the orbital separation of the binary at the ISCCO becomes small with increase of the compactness of each star, and as a result, the orbital angular velocity at the ISCCO increase…
We study the motion of a test particle in a stationary, axially and reflection symmetric spacetime of a central compact object, as affected by interaction with a test radiation field of the same symmetries. Considering the radiation flux with fixed but arbitrary (non-zero) angular momentum, we extend previous results limited to an equatorial motion within a zero-angular-momentum photon flux in the Kerr and Schwarzschild backgrounds. While a unique equilibrium circular orbit exists if the photon flux has zero angular momentum, multiple such orbits appear if the photon angular momentum is suffic…
Dynamical horizons are considered in full, non-linear general relativity. Expressions of fluxes of energy and angular momentum carried by gravitational waves across these horizons are obtained. Fluxes are local, the energy flux is positive and change in the horizon area is related to these fluxes. The flux formulae also give rise to balance laws analogous to the ones obtained by Bondi and Sachs at null infinity and provide generalizations of the first and second laws of black hole mechanics.
The angular momentum of a star is an important astrophysical quantity related to its internal structure, formation and evolution. On average, helioseismology yields S = 1.92 10^41 kg m^2 s^-1 for the angular momentum of the Sun. We show how it should be possible to measure or, at least, constrain it in a near future by using the gravitomagnetic Lense-Thirring effect predicted by general relativity for the orbit of a test particle moving around a central rotating body. We also discuss the present-day situation in view of the latest determinations of the supplementary perihelion precession of Me…
A covariant formula for conserved currents of energy, momentum and angular-momentum is derived from a general form of Noethers theorem applied directly to the Einstein-Hilbert action of classical general relativity. Energy conservation in a closed big-bang cosmology is discussed as a special case. Special care is taken to distinguish between kinematic and dynamic expressions.
We numerically investigate limits of a two-parameter family of stationary solutions to the Einstein-Vlasov system. The solutions are toroidal and have non-vanishing angular momentum. As one tunes to more relativistic solutions (measured for example by an increasing redshift) there exists a sequence of solutions which approaches the extreme Kerr black hole family. Solutions with angular momentum larger than the square of the mass are also investigated, and in the relativistic limit the near-field geometry of such solutions is observed to become conical in the sense that there is a deficit angle…
We study to what extent the effective-one-body description of the dynamical state of a nonspinning, coalescing binary black hole (considered either at merger, or after ringdown) agrees with numerical relativity results. This comparison uses estimates of the integrated losses of energy and angular momentum during ringdown, inferred from recent numerical-relativity data. We find that the values, predicted by the effective-one-body formalism, of the energy and angular momentum of the system agree at the per mil level with their numerical-relativity counterparts, both at merger and in the final st…
In this work we apply the Landau-Lifshitz pseudotensor flux formalism to the calculation of the total mass and the total angular momentum during the evolution of a binary black hole system. We also compare its performance with the traditional integrations for the global quantities. It shows that the advantage of the pseudotensor flux formalism is the smoothness of the numerical value of the global quantities, especially of the total angular momentum. Although the convergence behavior of the global quantities with the pseudotensor flux method is only comparable with the ones with the traditiona…
A four-dimensional asymptotic expansion scheme is used to study the next order effects of the nonlinearity near a spinning dynamical black hole. The angular momentum flux and energy flux formula are then obtained by asymptotic expansion and the compatibility of the coupling Newman-Penrose equations. After constructing the reference frame in terms of the compatible constant spinors, the energy-momentum flux is derived and it is related to the black hole area growth. Directly from the flux formula of the spinning dynamical horizon, we find that the physically reasonable condition on the positivi…
In this thesis, multipole expansions of mass, momentum and stress density will be made for a body in Newtonian mechanics. Using these definitions; momentum, angular momentum, center of mass, force and torque are defined for $N$ gravitationally interacting isolated bodies. Equations of motions of such a system are derived. Definitions of momentum, angular momentum, center of mass, force and torque are made in a relativistic theory. Dynamical (gravitational) skeleton is defined and the multipole moments of the dynamical skeleton are found. Equations of motion for a test body moving in a gravitat…
Using accurate numerical relativity simulations of (nonspinning) black-hole binaries with mass ratios 1:1, 2:1 and 3:1 we compute the gauge invariant relation between the (reduced) binding energy $E$ and the (reduced) angular momentum $j$ of the system. We show that the relation $E(j)$ is an accurate diagnostic of the dynamics of a black-hole binary in a highly relativistic regime. By comparing the numerical-relativity $E^{\rm NR} (j)$ curve with the predictions of several analytic approximation schemes, we find that, while the usual, non-resummed post-Newtonian-expanded $E^{\rm PN} (j)$ relat…
The angular momentum flux from an inspiralling binary system of compact objects moving in quasi-elliptical orbits is computed at the third post-Newtonian (3PN) order using the multipolar post-Minkowskian wave generation formalism. The 3PN angular momentum flux involves the instantaneous, tail, and tail-of-tails contributions as for the 3PN energy flux, and in addition a contribution due to non-linear memory. We average the angular momentum flux over the binary's orbit using the 3PN quasi-Keplerian representation of elliptical orbits. The averaged angular momentum flux provides the final input …
Owing to its transformation property under local boosts, the Brown-York quasilocal energy surface density is the analogue of E in the special relativity formula: E^2-p^2=m^2. In this paper I will motivate the general relativistic version of this formula, and thereby arrive at a geometrically natural definition of an `invariant quasilocal energy', or IQE. In analogy with the invariant mass m, the IQE is invariant under local boosts of the set of observers on a given two-surface S in spacetime. A reference energy subtraction procedure is required, but in contrast to the Brown-York procedure, S i…
We present a code to construct initial data for binary neutron star systems in which the stars are rotating. Our code, based on a formalism developed by Tichy, allows for arbitrary rotation axes of the neutron stars and is able to achieve rotation rates near rotational breakup. We compute the neutron star angular momentum through quasi-local angular momentum integrals. When constructing irrotational binary neutron stars, we find a very small residual dimensionless spin of $\sim 2\times 10^{-4}$. Evolutions of rotating neutron star binaries show that the magnitude of the stars' angular momentum…
The post-post-Newtonian (2PN) accurate mass quadrupole moment, for compact binaries of arbitrary mass ratio, moving in general orbits is obtained by the multi-polar post Minkowskian approach of Blanchet, Damour, and Iyer (BDI). Using this, for binaries in general orbits, the 2PN contributions to the gravitational waveform, and the associated far-zone energy and angular momentum fluxes are computed. For quasi-elliptic orbits, the energy and angular momentum fluxes are averaged over an orbital period, and employed to determine the 2PN corrections to the rate of decay of the orbital elements.
Using the ellipsoidal model for the density configuration, we calculate the equilibrium sequence of the corotating binary stars of the polytropic equation of state in the first post-Newtonian approximation of general relativity. After we calibrate this model by comparing with previous numerical results, we perform the stability analysis by calculating the energy and the angular momentum of the system as a function of the orbital separation. We find that the orbital angular velocity at the energy and/or momentum minimum increases with the increase of the compactness of each star, and this fact …
Classical black holes and event horizons are highly non-local objects, defined in terms of the causal past of future null infinity. Alternative, (quasi)local definitions are often used in mathematical, quantum, and numerical relativity. These include apparent, trapping, isolated, and dynamical horizons, all of which are closely associated to two-surfaces of zero outward null expansion. In this paper we show that three-surfaces which can be foliated with such two-surfaces are suitable boundaries in both a quasilocal action and a phase space formulation of general relativity. The resulting forma…
How does one measure the angular momentum carried away by gravitational radiation during the merger of a binary black hole? This has been a subtle issue since the 1960's due to the discovery of ``supertranslation ambiguity": the angular momentum recorded by two distant observers of the same system may not be the same. In this talk, I shall describe how the theory of quasilocal mass and optimal isometric embedding identifies a new definition of angular momentum that is free of any supertranslation ambiguity. This is based on joint work with Po-Ning Chen, Jordan Keller, Ye-Kai Wang, and Shing-Tu…
First post-Newtonian (1PN) hydrostatic equations for an irrotational fluid which have been recently derived are solved for an incompressible star. The 1PN configurations are expressed as a deformation of the Newtonian irrotational Riemann ellipsoid using Lagrangian displacement vectors introduced by Chandrasekhar. For the 1PN solutions, we also calculate the luminosity of gravitational waves in the 1PN approximation using the Blanchet-Damour formalism. It is found that the solutions of the 1PN equations exhibit singularities at points where the axial ratios of semi-axes are 1:0.5244:0.6579 and…
Mapping between black-hole perturbation theory and numerical relativity II: gravitational-wave momentum
本文提出 NR 与 BHPT 引力波动量之间的双参数时间无关映射，验证其在并合前的有效性，并分析质量比依赖、有限尺寸修正及遗迹属性意义。
A four-dimensional asymptotic expansion scheme is used to study the next order effects of the nonlinearity near a spinning dynamical black hole. The angular momentum flux and energy flux formula are then obtained by constructing the reference frame in terms of the compatible constant spinors and the compatibility of the coupling leading order Newman-Penrose equations. By using the slow rotation and small-tide approximation for a spinning black hole, we chose the horizon cross-section is spherical symmetric. It turns out the flux formula is rather simple and can be compared with the known resul…
A set of exact quasi-local conservation equations is derived from the Einstein's equations using the first-order Kaluza-Klein formalism of general relativity in the (2,2)-splitting of 4-dimensional spacetime. These equations are interpreted as quasi-local energy, momentum, and angular momentum conservation equations. In the asymptotic region of asymptotically flat spacetimes, it is shown that the quasi-local energy and energy-flux integral reduce to the Bondi energy and energy-flux, respectively. In spherically symmetric spacetimes, the quasi-local energy becomes the Misner-Sharp energy. Moreo…
The total flux of outgoing radiation in a strong gravitational field decreases due to backscattering if the sources are close to an apparent horizon. It can cause detectable changes in the shape of signals. Backscattering could well be of relevance to astrophysics and would constitute a new test of the validity of general relativity. An explicit bound for this effect is derived for scalar fields.
The Ashtekar-Streubel fluxes give a proposed definition of the angular momentum emitted by an isolated gravitationally radiating system. This was based on identifying a "phase space of radiative modes," independent of any internal degrees of freedom, and using the Hamiltonian functions conjugate to the action of the Bondi-Metzner-Sachs (BMS) group as the energy-momentum, supermomentum and angular momentum. However, there are some difficulties in formulating this phase space so as to get the proper degrees of freedom. I consider how to address this point, and also to identify circumstances in w…
Bekenstein bounds for the entropy of a body imply an universal inequality between size, energy, angular momentum and charge. We prove this inequality in Electromagnetism. We also prove it, for the particular case of zero angular momentum, in General Relativity. We further discuss the relation of these inequalities with inequalities between size, angular momentum and charge recently studied in the literature.
Binary black-hole systems are expected to be important sources of gravitational waves for upcoming gravitational-wave detectors. If the spins are not colinear with each other or with the orbital angular momentum, these systems exhibit complicated precession dynamics that are imprinted on the gravitational waveform. We develop a new procedure to match the precession dynamics computed by post-Newtonian (PN) theory to those of numerical binary black-hole simulations in full general relativity. For numerical relativity NR) simulations lasting approximately two precession cycles, we find that the P…
A new quasigroup approach to conservation laws in general relativity is applied to study asymptotically flat at future null infinity spacetime. The infinite-parametric Newman-Unti group of asymptotic symmetries is reduced to the Poincaré quasigroup and the Noether charge associated with any element of the Poincaré quasialgebra is defined. The integral conserved quantities of energy-momentum and angular momentum are linear on generators of Poincaré quasigroup, free of the supertranslation ambiguity, posess the flux and identically equal to zero in Minkowski spacetime.
We review a number of results recently obtained in the area of constructing rotating solitons in a four dimensional asymptotically flat spacetime. Various models are examined, special attention being paid to the monopole-antimonopole and gauged skyrmion configurations, which have a nonvanishing total angular momentum. For all known examples of rotating solitons, the angular momentum is fixed by some conserved charge of the matter fields.
The formalism of electric - magnetic duality, first pioneered by Reinich and Wheeler, extends General Relativity to encompass non-Abelian fields. Several energy Tensors T^uv with non-vanishing trace matter are developed solely as a function of the field strength tensor F^uv, including the Euler tensor, and tensors for matter in flux, pressure in flux, and stationary pressure. The spacetime metric g_uv is not only a solution to the second-order Einstein equation based on T^uv, but is also constrained by a third-order equation involving the Bianchi identity together with the gravitational energy…
New Hamiltonian formalism and quasi-local conservation equations of general relativity
本文用(2,2)形式体系描述3+1维时空的爱因斯坦引力，从散度型方程导出准局域守恒量与通量，证明其余方程为哈密顿演化方程，并在渐近区域退化为Bondi量。
Angular Momentum Flux in Scalar-Tensor Theories up to 1.5 post-Newtonian Order
该研究利用多极矩计算了无质量标量-张量理论中非圆无自旋致密双星最高至1.5PN阶的角动量通量，分解了通量的构成并推导了尾效应与非线性记忆效应的表达式，为偏心双星的辐射反作用力和波形模板构建提供了基础。
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