Boundary Energy-Momentum-News Complex

• Boundary Energy-Momentum-News Complex is a null-infinity formulation that unifies the Coulombic mass aspect with radiative shear and news data.
• It employs Bondi-Sachs and Carroll-covariant frameworks to encode asymptotic gravitational data via shear functions and conformal metrics.
• The approach yields explicit energy-momentum splits, flux balance laws, and variational definitions, enhancing both theoretical and numerical analyses.

The boundary energy-momentum-news complex is a null-infinity formulation of asymptotically flat gravitation in which the non-radiative charge sector and the radiative shear/news sector are treated together. In Bondi-Sachs language, the asymptotic data are encoded by the mass aspect and by shear functions whose retarded-time derivatives are the news; in Carroll-covariant formulations at future null infinity, the relevant boundary variables are the Carroll metric data $(\tau_\mu,h_{\mu\nu})$, the shear $C_{\mu\nu}$, and their responses obtained from a renormalised action. Across these formulations, the central structural point is that Bondi energy-momentum is not exhausted by a Coulombic mass aspect alone: radiative data enter either through explicit boundary terms, through conformally invariant flux laws on arbitrary cuts of $\scri$, or through variational Ward identities whose content is the Bondi loss equations (Maluf et al., 2015, Frauendiener et al., 2021, Hartong et al., 8 May 2025, Hartong et al., 8 Jul 2026).

1. Asymptotic data at future null infinity

In retarded coordinates $(u,r,\theta,\phi)$, with $u=t-r$, the radiative vacuum Bondi-Sachs line element can be written as

$$ds^2 = -V e^{2\beta} du^2 - 2 e^{2\beta} du\,dr + r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$

with asymptotic expansion

$$\beta = -\frac{c^2+d^2}{4r^2}+O(r^{-3}),\qquad U^A=O(r^{-2}),\qquad V=r-2M(u,\theta,\phi)+O(r^{-1}),$$

$$h_{AB}=q_{AB}+\frac{1}{r}c_{AB}(u,\theta,\phi)+O(r^{-2}),$$

where $q_{AB}=\mathrm{diag}(1,\sin^2\theta)$, $M(u,\theta,\phi)$ is the mass aspect, and

$C_{\mu\nu}$0

The two free Bondi-Sachs functions $C_{\mu\nu}$1 and $C_{\mu\nu}$2 determine the radiative shear, and their $C_{\mu\nu}$3-derivatives are the news (Maluf et al., 2015).

A Carroll-covariant reformulation replaces the unit-sphere presentation by boundary data intrinsic to $C_{\mu\nu}$4. In Bondi-like coordinates $C_{\mu\nu}$5, the metric takes the form

$C_{\mu\nu}$6

or, in the more explicit Carroll-covariant Bondi-Sachs gauge,

$C_{\mu\nu}$7

The boundary one-form $C_{\mu\nu}$8 and

$C_{\mu\nu}$9

define the Carroll structure, while the subleading symmetric trace-free term defines the shear,

$\scri$0

equivalently summarized by

$\scri$1

In this setting, $\scri$2 is the Carroll-covariant radiative data, on the same footing as the boundary metric variables (Hartong et al., 8 May 2025, Hartong et al., 8 Jul 2026).

A recurring theme is that the choice of Bondi gauge is not fundamental. A manifestly conformally invariant and gauge independent formulation on arbitrary cuts of $\scri$3 uses conformally invariant GHP operators and replaces gauge-fixed unit-sphere constructions by intrinsic densities, asymptotic translations, and a cut-independent Lorentzian structure on the translation space (Frauendiener et al., 2021).

2. Bondi-Sachs 4-momentum and the TEGR radiation term

The usual Bondi-Sachs 4-momentum is built from the mass aspect alone: $\scri$4 with $\scri$5 and $\scri$6. In this standard form, the charge is determined by the mass aspect only (Maluf et al., 2015).

In the teleparallel equivalent of general relativity (TEGR), the gravitational energy-momentum for an asymptotically flat tetrad is defined by the boundary integral

$\scri$7

For the time component, the resulting boundary expression is

$\scri$8

where

$\scri$9

$(u,r,\theta,\phi)$0

This yields the split

$(u,r,\theta,\phi)$1

Defining the news functions

$(u,r,\theta,\phi)$2

one may write

$(u,r,\theta,\phi)$3

The paper interprets the standard term $(u,r,\theta,\phi)$4 as the energy of the isolated source, and the extra piece $(u,r,\theta,\phi)$5 as the energy of gravitational radiation. The final boundary energy-momentum-news split is

$(u,r,\theta,\phi)$6

$(u,r,\theta,\phi)$7

In this TEGR construction, the total 4-momentum is explicitly split into “Coulomb” and radiative parts, and the news is identified as the carrier of gravitational-wave energy (Maluf et al., 2015).

3. Conformally invariant Bondi energy-momentum on arbitrary cuts

A different formulation addresses the problem of computing Bondi-Sachs energy-momentum on a cut of $(u,r,\theta,\phi)$8 that is not presented in a Bondi system. The construction uses a conformally invariant version of the GHP formalism, with GHP scalars carrying conformal weight $(u,r,\theta,\phi)$9 and GHP weights $u=t-r$0, and intrinsic derivative operators

$u=t-r$1

$u=t-r$2

Their commutators close on the conformal densities

$u=t-r$3

with Jacobi identities implying

$u=t-r$4

The quantity $u=t-r$5 combines the cut’s Gauss curvature with a GHP correction, while the co-curvature $u=t-r$6 is defined by

$u=t-r$7

Geometrically, $u=t-r$8 is Penrose-Geroch’s “gauge field” encoding how one must twist the cuts to restore unit-sphere geometry (Frauendiener et al., 2021).

In this framework, an asymptotic translation is a conformal density

$u=t-r$9

satisfying

$$ds^2 = -V e^{2\beta} du^2 - 2 e^{2\beta} du\,dr + r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$0

The space $$ds^2 = -V e^{2\beta} du^2 - 2 e^{2\beta} du\,dr + r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$1 of solutions is real 4-dimensional, and a Lorentzian quadratic form is defined by

$$ds^2 = -V e^{2\beta} du^2 - 2 e^{2\beta} du\,dr + r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$2

On $$ds^2 = -V e^{2\beta} du^2 - 2 e^{2\beta} du\,dr + r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$3, this quadratic form has signature $$ds^2 = -V e^{2\beta} du^2 - 2 e^{2\beta} du\,dr + r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$4, reducing in Bondi gauge to the standard Minkowski inner product of the $$ds^2 = -V e^{2\beta} du^2 - 2 e^{2\beta} du\,dr + r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$5 harmonics (Frauendiener et al., 2021).

The mass-aspect $$ds^2 = -V e^{2\beta} du^2 - 2 e^{2\beta} du\,dr + r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$6 is then defined by

$$ds^2 = -V e^{2\beta} du^2 - 2 e^{2\beta} du\,dr + r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$7

where $$ds^2 = -V e^{2\beta} du^2 - 2 e^{2\beta} du\,dr + r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$8 is the $$ds^2 = -V e^{2\beta} du^2 - 2 e^{2\beta} du\,dr + r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$9 component of the rescaled Weyl tensor, $$\beta = -\frac{c^2+d^2}{4r^2}+O(r^{-3}),\qquad U^A=O(r^{-2}),\qquad V=r-2M(u,\theta,\phi)+O(r^{-1}),$$0 is the shear, $$\beta = -\frac{c^2+d^2}{4r^2}+O(r^{-3}),\qquad U^A=O(r^{-2}),\qquad V=r-2M(u,\theta,\phi)+O(r^{-1}),$$1 is the conformal news density, and $$\beta = -\frac{c^2+d^2}{4r^2}+O(r^{-3}),\qquad U^A=O(r^{-2}),\qquad V=r-2M(u,\theta,\phi)+O(r^{-1}),$$2 is the conformal density with $$\beta = -\frac{c^2+d^2}{4r^2}+O(r^{-3}),\qquad U^A=O(r^{-2}),\qquad V=r-2M(u,\theta,\phi)+O(r^{-1}),$$3. The Bondi news scalar is

$$\beta = -\frac{c^2+d^2}{4r^2}+O(r^{-3}),\qquad U^A=O(r^{-2}),\qquad V=r-2M(u,\theta,\phi)+O(r^{-1}),$$4

and the symmetric trace-free news tensor on a cut is

$$\beta = -\frac{c^2+d^2}{4r^2}+O(r^{-3}),\qquad U^A=O(r^{-2}),\qquad V=r-2M(u,\theta,\phi)+O(r^{-1}),$$5

The Bondi-Sachs energy-momentum becomes a surface integral on an arbitrary cut $$\beta = -\frac{c^2+d^2}{4r^2}+O(r^{-3}),\qquad U^A=O(r^{-2}),\qquad V=r-2M(u,\theta,\phi)+O(r^{-1}),$$6: $$\beta = -\frac{c^2+d^2}{4r^2}+O(r^{-3}),\qquad U^A=O(r^{-2}),\qquad V=r-2M(u,\theta,\phi)+O(r^{-1}),$$7 For any two cuts $$\beta = -\frac{c^2+d^2}{4r^2}+O(r^{-3}),\qquad U^A=O(r^{-2}),\qquad V=r-2M(u,\theta,\phi)+O(r^{-1}),$$8 bounding a slab of $$\beta = -\frac{c^2+d^2}{4r^2}+O(r^{-3}),\qquad U^A=O(r^{-2}),\qquad V=r-2M(u,\theta,\phi)+O(r^{-1}),$$9,

$$h_{AB}=q_{AB}+\frac{1}{r}c_{AB}(u,\theta,\phi)+O(r^{-2}),$$0

In a Bondi system this reproduces

$$h_{AB}=q_{AB}+\frac{1}{r}c_{AB}(u,\theta,\phi)+O(r^{-2}),$$1

This formulation makes explicit that neither the definition of the mass aspect nor the normalization of asymptotic translations requires a preferred Bondi presentation of $$h_{AB}=q_{AB}+\frac{1}{r}c_{AB}(u,\theta,\phi)+O(r^{-2}),$$2 (Frauendiener et al., 2021).

4. Variational definition from renormalised action and Carroll geometry

A holographic-type renormalisation of the Einstein-Hilbert action near future null infinity yields a fully covariant variational definition of the boundary energy-momentum-news complex. Starting from

$$h_{AB}=q_{AB}+\frac{1}{r}c_{AB}(u,\theta,\phi)+O(r^{-2}),$$3

one regulates the variational problem at $$h_{AB}=q_{AB}+\frac{1}{r}c_{AB}(u,\theta,\phi)+O(r^{-2}),$$4 and adds counterterms. In one presentation these are

$$h_{AB}=q_{AB}+\frac{1}{r}c_{AB}(u,\theta,\phi)+O(r^{-2}),$$5

$$h_{AB}=q_{AB}+\frac{1}{r}c_{AB}(u,\theta,\phi)+O(r^{-2}),$$6

and, for $$h_{AB}=q_{AB}+\frac{1}{r}c_{AB}(u,\theta,\phi)+O(r^{-2}),$$7,

$$h_{AB}=q_{AB}+\frac{1}{r}c_{AB}(u,\theta,\phi)+O(r^{-2}),$$8

The full renormalised action is

$$h_{AB}=q_{AB}+\frac{1}{r}c_{AB}(u,\theta,\phi)+O(r^{-2}),$$9

A complementary summary writes the cutoff action schematically as

$q_{AB}=\mathrm{diag}(1,\sin^2\theta)$0

and states that in $q_{AB}=\mathrm{diag}(1,\sin^2\theta)$1 one needs a log counterterm $q_{AB}=\mathrm{diag}(1,\sin^2\theta)$2 (Hartong et al., 8 May 2025, Hartong et al., 8 Jul 2026).

On shell, after removing the cutoff, the boundary variation localises to

$q_{AB}=\mathrm{diag}(1,\sin^2\theta)$3

or equivalently

$q_{AB}=\mathrm{diag}(1,\sin^2\theta)$4

Here $q_{AB}=\mathrm{diag}(1,\sin^2\theta)$5 is the Carrollian volume density, $q_{AB}=\mathrm{diag}(1,\sin^2\theta)$6 is the boundary energy current, $q_{AB}=\mathrm{diag}(1,\sin^2\theta)$7 the stress tensor, and $q_{AB}=\mathrm{diag}(1,\sin^2\theta)$8 the response to the shear. In 4D, the shear is on par with the Carroll metric data, and their combined response defines the boundary energy-momentum-news complex (Hartong et al., 8 May 2025).

The news tensor is identified by

$q_{AB}=\mathrm{diag}(1,\sin^2\theta)$9

with

$M(u,\theta,\phi)$0

or equivalently

$M(u,\theta,\phi)$1

Thus $M(u,\theta,\phi)$2 is a spatial symmetric trace-free tensor and, in Newman-Penrose language, reduces to $M(u,\theta,\phi)$3 (Hartong et al., 8 May 2025, Hartong et al., 8 Jul 2026).

5. Ward identities, trace relation, and Carroll-boost anomaly

The variational definition leads directly to Ward-type identities. Under boundary diffeomorphisms $M(u,\theta,\phi)$4, one obtains

$M(u,\theta,\phi)$5

with

$M(u,\theta,\phi)$6

A Carroll-covariant form is

$M(u,\theta,\phi)$7

Projecting along $M(u,\theta,\phi)$8 and $M(u,\theta,\phi)$9 yields generalised Bondi-mass and angular-momentum loss equations (Hartong et al., 8 May 2025, Hartong et al., 8 Jul 2026).

In the split form summarized for $C_{\mu\nu}$00, the energy and momentum equations are

$C_{\mu\nu}$01

$C_{\mu\nu}$02

The corresponding Carroll-covariant Bondi flux-balance laws are

$C_{\mu\nu}$03

$C_{\mu\nu}$04

In the usual retarded-time gauge one recovers

$C_{\mu\nu}$05

and, in standard Bondi coordinates,

$C_{\mu\nu}$06

These are the familiar Bondi mass and momentum loss laws (Hartong et al., 8 May 2025, Hartong et al., 8 Jul 2026).

Weyl invariance gives the trace Ward identity

$C_{\mu\nu}$07

When $C_{\mu\nu}$08, the stress tensor is traceless (Hartong et al., 8 May 2025).

The Carroll-boost relation is more subtle. One formulation states that boost invariance would imply

$C_{\mu\nu}$09

but in three bulk dimensions and in four bulk dimensions one instead finds a genuine anomaly,

$C_{\mu\nu}$10

which cannot be removed by local counterterms, satisfies Wess-Zumino consistency, and is responsible for the nontrivial central extensions observed in the BMS$C_{\mu\nu}$11 and BMS$C_{\mu\nu}$12 algebras. The 2026 summary presents the classical relation as exact and allows an anomalous right-hand side $C_{\mu\nu}$13 in quantum gravity/gluing. The common point is that boost covariance is not purely kinematical once the shear-response sector is included (Hartong et al., 8 May 2025, Hartong et al., 8 Jul 2026).

6. Unified BMS fluxes and numerical realisation

A numerically oriented formulation treats the news sector as a single complex controlling the fluxes associated with the full Bondi-Metzner-Sachs symmetry algebra. In Bondi-Sachs coordinates $C_{\mu\nu}$14, introducing $C_{\mu\nu}$15, the conformal metric $C_{\mu\nu}$16 is smooth at $C_{\mu\nu}$17, and the asymptotic radiation content is encoded in the shear coefficient $C_{\mu\nu}$18. A 4-D transformation to inertial coordinates $C_{\mu\nu}$19 is then constructed so that $C_{\mu\nu}$20 is null, shear-free and divergence-free, the 2-metric on $C_{\mu\nu}$21 is the unit sphere, and the generators are affinely parametrized by $C_{\mu\nu}$22. The conformal factor $C_{\mu\nu}$23 is determined by the elliptic uniformization equation

$C_{\mu\nu}$24

and then propagated along null generators by

$C_{\mu\nu}$25

In the inertial frame, the Bondi news is

$C_{\mu\nu}$26

with $C_{\mu\nu}$27 the strain on a cross-section of $C_{\mu\nu}$28 (Handmer et al., 2016).

For a general BMS generator

$C_{\mu\nu}$29

Geroch’s linkage construction yields a local flux density $C_{\mu\nu}$30, whose retarded-time derivative satisfies

$C_{\mu\nu}$31

In Newman-Penrose form,

$C_{\mu\nu}$32

Because every BMS generator determines a scalar $C_{\mu\nu}$33, all fluxes are sourced by the same radiative field $C_{\mu\nu}$34, or equivalently by the Bondi news $C_{\mu\nu}$35. In particular, for translations,

$C_{\mu\nu}$36

while corresponding formulas are given for angular momentum, boosts, and supermomentum. This yields a single “news complex” covering energy, momentum, supermomentum, and angular momentum (Handmer et al., 2016).

The numerical implementation in the Spectral Einstein Code (SpEC) uses a Cauchy binary-black-hole evolution to provide data on a timelike worldtube, a spectral characteristic solver in $C_{\mu\nu}$37 and spherical harmonics up to $C_{\mu\nu}$38, spectral re-expansion on $C_{\mu\nu}$39 in inertial coordinates, and a 4th-order time integrator. Convergence tests on a generic precessing binary show spectral spatial convergence until the 4th-order time integrator error dominates, and all linkage fluxes exhibit clean 4th-order convergence once the initial junk radiation has passed. Within this framework, the boundary energy-momentum-news complex is not only geometrically defined but also directly computable in numerical relativity (Handmer et al., 2016).