---
title: Boundary Energy-Momentum-News Complex
url: https://www.emergentmind.com/topics/boundary-energy-momentum-news-complex
type: topic
---

# Boundary Energy-Momentum-News Complex

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 [1503.03695, 2104.13646, 2505.05432, 2607.07872].

## 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
\[
c_{AB}\,dx^A dx^B
=
c(u,\theta,\phi)\,[d\theta^2-\sin^2\theta\,d\phi^2]
+2\,d(u,\theta,\phi)\,\sin\theta\,d\theta\,d\phi.
\]
The two free Bondi-Sachs functions \(c\) and \(d\) determine the radiative shear, and their \(u\)-derivatives are the news [1503.03695].

A Carroll-covariant reformulation replaces the unit-sphere presentation by boundary data intrinsic to \(\scri^+\). In Bondi-like coordinates \((r,u,x^A)\), the metric takes the form
\[
ds^2=-2\,e^\beta\tau_\mu dx^\mu\,dr+\Pi_{\mu\nu}(r,x)\,dx^\mu dx^\nu,
\]
or, in the more explicit Carroll-covariant Bondi-Sachs gauge,
\[
ds^2=
-\,2\,e^{\beta(r,x)}\tau_\mu(x)\,dr\,dx^\mu
+\bigl[-e^{2\beta}S(r,x)\tau_\mu\tau_\nu+\Pi_{\mu\nu}(r,x)\bigr]dx^\mu dx^\nu.
\]
The boundary one-form \(\tau_\mu dx^\mu\) and
\[
h_{\mu\nu}=\lim_{r\to\infty}\frac1r\Pi_{\mu\nu}
\]
define the Carroll structure, while the subleading symmetric trace-free term defines the shear,
\[
C_{\mu\nu}
=
\lim_{r\to\infty}\Bigl[r^{-1}\Pi_{\mu\nu}
-\frac1d\,h_{\mu\nu}h^{\rho\sigma}\Pi_{\rho\sigma}\Bigr],
\]
equivalently summarized by
\[
\Pi_{\mu\nu}(r,x)=r^2h_{\mu\nu}(x)+r\,C_{\mu\nu}(x)+\cdots.
\]
In this setting, \(C_{\mu\nu}\) is the Carroll-covariant radiative data, on the same footing as the boundary metric variables [2505.05432, 2607.07872].

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 \(\scri\) 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 [2104.13646].

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

The usual Bondi-Sachs 4-momentum is built from the mass aspect alone:
\[
P^0_{BS}(u)=\frac{1}{4\pi}\int_{S^2}M(u,\theta,\phi)\,d\Omega,
\qquad
P^i_{BS}(u)=\frac{1}{4\pi}\int_{S^2}M(u,\theta,\phi)\,\hat x^i\,d\Omega,
\]
with \(d\Omega=\sin\theta\,d\theta\,d\phi\) and \(\hat x^i=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)\). In this standard form, the charge is determined by the mass aspect only [1503.03695].

In the teleparallel equivalent of general relativity (TEGR), the gravitational energy-momentum for an asymptotically flat tetrad is defined by the boundary integral
\[
P^a=-\oint_{S^2_{r\to\infty}} dS_k\,\Pi^{ak},
\qquad
\Pi^{ak}=-4k\,e\,\Sigma^{a0k},
\qquad
k=\frac{1}{16\pi}.
\]
For the time component, the resulting boundary expression is
\[
P^{(0)}(u)
=
\frac{1}{4\pi}\int_{S^2}
\Bigl[M(u,\theta,\phi)+\partial_u F\bigl(c,d,\partial_\theta c,\partial_\phi d\bigr)\Bigr]\,d\Omega,
\]
where
\[
F(c,d)=
-\frac14\Bigl\{l^2+\bar l^{\,2}+2(c^2+d^2)\Bigr\},
\]
\[
l=\partial_\theta c+2c\cot\theta+\frac{\partial_\phi d}{\sin\theta},
\qquad
\bar l=\partial_\theta d+2d\cot\theta-\frac{\partial_\phi c}{\sin\theta}.
\]
This yields the split
\[
P^0=P^0_{BS}+\Delta E_{\rm rad},
\qquad
\Delta E_{\rm rad}(u)
=
\frac{1}{4\pi}\int_{S^2}\partial_u F(c,d)\,d\Omega.
\]

Defining the news functions
\[
N_c=\partial_u c,\qquad N_d=\partial_u d,
\]
one may write
\[
\Delta E_{\rm rad}(u)
=
\frac{1}{32\pi}\int_{S^2}N_{AB}N^{AB}\,d\Omega,
\qquad
N_{AB}=\partial_u c_{AB},
\qquad
N_{AB}N^{AB}=2(N_c^2+N_d^2).
\]
The paper interprets the standard term \(\int M\,d\Omega\) as the energy of the isolated source, and the extra piece \(\partial_u F\) as the energy of gravitational radiation. The final boundary energy-momentum-news split is
\[
P^{0}(u)
=
\frac{1}{4\pi}\int_{S^2}M\,d\Omega
+
\frac{1}{4\pi}\int_{S^2}\partial_u F\,d\Omega
\equiv
P^0_{\rm BS}+\Delta E_{\rm rad},
\]
\[
P^{i}(u)
=
\frac{1}{4\pi}\int_{S^2}M\,\hat x^i\,d\Omega
+
\frac{1}{4\pi}\int_{S^2}\partial_u F\,\hat x^i\,d\Omega
\equiv
P^i_{\rm BS}+(\Delta P_{\rm rad})^i.
\]
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 [1503.03695].

## 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 \(\scri\) 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 \(w\) and GHP weights \((p,q)\), and intrinsic derivative operators
\[
\thorn'_c\,\eta=\thorn'\eta+(w+p+q)\rho'\eta,
\]
\[
\eth_c\,\eta=\eth\eta+(w+q)\tau\eta,
\qquad
\tilde\eth_c\,\eta=\eth'\eta+(w+p)\bar\tau\eta.
\]
Their commutators close on the conformal densities
\[
\cP=\thorn'\tau-\eth\rho',
\qquad
\cQ=K-\eth'\tau,
\qquad
\bar\cP=\overline{\cP},
\]
with Jacobi identities implying
\[
\thorn'_c\,\cQ+\tilde\eth_c\,\cP
=
\thorn'_c\,\cQ+\eth_c\,\bar\cP=0.
\]
The quantity \(\cQ\) combines the cut’s Gauss curvature with a GHP correction, while the co-curvature \(\B\) is defined by
\[
\tilde\eth_c\,\B+\eth_c\,\cQ=0,
\qquad
\thorn'_c\,\B=\eth_c\,\cP.
\]
Geometrically, \(\B\) is Penrose-Geroch’s “gauge field” encoding how one must twist the cuts to restore unit-sphere geometry [2104.13646].

In this framework, an asymptotic translation is a conformal density
\[
U:[1;0,0]
\]
satisfying
\[
\eth_c^2 U=\B\,U.
\]
The space \(\scr T\) of solutions is real 4-dimensional, and a Lorentzian quadratic form is defined by
\[
q[U]
=
2\,\cQ\,U^2
+
2\,U\,\eth_c\tilde\eth_c\,U
-
2\,\eth_cU\,\tilde\eth_cU.
\]
On \(\scr T\), this quadratic form has signature \((-,+,+,+)\), reducing in Bondi gauge to the standard Minkowski inner product of the \(\ell=0,1\) harmonics [2104.13646].

The mass-aspect \(\fM\) is then defined by
\[
A\,\fM
=
-\,A\,\psi_2
+\sigma\,N
+\tilde\eth_c^{\,2}\sigma,
\]
where \(\psi_2\) is the \(n-m\) component of the rescaled Weyl tensor, \(\sigma\) is the shear, \(N\) is the conformal news density, and \(A\) is the conformal density with \(N^a=-\nabla^a\Omega=A\,n^a\). The Bondi news scalar is
\[
\News=N+\overline\B,
\qquad
\thorn'_c\News=A\psi_4,
\qquad
\eth_c\News=A\psi_3,
\]
and the symmetric trace-free news tensor on a cut is
\[
N_{AB}=2\,\Re\!\bigl(\News\,m_A m_B\bigr),
\qquad
N_{AB}N^{AB}=2\,\News\,\overline\News.
\]

The Bondi-Sachs energy-momentum becomes a surface integral on an arbitrary cut \(\cC\):
\[
4\pi\,m_\cC[U]
=
\int_\cC U\,\fM\,\dd^2\cS.
\]
For any two cuts \(\cC_1,\cC_2\) bounding a slab of \(\scri\),
\[
m_{\cC_2}[U]-m_{\cC_1}[U]
=
-\frac1{4\pi}\int_{\scri_1^2}U\,\News\,\overline\News\,\dd^3V.
\]
In a Bondi system this reproduces
\[
\frac{\dd P^a}{\dd u}
=
-\frac1{4\pi}\int_{S^2}N_{BC}N^{BC}\,U^a\,\dd\Omega.
\]
This formulation makes explicit that neither the definition of the mass aspect nor the normalization of asymptotic translations requires a preferred Bondi presentation of \(\scri\) [2104.13646].

## 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
\[
S_{\rm EH}=\frac1{16\pi G}\int_{\cM}d^{d+2}x\sqrt{-g}\,R,
\]
one regulates the variational problem at \(r=\Lambda\) and adds counterterms. In one presentation these are
\[
S_{\rm ext}
=
2\,\frac1{16\pi G}\int_{r=\Lambda}d^{d+1}x\sqrt{-g}\,\bigl(K_{MN}N^MN^N\bigr),
\]
\[
S_{\rm norm}
=
-\,d\,\frac1{16\pi G}\int_{r=\Lambda}d^{d+1}x\sqrt{-g}\,\frac{N^2}{r},
\qquad
N^2=g^{rr},
\]
and, for \(d=2\),
\[
S_{\rm int}
=
-\,\frac1{16\pi G}\int_{r=\Lambda}d^3x\sqrt{-g}\,\frac1r\,R[C].
\]
The full renormalised action is
\[
S_{\rm ren}=S_{\rm EH}+2\,S_{\rm ext}-d\,S_{\rm norm}-S_{\rm int}.
\]
A complementary summary writes the cutoff action schematically as
\[
S
=
\frac{1}{16\pi G}\int_{r\le R}d^{d+2}x\sqrt{-g}\,R
+
\frac{1}{8\pi G}\int_{r=R}d^{d+1}x\sqrt{\gamma}\bigl(K+\cdots\bigr),
\]
and states that in \(d=2\) one needs a log counterterm \(\sim \ln R\,C^{ab}C_{ab}\) [2505.05432, 2607.07872].

On shell, after removing the cutoff, the boundary variation localises to
\[
\delta S_{\rm ren}\big|_{\scri^+}
=
2\int_{\scri^+}d^{d+1}x\;e\,
\Bigl(
T^\mu\,\delta\tau_\mu
+\tfrac12\,T^{\mu\nu}\,\delta h_{\mu\nu}
+\tfrac12\,S^{\mu\nu}\,\delta C_{\mu\nu}
\Bigr),
\]
or equivalently
\[
\delta S
=
\int_{\scri^+}d^{d+1}x\;e\,
\Bigl[
T^\mu\,\delta\tau_\mu
+\tfrac12\,T^{\mu\nu}\,\delta h_{\mu\nu}
+\tfrac12\,S^{\mu\nu}\,\delta C_{\mu\nu}
\Bigr].
\]
Here \(e=\det(\tau_\mu,e^a{}_\mu)\) is the Carrollian volume density, \(T^\mu\) is the boundary energy current, \(T^{\mu\nu}\) the stress tensor, and \(S^{\mu\nu}\) 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 [2505.05432].

The news tensor is identified by
\[
S^{\mu\nu}
=
\frac12\,h^{\mu\rho}h^{\nu\sigma}N_{\rho\sigma},
\]
with
\[
N_{\mu\nu}
=
-\,\cL_vC_{\mu\nu}-\frac12\,K\,C_{\mu\nu},
\]
or equivalently
\[
N_{\mu\nu}
=
-\Bigl(\mathcal L_v+\frac12K\Bigr)C_{\mu\nu}.
\]
Thus \(N_{\mu\nu}\) is a spatial symmetric trace-free tensor and, in Newman-Penrose language, reduces to \(N_{AB}=\partial_u C_{AB}\) [2505.05432, 2607.07872].

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

The variational definition leads directly to Ward-type identities. Under boundary diffeomorphisms \(\chi^\mu\), one obtains
\[
0
=
-\,e^{-1}\partial_\mu\Bigl[e\bigl(T^\mu{}_\nu+S^{\mu\rho}C_{\rho\nu}\bigr)\Bigr]
+T^\mu\partial_\nu\tau_\mu
+\frac12T^{\mu\rho}\partial_\nu h_{\mu\rho}
+\frac12S^{\mu\rho}\partial_\nu C_{\mu\rho},
\]
with
\[
T^\mu{}_\nu=T^\mu\tau_\nu+T^{\mu\rho}h_{\rho\nu}.
\]
A Carroll-covariant form is
\[
0=\nabla_\mu T^\mu{}_\nu
+T^\mu{}_\rho\,F^\rho{}_{\nu\mu}
-\frac12\,S^{\rho\sigma}\nabla_\nu C_{\rho\sigma}.
\]
Projecting along \(v^\nu\) and \(h^\nu{}_\kappa\) yields generalised Bondi-mass and angular-momentum loss equations [2505.05432, 2607.07872].

In the split form summarized for \(d=2\), the energy and momentum equations are
\[
-\bigl(\mathcal L_v-\tfrac{d+1}{d}K\bigr)E
-\frac12\,S^{ab}N_{ab}
+(\nabla_a+a_a)P^a=0,
\]
\[
-\bigl(\mathcal L_v-K\bigr)P_a
+\nabla_b\tilde T^b{}_a
+\frac1d\,\nabla_a(T^{bc}h_{bc})
+E\,F_{a\mu}v^\mu
-\frac12\,S^{bc}(\nabla_b-a_b)C_{ca}=0.
\]
The corresponding Carroll-covariant Bondi flux-balance laws are
\[
\bigl(\partial_u-\tfrac32K\bigr)E
=
-\frac12\,N^{ab}S_{ab}
+(\nabla_a+a_a)P^a,
\]
\[
\bigl(\partial_u-K\bigr)P_a
=
-\nabla^b\tilde T_{ba}
-E\,F_{a\mu}v^\mu
+\frac12\,S^{bc}(\nabla_b-a_b)C_{ca}.
\]
In the usual retarded-time gauge one recovers
\[
\partial_u M_B
=
-\,\frac1{32\pi G}\int_{S^2}N^{AB}N_{AB},
\qquad
\partial_u J_A=\cdots,
\]
and, in standard Bondi coordinates,
\[
\partial_u M
=
-\frac12\,\gamma^{AC}\gamma^{BD}N_{AB}N_{CD},
\qquad
\partial_u N_A
=
-\,D^B\bigl(\tfrac12\,C_{BC}N^C{}_A\bigr).
\]
These are the familiar Bondi mass and momentum loss laws [2505.05432, 2607.07872].

Weyl invariance gives the trace Ward identity
\[
0
=
T^\mu\tau_\mu
+
T^{\mu\nu}h_{\mu\nu}
+
\frac12\,S^{\mu\nu}C_{\mu\nu}
\qquad\Longrightarrow\qquad
T^\mu{}_\mu=-\frac12N_{\mu\nu}C^{\mu\nu}.
\]
When \(C_{\mu\nu}=0\), the stress tensor is traceless [2505.05432].

The Carroll-boost relation is more subtle. One formulation states that boost invariance would imply
\[
0=h^\rho{}_\sigma T^\sigma-(\cD_\mu-a_\mu)S^{\mu\rho},
\]
but in three bulk dimensions and in four bulk dimensions one instead finds a genuine anomaly,
\[
h^\rho{}_\sigma T^\sigma-(\cD_\mu-a_\mu)S^{\mu\rho}
=
\mathcal A^\rho_{\rm B},
\]
which cannot be removed by local counterterms, satisfies Wess-Zumino consistency, and is responsible for the nontrivial central extensions observed in the BMS\(_3\) and BMS\(_4\) algebras. The 2026 summary presents the classical relation as exact and allows an anomalous right-hand side \(\sim\mathcal A_a\) in quantum gravity/gluing. The common point is that boost covariance is not purely kinematical once the shear-response sector is included [2505.05432, 2607.07872].

## 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 \((u,r,x^A)\), introducing \(\ell=1/r\), the conformal metric \(\hat g_{\mu\nu}=\ell^2g_{\mu\nu}\) is smooth at \(\scri^+=\{\ell=0\}\), and the asymptotic radiation content is encoded in the shear coefficient \(c_{AB}\). A 4-D transformation to inertial coordinates \((\tilde u,\tilde\ell,\tilde x^A)\) is then constructed so that \(\scri^+\) is null, shear-free and divergence-free, the 2-metric on \(\scri^+\) is the unit sphere, and the generators are affinely parametrized by \(\tilde u\). The conformal factor \(\omega\) is determined by the elliptic uniformization equation
\[
{\cal R}[H_{AB}]
=
2\bigl(\omega^2+D^AD_A\ln\omega\bigr),
\]
and then propagated along null generators by
\[
2\,\hat n^\alpha\partial_\alpha\ln\omega
=
-\,e^{-2H}D_AL^A
\qquad\text{on }\scri^+.
\]
In the inertial frame, the Bondi news is
\[
N=\partial_{\tilde u}h,
\qquad
N_{AB}=\partial_u C_{AB},
\qquad
C_{AB}=\lim_{r\to\infty}r\,(g_{AB}-q_{AB}),
\]
with \(h\) the strain on a cross-section of \(\scri^+\) [1605.04332].

For a general BMS generator
\[
\xi[\alpha,f^A]
=
\Bigl(\alpha(\tilde x^A)+\frac12\tilde u\,D_Af^A(\tilde x^B)\Bigr)\partial_{\tilde u}
+
f^A(\tilde x^B)\partial_A,
\]
Geroch’s linkage construction yields a local flux density \(F_\xi\), whose retarded-time derivative satisfies
\[
\partial_{\tilde u}F_\xi
=
-\,\Omega^{-1}\,\tilde C_{\alpha\beta\gamma\delta}\,
\tilde n^\beta\tilde n^\delta\,X^{\alpha\gamma}\bigl|\scri^+\bigr.
\]
In Newman-Penrose form,
\[
\partial_{\tilde u}F_\xi
=
\Psi\,\overline{\mathcal X}
+
\overline{\Psi}\,{\mathcal X}.
\]
Because every BMS generator determines a scalar \(\mathcal X\), all fluxes are sourced by the same radiative field \(\Psi\), or equivalently by the Bondi news \(N\). In particular, for translations,
\[
F[\alpha]=-\partial_uM[\alpha]
=
\frac1{4\pi}\oint \alpha\,|N|^2\,d\Omega,
\]
while corresponding formulas are given for angular momentum, boosts, and supermomentum. This yields a single “news complex” covering energy, momentum, supermomentum, and angular momentum [1605.04332].

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 \(\ell\) and spherical harmonics up to \(L\sim17\), spectral re-expansion on \(\scri^+\) 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 [1605.04332].

Source: https://www.emergentmind.com/topics/boundary-energy-momentum-news-complex