The Energy-Momentum-News Complex near Future Null Infinity

Abstract: We study asymptotically flat vacuum solutions of general relativity in three and four dimensions, with an emphasis on the geometric structures that emerge near null infinity. We construct asymptotic solutions to the three- and four-dimensional Einstein equations near future null infinity, which is a conformal Carroll manifold, starting from the most general Carroll metric data allowed by the Einstein equations. We use a Carroll-covariant version of Bondi--Sachs gauge, whose residual transformations act on the boundary Carroll geometry and shear as boundary diffeomorphisms, Weyl transformations and Carroll boosts. We then define a boundary energy-momentum-news complex at future null infinity by varying a suitably renormalised action with respect to the boundary Carroll metric data and shear. This involves adding boundary terms to the Einstein--Hilbert action on a cut-off surface near future null infinity. The boundary energy-momentum-news complex obeys two relations due to the boundary diffeomorphism and Weyl gauge invariance of the renormalised action. A third relation, due to the Carroll boost, is anomalous, and the corresponding anomaly is obtained from the variation of the renormalised action. Together, these Ward-type identities obeyed by the boundary energy-momentum-news complex lead to a Carroll-covariant generalisation of the Bondi loss equations.

• The paper introduces a Carroll-covariant reformulation of the Bondi–Sachs framework that defines a unified energy-momentum-news complex at future null infinity.
• It employs holographic renormalization and a variational principle to extract boundary data, including energy currents, momentum-stress tensors, and news tensors.
• The findings have significant implications for gravitational wave physics and flat space holography, highlighting the role of Carroll boosts and associated anomalies.

Detailed Technical Essay on "The Energy-Momentum-News Complex near Future Null Infinity" (2607.07872)

Overview and Objectives

This paper systematically investigates the energy-momentum structure present at future null infinity ($\mathcal{I}^+$) in asymptotically flat three- and four-dimensional vacuum solutions to Einstein's equations, with a particular focus on geometric, gauge, and variational aspects underpinning current approaches to asymptotic symmetries and holography in flat spacetime. Motivated by advances in celestial and Carrollian holography, the authors reformulate the Bondi–Sachs analysis in a Carroll-covariant language, extract the boundary data, and rigorously define a boundary energy-momentum-news complex by variational methods including holographic renormalisation. They identify gauge and diffeomorphism-induced Ward identities, analyze the appearance and nature of the Carroll boost anomaly, and explicitly connect these features to generalizations of the Bondi mass-loss equations.

Carrollian Geometry and Gauge Structure at Future Null Infinity

The geometry of $\mathcal{I}^+$ in asymptotically flat spacetimes is intrinsically Carrollian. The boundary manifold at null infinity possesses a degenerate metric structure described by a nowhere-vanishing clock one-form $\tau_\mu$ and a spatial metric $h_{\mu\nu}$ orthogonal to the null direction, together with their respective "inverses" ($v^\mu$, $h^{\mu\nu}$). The Carrollian symmetry algebra includes not just diffeomorphisms and Weyl transformations but also Carroll boosts, which involve a redundancy between $\tau_\mu$ and $h^{\mu\nu}$ (parameterized by a spatial covector $\lambda_\mu$).

The authors show, using careful gauge fixing of Einstein's equations in a Bondi–Sachs-like setup, that the residual bulk diffeomorphisms at future null infinity act as boundary diffeomorphisms, Weyl transformations, and Carroll boosts of the induced Carrollian geometry. Specifically, they achieve a Carroll-covariant version of the Bondi–Sachs gauge that maintains manifest Carroll invariance and allows for arbitrary conformal Carrollian structures at $\mathcal{I}^+$, up to physical constraints imposed by the equations of motion (see Section 2).

An important result is the identification of a constraint on the boundary data: the extrinsic curvature of the spatial metric at $\mathcal{I}^+$0 must be pure trace. Unlike AdS/CFT, where the boundary metric can be arbitrary, this reflects a genuine, non-removable constraint on the solution space imposed by flatness at null infinity. This impacts the admissibility of variations and therefore the definition of conserved quantities and their fluxes.

Asymptotic Solution Space and Boundary Data

By solving the vacuum Einstein equations near $\mathcal{I}^+$1 in a radial $\mathcal{I}^+$2 expansion, the authors classify the boundary data into primary Carrollian structure $\mathcal{I}^+$3 and an additional symmetric, traceless spatial tensor $\mathcal{I}^+$4, which plays the role of the boundary shear. The identification of $\mathcal{I}^+$5 as boundary data is justified both by solving the equations of motion and by the form of the variational principle: variations of the gravitational action at large $\mathcal{I}^+$6 yield terms linear in variations of both the Carrollian structure and the shear. This marks a departure from standard Lorentzian settings, where (subleading) "radiative data" is often treated differently from the (leading) metric data.

They emphasize that the appearance of nontrivial shear as boundary data is closely tied both to gravitational radiation and to the transformation laws under Carroll boosts. The shear transforms inhomogeneously and, crucially, its transformation contains a term proportional to the Carroll boost parameter times the acceleration: a source of the subsequent boost anomaly.

Holographic Renormalisation and the Energy-Momentum-News Complex

A cornerstone of the paper is the precise definition of a boundary energy-momentum-news complex. The authors execute a Carroll-covariant version of holographic renormalisation: the on-shell gravitational action is supplemented with suitable boundary counterterms on a regulated surface at large $\mathcal{I}^+$7, and variations are performed in a gauge-preserving and Carroll-covariant manner. The result is a finite, cutoff-independent variational formula:

$\mathcal{I}^+$8

where $\mathcal{I}^+$9 is an energy current, $\tau_\mu$0 is a momentum-stress tensor (with ambiguities due to gauge and the boundary metric constraint), and $\tau_\mu$1 is the news tensor conjugate to the boundary shear $\tau_\mu$2. This energy-momentum-news complex combines information about asymptotic energy, momentum, and radiation fluxes.

Essentially, the boundary energy-momentum complex generalizes the Bondi mass and angular momentum aspects, while the news tensor encodes radiative gravitational degrees of freedom. The careful handling of the allowed variations—restricted by the boundary constraint and the form of the asymptotic expansion—is necessary for the consistency of the variational principle and the resulting conservation and balance laws.

Ward Identities, Carroll Boost Anomaly, and Bondi Loss Equations

A central technical achievement is the precise identification of three types of Ward identities for the energy-momentum-news complex:

1. Diffeomorphism Invariance: Yields a generalization of the divergence (conservation law) of $\tau_\mu$3 and $\tau_\mu$4, which implies a covariantized version of the Bondi mass/flux balance equation.
2. Weyl Invariance: Imposes scaling constraints on the trace structure of the energy-momentum tensor components, determined by the conformal weight assignments under Weyl transformations.
3. Carroll Boost Anomaly: The variation of the renormalized action under boundary Carroll boosts does not vanish but produces a well-defined anomaly, computed explicitly in terms of the boundary data and shown to satisfy Wess–Zumino consistency. This boost anomaly modifies the canonical form of the conservation laws and is fundamentally tied to the presence of boundary shear.

Together, these identities encode the Carroll-covariant Bondi mass loss equation. That is, the familiar statement that outgoing gravitational radiation reduces the Bondi mass at $\tau_\mu$5 is reformulated as a Ward identity within the Carroll-invariant boundary field theory, with additional structure reflecting the shear/news sector and the anomaly.

Logarithmic Terms and Non-Smooth Null Infinity

The treatment of logarithmic terms in the asymptotic expansion (notably $\tau_\mu$6 corrections to the metric) is dealt with in detail. The authors show these terms generically appear when "peeling" does not hold strictly (as guaranteed in AdS or certain smooth null-infinity settings) and are necessary in four dimensions at subleading order [see also Barnich & Troessaert]. They demonstrate that such logs do not affect the derivation of the Weyl Ward identity nor the assignment of Weyl weights to the boundary currents, but instead serve primarily to remove artificial constraints on the $\tau_\mu$7 part of the shear.

Practical and Theoretical Implications

On the practical side, the explicit construction of the energy-momentum-news complex with all anomalous, gauge, and shear contributions enables systematic computation of fluxes and, thus, of physically measurable quantities such as mass and angular momentum loss at null infinity—not restricted to the standard Bondi frame but for completely arbitrary Carrollian backgrounds. This level of generality is crucial for applications to precision waveform modeling and the ongoing study of gravitational wave observations, which are sensitive to asymptotic symmetry structures.

On the theoretical side, the results lay rigorous groundwork for Carrollian and celestial holographic correspondences in asymptotically flat spacetimes. The identification of “holographic currents," their anomalies, and their Ward identities brings the asymptotically flat setting much closer in structure to the AdS/CFT paradigm—though significant differences remain (for example, the aforementioned constraint on boundary geometry and the lack of a well-defined Carrollian QFT). Furthermore, the treatment of anomalies, improvement terms, and the role of boundary shear deepens the understanding of asymptotic symmetry enhancement (BMS, superrotations/supertranslations), their algebraic structure, and potential dual field theories.

Outlook and Future Developments

While the paper achieves a high degree of technical completeness, it also highlights several outstanding theoretical challenges:

• Intrinsic Carrollian QFTs: The lack of well-defined quantum field theories on arbitrary Carrollian backgrounds (with interactions, anomalies, and radiative structure comparable to their Lorentzian counterparts) is a significant open problem.
• Non-smooth null infinity: The necessity of logarithmic terms, reflecting the breakdown of smoothness in physically relevant spacetimes but not directly tied to anomalies (as in AdS), points to a more nuanced function space for asymptotic data and potential refinements of the holographic dictionary.
• Dual holographic description: The explicit form of the energy-momentum-news complex and its connection to scattering data feed into the broader program of celestial and Carrollian holography, where a definitive dual description of gravity in flat space is still lacking.
• Generalizations to higher dimensions and inclusion of matter: The methods established here are extendable, but new challenges—especially concerning the nature and regularity of the asymptotic expansions—remain.

Conclusion

This work gives a comprehensive, Carroll-covariant, and variationally principled treatment of energy-momentum structures at null infinity, offering a unified framework for the Bondi mass loss, boundary anomalies, and the geometric structure of radiative gravitational data in both three and four spacetime dimensions. The precise connection between boundary geometry, symmetry, and flux—which accommodates radiative (news) sectors and anomalies—provides valuable tools for both gravitational wave physics and the pursuit of flat space holography, and underscores the distinct geometric and variational subtleties of asymptotically flat spacetimes compared to the AdS/CFT context.