Null Brown-York Stress Tensor

• The null Brown-York stress tensor is a quasi-local gravitational stress tensor defined on null boundaries, addressing the challenges of degenerate induced metrics.
• It employs a null-adapted boundary term and Carrollian geometric structure to derive conserved charges like energy, momentum, and angular momentum in diverse gravitational contexts.
• Its formulation extends to higher-derivative and scalar-tensor theories while ensuring consistency with BMS symmetries, fluid/gravity correspondence, and quantum aspects.

The Null Brown–York stress tensor is a generalization of the quasi-local gravitational stress tensor, originally defined for timelike or spacelike boundaries, to the context of null (lightlike) boundaries in general relativity and related gravitational theories. This construction addresses the subtleties that arise due to the degenerate nature of the induced metric on null hypersurfaces. The null Brown–York tensor provides a means to define quasi-local charges—such as energy, momentum, angular momentum—and plays a fundamental role in the paper of black hole thermodynamics, gravitational radiation near null infinity, flat space holography, and the fluid/gravity correspondence.

1. Variational Principles, Degenerate Geometry, and Definition on Null Hypersurfaces

On timelike or spacelike boundaries the Brown–York stress tensor is defined by varying the (renormalized) gravitational action with respect to the induced boundary metric, facilitated by the addition of a Gibbons–Hawking–York (GHY) term to ensure a well-posed variational principle. For null hypersurfaces, the boundary geometry is degenerate (the induced metric γ{ab} satisfies γ{ab} l^b = 0 for a null vector l^a tangent to the generators), so the standard approach must be modified.

The null boundary is characterized by:

• A degenerate induced metric q_{ab}
• A preferred null generator l^a satisfying l^a q_{ab} = 0
• An Ehresmann connection k_a defined via l^a k_a = 1 (often called the rigging vector)
• Carrollian geometry structure, consisting of (q_{ab}, l^a, k_a) (Chandrasekaran et al., 2021, Adami et al., 30 Apr 2024)

To formulate a well-posed variational problem, one introduces a null-adapted boundary term (analogous to the GHY term) in the action, built from the inaffinity (surface gravity) κ and expansion θ:

$$S_{\text{null bnd}} = -\frac{1}{8\pi G} \int_\mathcal{N} \eta\, \kappa$$

where η is the volume form intrinsic to the null hypersurface.

The null Brown–York stress tensor is then defined by taking a functional derivative of the on-shell gravitational action with respect to variations of the Carrollian structure (q_{ab}, l^a) rather than a non-degenerate induced metric (Chandrasekaran et al., 2021). The resulting mixed-index tensor takes the form:

$$T^i{}_j = -\frac{1}{8\pi G} (W^i{}_j - W\, \delta^i{}_j)$$

where W^i{}_j is the "shape operator" (Weingarten tensor) associated with the extrinsic geometry of the null hypersurface, effectively encoding how l^a "bends" in the ambient spacetime. This formalism is independent of the choice of auxiliary rigging vector and is compatible with the constraints imposed by the null geometry (Chandrasekaran et al., 2021, Jafari, 2019, Bhambure et al., 11 Dec 2024).

2. Canonical, Covariant, and Conservation Structure

On null boundaries, the canonical structure adapts the Hamilton–Jacobi analysis, replacing the role of canonical momenta conjugate to the induced metric with variations conjugate to:

• the intrinsic 2-metric (on spacelike cross-sections)
• "shift" vector components arising from the coordinate decomposition adapted to null hypersurfaces
• an additional scalar (arising from the lapse-like function ensuring null character)

The null Brown–York tensor thus provides:

• a spatial stress s^{ab} (projected onto spatial cross-sections)
• a momentum density j^a (conjugate to the shift)
• an energy density ε (conjugate to the null-lapse variation)

The conservation law for the null Brown–York stress tensor is written with respect to the null-adapted (rigged) connection D_a, compatible with the Carroll structure:

$D_i T^i{}_j = 0$

This conservation is not postulated but arises directly from the null constraint equations: the Raychaudhuri equation (projected along l^a), and the Damour–Navier–Stokes equation (projected onto the spatial cross-section) (Chandrasekaran et al., 2021, Jafari, 2019, Ciambelli, 29 Jan 2025). At null infinity, the conservation law of this stress tensor reproduces the Bondi mass-loss and angular momentum flux equations (Adami et al., 30 Apr 2024, Ciambelli, 29 Jan 2025).

3. Null Charges, BMS Symmetries, and Holography

The null Brown–York stress tensor enables the definition of charges for vector fields ξ^a tangent to the null surface:

$Q_\xi = - \int_S T^i{}_j \xi^j \eta_i$

where the integral is over a codimension-2 cross-section S (such as a cut of null infinity). This structure naturally encodes the charges (Bondi mass, angular momentum, supertranslation, and superrotation charges) for the asymptotic symmetry algebra, in particular the BMS algebra of asymptotically flat spacetimes (Bhambure et al., 11 Dec 2024, Adami et al., 30 Apr 2024).

For general symmetry transformations, the charges computed from the null Brown–York tensor match the Wald–Zoupas canonical charges up to a possible "anomalous" contribution, which depends on the transformation's action on the fixed Carroll structure. In particular, under BMS transformations (supertranslations and superrotations), the deviation appears in the non-integrable (news-dependent) part of the flux, which can be interpreted as a scaling anomaly remediable by suitable boundary counterterms (Bhambure et al., 11 Dec 2024, Chandrasekaran et al., 2021).

In three-dimensional asymptotically flat holography, the null Brown–York tensor and its associated current give rise to the data of a Carrollian fluid living on the celestial circle at null infinity. Imposing a well-posed variational principle produces a boundary Schwarzian action, governing the dynamics of reparameterizations as in codimension-2 holography (Adami et al., 30 Apr 2024).

4. Physical Applications: Black Holes, Horizons, and Fluid/Gravity Correspondence

The null Brown–York stress tensor provides quasi-local definitions of energy, angular momentum, and fluxes in a variety of physically relevant contexts:

• In stationary black hole spacetimes (e.g., Kerr, Kerr–Newman), the null Brown–York tensor, evaluated at infinity, reproduces the correct ADM mass and angular momentum (Bhambure et al., 11 Dec 2024, Zhang, 2020).
• Near event horizons, the change in Brown–York energy of a surface just outside the horizon reduces to a proportionality with the change in its area, $dE_{BY} = a\, dA$, where a is the observer’s proper acceleration. This connects to black hole thermodynamics and the identification of entropy as horizon area (Mäkelä, 2021).
• In the context of the fluid/gravity correspondence, the null Brown–York stress tensor on (stretched) null boundaries subject to Petrov-type conditions yields the incompressible Navier–Stokes equations in the near-horizon limit (Huang et al., 2011).
• In asymptotically flat spacetimes, at null infinity, its conservation law encapsulates the Bondi mass-loss formula and matches the symplectic structure of the radiative (Ashtekar–Streubel) phase space (Ciambelli, 29 Jan 2025).
• In gauge/gravity duality, the vanishing of the appropriate component (e.g., $T_{tt}$) of the Brown–York stress tensor on the entangling surface leads to the Ryu–Takayanagi minimal-area prescription for holographic entanglement entropy (Bhattacharyya et al., 2013).

5. Extensions: Higher Derivative & Modified Gravity, Regularization, and Anomaly Structure

Formulation of a null Brown–York stress tensor in higher-derivative or scalar–tensor theories requires extra care in defining the boundary terms and in implementing the variational principle. In curvature-squared theories, the action is recast using auxiliary tensor fields to ensure that the appropriate generalized Gibbons–Hawking–York boundary term "tames" all variations, allowing the definition of a covariant, finite (and, by construction, divergence-free) stress tensor even in asymptotically AdS or Lifshitz backgrounds (Hohm et al., 2010). In scalar–tensor gravity, conformal frame transformations dictate how the null Brown–York tensor and its quasi-local charges transform, with black hole thermodynamic quantities remaining invariant while the fluid/gravity related transport coefficients may not (Bhattacharya et al., 2023).

Divergences in the null Brown–York tensor (or its associated charges as evaluated at infinity) can be regularized either by subtracting a reference action (embedding the boundary in flat space or an AdS background) or, more efficiently in the null case, by the addition of boundary counterterms constructed intrinsically from the Carrollian data. These may be non-analytic in the degenerate metric or constructed using auxiliary fields in higher-derivative theories; their variation is precisely adjusted to cancel non-physical divergences while preserving the physical finite part (Jafari, 2019, Bhambure et al., 11 Dec 2024).

6. Relations to Carrollian Field Theory, Phase Space, and Quantum Aspects

At null infinity, the null Brown–York stress tensor naturally projects to a Carrollian stress tensor, defined on the degenerate Carrollian geometry of the boundary. Conservation of this stress tensor (with respect to a torsionful affine connection) reproduces the Bondi mass and angular momentum flux (Adami et al., 30 Apr 2024). In the bulk, the covariant phase space on finite null hypersurfaces reduces to the canonical Ashtekar–Streubel symplectic form at null infinity:

$$\theta_{\text{can}} \rightarrow \frac{1}{32\pi G} \int du\, d^2\Omega\, N^{AB}\, \delta C_{AB}$$

where $N_{AB}$ is the Bondi News and $C_{AB}$ is the shear (Ciambelli, 29 Jan 2025).

In quantum field theory in curved spacetime with null boundaries (e.g., radiative collapse forming a black hole), regulated stress tensors are calculated by subtraction schemes reminiscent of Brown–York constructions—such as subtracting the Unruh state from the in-state result across a null shell (Siahmazgi et al., 2021). In the high-energy limit of string theory, the tensionless (null string) worldsheet necessitates the use of vector density Lagrange multipliers, sharing conceptual similarity with the degeneracy management in the null Brown–York tensor (Davydov et al., 2022).

7. Summary Table: Key Components Across Contexts

Context	Geometric Structure	Null Brown–York Stress Tensor Definition
Classical GR, null boundary	Carrollian (q_{ab}, l^a)	$$T^i{}_j = -\frac{1}{8\pi G}(W^i{}_j - W \delta^i_j)$$
Higher-derivative gravity (AdS/Lifshitz)	Auxiliary field reformulation	Modified GHY term, covariantized stress tensor
Scalar–tensor gravity	Conformal (Jordan/Einstein frames)	Frame-dependent stress tensor, matched boundary terms
Asymptotic flatness, null infinity	Carroll structure w/ torsion	$D_i T^i{}_j = 0$, Bondi mass/ang. momentum consistency
Fluid/gravity correspondence	Near-horizon null membrane	BY tensor + Petrov condition yields Navier–Stokes

The null Brown–York stress tensor provides a robust, covariant, and finite construction for energy–momentum on null hypersurfaces, underpinning both classical and quantum gravitational physics in contexts ranging from black hole horizons to holographic duality and the radiative dynamics at null infinity. Its formulation, grounded in rigorous variational principles and adapted boundary terms, enables consistent definitions of conserved charges, establishes relations to Carrollian field theory structures, and informs the ongoing development of flat space holography and gravitational phase space theory.