Bulk Supertranslations: Extending BMS into the Interior

• Bulk supertranslations are large diffeomorphisms that extend traditional BMS symmetries from the boundaries of asymptotically flat spacetimes into the interior, defining a continuum of distinct vacuum structures.
• They are realized through smooth adjustments of null hypersurfaces in various geometries such as Minkowski, Schwarzschild, and Kerr, and are linked to observable phenomena like gravitational memory and soft hair.
• The symmetry algebra features both abelian and logarithmic extensions, with associated Hamiltonian surface charges that resolve ambiguities in angular momentum and capture infrared aspects of gravity.

Bulk supertranslations are large diffeomorphisms that smoothly extend the familiar BMS (Bondi-Metzner-Sachs) supertranslation symmetries from the boundaries of asymptotically flat spacetimes—such as null infinity, spatial infinity, or timelike infinity—into the interior (bulk) of the spacetime manifold. They generalize the concept of asymptotic symmetries and give rise to a rich structure of physically distinct vacuum states, associated surface charges, and gravitational memory effects. These structures are significant both at the level of classical general relativity and in the quantum field, especially in the context of black hole soft hair and infrared dynamics.

1. Geometric and Coordinate-Independent Definition

Let $(M, g)$ denote a Lorentzian manifold of arbitrary dimension $d$, equipped with a smooth foliation by null hypersurfaces $\Sigma_u = \{ x \in M\,|\, u(x) = \text{const} \}$. A bulk supertranslation is defined as a diffeomorphism that pushes the slice $\Sigma_u$ to $\Sigma_{u+f}$, where $f$ is a smooth function on $M$ such that the new hypersurfaces $u+f=\text{const}$ remain null: $g^{ab} \partial_a(u+f)\,\partial_b(u+f) = 0$ (Mao, 23 Dec 2025). The infinitesimal generator $\xi^a[f]$ of a supertranslation satisfies $\mathcal{L}_\xi u = f$, with leading component $f\,n^a$, $n^a$ being the normalized null normal, and correction terms determined by the requirement that the transformed foliation remains null.

This definition is fully coordinate-independent and encompasses null infinity, horizons, and extends to generic subregions of the bulk, providing a unified framework for asymptotic and near-horizon symmetries (Mao, 23 Dec 2025).

2. Bulk Extension: Technical Realizations and Examples

Bulk supertranslations can be explicitly realized in multiple settings:

• Minkowski Space: In spherical coordinates $(t, r, \theta, \phi)$, the generator $\xi = T(\theta,\phi)\,s(r)\,\partial_t$ defines a smooth global diffeomorphism, interpolating between the identity near the origin and a pure supertranslation at large $r$. The associated metric is globally smooth and isometric to the original, but carries distinct supertranslation charges (1711.02670).
• Generic Spacetimes: In the Bondi–Sachs framework, solving the null-condition PDE for $f$ induces a one-parameter family of outgoing null hypersurfaces. In Schwarzschild space, the presence of curvature modifies the solution, potentially introducing novel gravitational memory phenomena associated with the geometry (e.g., "curvature-induced" memory or modified photon sphere behavior) (Mao, 23 Dec 2025).
• Black Hole Spacetimes: Bulk BMS supertranslations can be extended throughout the spacetime exterior to a black hole horizon by explicit construction, and yield soft hair modes or encode multipolar structure, as demonstrated for transitions between Schwarzschild and Kerr-based asymptotics (Ghosh et al., 1 Sep 2025).

3. Symmetry Algebra, Charges, and Central Extensions

The algebra of bulk supertranslations is governed by the Lie bracket of their generating vector fields. In the case where the supertranslation function $f$ is constant along null generators, the algebra is abelian. More generally, allowing $v$-dependence on the affine parameter yields a (Diff($\mathbb{R}$)) extension in each generator: $$[\xi[f_1], \xi[f_2]] = \xi[f_1 \partial_v f_2 - f_2 \partial_v f_1]$$ where $v$ is the affine parameter and $n^a = \partial/\partial v$ (Mao, 23 Dec 2025). For extended BMS algebras at spatial infinity, "logarithmic supertranslations" appear as conjugate partners to the pure supertranslations, resulting in a centrally extended, abelian ideal: $$[Q[T], Q^{\rm log}[T_{\rm log}]] = 2 \oint_{S^2} d^2 x\, T(x)\,T_{\rm log}(x)$$ with the mixed bracket providing a nontrivial central term (Fuentealba et al., 2022, Henneaux, 2023). Redefinition of Lorentz charges decouples the Poincaré algebra from the supertranslation sector, allowing for unambiguous definitions of angular momentum and center-of-mass (Fuentealba et al., 2022).

4. Physical Charges, Hamiltonian Formalism, and Memory

Bulk supertranslations admit associated Hamiltonian surface charges, constructed via the Regge–Teitelboim or Iyer–Wald procedures. For ordinary supertranslations at spatial infinity or null infinity, the charges take the schematic form: $$Q_f = \int_{S^2_\infty} f(n)\,\text{(asymptotic field combinations)}\,d\Omega$$ where $f$ encodes supertranslation modes (even/odd functions on $S^2$ for pure/logarithmic cases) (Fuentealba et al., 2022, Henneaux, 2023). In the light-ray operator framework, one realizes these as

$$E[f] = \int_{\Sigma_u} d^{d-1}x\,\sqrt{\gamma}\,f(v,x^A)\,T_{vv}(v,x^A)$$

with the algebra reflecting the underlying symmetry structure (Mao, 23 Dec 2025).

Bulk supertranslations are intimately linked to the gravitational memory effect: passage of gravitational radiation shifts the foliation $\Sigma_u$ to $\Sigma_{u+f_{\text{mem}}}$, with the induced change in shear (Bondi data $C_{AB}$) given by

$$\Delta C_{AB}(x^A) = 2 D_A D_B f_{\mathrm{mem}}(x^A)$$

and corresponding permanent displacement of test particles or light rays—observable in the bulk far from null infinity (Mao, 23 Dec 2025, Bart, 2019).

5. Role in Initial Data, Gluing, and Vacuum Structure

The extended Corvino–Schoen gluing theorem allows generic asymptotically flat initial data to be matched (in an annular transition region) to a member of a generalized Kerr family characterized by arbitrary supertranslation (including logarithmic) charges (Henneaux, 2023). This guarantees the existence of bulk extensions for any prescribed set of boundary charges, and allows for construction of bulk fields carrying physically distinct BMS "vacuum" data, corroborated by direct calculation of the charges (Henneaux, 2023).

All spacetimes related by bulk supertranslations are globally (locally) isometric but carry distinct values for the nontrivial BMS-invariant charges, and thus represent physically distinguishable vacua even in flat space (1711.02670).

6. Timelike Infinity, Supertranslation Hair, and Kerr Geometry

At future timelike infinity, a consistent definition of asymptotic flatness permits the extension of supertranslation symmetry, together with finite, well-defined supertranslation and Lorentz charges that can be computed as integrals on any 2-sphere surrounding timelike sources. In this setting, bulk supertranslations encode the net "supertranslation hair" of the gravitational field contributed by bound systems such as black holes, directly generalizing null-infinity BMS structure (Chakraborty et al., 2021).

Explicit constructions demonstrate that by composing appropriate supertranslations in the bulk, the Kerr geometry can be generated from Schwarzschild, with the infinite tower of even-$\ell$ modes corresponding to the Kerr mass multipole moments and soft hair (Ghosh et al., 1 Sep 2025).

7. Logarithmic Extensions and Central Structure

Boundary conditions at spatial infinity that admit logarithmic terms in the asymptotic expansion ("logarithmic supertranslations") extend the bulk BMS symmetry algebra and generate new “slow-fall-off” modes with well-defined Hamiltonian charges. The central extension in the algebra (see section 3) allows a full decoupling of Lorentz and supertranslation sectors and removes ambiguities in the definition of angular momentum, resolving a significant foundational issue in asymptotically flat general relativity (Fuentealba et al., 2022, Henneaux, 2023).

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In summary, bulk supertranslations constitute a robust, coordinate-invariant extension of the BMS supertranslation symmetries from the boundary to the full spacetime. They are manifested in the structure of asymptotically flat gravitational vacua, the algebra of large diffeomorphisms, the emergence of physically distinguishable soft hair and multipole moments, the memory effect, and the canonical structure of general relativity (Mao, 23 Dec 2025, Ghosh et al., 1 Sep 2025, Chakraborty et al., 2021, Fuentealba et al., 2022, Henneaux, 2023, 1711.02670, Bart, 2019).