Near-Horizon Supertranslations

• Near-horizon supertranslations are large diffeomorphisms that shift the null coordinate at black-hole horizons, yielding an infinite-dimensional Abelian sector analogous to BMS supertranslations at null infinity.
• Using Gaussian null coordinates, they lead to degenerate near-horizon solutions of Einstein’s equations whose zero modes encode thermodynamic properties like temperature and entropy.
• Reformulations in Carrollian and higher-curvature frameworks uncover connections to soft hair, horizon memory effects, and potential physical versus gauge interpretations.

Near-horizon supertranslations are large diffeomorphisms that preserve a prescribed near-horizon boundary structure at a black-hole horizon and act as angle-dependent shifts of the null horizon coordinate. In the non-extremal four-dimensional setting, they are naturally formulated in Gaussian null coordinates, generate an infinite-dimensional Abelian sector analogous to BMS supertranslations at null infinity, and relate infinitely many degenerate near-horizon solutions of Einstein’s equations order by order in the radial expansion away from the horizon (Cai et al., 2016). In subsequent developments, the same structure has been embedded in broader horizon symmetry algebras, reformulated in Carrollian and bulk-geometric terms, extended to higher-curvature gravity, and contrasted with quasi-local frameworks in which smooth horizon supertranslations can instead be treated as gauge redundancies (Donnay et al., 2015).

1. Horizon asymptotic symmetries and the emergence of supertranslations

The modern near-horizon literature begins from the observation that non-extremal black-hole horizons admit an asymptotic symmetry analysis closely paralleling the BMS analysis at null infinity. In a horizon-adapted gauge, Donnay, Giribet, González, and Pino identified asymptotic Killing vectors generated by supertranslations and superrotations, with algebra

$$[Y_m,Y_n]=(m-n)Y_{m+n},\qquad i[Y_m,T_n]=-n\,T_{m+n},\qquad i[T_m,T_n]=0,$$

that is, a semidirect sum of a Witt algebra with an Abelian current algebra (Donnay et al., 2015). In stationary cases, the zero modes reproduce horizon thermodynamic data: for BTZ and Kerr, the supertranslation zero mode gives $T_H S_H$, while the superrotation zero mode gives the angular momentum $J$ (Donnay et al., 2015).

A complementary asymptotic analysis for generic Killing horizons in Gaussian null coordinates made explicit the distinction between subextremal and extremal cases. Writing the near-horizon metric as

$$ds^2 = L(x)^2[- r^n F(r,x) dv^2 + 2 dv dr] + \gamma_{IJ}(r,x)(dx^I - r h^I(r,x) dv)(dx^J - r h^J(r,x) dv),$$

with $n=1$ for subextremal horizons and $n=2$ for extremal horizons, one finds that the arbitrary function $f(x)$ multiplying $\partial_v$ persists in both cases, but its allowed $v$-dependence differs: polynomial in $v$ for extremal horizons and exponential $T_H S_H$0 for subextremal ones (Akhmedov et al., 2017). This difference was related to the redshift effect at non-extremal horizons (Akhmedov et al., 2017). In the spherical subextremal sector, the pure supertranslation generator takes the form $T_H S_H$1 plus the compensating $T_H S_H$2- and angular pieces required by the boundary conditions (Akhmedov et al., 2017).

These analyses established the basic horizon analogue of the BMS paradigm: an infinite-dimensional family of residual diffeomorphisms survives the gauge fixing at the horizon, and the supertranslation sector is Abelian. The central question then becomes whether these transformations label physically distinct horizon configurations or merely different descriptions of the same intrinsic geometry.

2. Gaussian null coordinates and the near-horizon supertranslation generator

A concrete four-dimensional realization is provided by the near-horizon construction in Gaussian null coordinates $T_H S_H$3, with horizon at $T_H S_H$4, gauge conditions

$T_H S_H$5

and near-horizon fall-offs

$T_H S_H$6

$T_H S_H$7

$T_H S_H$8

where $T_H S_H$9 is the round unit-$J$0 metric and $J$1 is the surface gravity (Cai et al., 2016). For the Schwarzschild background one may take $J$2 and $J$3, giving

$J$4

near the horizon (Cai et al., 2016).

The horizon supertranslation is generated by an arbitrary function $J$5 on the horizon cross-section. An exact form of the vector field preserving both the Gaussian null gauge and the boundary conditions is

$J$6

with asymptotic expansion

$J$7

(Cai et al., 2016). The supertranslation algebra is Abelian,

$J$8

and the leading transformation laws on the near-horizon fields are

$J$9

$$ds^2 = L(x)^2[- r^n F(r,x) dv^2 + 2 dv dr] + \gamma_{IJ}(r,x)(dx^I - r h^I(r,x) dv)(dx^J - r h^J(r,x) dv),$$0

(Cai et al., 2016). These formulas are the basic kinematical statement of near-horizon supertranslations: they preserve the horizon gauge but shift the subleading horizon data nontrivially.

3. Einstein equations, degenerate black-hole solutions, and late-time vacua

The nontrivial content of near-horizon supertranslations appears when the Einstein equations are imposed order by order in $$ds^2 = L(x)^2[- r^n F(r,x) dv^2 + 2 dv dr] + \gamma_{IJ}(r,x)(dx^I - r h^I(r,x) dv)(dx^J - r h^J(r,x) dv),$$1. Solving $$ds^2 = L(x)^2[- r^n F(r,x) dv^2 + 2 dv dr] + \gamma_{IJ}(r,x)(dx^I - r h^I(r,x) dv)(dx^J - r h^J(r,x) dv),$$2 in the ansatz above yields, at leading order, the stationarity condition

$$ds^2 = L(x)^2[- r^n F(r,x) dv^2 + 2 dv dr] + \gamma_{IJ}(r,x)(dx^I - r h^I(r,x) dv)(dx^J - r h^J(r,x) dv),$$3

together with algebraic and differential relations among $$ds^2 = L(x)^2[- r^n F(r,x) dv^2 + 2 dv dr] + \gamma_{IJ}(r,x)(dx^I - r h^I(r,x) dv)(dx^J - r h^J(r,x) dv),$$4, $$ds^2 = L(x)^2[- r^n F(r,x) dv^2 + 2 dv dr] + \gamma_{IJ}(r,x)(dx^I - r h^I(r,x) dv)(dx^J - r h^J(r,x) dv),$$5, and $$ds^2 = L(x)^2[- r^n F(r,x) dv^2 + 2 dv dr] + \gamma_{IJ}(r,x)(dx^I - r h^I(r,x) dv)(dx^J - r h^J(r,x) dv),$$6 (Cai et al., 2016). The leading solutions are

$$ds^2 = L(x)^2[- r^n F(r,x) dv^2 + 2 dv dr] + \gamma_{IJ}(r,x)(dx^I - r h^I(r,x) dv)(dx^J - r h^J(r,x) dv),$$7

$$ds^2 = L(x)^2[- r^n F(r,x) dv^2 + 2 dv dr] + \gamma_{IJ}(r,x)(dx^I - r h^I(r,x) dv)(dx^J - r h^J(r,x) dv),$$8

$$ds^2 = L(x)^2[- r^n F(r,x) dv^2 + 2 dv dr] + \gamma_{IJ}(r,x)(dx^I - r h^I(r,x) dv)(dx^J - r h^J(r,x) dv),$$9

$n=1$0

with $n=1$1 functions on the sphere and $n=1$2 transients that decay at late advanced time (Cai et al., 2016). The subleading fields are then fixed by

$n=1$3

$n=1$4

and the corresponding $n=1$5 expression (Cai et al., 2016).

The degeneracy becomes especially transparent after introducing a scalar potential $n=1$6 such that

$n=1$7

for which

$n=1$8

Under a supertranslation $n=1$9,

$n=2$0

so one exact near-horizon solution is mapped to another (Cai et al., 2016). The infinitely degenerate sector is therefore labeled by functions $n=2$1, or equivalently by their harmonic coefficients. The choice $n=2$2 and $n=2$3 defines the stationary “physical vacuum,” while nonzero $n=2$4 represent decaying transients (Cai et al., 2016).

The same structure persists with cosmological constant $n=2$5. The near-horizon ansatz and the supertranslation vector fields are unchanged, while the solved coefficients become

$n=2$6

and similarly for $n=2$7 and $n=2$8 (Cai et al., 2016). At higher order, the radial hierarchy remains solvable through transport equations of the form

$n=2$9

with general solution

$f(x)$0

showing that the near-horizon expansion can be continued order by order (Cai et al., 2016).

4. Conserved charges, soft hair, and memory-type effects

The covariant phase-space/Barnich–Brandt construction associates to the supertranslation parameter $f(x)$1 the horizon charge

$f(x)$2

with $f(x)$3 for Schwarzschild and $f(x)$4 (Cai et al., 2016). These charges satisfy

$f(x)$5

so supertranslations do not change the energy; they generate soft deformations carrying soft gravitons as Goldstone modes (Cai et al., 2016). In this sense, black holes possess soft supertranslation hair: distinct classical configurations related by large diffeomorphisms with nontrivial charges.

The Rindler horizon provides a flat-space analogue with the same logic. In Gaussian null coordinates,

$f(x)$6

the same near-horizon boundary conditions apply, and the horizon supertranslation charge becomes

$f(x)$7

(Cai et al., 2016). This universality underlies the Rindler memory construction of Kolekar and Louko: a non-planar shock wave implants supertranslation hair on the Rindler horizon, inducing the angle-dependent shift

$f(x)$8

on $f(x)$9, together with a classical memory $\partial_v$0 for uniformly accelerated observers (Kolekar et al., 2017). In quantum field theory on this background, the same shock yields nonvanishing Bogoliubov coefficients $\partial_v$1 and $\partial_v$2, so the supertranslation memory is accompanied by a quantum memory that modulates entanglement between the left and right Rindler wedges (Kolekar et al., 2017).

These developments motivate a horizon-memory interpretation parallel to null-infinity memory, although the fully dynamical horizon formula remains less settled. In the stationary near-horizon construction, the relevant information is stored in the shifted fields $\partial_v$3 and $\partial_v$4, and matter or radiation crossing the horizon is expected to move the system between degenerate vacua (Cai et al., 2016).

5. Bulk, Carrollian, and higher-curvature reformulations

A coordinate-independent bulk reformulation treats supertranslations as deformations of a null foliation $\partial_v$5. In Newman–Unti coordinates $\partial_v$6, with $\partial_v$7, $\partial_v$8, $\partial_v$9, a bulk supertranslation is

$v$0

subject to the nullness constraint

$v$1

At a black-hole horizon, smoothness and regularity pick $v$2 that is smooth on the horizon cross-section and constant along the horizon null generators, so the action reduces to

$v$3

(Mao, 23 Dec 2025). In this framework, the same bulk profile $v$4 interpolates between BMS supertranslations at null infinity and near-horizon supertranslations at $v$5, while the Abelian supertranslation algebra is realized by light-ray operators on the null hypersurface (Mao, 23 Dec 2025).

A second reformulation identifies the horizon as a Carrollian geometry obtained from an ultra-relativistic $v$6 limit of the stretched horizon. In null Gaussian coordinates,

$v$7

the Raychaudhuri and Damour equations become Carrollian conservation laws, and the horizon symmetries are Carrollian Killing vectors including BMS-like supertranslations and superrotations (Donnay et al., 2019). The corresponding Carrollian charge is

$v$8

while the generalized angular momentum is recovered from the superrotation sector (Donnay et al., 2019). This formulation makes explicit the role of horizon expansion, shear, and twist in the charge algebra and flux balance.

In higher-curvature gravity, near-horizon supertranslations survive and their charges become a direct generalization of the Wald entropy functional. In Lovelock theory,

$v$9

where the $v$0 are the induced Lovelock densities on the horizon cross-section, and the zero mode satisfies

$v$1

(Chernicoff et al., 26 Jun 2025). In four dimensions, the same computation reduces to evaluating the Jackiw–Teitelboim action on the two-dimensional spacelike sections of the event horizon (Chernicoff et al., 26 Jun 2025). A related AdS$v$2 realization in New Massive Gravity exhibits the algebra

$v$3

and the zero mode of the supertranslation charge reproduces the black-hole entropy, with subleading horizon data contributing in a way absent in Einstein gravity (Donnay et al., 2020).

6. Pure-gauge interpretations, singular modes, and open problems

A major point of contention is whether horizon supertranslations label physical soft hair or only gauge redundancy. In the exact nonlinear analysis of vacuum non-expanding horizons, smooth horizon supertranslations act trivially on a complete set of gauge-invariant free data,

$v$4

so they were argued to be pure gauge on generic non-expanding horizons, in contrast to BMS supertranslations at null infinity (Sousa et al., 2017). The difference is tied to the boundary conditions and the notion of horizon data: the Gaussian-null “soft hair” constructions and the non-expanding-horizon analysis are not imposing the same phase-space structure.

Dynamical and quasi-local treatments sharpen this issue. For transitions between isolated-horizon phases, a supertranslation is said to be induced when

$v$5

that is, when the preferred-slice condition fails to be preserved through the intervening dynamical horizon (Chatterjee et al., 2020). In the weakly isolated horizon framework, joining two horizons that differ by supertranslation hair requires an intermediate non-WIH null phase with stress-energy flux that, under the assumptions made, violates the dominant energy condition (Ghosh et al., 2020). These results do not negate the existence of symmetry generators; they delimit when horizon supertranslations can be treated as physically operative transitions rather than slice changes.

A further subtlety arises for singular supertranslations. On the future horizon of Schwarzschild, the first-order Holst-action analysis of standard and dual supertranslation charges shows that singular functions on the sphere can generate a central term in the Dirac bracket algebra. For a simple pole $v$6,

$v$7

while the remaining brackets vanish (Akhoury et al., 2022). The anomaly can be canceled by introducing a gravitational Chern–Simons theory on the horizon, implying that consistency of the asymptotic symmetry algebra requires additional horizon structure when singular modes are admitted (Akhoury et al., 2022).

Finally, the relation to evaporation remains unsettled. In one asymptotic-BMS analysis of black-hole evaporation, classical supertranslation hair was argued to act as a diffeomorphism near and outside the horizon and therefore not to modify Hawking’s local pair-creation calculation by itself; in that view, any observable relation between incoming $v$8 and outgoing $v$9 requires nontrivial interior quantum dynamics rather than classical near-horizon hair alone (Gomez et al., 2017). This leaves near-horizon supertranslations in a characteristic position: they are mathematically robust as a horizon symmetry sector, physically compelling as a soft-hair construction, and still method-dependent in their interpretation once one passes from stationary near-horizon kinematics to fully dynamical black-hole physics.