Kerr/CFT, Celestial, and Flat-Space Black-Hole Interfaces

Extremal Kerr supplies an exact near-horizon geometry and a striking symmetry-based entropy match, while celestial amplitudes and null-infinity symmetries reorganize asymptotically flat scattering. These are related interfaces, not one established duality. Kerr/CFT depends on boundary conditions, a Virasoro charge algebra, and a Cardy interpretation; hidden conformal symmetry concerns a low-frequency wave equation; celestial holography concerns an invertible scattering basis. This chapter derives each controlled statement and keeps the stronger microscopic claims conditional as of 2026-08-20.

Helpful background. Gauge Theory and String-Theory Preparation supplies symmetry algebras and scattering; Quantum Gravity frames microscopic duality; Quantum Information supplies completeness and entropy tests; and Black-Hole Geometry and Mechanics supplies extremality, horizon area, and Kerr coordinates.

The NHEK thread

For extremal Kerr with the azimuthal orientation chosen so $J\gt0$, $J=GM^2$ in the site’s units $c=1$, and the two horizon roots coincide. A correlated scaling zooms into the throat while co-rotating with the horizon. The resulting near-horizon extremal Kerr geometry has enhanced $\mathrm{SL}(2,\mathbb R)\times \mathrm{U}(1)$ isometry and remains an exact vacuum solution. Bardeen and Horowitz 1999, §§ II–III, pp. 3–9 derive the limit and its global structure.

NHEK is the chapter’s recurring geometric object. It supports an asymptotic-symmetry calculation after boundary falloffs are chosen, determines the Frolov–Thorne scaling of near-horizon modes, and provides a comparison point for low-energy scattering. It does not replace the asymptotically flat exterior. Matching a throat mode to an observable at infinity still requires an overlap region and ultraviolet boundary data.

Under the original Kerr/CFT boundary conditions, the diffeomorphisms enhancing the rotational $\mathrm{U}(1)$ have a Virasoro algebra with

$$c_L=12J$$

in $\hbar=1$ units, retaining $G$ in the metric and area. Together with $T_L=1/(2\pi)$, the canonical Cardy expression gives

$$S_{\mathrm{Cardy}}=\frac{\pi^2}{3}c_LT_L=2\pi J=S_{\mathrm{BH}}.$$

The equality is exact at the displayed semiclassical order. Its microscopic interpretation assumes a suitable chiral quantum theory and Cardy regime; the charge algebra alone does not construct that theory.

Read the chapter in this order

1. Near-Horizon Extremal Kerr Geometry derives the NHEK scaling, isometries, and matching boundary.
2. Kerr/CFT Asymptotic Symmetries and Central Charge computes charges for declared falloffs and checks finiteness and integrability.
3. Cardy Entropy for Extremal Kerr separates the algebraic calculation from the assumptions needed for a microscopic count.
4. Hidden Conformal Symmetry of Kerr treats the near-region, low-frequency scalar equation and distinguishes it from a spacetime isometry.
5. Celestial Amplitudes and Black-Hole Scattering transforms massless and massive scattering data to boost eigenstates and identifies the missing black-hole dynamics.
6. Requirements for Flat-Space Holography applies one state-space, unitarity, reconstruction, and entropy matrix to null-infinity, celestial, Carrollian, and flat-limit proposals.

The first four pages test an extremal or low-energy Kerr sector. The last two ask for a complete asymptotically flat theory. No argument in the first sector automatically supplies the state space or finite-energy black-hole S-matrix demanded in the second.

Results and limitations through 2026-08-20

The formulation axes are extremality, near-horizon scaling, boundary falloffs, charge integrability, quantum state, Cardy regime, wave frequency, asymptotic boundary conditions, and scattering basis.

Evidence or position	Controlled result	Limitation or adverse discriminator
NHEK geometry, Bardeen and Horowitz 1999, §§II–III, pp. 3–9	A precise extremal scaling yields an exact throat with enhanced $\mathrm{SL}(2,\mathbb R)\times \mathrm{U}(1)$ symmetry	The limit discards the asymptotically flat region and generic nonextremal excitations
Kerr/CFT entropy, Guica, Hartman, Song, and Strominger 2009, §§4–7, pp. 8–17	Declared falloffs give $c_L=12J$, and the Frolov–Thorne temperature gives the Bekenstein–Hawking entropy through Cardy’s formula	Existence, spectrum, unitarity, and Cardy applicability of a microscopic chiral CFT are not independently constructed
Hidden conformal symmetry, Castro, Maloney, and Strominger 2010, §§2–5, pp. 3–12	The near-region, low-frequency scalar equation and absorption data organize into left/right conformal structures	The generators are not global spacetime isometries and the approximation fails at generic frequency or radius
Celestial basis, Pasterski and Shao 2017, §§ 3–4	Complete principal-series bases for massless and massive scalars organize scattering covariantly	The basis does not provide a nonperturbative black-hole formation-and-decay amplitude or complete flat-space dual

Controlled geometry, charges, wave-equation symmetries, and basis transforms support a set of interfaces. Competing positions interpret them as sectors of a microscopic dual, hints of a broader flat-space theory, or powerful reorganizations without a complete dual. The strong unification remains conjectural. An unresolved limitation is the absence of one state space and observable map that contains NHEK microstates, generic Kerr scattering, evaporation, and null-infinity data together.

Checks that prevent overclaiming

For asymptotic symmetries, changing falloffs can change the allowed generators, central term, and integrability. Charges must be finite and conserved on the chosen phase space. For the entropy match, the same angular-momentum and temperature conventions must enter both the Frolov–Thorne limit and Cardy formula. Agreement of one leading entropy is compelling but is not a protected degeneracy unless a quantum spectrum is supplied.

For hidden conformal symmetry, independently check $\omega M\ll1$ and the near-region condition before using hypergeometric solutions. For celestial amplitudes, verify the inverse Mellin contour and massive sector; large celestial dimension alone is not evidence for a horizon. A complete flat-space proposal must additionally reproduce BMS charges, a unitary S-matrix, the Kerr or Schwarzschild density of states, and a controlled bulk-locality limit.

What could change these conclusions

The conclusions could change if a theorem constructs the Kerr/CFT state space under explicit boundary conditions, if an independent microscopic calculation reproduces entropy and finite corrections, or if a counterexample invalidates charge integrability or Cardy use. The conclusions also depend on whether a complete celestial or flat-space reconstruction, massive-sector result, black-hole scattering amplitude, or new observational discriminator changes the relation between the throat and asymptotic descriptions.

Review the chapter

Assume the NHEK charge calculation gives $c_L=12J$ and the Frolov–Thorne limit gives $T_L=1/(2\pi)$. Compute the Cardy entropy and state exactly what the match proves and does not prove.

Solution

Substitution gives $S=(\pi^2/3)(12J)(1/2\pi)=2\pi J$, equal to the extremal Kerr Bekenstein–Hawking entropy in $G=\hbar=1$ units. This checks the normalization of the central term and thermal identification under the chosen falloffs. It does not prove that a unique unitary chiral CFT exists, that its spectrum lies in the Cardy regime, or that its degeneracy counts individual Kerr microstates. Those are additional hypotheses. Finite corrections and agreement for more observables would discriminate a microscopic theory from a leading symmetry-based entropy match.

References

• Bardeen, James M., and Gary T. Horowitz. “The Extreme Kerr Throat Geometry: A Vacuum Analog of AdS2 × S2.” Physical Review D 60 (1999) 104030. arXiv:hep-th/9905099.
• Castro, Alejandra, Alexander Maloney, and Andrew Strominger. “Hidden Conformal Symmetry of the Kerr Black Hole.” Physical Review D 82 (2010) 024008. arXiv:1004.0996.
• Guica, Monica, Thomas Hartman, Wei Song, and Andrew Strominger. “The Kerr/CFT Correspondence.” Physical Review D 80 (2009) 124008. arXiv:0809.4266.
• Pasterski, Sabrina, and Shu-Heng Shao. “A Conformal Basis for Flat Space Amplitudes.” Physical Review D 96 (2017): 065022. DOI. Open PDF.