Celestial Amplitudes and Black-Hole Scattering

Celestial amplitudes rewrite a four-dimensional flat-space S-matrix in boost eigenstates that transform as correlators on the celestial sphere. The Mellin variables retain energy information through conformal dimensions and contours; the angular positions retain null directions. Soft theorems and asymptotic symmetries become two-dimensional Ward identities. These are controlled reorganizations of scattering data, not yet a nonperturbative calculation of black-hole formation, evaporation, or interior dynamics.

Required background. Gauge Theory and String-Theory Preparation provides S-matrix, infrared, and conformal-representation conventions.

Helpful background. Quantum Gravity supplies the flat-gravity problem; Quantum Information supplies unitarity and completeness language; Black-Hole Geometry and Mechanics provides horizon formation criteria; and Gravitational Radiation and Waveforms provides null-infinity radiation and memory.

From momentum eigenstates to celestial data

For a future-directed massless momentum $p^\mu=\omega q^\mu(z,\bar z)$ with $\omega\gt0$, choose the null direction

$$q^\mu=(1+z\bar z,\ z+\bar z,\ -i(z-\bar z),\ 1-z\bar z).$$

It obeys $q^2=0$ in the site’s signature; in this normalization the physical energy is $p^0=\omega(1+z\bar z)$. Incoming/outgoing signs are supplied separately in the momentum-conservation delta function. In fixed energy units, define a celestial transform

$$\widetilde{\mathcal A}(\Delta_i,J_i;z_i,\bar z_i) =\prod_i\int_0^\infty d\omega_i\, \omega_i^{\Delta_i-1} \mathcal A(\omega_i,J_i;z_i,\bar z_i).$$

The complex coordinates specify directions on the sphere, helicity becomes two-dimensional spin $J_i$, and the Mellin dimension $\Delta_i$ labels the energy-scaling basis. For four-dimensional massless scalar states the complete normalizable basis uses $\Delta=1+i\nu$, $\nu\in\mathbb R$. The inverse Mellin contour and infrared/distributional prescription are part of the definition; discarding them would discard the energy dependence needed to reconstruct the S-matrix.

Massive external states are represented by conformal-primary wavefunctions built by integrating plane waves over a unit timelike hyperboloid. Pasterski, Shao, and Strominger 2017, §§ 2–3 construct the massive scalar wavefunctions and a three-point transform. The completeness proof and massless Mellin basis are given separately by Pasterski and Shao 2017, §§ 3–4. This matters for black-hole questions because massive asymptotic data cannot be represented by null directions alone. An evaporating black hole is generally an intermediate resonance, rather than an additional stable massive external particle.

Results and limitations through 2026-08-20

The formulation axes are asymptotic flatness, external-state basis, infrared prescription, Mellin contour, unitarity domain, and whether a black-hole intermediate state is calculated or only inferred.

Evidence or position	Controlled result	Limitation or adverse discriminator
Pasterski and Shao 2017, §§ 3–4	Principal-series conformal-primary wavefunctions provide complete massive and massless scalar bases with a Klein–Gordon inner product	Basis completeness does not supply interacting dynamics, infrared finiteness, or a black-hole sector
Strominger 2014, §§2–4, pp. 3–12	Diagonal BMS invariance and soft-graviton relations constrain gravitational scattering at null infinity	Soft Ward identities constrain zero-energy limits but do not reconstruct finite-energy strong-gravity amplitudes
Perturbative celestial transforms	Three- and four-point amplitudes, soft currents, and operator-product limits can be computed with explicit contours	Distributional support, infrared divergences, and nonstandard reality or unitarity properties complicate a conventional Euclidean-CFT interpretation
Black-hole formation proposals	High-energy impact-parameter amplitudes suggest horizons and eikonal regimes in momentum space	No controlled celestial observable presently computes formation, a complete intermediate-state spectrum, evaporation, and unitary decay in one framework; this remains unresolved

The controlled evidence establishes a useful kinematic basis and exact symmetry constraints. Competing positions differ on whether these ingredients are the beginning of a complete celestial dual, a boundary reformulation of the S-matrix, or one sector of a broader flat-space holography. Strong claims about black-hole interiors remain conjectural and model-dependent.

What energy and angular information survive

The celestial dimension is not a particle energy. Instead, the full dependence on energy is encoded in the function of complex $\Delta_i$ and recovered by inverse Mellin integration. Angular momentum is split between celestial spin and orbital dependence on $z_i$. Four-momentum conservation appears as support conditions coupling all insertion points and dimensions, so a generic celestial object is a distribution rather than an ordinary smooth correlator.

A simple independent check is to transform a soft factor. Multiplication by a power of $\omega$ shifts $\Delta$, while the angular soft factor becomes a meromorphic current insertion. Inverse transformation must recover the original energy scaling and helicity weight. Failure of either test signals an inconsistent contour or normalization.

For a black-hole intermediate state, a credible observable must retain total center-of-mass energy, impact parameter or partial waves, angular momentum, and all allowed decay channels. A celestial partial-wave or conformal-block decomposition is a candidate language, but interpreting one block as a black hole requires dynamical evidence such as a pole, density, or controlled saddle—not merely large dimension.

Infrared, unitarity, and black-hole limitations

Four-dimensional gravitational S-matrix elements between Fock states suffer infrared divergences. A physical construction must specify inclusive observables, dressed states, or another infrared prescription before Mellin transformation. The celestial inner product and reality conditions also differ from those of a standard unitary Euclidean CFT; complex principal-series dimensions are expected in the scattering basis.

Black-hole formation is nonperturbative in the regime where the Schwarzschild radius exceeds the impact parameter. Perturbative soft constraints remain true but are not sufficient to determine the finite-energy amplitude. Evaporation introduces a long multiparticle final state and potentially a sum over topologies or saddles. The evidence does not establish that a presently defined celestial correlator reconstructs the interior or proves unitary information recovery.

What could change these conclusions

The conclusions could change if a theorem supplies a unitary and complete celestial state space with an inverse map for an interacting gravitational S-matrix, if an independent calculation produces and reproduces an explicit black-hole formation-and-decay amplitude, or if a counterexample reveals loss of energy, angular momentum, or probability under the proposed transform. The conclusions also depend on whether a new infrared completion, observation-class definition, or massive-sector construction changes the boundary between kinematic reorganization and full holographic dynamics.

Exercises

Explain how the celestial transform retains energy information and determine two additional data needed before a large-dimension contribution can be called a black-hole intermediate state.

Solution

Energy is retained through the complete dependence on the Mellin dimensions and the chosen inverse contour; it is not stored as a single local eigenvalue. Applying the inverse transform reconstructs each external energy dependence. This is a useful check because a contour choice that fails to recover the original energy scaling is not an equivalent formulation. To identify a black-hole intermediate state, one also needs dynamical control of the total energy and angular-momentum channel, plus evidence for its formation and decay—such as a controlled saddle, spectral density, or unitary set of final-state amplitudes. Comparing candidate interpretations therefore requires both an invertible transform and a dynamical discriminator. Large conformal dimension alone is only a basis label and cannot establish a horizon.

References

• Pasterski, Sabrina, and Shu-Heng Shao. “A Conformal Basis for Flat Space Amplitudes.” Physical Review D 96 (2017): 065022. DOI. Open PDF.
• Pasterski, Sabrina, Shu-Heng Shao, and Andrew Strominger. “Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere.” Physical Review D 96 (2017) 065026. arXiv:1701.00049.
• Strominger, Andrew. “On BMS Invariance of Gravitational Scattering.” Journal of High Energy Physics 07 (2014) 152. arXiv:1312.2229.