Requirements for Flat-Space Holography

A nonperturbative flat-space holographic theory must do more than organize soft limits: it must define its state space and observables, implement asymptotic symmetries without losing finite-energy information, and reproduce unitary scattering and black-hole thermodynamics under explicit boundary conditions. This page formulates those tests for four-dimensional asymptotically flat gravity. Null-infinity, celestial, Carrollian, and flat-limit programs currently meet different subsets. None has yet established the complete dual required below.

Required background. Gauge Theory and String-Theory Preparation provides gauge constraints, scattering amplitudes, and large-gauge symmetry.

Helpful background. Quantum Gravity frames the nonperturbative question; Quantum Information supplies unitarity and factorization tests; Black-Hole Geometry and Mechanics supplies horizon thermodynamics; and Gravitational Radiation and Waveforms supplies Bondi radiation and memory.

Minimum data of a flat-space dual

Fix four-dimensional gravity with asymptotically flat Bondi boundary conditions and specified matter content. Near future null infinity, radiative data reside in the Bondi shear $C_{AB}(u,z,\bar z)$ and news

$$N_{AB}=\partial_u C_{AB}.$$

A candidate theory must state which boundary data are dynamical, their symplectic form, the allowed Bondi–Metzner–Sachs (BMS) transformations and charges, and the treatment of zero modes and infrared sectors. The gravitational scattering construction in Strominger 2014, §§ 2–4 fixes matching conditions and relates BMS charges to soft-graviton identities. Changing those boundary conditions can change the asymptotic algebra.

The scattering requirement is an operator

$$S:\mathcal H_{\mathrm{in}}\longrightarrow\mathcal H_{\mathrm{out}}, \qquad S^\dagger S=SS^\dagger=1,$$

with a declared infrared prescription. Its Hilbert spaces must contain massless radiation, massive particles associated with timelike infinity, bound states, and states energetic enough to form black holes. Symmetry Ward identities must constrain this operator, while completeness must ensure that every physical bulk state and channel is represented. A basis transform is valuable only if it has an inverse on this full domain.

Results and limitations through 2026-08-20

The comparison axes are dimension, boundary conditions, state space, observables, symmetry, unitarity, massive sectors, finite-energy black holes, and reconstruction. The date is the evidence cutoff, not a claim that all programs have equal maturity.

Evidence or position	Assumptions and controlled result	Limitation or adverse test	Date
Null-infinity S-matrix and BMS Ward identities, Strominger 2014, §§2–4, pp. 3–12	Four-dimensional asymptotic flatness and matched boundary data relate soft-graviton identities to diagonal BMS invariance	Exact soft constraints do not construct a nonperturbative Hilbert space, finite-energy amplitudes, or black-hole microstates	2014
Celestial transform, Pasterski and Shao 2017, §§ 3–4	Complete principal-series bases exist for massless and massive free scalar states; the massless transform retains energy information in Mellin variables	Distributional correlators, infrared completion, the interacting inner product, and a controlled black-hole intermediate sector require more data	2017
Carrollian null-boundary formulation	Fields live directly on null infinity and display BMS covariance together with retarded-time dependence directly	No unique interacting Carrollian theory is known to supply the complete gravitational S-matrix, massive timelike sectors, entropy, and interior reconstruction	2026 cutoff
Flat limit of AdS holography	A controlled AdS dual can motivate flat observables by sending the curvature radius to infinity with correlated operator dimensions and insertion kinematics	The limit can be singular or retain only perturbative scattering; finite Newton coupling, black-hole density of states, and nonperturbative completeness need separate control	2026 cutoff

Controlled calculations therefore establish charge algebras, soft Ward identities, and useful transforms. Competing positions interpret these as a partial dictionary, a reformulation of scattering, or evidence for a complete holographic dual. The last interpretation remains conjectural: satisfying symmetry constraints is necessary but does not establish unitarity, completeness, or black-hole dynamics.

Applying one requirement matrix

Apply the same tests to the null-infinity S-matrix and celestial candidates. Both can reproduce BMS constraints when matching, soft sectors, and normalizations are chosen consistently. The S-matrix formulation directly names the unitary object but does not by itself give a microscopic boundary theory. The celestial formulation reorganizes its matrix elements into boost eigenstates, as developed on Celestial Amplitudes and Black-Hole Scattering, but the transform cannot improve incomplete dynamics merely by changing basis.

For massive states, a celestial basis exists, yet their trajectories end at timelike rather than null infinity. A complete proposal must explain how timelike data join the null-boundary algebra. For locality, it must identify a regime in which boundary observables reconstruct approximately local bulk operators with causal commutators, then state the corrections. For factorization, it must specify whether the Hilbert space decomposes across asymptotic channels or is constrained by gravitational dressings and charges.

A Carrollian formulation preserves retarded time rather than Mellin-transforming it. This may make flux and causality more direct, whereas celestial variables make Lorentz covariance look like two-dimensional conformal covariance. These are formulation choices, not independent successes, unless each construction supplies a state space, inner product, dynamics, and inverse map. A flat limit of AdS has more microscopic input at finite radius, but the limit must commute with the large-$N$, energy, and infrared limits used to define scattering. Noncommuting limits are a decisive failure case for claims of equivalence.

The finite-energy black-hole test

Let $\mathcal N(M,J;\Delta M)$ be the dimensionless number of states in a specified narrow mass window $\Delta M$ and fixed angular-momentum sector, with an infrared regulator or resonance prescription stated. The window must be broad compared with microscopic level spacings and narrow compared with scales over which the macroscopic entropy varies substantially. A candidate containing four-dimensional Kerr or Schwarzschild black holes must recover the leading semiclassical growth,

$$\log \mathcal N(M,J;\Delta M)=\frac{A(M,J)}{4G}+\text{subleading corrections},$$

while also giving a unitary description of formation and decay. Reproducing the area scaling from an assumed thermodynamic saddle is weaker than deriving the relevant degeneracy or coarse-grained density from the boundary state space. The theory must retain center-of-mass energy, impact parameter or partial waves, angular momentum, and multiparticle decay channels. It must distinguish a resonance, an inclusive semiclassical saddle, and an individual microstate.

This test also exposes a tension between observables and reconstruction. An asymptotic S-matrix may determine all gauge-invariant scattering data without assigning local interior operators. A stronger holographic claim must explain what interior questions are encoded, with what state dependence or error tolerance, and why the prescription respects asymptotic unitarity. No current framework has passed all of these requirements under controlled four-dimensional boundary conditions; that negative assessment is an unresolved limitation, not a no-go theorem.

What could change these conclusions

The conclusions could change if a theorem constructs a complete positive-norm boundary state space and invertible bulk map, if an independent calculation reproduces a finite-energy black-hole formation-and-decay amplitude together with the area-law density, or if a counterexample shows that BMS charges, probability, or massive data are lost. The comparison also depends on whether new boundary conditions, algebra classes, infrared dressings, or flat-limit prescriptions change which candidate is being tested.

Exercises

Compare the null-infinity S-matrix and celestial formulations using four tests: BMS charges, unitarity, Schwarzschild entropy scaling, and a controlled bulk-locality limit. Which conclusions are basis-independent?

Solution

Both formulations can encode the same BMS Ward identities because an invertible celestial transform changes basis without changing the constrained scattering operator. Unitarity is likewise basis-independent only after one supplies a positive inner product, complete transform domain, inverse contour, and common infrared prescription. Neither formulation presently derives Schwarzschild entropy from a microscopic state count or supplies a controlled interior-locality map. Therefore those two tests fail at the dynamical level, not because celestial variables are inferior. The comparison also shows why soft symmetry plus invertibility is insufficient: it preserves whatever scattering information was supplied but cannot create missing black-hole states or amplitudes.

References

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