Penrose's 1982 twistorial construction assigns to any closed spacelike 2-sphere S in a general spacetime a 2-surface twistor space T^alpha(S) and, via a curvature charge integral, a kinematical twistor A_{alpha beta}; at future null infinity it reproduces the Bondi-Sachs four-momentum and yields a new, supertranslation-ambiguity-free angular momentum. The construction has remained incomplete for four decades: no natural, conformally invariant, non-degenerate Hermitian pairing (twistor norm) or infinity-twistor structure is known on T^alpha(S) for general, 'contorted' 2-spheres; without it the ten complex components of A_{alpha beta} cannot be reduced to ten real energy-momentum and angular-momentum quantities, the quasi-local mass m(S) is ambiguous or complex, its norm-choice dependence is unresolved, no positivity theorem exists for the Penrose mass, and the quasi-local interpretation of the six angular-momentum quantities and their balance laws is open. Authoritative statements from 2009 (Szabados's review) and 2026 (Kmec-Mason-Ruzziconi) confirm these issues remain open for the Penrose definition, while positivity is proved only for neighboring Dougan-Mason-type constructions. This problem asks for the missing reality structure and its consequences, retaining the original generality of generic 2-spheres in general spacetimes.

Background

In general relativity the gravitational field carries energy, but because spacetime itself is dynamical there is no tensorial local energy density. The standard exact notions of gravitational energy are global: the ADM mass, defined at spatial infinity of an asymptotically flat initial data set, and the Bondi-Sachs four-momentum, defined on cuts of future null infinity I+\mathscr{I}^+. A quasi-local mass instead assigns energy-momentum (and ideally angular momentum) to an arbitrary closed spacelike 2-surface S\mathscr{S} bounding a finite region; finding a fully satisfactory definition is a recognized long-standing problem (Szabados 2009).

Penrose's 1982 proposal is the twistorial one. In Minkowski spacetime a twistor is a pair Zα=(λA,πA′)Z^{\alpha}=(\lambda^{A},\pi_{A'}) of spinor fields solving the twistor equation for a single (valence-one) spinor field, ∇A′AλB=−iεABπA′\nabla^{A'A}\lambda^{B}=-i\varepsilon^{AB}\pi_{A'}; the solution space is four complex dimensional and carries a conformally invariant Hermitian metric Hαβ′H_{\alpha\beta'} of signature (+,+,−,−)(+,+,-,-). In a general curved spacetime this equation has only the zero solution, so Penrose projected it tangentially to S\mathscr{S}: the resulting 2-surface twistor equation is a first-order elliptic system whose solution space Tα(S)\mathbb{T}^{\alpha}(\mathscr{S}) has, for a generic topological 2-sphere, exactly four complex dimensions (its Fredholm index is 4(1−g)4(1-g) for genus gg; 'exceptional' spheres admitting more solutions exist and break the construction). Pairing solutions λA,ωA\lambda^{A},\omega^{A} in the curvature charge integral AS[λ,ω]=i8πG∮SλAωBRABcd dxc∧dxdA_{\mathscr{S}}[\lambda,\omega]=\frac{i}{8\pi G}\oint_{\mathscr{S}}\lambda^{A}\omega^{B}R_{ABcd}\,\mathrm{d}x^{c}\wedge\mathrm{d}x^{d}, the integral over S\mathscr{S} of the pullback of the curvature 2-form (the overall constant depends on signature conventions), defines the kinematical twistor AαβA_{\alpha\beta}, a symmetric twistor with ten complex components built only from data on S\mathscr{S}.

In flat space the energy-momentum and self-dual angular momentum of a source are specific spinor parts of such a twistor - the self-dual angular momentum being the complex combination in which the spatial angular momentum and center-of-mass triples form its real and imaginary parts - and extracting them needs two auxiliary structures: the Hermitian metric Hαβ′H_{\alpha\beta'} and the infinity twistor IαβI^{\alpha\beta}, a fixed symmetric twistor encoding the conformal structure's points at infinity. A Hermiticity condition built from these reduces the ten complex components of AαβA_{\alpha\beta} to ten real quantities - four of energy-momentum, six of angular momentum and center of mass - and the mass is then a norm (or a determinant) of the kinematical twistor.

The unresolved difficulty is that in a general curved spacetime neither auxiliary structure is available. The pointwise Hermitian pairing of 2-surface twistors is constant on S\mathscr{S} precisely when S\mathscr{S} is non-contorted, meaning locally embeddable, with its intrinsic metric and extrinsic curvature, into a conformally flat (conformal Minkowski) spacetime; important surfaces, such as spacelike cross-sections of the horizon of the Kerr black hole (the rotating vacuum solution), are contorted. The candidate norms proposed in the 1982 paper and afterwards (for example averaging the pointwise pairing over S\mathscr{S}, with curvature-weighted alternatives) fail conformal invariance or degenerate in such cases, and no natural simple infinity twistor exists on Tα(S)\mathbb{T}^{\alpha}(\mathscr{S}); Helfer gave a counting argument that no such twistor can satisfy the Hermiticity condition generically, while Tod observed that only the combined 'bar-hook' Hαβ′=IαβHββ′H^{\alpha}{}_{\beta'}=I^{\alpha\beta}H_{\beta\beta'} is actually needed and might still be found. Consequences of the gap: the mass of a contorted surface is ambiguous or complex; the small-sphere limit - the limit in which S\mathscr{S} shrinks toward a point, where the leading term of a correct mass should reflect the local matter content, or the free gravitational field in vacuum - shows anomalous signs in vacuum; and the dependence of m(S)m(\mathscr{S}) on the choice of norm, the two standard definitions being the norm-mass and the determinant-mass, which already differ in de Sitter backgrounds, is unresolved.

What the construction does achieve, and why it is still regarded as containing essential physical insight: on cuts of I+\mathscr{I}^+ it reproduces the Bondi-Sachs four-momentum, and the remaining six real quantities give an angular momentum free of the famous supertranslation ambiguity - supertranslations being the translation subgroup of the Bondi-Metzner-Sachs (BMS) asymptotic symmetry group at null infinity, whose infinite dimensionality otherwise makes angular momentum origin-dependent. The 2-surface twistor space of a cut reduces the BMS algebra to a Poincare subalgebra, and the angular-momentum flux between cuts computed by Dray and Streubel coincides with the standard radiative flux. Neighboring quasi-local constructions partially answer other parts of the wish list: Dougan and Mason defined two closely related spinorial masses, via (anti-)holomorphic local twistors (twistor-valued fields on spacetime obeying a curvature-driven first-order transport equation), that are well defined in general with a positivity theorem for the energy; Brown-York and Wang-Yau type masses are based on isometric embeddings into flat space and have their own positivity results. None supplies the missing natural Hermitian structure on the 2-surface twistor space itself, and in 2026 the Oxford twistor group, extending the construction to higher-spin celestial charges, repeated that these issues remain open for the Penrose definition.

Problem Statement

Complete Penrose's 1982 twistorial construction into an unambiguous, real, physically interpretable quasi-local energy-momentum and angular momentum for general closed spacelike 2-spheres in general spacetimes. The single objective is that the 2-surface twistor space Tα(S)\mathbb{T}^{\alpha}(\mathscr{S}) and kinematical twistor AαβA_{\alpha\beta} of an arbitrary topological 2-sphere S\mathscr{S} - in the generic, contorted case, without assumptions that fix an asymptotic or Bondi-Sachs rest-frame - be equipped with the missing reality structure, and that the resulting quantities be shown to have the properties expected of gravitational energy and angular momentum. Throughout this Problem Statement, 'natural' - as applied to the pairing, the norm, or the infinity-twistor structure - means canonically determined by the intrinsic and extrinsic geometry of S\mathscr{S} together with the ambient conformal structure, with no additional auxiliary choice such as a conformal factor, a frame, or a preferred point at infinity. The requested results are the following four components of this one objective:

(1) Construct, or prove impossible within a precisely stated class, a natural conjugation or Hermitian pairing Tα(S)→Tα(S)\mathbb{T}^{\alpha}(\mathscr{S})\to\mathbb{T}_{\alpha}(\mathscr{S}) that is conformally invariant and non-degenerate for general 2-spheres in general spacetimes - equivalently the required 'bar-hook' combination of Hermitian metric and infinity twistor - resolving the known obstructions: conformal non-invariance of the averaged pairing, degeneracy on contorted surfaces such as Kerr horizon cross-sections, and Helfer's counting argument against a simple infinity twistor.

(2) With such a structure, formulate the Hermiticity condition that reduces the ten complex components of AαβA_{\alpha\beta} to ten real quantities (four of energy-momentum, six of angular momentum and center of mass) in the genuinely quasi-local regime, and determine the resulting quasi-local mass m(S)m(\mathscr{S}), resolving its dependence on the choice of norm, for example the discrepancy between the norm-mass and the determinant-mass.

(3) Establish positivity, or find precise obstructions: prove under appropriate energy conditions that m(S)m(\mathscr{S}) is non-negative (vanishing only for surfaces in flat spacetime), or exhibit within the twistor framework the surfaces or spacetimes for which this fails.

(4) Elucidate the six angular-momentum quantities at the quasi-local level: their transformation behavior under change of origin, their supertranslation and gauge behavior beyond the null-infinity case where it is already controlled, their relation to previously proposed angular-momentum fluxes, and the surface-level balance (flux) laws they satisfy.

Whenever component (1) is resolved constructively, components (1)-(4) form the objective jointly; if component (1) is instead resolved negatively for a class containing the generic contorted 2-spheres, then (2) and (4) do not arise and (3) becomes its obstruction branch.

The problem is posed for generic closed spacelike 2-spheres in general spacetimes, retaining the original generality of the construction; it is not restricted to non-contorted surfaces, cuts of I+\mathscr{I}^+, round spheres, or spheres in special spacetimes. In particular, a no-go result under (1) and any positivity obstruction under (3) must hold for a class that contains the generic contorted 2-spheres.

References

  1. Roger Penrose, Quasi-local mass and angular momentum in general relativity, Proc. R. Soc. Lond. A 381 (1982) 53-63, DOI 10.1098/rspa.1982.0058, https://doi.org/10.1098/rspa.1982.0058
  2. László B. Szabados, Quasi-Local Energy-Momentum and Angular Momentum in General Relativity, Living Reviews in Relativity 12 (2009) 4, DOI 10.12942/lrr-2009-4, https://doi.org/10.12942/lrr-2009-4
  3. K. P. Tod, Some examples of Penrose's quasi-local mass construction, Proc. R. Soc. Lond. A 388 (1983) 457-477, DOI 10.1098/rspa.1983.0092, https://doi.org/10.1098/rspa.1983.0092
  4. K. P. Tod, More on Penrose's quasi-local mass, Classical and Quantum Gravity 3 (1986) 1169-1189, DOI 10.1088/0264-9381/3/6/016, https://doi.org/10.1088/0264-9381/3/6/016
  5. L. J. Mason, A Hamiltonian interpretation of Penrose's quasi-local mass, Classical and Quantum Gravity 6 (1989) L7-L13, DOI 10.1088/0264-9381/6/2/001, https://doi.org/10.1088/0264-9381/6/2/001
  6. A. D. Helfer, Difficulties with quasilocal momentum and angular momentum, Classical and Quantum Gravity 9 (1992) 1001-1008, DOI 10.1088/0264-9381/9/4/014, https://doi.org/10.1088/0264-9381/9/4/014
  7. A. J. Dougan and L. J. Mason, Quasilocal mass constructions with positive energy, Physical Review Letters 67 (1991) 2119-2122, DOI 10.1103/PhysRevLett.67.2119, https://doi.org/10.1103/PhysRevLett.67.2119
  8. T. Dray and M. Streubel, Angular momentum at null infinity, Classical and Quantum Gravity 1 (1984) 15-26, DOI 10.1088/0264-9381/1/1/005, https://doi.org/10.1088/0264-9381/1/1/005
  9. J. David Brown and James W. York, Quasilocal energy and conserved charges derived from the gravitational action, Physical Review D 47 (1993) 1407-1419, DOI 10.1103/PhysRevD.47.1407, https://doi.org/10.1103/PhysRevD.47.1407
  10. Mu-Tao Wang and Shing-Tung Yau, Quasilocal Mass in General Relativity, Physical Review Letters 102 (2009) 021101, DOI 10.1103/PhysRevLett.102.021101, https://doi.org/10.1103/PhysRevLett.102.021101
  11. Ron Kelly and Paul Tod, Penrose's quasi-local mass for asymptotically anti-de Sitter space-times, arXiv:1505.00214 (2015), https://arxiv.org/abs/1505.00214
  12. Adam Kmec, Lionel Mason and Romain Ruzziconi, Quasi-Local Celestial Charges and Multipoles, arXiv:2604.13362 (2026), https://arxiv.org/abs/2604.13362
  13. Virinchi Rallabhandi, Spinorial quasilocal mass for spacetimes with negative cosmological constant, arXiv:2504.11971 (2025), https://arxiv.org/abs/2504.11971
  14. Harvey S. Reall, A third law of black hole mechanics for supersymmetric black holes and a quasi-local mass-charge inequality, arXiv:2410.11956 (2024), https://arxiv.org/abs/2410.11956
  15. Po-Ning Chen, Mu-Tao Wang, Ye-Kai Wang and Shing-Tung Yau, Supertranslation invariance of angular momentum, arXiv:2102.03235 (2021), https://arxiv.org/abs/2102.03235
  16. Po-Ning Chen, Jordan Keller, Mu-Tao Wang, Ye-Kai Wang and Shing-Tung Yau, Evolution of Angular Momentum and Center of Mass at Null Infinity, Communications in Mathematical Physics 386 (2021) 551-588, DOI 10.1007/s00220-021-04053-7, https://doi.org/10.1007/s00220-021-04053-7