Completing Dirac's coordinate-fixed Hamiltonian formulation of gravitation: strong-field solvability of the conformal-factor equation, the global bracket kernel, and spinor and point-particle sources

Dirac's 1959 paper 'Fixation of Coordinates in the Hamiltonian Theory of Gravitation' eliminates the supplementary conditions of canonical general relativity by completely fixing the coordinate system, splitting the spatial metric into a unimodular matrix and a positive conformal factor $\kappa$ governed by a nonlinear elliptic equation, and modifying the Poisson brackets through a reciprocal kernel $C_v{}^r$. The paper acknowledges four unresolved tasks: general (strong-field) solvability of the $\kappa$ equation, for which only a weak-field successive-approximation scheme is given; a global construction of $C_v{}^r$, of which only a locally nearly-flat approximation is computed; extension to spinor fields, which are excluded; and a consistent treatment of point particles, whose negative Newtonian field energy threatens the positivity of $\kappa$ and hence the spacelike character of the slices. The conformal (Lichnerowicz–York) method resolves the constant-mean-curvature case and the puncture description of point sources as additional asymptotically flat ends, but the general coupled problem — existence, uniqueness, and construction for non-constant mean curvature and strong data, spinorial matter, a globally controlled bracket kernel, and a literal point-particle treatment within the reduced formalism — remains open.

Background

General relativity admits a Hamiltonian (canonical) formulation in which the instantaneous state of the gravitational field is data on a spacelike hypersurface $x^0=t$: a Riemannian three-metric $\tilde g_{rs}$ together with its conjugate momentum. Because the theory is invariant under arbitrary reparametrizations of the coordinates, the canonical variables satisfy constraints — identities among them, called first-class in Dirac's terminology when their Poisson brackets (P.b., the bracket structure of Hamiltonian mechanics) with all constraints vanish. First-class constraints generate coordinate (gauge) transformations; after quantization they become supplementary conditions on the wave function, so that specifying an initial state is as hard as solving the full dynamics. To remove this obstacle, Dirac proposed in 1959 to fix the coordinate system completely. He imposed one condition per point of space fixing the hypersurface — vanishing of the momentum trace, i.e. a maximal (extremal-volume) slice — and three further conditions fixing the coordinates within the slice, replacing the natural three-dimensional harmonic conditions by a compatible variant (his equation (35)). These conditions turn the constraints into second-class ones, which are eliminated by replacing the Poisson brackets with what are now called Dirac brackets: the bracket is corrected by the matrix of P.b.'s of the second-class constraints and its inverse.

Two analytic objects then carry the whole reduced theory. First, Dirac splits the spatial metric into a unimodular matrix $g_{rs}$ (determinant one; five independent components per point) and a positive scalar conformal factor $\kappa$, and rescales the momentum accordingly. The Hamiltonian constraint becomes a nonlinear elliptic equation for $\kappa$ on the slice (his equations (30) and (36)): schematically, the Laplacian of $\kappa$ balances a term quadratic in the rescaled momentum, entering with a negative power of $\kappa$, against curvature and matter-density terms; in the standard normalization $\tilde g = \phi^4 g$ of the modern literature in four spacetime dimensions, the momentum-squared term of the equation carries the power $\phi^{-7}$, the scalar-curvature term $\phi$, and the mean-curvature-squared term $\phi^{5}$, with further powers for matter. Dirac writes that this is 'a difficult equation to solve generally' and solves it by successive approximation, valid for gravitational fields 'that are not too strong'. On a reference metric of flat type this is the equation the modern literature calls the Lichnerowicz equation — the scalar part of the conformal method for the Einstein constraint equations, introduced by Lichnerowicz in 1944 for the $n$-body problem and developed by York: one freely prescribes a conformal three-metric and a transverse (divergence-free) traceless momentum, and determines the physical data by solving the Lichnerowicz equation for the conformal factor together with a vector equation (the momentum constraint) for a momentum potential whose principal operator is built from the conformal Killing operator, the operator measuring the failure of a vector field to generate a pure rescaling of the metric (fields that do generate such rescalings are the conformal Killing fields, and they obstruct inversion); the inverse of that vector elliptic operator is exactly what Dirac's second object, the reciprocal kernel $C_v{}^r(x',x)$ entering his modified brackets, approximates. Dirac computes $C_v{}^r$ only in a locally nearly-flat expansion (obtaining a $1/|x-x'|$-type Green's function) and notes that successive approximations would refine it to any desired accuracy, without controlling its global properties or its boundary conditions.

The source adds two matter-side exclusions. Spinor fields are excluded outright, because they 'require a special treatment': spinors do not transform as tensors under coordinate changes, and their conformal rescaling differs from the integer powers of the tensorial rescaling, so the split into unimodular metric plus factor does not decouple the spinor contributions as it does for integer-spin matter. And point particles are singled out: the energy density of the Newtonian gravitational field is negative definite, so very close to a point source the effective mass density in the $\kappa$ equation becomes negative and can drive $\kappa$ out of the real positive range required for the hypersurface to remain spacelike — a consistency condition on the whole subsequent evolution and quantization.

The later theory of the conformal method (surveyed by Bartnik and Isenberg) settles a large part of this in special regimes. When the mean extrinsic curvature (the trace of the extrinsic curvature, the tensor measuring how the slice bends within spacetime, fixed to zero by Dirac's maximal-slicing condition) is constant over the slice (CMC data), the vector equation decouples and the solvability of the Lichnerowicz equation is completely classified by the Yamabe class of the conformal metric — whether its conformal class contains a representative of positive, zero, or negative scalar curvature: existence and uniqueness hold for essentially all such data on closed manifolds or on asymptotically Euclidean ones (manifolds that outside a compact set equal Euclidean space up to a metric falloff), and the constructive sub/supersolution (upper and lower barrier) iteration that proves it is the rigorous descendant of Dirac's successive approximations. For non-constant mean curvature the scalar and vector equations couple, and the general solvability question — existence, uniqueness, and construction for arbitrary prescribed, strong data — remains open, with results known only near CMC, in special far-from-CMC classes, and with non-existence and blow-up phenomena. Singular point sources are handled in the time-symmetric and puncture frameworks, where the points are reinterpreted as additional asymptotically flat ends rather than literal particles. The background thus makes precise the gap between the resolved special regimes and the general strong-field, general-matter setting that Dirac's four acknowledged tasks jointly demand.

Problem Statement

The problem is to complete the coordinate-fixed, reduced Hamiltonian formulation of gravitation that Dirac set up in 1959 by resolving the tasks the paper itself leaves open. They jointly serve one objective: a mathematically complete and practically usable reduced canonical formalism in which the initial state can be freely specified (no supplementary conditions on the wave function) and evolved beyond the weak-field approximation, on an asymptotically flat slice (the setting of the source: the falloff conditions at infinity make the surface terms vanish, and the weak-field limit is Newtonian).

1. Conformal-factor equation. Establish a general solvability and construction theory for positive solutions $\kappa$ of the nonlinear elliptic equation (Dirac's equations (30)/(36), the Lichnerowicz-type Hamiltonian constraint) for the conformal-method data class defined in the Background — a prescribed conformal three-metric, transverse-traceless momentum, and mean curvature on an asymptotically flat slice: prove existence, uniqueness, and constructive solution methods on the data classes characterized by independent, checkable conditions on the conformal data — the Yamabe class of the conformal three-metric and the magnitudes of the transverse-traceless momentum and mean curvature — including strong momentum sources and non-constant mean curvature beyond the paper's successive-approximation scheme (stated to be valid only for fields that are not too strong) — this non-constant-mean-curvature demand deliberately generalizes Dirac's maximal-slicing condition, for which the mean curvature vanishes and the equation reduces to the already classified zero-mean-curvature CMC case; delineate the data classes where positive solutions provably fail to exist, consistent with the known maximum-principle, near-CMC, and far-from-CMC non-existence and blow-up results; and control the global properties and boundary/asymptotic conditions of $\kappa$.

2. Reciprocal bracket kernel. Construct the reciprocal kernel $C_v{}^r(x',x)$ entering the modified (Dirac) brackets as a globally defined object — the Green's function of the vector elliptic operator of the momentum constraint on the slice — with controlled behavior at infinity on the stated asymptotically flat slice, where the conformal Killing operator has trivial kernel, and — as a separately scoped closed-slice sub-case — on compact manifolds where conformal Killing fields obstruct invertibility, so that the modified brackets define a consistent Poisson structure on the reduced phase space; the paper computes only the locally nearly-flat first approximation.

3. Spinor matter. Extend the gauge-fixing and bracket-modification procedure to spinor fields, which the source excludes because they 'require a special treatment'.

4. Point particles. Give a consistent treatment of point particles within the reduced formalism — resolving the short-distance singularities and the negative Newtonian field-energy density that can drive $\kappa$ out of the real positive range keeping the hypersurface spacelike. The treatment sought is a literal one of the point sources, not a reinterpretation of them as additional asymptotically flat ends.

For items 3–4 the task is to determine whether the gauge-fixed constraints close and the modified brackets define a consistent reduced bracket structure for the spinor extension, and whether a regularization exists under which $\kappa$ remains strictly positive for the point-particle treatment, exhibiting one if it does. A rigorous no-go result on items 3–4 — showing that no consistent spinor bracket closure or no admissible point-particle regularization exists — resolves those items; for items 1–2, non-existence on a data sub-class is the sub-deliverable already required within item 1, and their completion still requires the positive solvability and construction theory. Constant or near-constant mean curvature, time-symmetric data, and weak fields are sub-cases of the stated general question.

References

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