The physical status of Thorne's C-energy: the torus–Schwarzschild mass relation, comparison with other energy prescriptions, and generalization to non-asymptotically-flat spacetimes (the open questions of Thorne 1965)

Kip Thorne's 1965 paper introducing C-energy — a localizable, conserved energy for spacetimes with whole-cylinder symmetry — closes with three explicitly open questions: (1) the relation, in full general relativity, between the C-energy of a massive thin-ring torus and its Schwarzschild mass, an equality Thorne proved only in the Newtonian (light-ring) limit; (2) the relation between C-energy and other energy concepts of general relativity, especially the localizable energies of Bergmann, Komar and Møller; and (3) whether C-energy-like, geometrically intrinsic conserved quantities can be formulated for other non-asymptotically-flat spacetimes.

Background

A fundamental feature of general relativity is that the total energy of an isolated system is well defined only when spacetime is asymptotically flat, i.e. when its geometry approaches that of special relativity far from all sources: the ADM energy (the total mass measured at spatial infinity, named after Arnowitt, Deser and Misner) and the Bondi energy (the mass measured at null infinity, after the emission of gravitational waves). For systems that are not asymptotically flat — a closed universe, or the idealization of an infinitely long cylinder — no satisfactory definition of total energy existed in 1965, and no local gravitational-energy density exists in general (the equivalence principle forbids a generally covariant tensorial local energy of the gravitational field). Several tentative definitions of localizable energy had been proposed, notably by Bergmann, by Komar and by Møller; each had been found wanting in some respect, typically through coordinate dependence or disagreement with physical expectations.

Thorne (1965) analyzed spacetimes with whole-cylinder symmetry — invariance under rotations about, and translations along, a symmetry axis, which is the exact symmetry of an infinite cylinder and of cylindrical waves. In adapted 'standard' cylindrical coordinates $(t,r,\phi,z)$ the metric of any such spacetime is determined by two potentials $\psi$ and $\gamma$ (and a radial function $\alpha$); in the vacuum and electromagnetic cases one can pass to the hyperbolic canonical form

$$ds^2=e^{2(\gamma-\psi)}(dt^2-dr^2)-e^{2\psi}dz^2-r^2e^{-2\psi}d\phi^2.$$

Thorne showed that the scalar $E=\gamma/4G$ (in his conventions) is the potential of a covariant conserved C-energy flux vector $P^\mu$, obeying $P^\mu{}_{;\mu}=0$, constructed from $E$ and the two Killing vectors of the symmetry (the infinitesimal generators of the rotations and the translations). C-energy is localizable and locally measurable: the component of $P^\mu$ along an observer's world line is the C-energy density that observer measures, and near the symmetry axis of a static weak-field system it reduces to the proper mass density. In vacuo and with electromagnetic sources, C-energy on a spacelike hypersurface is minimized when the system is static, and C-energy is transported by Einstein–Rosen waves — the exact cylindrical gravitational-wave solutions of Einstein and Rosen (1935).

Thorne tested whether C-energy 'deserves the title of energy'. For a static cylinder of small mass per unit length, the total C-energy per unit length equals the proper mass per unit length, whereas the mass per unit length attached to the Levi-Civita solution (the static cylindrically symmetric vacuum metric) need not agree. The decisive test is a toroidal body with major radius much larger than its minor radius (a thin ring): near the ring the geometry approximately has whole-cylinder symmetry, so the C-energy is well defined there, while far away the field is spherical and one can measure the Schwarzschild mass — the mass inferred from the Keplerian motion of distant orbiting particles, i.e. the asymptotic (ADM-type) mass. In the Newtonian approximation Thorne proved that the total C-energy of a light thin-ring torus equals its Schwarzschild mass. But he showed that the C-energy stored in the gravitational field outside the ring becomes comparable to the ring's own rest mass exactly when the ring is massive enough to produce large gravitational redshift — the regime where Newtonian theory fails — so the relation in the full Einstein theory remained unknown.

Two further features of the 1965 construction motivate item (4) below. First, there exist nonsingular hypersurfaces on which $\alpha'^2-\dot{\alpha}^{2}=0$ (with $\alpha$ the radial function of the standard metric) and where every observer measures an infinite C-energy density although the geometry is smooth. Second, in a note added in proof Thorne observed that redefining the C-energy potential as $E^{(\mathrm{new})}=(1/8G)(1-e^{-8GE})$ removes these divergences, and referred to his (unpublished) 1965 Princeton PhD thesis for a development of this viewpoint, leaving its physical status open.

The paper's conclusions section states verbatim: 'Three major questions remain to be investigated: (1) What is the relationship between the C energy of a massive, thin-ring torus and its Schwarzschild mass? (2) What is the relationship between C energy and other energy concepts of general relativity, especially the localizable energies of Bergmann, Komar, and Møller? (3) Can energy-like concepts similar to C energy be formulated for other nonasymptotically flat systems?' These three questions share one objective — deciding the physical status of C-energy as an energy concept. The later literature relevant to them concerns quasilocal energy, meaning energy assigned to a finite spatial region bounded by a closed two-surface rather than to infinity; Hamiltonian and 2+1-dimensional reductions, in which factoring out the symmetry direction turns the four-dimensional theory into an effective gravity theory in two space dimensions; pseudotensor energy prescriptions, which define coordinate-dependent, non-tensorial local energy densities; and hoop-conjecture studies of black-hole horizon formation by compacted mass.

Problem Statement

Settle the physical status of Thorne's C-energy by answering the three questions the 1965 paper left open together with the note-in-proof item (4), which together decide whether C-energy deserves the title 'energy':

(1) Torus–Schwarzschild relation. For a static torus whose major radius is much larger than its minor radius (a massive thin ring), whose near-ring region is describable by whole-cylinder symmetry, and whose distant exterior field is spherical, determine the relationship in full general relativity between the total C-energy of the torus and its Schwarzschild mass (the mass measured from the Keplerian motion of sufficiently distant particles, i.e. the asymptotic mass). The equality is established only in the Newtonian limit of a light ring; the open regime is where the gravitational binding energy in the exterior field is comparable to the ring's rest mass, so that the full Einstein theory is required. Determine whether the two quantities are equal, or find the precise relation connecting them, and state the conditions (mass per unit length, compactness, redshift) under which any identification holds or fails.

(2) Relation to other energy concepts. Determine how C-energy is related to the other proposed energy concepts of general relativity — especially the localizable energies of Bergmann, Komar and Møller, and by extension the Hamiltonian and quasilocal energy definitions: in cylindrical spacetimes, are they equivalent, and if not, in what precise sense do they differ, and which (if any) is the physical energy?

(3) Generalization. Determine whether energy-like, geometrically intrinsic, conserved quantities analogous to C-energy can be formulated for other classes of non-asymptotically-flat spacetimes. Fix at least one concrete class of such spacetimes, and determine whether a geometrically intrinsic, conserved quantity analogous to C-energy exists for that class; state the conditions under which it does.

(4) Redefined C-energy (from the note added in proof). For Thorne's redefined C-energy potential $E^{(\mathrm{new})}=(1/8G)(1-e^{-8GE})$, introduced to remove the divergences of the observer-measured C-energy density on the hypersurfaces where $\alpha'^2-\dot{\alpha}^{2}=0$, determine whether it is conserved and localizable in the same sense as $E=\gamma/4G$ and whether it is finite everywhere, and state its physical status as an energy.

The common objective of (1)–(4) is the coherence of C-energy as an energy concept: (1) tests it against the asymptotically measured mass of a compact source, (2) tests it against competing energy prescriptions, (3) tests the scope of the construction, and (4) tests the consistency of the note-in-proof redefinition.

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