Completing the Maxwell-like formulation of post-Newtonian gravity: the explicit 1PN energy-conservation law

The equations of the first post-Newtonian (1PN) approximation to general relativity can be written in a Maxwell-like 'gravitoelectromagnetic' form built from a gravitoelectric field and a gravitomagnetic field. Kaplan, Nichols, and Thorne [Phys. Rev. D 80, 124014 (2009)] compacted the Damour-Soffel-Xu 1PN formalism in fully harmonic gauge and extended it to the Landau-Lifshitz gravitational momentum density, momentum flux, and momentum-conservation law in Maxwell-like form, accurate through 1PN order, for perfect fluids and for compact bodies described by masses and spins. The paper explicitly declined to derive the corresponding 1PN energy-conservation law (calling it 'a very complex and delicate task'), wrote only its Newtonian-order version, and noted that the gravitational energy density and flux are gauge-dependent. The resulting task is to complete the Maxwell-like formulation of 1PN gravity by deriving the explicit 1PN matter-plus-field energy conservation law in Maxwell-like form — the single task the source paper explicitly deferred.

Background

In the post-Newtonian (PN) approximation, general relativity is expanded in powers of $1/c^2$ for weak-field, slow-motion systems such as compact binaries (pairs of neutron stars or black holes) during their early inspiral; '1PN order' denotes terms of relative order $1/c^2$ beyond Newtonian gravity, and 2PN the next order. Because the resulting equations are easier to interpret when they resemble electromagnetism, a long tradition rewrites them in 'gravitoelectromagnetic' (GEM) or Maxwell-like form: a gravitoelectric field $\boldsymbol{g}$ (which reduces to the Newtonian gravitational acceleration) and a gravitomagnetic field $\boldsymbol{H}$ (which encodes frame-dragging, i.e. the dragging of inertial frames by moving or spinning matter, together with spin effects) obey Maxwell-like divergence and curl equations, and the geodesic equation resembles the Lorentz force. Damour, Soffel, and Xu (DSX), in their relativistic celestial-mechanics program, showed that the full 1PN theory, including its nonlinear field contributions, can be put in this form using a scalar potential $w$ and a vector potential $w_i$; here 'gauge' means the coordinate choice within the PN metric, and quantities built from the potentials (such as energy densities) generally change under a change of gauge. Imposing the harmonic-gauge condition $w_{,t}+w_{i,i}=0$ makes the 1PN field equations take a particularly simple Maxwell-like form.

General relativity has no tensorial local energy-momentum for the gravitational field itself: any local 'field energy density' is a pseudotensor that depends on the coordinates. The standard construction of Landau and Lifshitz defines a complex $t_{\mathrm{LL}}^{\alpha\beta}$ such that matter plus field satisfies the exactly conserved law $[(-g)(T^{\alpha\beta}+t_{\mathrm{LL}}^{\alpha\beta})]_{,\beta}=0$, where $g$ is the metric determinant and $T^{\alpha\beta}$ the matter stress-energy tensor. Its spatial components give momentum density and momentum flux (stress); its time components give energy density and energy flux. Chandrasekhar (1969) evaluated the time-time and time-space components through 1PN order in conventional PN variables, yielding the standard-form 1PN energy conservation for a self-gravitating perfect fluid, and Chandrasekhar and Nutku (1969) carried the PN hydrodynamics to second order.

Kaplan, Nichols, and Thorne (2009) summarized the DSX Maxwell-like 1PN formalism in a compact, practitioner-friendly harmonic-gauge presentation and then extended it to momentum. To 1PN accuracy, the gravitational momentum density is $(-g)t_{\mathrm{LL}}^{0j}=-(1/4\pi)\,\boldsymbol{g}\times\boldsymbol{H}+(3/4\pi)\dot{U}_N\boldsymbol{g}$ (up to 2PN remainders), where $U_N$ is the Newtonian potential — identical to the electromagnetic momentum density $(1/4\pi)\boldsymbol{E}\times\boldsymbol{B}$ except for an overall sign and the $\dot U_N$ term — and the momentum flux likewise takes the electromagnetic form of field-line tensions $|\boldsymbol{g}|^2/8\pi$ and $|\boldsymbol{H}|^2/8\pi$ with transverse pressures of the same magnitude, plus $\dot U_N$ terms. They cast the 1PN momentum conservation into a Lorentz-force-like Maxwell analogy and specialized the vacuum potentials to compact bodies retaining only masses and spins. For energy they wrote only the Newtonian-order Maxwell-like conservation law (their Eqs. (5.1)-(5.2)), declining to expand the 1PN law $[(-g)(T^{0\mu}+t_{\mathrm{LL}}^{0\mu})]_{,\mu}=0$ because it is 'a very complex and delicate task', and noting that even the Newtonian energy density and flux are gauge-dependent — indeed the passage between their two Newtonian versions adds a divergence-free piece to the flux, which strictly takes it out of harmonic gauge.

The same Caltech/Cornell research program immediately applied these momentum tools: Keppel, Nichols, Chen, and Thorne analyzed momentum flow in inspiraling spinning binaries at 1.5PN order (including spin-orbit effects and radiation reaction, the dissipative back-reaction by which the binary loses orbital energy and angular momentum to emitted gravitational waves, as well as a discussion of the gauge dependence of momentum localization), and Lovelace et al. computed Landau-Lifshitz energy and momentum densities in full numerical-relativity simulations of head-on mergers, comparing two gauge conditions. The program then pivoted to a different, fully nonlinear visualization formalism based on the electric and magnetic parts of the Weyl curvature tensor (the trace-free part of the Riemann curvature tensor, which carries the free gravitational field; its eigenvector fields are the tidal 'tendex' and frame-dragging 'vortex' lines). In parallel, Costa and Natário and coauthors systematically classified the gravitoelectromagnetic analogies and their exact, gauge-aware content.

Problem Statement

Complete the Maxwell-like (gravitoelectromagnetic) formulation of 1PN gravity left unfinished by Kaplan, Nichols, and Thorne. The scientific objective is the missing energy-side counterpart of the paper's Maxwell-like momentum formalism: the explicit 1PN matter-plus-field energy conservation law in Maxwell-like form, in the conventions of the source paper (fully harmonic gauge, DSX potentials, gravitoelectric field $\boldsymbol{g}$, gravitomagnetic field $\boldsymbol{H}$, Newtonian potential $U_N$).

Expand the exact local conservation law $[(-g)(T^{\alpha\beta}+t_{\mathrm{LL}}^{\alpha\beta})]_{,\beta}=0$ through 1PN order for the $\alpha=0$ components, for a perfect fluid (and, where possible, for a system of compact bodies), and cast the resulting energy density and energy flux entirely in Maxwell-like language — a gravitational analogue of electromagnetic energy conservation (Poynting's theorem) expressed through $\boldsymbol{g}$, $\boldsymbol{H}$, $U_N$ and their time derivatives. The result must reduce, at Newtonian order, to the conservation law written in the source paper [its Eqs. (5.1)-(5.2)], and must be demonstrably equivalent, after translation of variables, to the standard-form 1PN energy conservation of Chandrasekhar. Any freedom in the localization (for example divergence-free additions to the energy flux, exhibited even at Newtonian order in the source) must be exhibited and fixed by an explicit convention.

The source paper's passing remarks on the gauge dependence of the gravitational energy density and flux, and its deliberate, numerically justified truncation of compact bodies to masses and spins, point to natural further directions (a gauge-translation structure for the Maxwell-like quantities, and higher-PN or higher-multipole extensions); these are background context for the problem, not part of the completion boundary. The single task the source explicitly deferred is the derivation of the 1PN energy-conservation law above.

Application context (secondary): the completed formalism is meant to be exercised as an intuition-building and diagnostic tool for compact-binary dynamics, including quantitative comparison of the near-zone energy and momentum flow with numerical-relativity data, continuing the momentum-flow program; this application already partially exists for momentum and is not by itself the completion boundary.

References

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