Gravity’s Preferred Frame: A Force-Based Route to Relativistic N-Body Problems

For more than a century, physicists have described gravity not as a force but as the curvature of spacetime itself. Yet a pair of mechanical engineers at North Carolina State University argues that the old Newtonian language of forces between bodies can be resurrected in a fully relativistic form, and that doing so may crack one of the hardest computational problems in gravitational physics: the relativistic N-body problem. In a new open-access paper in the journal Celestial Mechanics and Dynamical Astronomy, Larry Silverberg and Jeffrey Eischen extend their previously developed mechanics-based formulation of relativistic gravitation from two bodies to systems of arbitrary size, using a deceptively simple idea they call the gravitational frame.

The difficulty with the standard approach is well known. In general relativity, solving a system of interacting bodies means finding the spacetime metric while simultaneously computing how the bodies move through that metric. The two problems are coupled, and the coupling grows ferociously with each additional body. Beyond the two-body case, explicit solutions become impractical, and researchers typically resort to large-scale numerical simulations of the Einstein field equations, slicing time and continually updating the metric as the system evolves. Silverberg and Eischen take a different path. Rather than reorganizing the geometry of general relativity, as the ADM or teleparallel formulations do, they return to the classical tradition in which gravity is expressed through force relations between bodies, updated to match relativistic predictions.

Their formulation, which they call general mechanics, rests on what they term the parity hypothesis: that a force-based description of gravity can be mathematically faithful to the trajectories predicted by general relativity, at least in the absence of external radiation sources and gravitational waves. The centerpiece is a relativistic universal law of gravitation that modifies the familiar inverse-square law by the factor 1 + 3(h_R/cr)^2, where h_R is the specific relativistic angular momentum of the pair, c is the speed of light, and r is the separation of the bodies. This factor injects a rotational component of energy that becomes significant only when motion is relativistic. In the slow, weak-field limit it vanishes and Newton’s law reappears. When retained, the law reproduces the classic relativistic signatures, including perihelion precession, light deflection, and the photon sphere, in exact agreement with the Schwarzschild two-body solution.

The new contribution is the gravitational frame itself. For each interacting pair of bodies, the authors define the gravitational frame as the reference frame in which the pair’s total relativistic linear momentum vanishes, mathematically the familiar center-of-mass frame but reinterpreted as something far more consequential. In their view, this is not merely a convenient coordinate choice but the frame in which nature actually formulates gravitational interaction. The law of gravity, they argue, must first be evaluated in this pairwise frame, and only then Lorentz-transformed into whatever global frame the problem requires. The distinction is subtle but, they contend, physically essential.

The authors support the hypothesis with an energetic argument. The one-body formulation of the two-body problem, the setup Schwarzschild himself used, is independent of the velocity of the system’s mass center, so it corresponds to a whole family of possible two-body problems differing in their net linear momentum. Comparing the relativistic kinetic energies of the one-body and two-body descriptions, the two are equal only when the pair’s relativistic linear momentum is zero, that is, only in the gravitational frame. Among all admissible two-body solutions, only this one preserves energetic equivalence between the two formulations, which the authors take as strong evidence that nature selects this frame. They also note that the historic tests of general relativity, from Mercury’s perihelion to Eddington’s 1919 eclipse expedition, were implicitly performed in exactly such frames, where a dominant source and a light test body leave the pair with effectively zero net momentum.

To extend the idea to N bodies, the authors apply the gravitational law pairwise, computing each interaction in the gravitational frame of the relevant pair and then transforming the resulting accelerations into a common global frame using Lorentz transformations. The paper also defends the need for a global frame at all, using a matrix argument showing that frames sharing Lorentz transformability to a common global reference yield consistent physics, a point illustrated through the twin paradox. Evolution proceeds with respect to proper time in flat Minkowski spacetime rather than the coordinate-time slicing of curved-spacetime relativity, with radiation and gravitational-wave degrees of freedom set aside so that the conservation laws of mechanics survive.

The real test comes when the source is not a single point mass but a rotating, extended body. In general relativity, a spinning source drags spacetime around with it, the frame-dragging effect described by the Kerr metric, so that a body falling radially inward develops an azimuthal drift. A naive force law applied to the whole source as one object would miss this entirely, predicting purely radial infall. Silverberg and Eischen show that when the source is instead treated as an aggregate of moving constituents, each interacting pairwise in its own gravitational frame, the tangential deflections emerge naturally. No Kerr-like metric term, no extra transverse force, and no separately imposed frame-dragging term is added; the rotational behavior arises from the frame-selection rule itself.

The numerical experiments are striking. In one set, a test body starting at three Schwarzschild radii and moving inward at 0.8c approaches a source with an artificially pinned upward velocity. With a stationary source the body falls straight in; at 0.5c it deflects upward, and at 0.9c the deflection becomes pronounced. In the flagship example, the source is a rigid ring of 800 point sources, with total mass equal to the Sun’s, rotating at 0.95c at a radius of one-tenth the Schwarzschild radius. A test body aimed radially inward is swept sideways by roughly 0.044 Schwarzschild radii as it crosses the ring, deflected in the direction of rotation, exactly the qualitative signature of frame dragging. Near the photon sphere at 1.5 Schwarzschild radii, a light-speed test body orbiting a rotating ring shows increased attraction as the ring spins faster, with small but measurable differences between prograde and retrograde configurations, the retrograde orbit radius slightly larger than the prograde one, consistent in character with Kerr-type behavior.

The authors are careful about scope. They emphasize that their rotating-ring examples are behavioral tests, not quantitative comparisons with the full Kerr solution, which is a three-dimensional, axisymmetric black-hole exterior with an intrinsic spin parameter, horizon conditions, and a fixed multipole structure. In their formulation, rotation is not imposed through a spin parameter at all but constructed from the motion of constituent masses, much as rigid-body motion is built from moving point masses in classical mechanics. A decisive quantitative comparison with Kerr, they acknowledge, would require extending the planar formulation to fully three-dimensional rotating sources and carefully translating Kerr’s spin, horizon, and multipole assumptions into the mechanics framework. They also distinguish their exact two-body law from post-Newtonian approximations, which are systematically accurate only in the weak-field, slow-motion limit and become strained as velocities approach the speed of light.

Still, the implications are tantalizing. If the gravitational-frame construction holds up under deeper scrutiny, relativistic N-body dynamics, from merging compact binaries to accretion disks threading spinning black holes, could one day be attacked with the same force-based, pairwise computational machinery that has served celestial mechanics since Newton, integrated here with a fourth-order Runge-Kutta scheme in proper time. The authors argue that this transition mirrors the shift toward modern N-body computational methods that transformed other fields of mechanics over the past half-century. Whether general mechanics can ultimately match the precision of full numerical relativity remains an open question, but the paper offers a provocative demonstration that frame dragging, strong-field precession, and spin-enhanced attraction can emerge from a force law and a well-chosen frame, without a single line of curved-spacetime geometry.

Subject of Research: A mechanics-based, force-law formulation of relativistic gravitation extended to N-body systems via pairwise gravitational frames

Article Title: The gravitational frame for solving relativistic N-body problems

Article References: Silverberg, L. M., & Eischen, J. W. (2026). The gravitational frame for solving relativistic N-body problems. Celestial Mechanics and Dynamical Astronomy, 138(5), Article 60. https://doi.org/10.1007/s10569-026-10326-x

Image Credits: AI Generated

DOI: 10.1007/s10569-026-10326-x

Keywords: general relativity, N-body problem, gravitational frame, frame dragging, Schwarzschild solution, Kerr metric, celestial mechanics, special relativity, Lorentz transformation, relativistic gravitation, numerical simulation, rotating black holes