The gravitational two-body problem in general relativity after Bach and Weyl (1922): exact dynamics of two attracting masses and physically acceptable static equilibria

Bach and Weyl (1922) constructed the first explicit static, axially symmetric two-body solutions of Einstein's vacuum equations and showed that static equilibrium of two masses requires a supporting stress, realized as a singularity on the axis between the bodies; the physical nature of that stress, the interior of the paper's infinitely thin ring source, and above all the true dynamical two-body problem — the motion of two attracting gravitational masses — were left open. A century of later work has settled much of the static mechanics: the support is a conical singularity (a strut) with a definite tension and consistent thermodynamics; strut-free static equilibrium of two neutral black holes is not known in asymptotically flat four-dimensional vacuum general relativity, with balance achievable only via extremal charge, external or non-asymptotically-flat backgrounds, exotic (negative-mass or double-sheeted) sources, or scalar hair; and the Bach–Weyl ring's physical interpretation remains unsettled. The dynamical core, however, stands: no exact solution of Einstein's equations describing two gravitating masses in mutual motion is known, and binary dynamics is determined only through post-Newtonian, effective-one-body, self-force, and numerical approximations. The problem asks for a physically complete exact treatment of the two-body problem in general relativity: an exact solution for two attracting masses in motion, and physically acceptable, energy-condition-obeying source models for the static configurations.

Background

Soon after general relativity was formulated, Hermann Weyl initiated the systematic study of static, axially symmetric solutions of Einstein's vacuum equations (Weyl 1917, 1919). In canonical cylindrical coordinates $(t,\rho,z,\varphi)$ adapted to the static and axial symmetries, the metric depends on two functions $\psi(\rho,z)$ and $\gamma(\rho,z)$, and the vacuum field equations reduce to a single linear equation for $\psi$ — the same axisymmetric Laplace equation as in Newtonian gravity — while $\gamma$ then follows by quadrature. Every axisymmetric Newtonian potential therefore seeds a relativistic candidate, but the correspondence is treacherous: the Schwarzschild spacetime corresponds not to a point mass but to a finite rod on the symmetry axis, and the geometry remembers the rod through $\gamma$. In 1922 Rudolf Bach (a pseudonym of the engineer Rudolf Förster, who was barred by his employer from publishing under his own name) and Weyl pushed this framework to multi-body constructions: two particles or two rods on the axis — the latter giving the double-Schwarzschild (Bach–Weyl) solution — and an infinitely thin circular ring. A necessary regularity condition on the axis, elementary flatness (the metric function $\gamma$ must vanish as $\rho\to 0$ on matter-free parts of the axis, so that small circles around the axis have their Euclidean circumference), fails on the segment between the two bodies: there $\gamma$ tends to a nonzero constant and the spacetime contains a conical singularity — a line defect where the ratio of circumference to radius differs from $2\pi$ — commonly called a strut, which holds the bodies apart. Weyl's supplement interpreted this support through stress components of the energy-momentum tensor living in a nominally matter-free region — the framework had only ever assumed the condition $T^{1}{}_{1}+T^{2}{}_{2}=0$ on the principal stresses, leaving the individual stresses and the energy density undetermined — and defined the attractive force between the bodies as proportional to that constant. The ring's interior was left unmodeled, and everything obtained was static, axial, and vacuum.

Later work splits into an equilibrium half and a dynamics half. On equilibrium, the strut was identified as a conical singularity with a definite tension (a discussion going back to Einstein and Rosen's 1936 attack on the relativistic two-body problem) and the construction was generalized to $N$ collinear black holes by Israel and Khan (1964); Herdeiro, Kleihaus, Kunz and Radu (2010) showed that the thermodynamics of such strut-balanced black holes is consistent once the strut energy is included, with the thermodynamic mass differing from the ADM mass (the total mass read off from the falloff of the metric at spatial infinity). Balance without a strut is possible for extremally charged black holes (the Majumdar–Papapetrou class, where electrostatic repulsion cancels gravitational attraction) but not, as far as is known, for two neutral black holes in four-dimensional vacuum general relativity in an asymptotically flat spacetime (a spacetime whose metric approaches the flat Minkowski metric at spatial infinity; Herdeiro and Radu 2023 state that no such solution is known); spin does not help, since the repulsive spin-spin interaction of the double-Kerr solution (an exact stationary superposition of two rotating Kerr black holes) cannot balance two horizon-covered objects; and the known workarounds — external fields that destroy asymptotic flatness, negative-mass or double-sheeted ring sources (a supporting layer of negative mass density, or a source living on two identical sheets of space glued along the ring; Semerák et al. 2019), phantom scalar fields (scalar fields of negative kinetic energy, which violate the standard energy conditions; Egorov et al. 2016), or scalar hair (a nontrivial scalar field carried by black holes) with a non-positive potential (Herdeiro and Radu 2023) — each abandon some part of the original physical requirement of neutral sources satisfying the standard energy conditions (inequalities on the stress-energy tensor, such as non-negative energy density as measured by any observer). On dynamics, nothing exact exists: no solution of Einstein's equations is known that describes two gravitating masses moving under their mutual attraction in an asymptotically flat spacetime. The only explicit radiative solutions with two moving sources are the boost-rotation symmetric spacetimes (uniformly accelerated pairs), which again require conical strings or negative masses, as surveyed by Bičák (2000). The motion of real binaries — the central computation of gravitational-wave astronomy — is obtained entirely through approximation: the post-Newtonian expansion in powers of $v/c$ and $Gm/(rc^{2})$ (Blanchet 2014), the effective one-body formalism that maps the conservative two-body dynamics onto geodesic motion in a deformed Schwarzschild metric (Buonanno and Damour 1999), the gravitational self-force (treating one body as a small perturbation of the other's field), and numerical relativity, which since Pretorius's 2005 breakthrough evolves the full Einstein equations for merging binaries on supercomputers. The exact solution that Weyl's generation would have recognized as solving the two-body problem has never been found; even the static Bach–Weyl two-center field, the relativistic analogue of the Newtonian Euler two-center problem (two fixed attracting centers, integrable thanks to an extra conserved quantity), admits no hidden constant of motion (Mirshekari and Will 2010). The physical interpretation of the original ring source — whose vicinity is directionally singular in a way that still resists interpretation (Semerák 2016) — is likewise unfinished.

Problem Statement

The question is the gravitational two-body problem in general relativity as left open by Bach and Weyl (1922), with one overarching objective — a physically complete exact solution for two attracting masses — and two linked parts of the question whose relation is stated below.

(1) The dynamical two-body problem (the 'true two-body problem' of the source). Determine the motion of two attracting gravitational masses and their exact gravitational field. Determine whether a closed-form exact solution of Einstein's field equations exists in a four-dimensional asymptotically flat spacetime (the metric approaches the flat Minkowski metric at spatial infinity), describing two gravitating bodies (black holes or material sources) in mutual dynamical interaction — bound orbital motion with inspiral driven by gravitational-radiation damping (the slow orbital shrinkage as orbital energy is carried away by gravitational waves), or unbound scattering — where 'closed form' means that the metric functions, the worldlines of both bodies, and the asymptotic news (the emitted radiation field at future null infinity) are given by explicit expressions built finitely from a fixed list of functions — the elementary functions (algebraic, exponential, logarithmic, trigonometric and hyperbolic, together with their inverses) and the elliptic, hypergeometric, Bessel and Painlevé special functions; a quantity does not count as closed-form if it is left as an unevaluated integral or quadrature, defined only implicitly, given as a convergent series with unsummed coefficients, or produced by a perturbative truncation or a numerical evolution; if such a closed-form exact solution exists, construct it, and otherwise establish the obstruction — that is, establish that no exact solution admits the required closed form; existence in the ordinary PDE sense (for example the maximal development of prescribed initial data) is not the required closed form. The two regimes are alternatives: a closed-form exact solution of either bound inspiral or unbound scattering settles the dynamical question, and correspondingly the obstruction branch must rule out a closed-form solution in both regimes, since a single closed-form solution of either regime would already settle it. No symmetry is assumed — neither axial symmetry nor time independence (stationarity) — since the source's results were static and its force expression does not generalize.

(2) The equilibrium and source half (the 'exact nature of the stresses'). For the known static axisymmetric two-body solutions, determine which physically acceptable matter can hold two neutral masses in static equilibrium, by a general characterization of the admissible matter or by explicit stress-energy distributions or interior solutions matched to the Weyl-class exteriors (the static, axially symmetric vacuum metrics of Weyl's form, which include the Bach–Weyl two-body solutions), satisfying the standard energy conditions (inequalities on the stress-energy tensor, such as non-negative energy density as measured by any observer), with the symmetry axis regular (no conical strut). If no such physically acceptable matter exists, establish that no-go.

Relationship: (2) is the static limit of the same physical question as (1) — what, physically, does it take for two masses to coexist or move in general relativity without unphysical supports. A solution of (1) with regular sources already exhibits the static configurations of (2); conversely, the balance mechanisms examined in the static two-body literature — charge, scalar fields, material rings — are the matter models available under (2)'s requirements that the two masses stay neutral and the matter satisfy the standard energy conditions, and they delineate the matter models that can appear within (1).

References

1. R. Bach and H. Weyl (1922). 'Neue Lösungen der Einsteinschen Gravitationsgleichungen. B. Explizite Aufstellung statischer axialsymmetrischer Felder. Mit einem Zusatz über das statische Zweikörperproblem von H. Weyl.' Mathematische Zeitschrift 13, 134–145. DOI: 10.1007/BF01485284. URL: https://doi.org/10.1007/BF01485284
2. R. Bach and H. Weyl (2012). 'Republication of: New solutions to Einstein's equations of gravitation. B. Explicit determination of static, axially symmetric fields. By Rudolf Bach. With a supplement on the static two-body problem. By H. Weyl.' General Relativity and Gravitation 44, 817–832. DOI: 10.1007/s10714-011-1312-5. URL: https://doi.org/10.1007/s10714-011-1312-5
3. H. Weyl (1917; English translation 2012). 'Republication of: 3. On the theory of gravitation. By Hermann Weyl.' General Relativity and Gravitation 44, 779–810. DOI: 10.1007/s10714-011-1310-7. URL: https://doi.org/10.1007/s10714-011-1310-7
4. H. Weyl (1919; English translation 2012). 'Republication of: 5. Comment on the axially symmetric solutions to Einstein's equations of gravitation. By Hermann Weyl.' General Relativity and Gravitation 44, 811–815. DOI: 10.1007/s10714-011-1311-6. URL: https://doi.org/10.1007/s10714-011-1311-6
5. G. Neugebauer, D. Petroff and B. Mashhoon (2012). 'Editorial notes to: 3. On the theory of gravitation. By Hermann Weyl and to: 5. Comment on the axially symmetric solutions to Einstein's equations of gravitation. By Hermann Weyl and to: New solutions to Einstein's equations of gravitation. B. Explicit determination of static, axially symmetric fields. By Rudolf Bach. With a supplement on the static two-body problem. By H. Weyl.' General Relativity and Gravitation 44, 771–777. DOI: 10.1007/s10714-011-1309-0. URL: https://doi.org/10.1007/s10714-011-1309-0
6. A. Einstein and N. Rosen (1936). 'Two-Body Problem in General Relativity Theory.' Physical Review 49, 404–405. DOI: 10.1103/PhysRev.49.404.2. URL: https://doi.org/10.1103/PhysRev.49.404.2
7. W. Israel and K. A. Khan (1964). 'Collinear particles and Bondi dipoles in general relativity.' Il Nuovo Cimento 33, 331–344. DOI: 10.1007/BF02750196. URL: https://doi.org/10.1007/BF02750196
8. J. Bičák (2000). 'Selected solutions of Einstein's field equations: their role in general relativity and astrophysics.' In B. G. Schmidt (ed.), Einstein's Field Equations and Their Physical Implications, Lecture Notes in Physics 540, 1–126. DOI: 10.1007/3-540-46580-4_1. arXiv:gr-qc/0004016. URL: https://arxiv.org/abs/gr-qc/0004016
9. C. Herdeiro, B. Kleihaus, J. Kunz and E. Radu (2010). 'On the Bekenstein-Hawking area law for black objects with conical singularities.' Physical Review D 81, 064013. DOI: 10.1103/PhysRevD.81.064013. arXiv:0912.3386. URL: https://arxiv.org/abs/0912.3386
10. A. Buonanno and T. Damour (1999). 'Effective one-body approach to general relativistic two-body dynamics.' Physical Review D 59, 084006. DOI: 10.1103/PhysRevD.59.084006. arXiv:gr-qc/9811091. URL: https://arxiv.org/abs/gr-qc/9811091
11. L. Blanchet (2014). 'Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries.' Living Reviews in Relativity 17, 2. DOI: 10.12942/lrr-2014-2. URL: https://doi.org/10.12942/lrr-2014-2
12. F. Pretorius (2005). 'Evolution of Binary Black Hole Spacetimes.' Physical Review Letters 95, 121101. DOI: 10.1103/PhysRevLett.95.121101. arXiv:gr-qc/0507014. URL: https://arxiv.org/abs/gr-qc/0507014
13. O. Semerák (2016). 'Static axisymmetric rings in general relativity: How diverse they are.' Physical Review D 94, 104021. DOI: 10.1103/PhysRevD.94.104021. arXiv:1611.03299. URL: https://arxiv.org/abs/1611.03299
14. M. Basovník and O. Semerák (2016). 'On geometry of deformed black holes: II. Schwarzschild hole surrounded by a Bach-Weyl ring.' Physical Review D 94, 044007. DOI: 10.1103/PhysRevD.94.044007. arXiv:1608.05961. URL: https://arxiv.org/abs/1608.05961
15. D. Lynden-Bell and J. Katz (2012). 'Toroidal metrics: gravitational solenoids and static shells.' Classical and Quantum Gravity 29, 115010. DOI: 10.1088/0264-9381/29/11/115010. arXiv:1205.1616. URL: https://arxiv.org/abs/1205.1616
16. O. Semerák, M. Basovník and P. Kotlařík (2019). 'Schwarzschild binary supported by an Appell ring.' Physical Review D 99, 064050. DOI: 10.1103/PhysRevD.99.064050. arXiv:1905.07638. URL: https://arxiv.org/abs/1905.07638
17. C. A. R. Herdeiro and E. Radu (2023). 'Two Schwarzschild-like black holes balanced by their scalar hair.' Physical Review D 107, 064044. DOI: 10.1103/PhysRevD.107.064044. arXiv:2302.00016. URL: https://arxiv.org/abs/2302.00016
18. M. Astorino and M. Torresan (2025). 'Rotating and swirling binary black hole system balanced by its gravitational spin-spin interaction.' Physical Review D 112, 064034. DOI: 10.1103/3954-766n. arXiv:2502.08706. URL: https://arxiv.org/abs/2502.08706
19. A. I. Egorov, P. E. Kashargin and S. V. Sushkov (2016). 'Scalar multi-wormholes.' Classical and Quantum Gravity 33, 175011. DOI: 10.1088/0264-9381/33/17/175011. arXiv:1603.09552. URL: https://arxiv.org/abs/1603.09552
20. S. Mirshekari and C. M. Will (2010). 'Carter-like constants of motion in the Newtonian and relativistic two-center problems.' Classical and Quantum Gravity 27, 235021. DOI: 10.1088/0264-9381/27/23/235021. arXiv:1009.2839. URL: https://arxiv.org/abs/1009.2839
21. L. Blanchet (2024). 'Post-Newtonian theory for gravitational waves.' Living Reviews in Relativity 27, 4. DOI: 10.1007/s41114-024-00050-z. arXiv:1310.1528. URL: https://arxiv.org/abs/1310.1528