K., Rotating universes in general relativity theory · 1950-mathphys-1
Astronomy and Astrophysics · ICM 1950: K., Rotating universes in general relativity theory
Problem statement
Consider a four-dimensional spacetime satisfying Einstein’s field equations with cosmological constant and pressureless matter, whose stress-energy tensor is , where and is the unit timelike four-velocity of the matter. Assume that the spacetime is spatially homogeneous, that the space of matter worldlines is compact, that is not constant in time (so the universe expands or contracts), and that the matter congruence has everywhere nonzero vorticity vector . At each event, let the quadric of expansion be the quadric in the three-dimensional rest space orthogonal to determined by the symmetric spatial expansion tensor of the matter congruence; in this class of rotating universes it is never rotationally symmetric about , so expansion is necessarily accompanied by anisotropic deformation. Construct a quantitative model of the gravitational formation and subsequent evolution of a matter condensation in such a spacetime, and determine whether this anisotropic expansion generically produces an oblong body rotating about a minor axis whose differential rotation bends it into a spiral with morphology and kinematics consistent with observed spiral galaxies.
Lean 4
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Progress & source assessment
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Unresolved.
The exact problem asks for a quantitative nonlinear model of condensation and spiral formation inside the spatially homogeneous, compact, rotating, expanding dust spacetimes described by Gödel, and for a comparison with observed spiral-galaxy morphology and kinematics. No citable result found through July 31, 2026 supplies that complete calculation or a proof that the mechanism succeeds or fails under all of these hypotheses.
Key references and progress:
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Kurt Gödel, “Rotating Universes in General Relativity Theory,” Proceedings of the International Congress of Mathematicians, Cambridge, Massachusetts, 1950, vol. 1, pp. 183–189 (1950). Gödel proves the relevant background result that, for his spatially homogeneous finite rotating solutions with nonconstant density, the expansion quadric cannot be rotationally symmetric about the vorticity vector. He presents the spiral-galaxy mechanism only as a possibility and explicitly leaves its quantitative comparison with observation to future work. Direct source: https://www.mathunion.org/icm-proceedings.
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Li-Xin Li, “Effect of the Global Rotation of the Universe on the Formation of Galaxies,” General Relativity and Gravitation 30 (1998), 497–507. DOI: 10.1023/A:1018867011142; arXiv:astro-ph/9703082. Li gives the strongest directly relevant partial model located: an approximately Friedmann-like rotating expanding dust model, with shear neglected, spherical primordial fluctuations, and simplified equatorial/polar collapse. It derives a global-rotation contribution to galaxy angular momentum, approximately J proportional to M^(5/3), and argues heuristically that different collapse regimes could resemble elliptical galaxies, spiral galaxies, or walls. This is not a solution of the exact problem: it does not use Gödel’s full compact homogeneous class, does not model the expansion quadric’s general anisotropy, and does not calculate the subsequent differential-rotation winding or quantitatively fit spiral morphology and kinematics.
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Yuri N. Obukhov, “On Physical Foundations and Observational Effects of Cosmic Rotation,” review preprint (2000), arXiv:astro-ph/0008106. The review records the historical cosmic-rotation and galaxy-formation proposals, including primordial turbulence and global rotation, and identifies the construction of a self-consistent dynamical evolution with matter irregularities relevant to galaxy formation as a remaining problem. It surveys related partial programs rather than establishing the requested result.
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Daniela Saadeh, Stephen M. Feeney, Andrew Pontzen, Hiranya V. Peiris, and Jason D. McEwen, “How Isotropic Is the Universe?,” Physical Review Letters 117, 131302 (2016). DOI: 10.1103/PhysRevLett.117.131302; arXiv:1605.07178. Using Planck temperature and polarization data, this paper constrains the present normalized vorticity parameter to (omega/H)_0 < 5.2 x 10^(-11) at 95% confidence in the tested Bianchi models and strongly disfavors anisotropic expansion. This is an observational constraint on the background, not a theorem about nonlinear condensation or a complete answer to the stated problem; it also does not establish that every Gödel-type compact solution is excluded.
Thus the literature contains substantive partial modeling and strong observational constraints, but the exact problem remains unresolved rather than resolved by either a proof, a counterexample, or a complete classification.
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