Braeck, Grøn and Farup (Universe 3 (2017) 38) introduced the causal mass of an observer — the mass inside the particle horizon integrated along the past light cone — and computed it for the flat ΛCDM model, finding that its Schwarzschild radius is about three times the particle-horizon radius. From this they conjectured (the BC-condition) that a universe exhibits perfect inertial dragging whenever the Schwarzschild radius of its causal mass equals or exceeds the particle-horizon radius, and they left three issues open: the conjecture is not a theorem of general relativity; the actual dragging angular velocity produced by the ΛCDM causal mass-energy-current distribution has never been computed from first principles beyond the Schwarzschild-radius heuristic; and the global role of boundary conditions — replacing an asymptotically empty Minkowski boundary by a mass shell at its own Schwarzschild radius — in defining relative rotation and inertial dragging is unsettled. The conjecture and the two companion questions remain open: Schmid's exact-dragging theorems hold only for linear perturbations of FLRW universes, and the 2023 follow-up by Braeck settles relative rotation only in asymptotically flat spacetimes. A complete answer proves or refutes the causal-mass criterion in full general relativity, or computes the dragging angular velocity of the ΛCDM causal mass distribution from first principles, fixing along the way an operational, coordinate-invariant notion of perfect dragging in a general cosmology.

Background

Inertial dragging (frame dragging) is the effect in general relativity (GR) whereby rotating or accelerated matter partially drags along the local inertial frames — the frames in which free particles move without acceleration and the plane of a Foucault pendulum does not rotate. The weak-field, slow-rotation limit is the Lense–Thirring effect measured around the Earth. Mach's principle is the programmatic idea, going back to Mach and Einstein, that the local inertial frames should be fully determined by the motion of the cosmic matter distribution rather than by an absolute space.

The quantitative touchstone is the 1966 calculation of Brill and Cohen (Phys. Rev. 143 (1966) 1011) for a slowly rotating massive spherical shell of radius RR and Schwarzschild radius rSr_S. The inertial dragging coefficient KIN=Ω/ωK_{\mathrm{IN}}=\Omega/\omega, the ratio of the angular velocity Ω\Omega of an inertial frame at the center to the angular velocity ω\omega of the shell, tends to 1 as R→rSR\to r_S: when the shell's radius equals its own Schwarzschild radius, the interior inertial frames co-rotate exactly with the shell, a situation called perfect inertial dragging. In that limit the interior inertial properties no longer depend on frames at infinity but are fixed by the shell itself — the Machian ideal. Later shell and perturbative studies (Lindblom and Brill on collapsing rotating dust shells, Lewis, Klein on rotational perturbations of expanding dust universes) found the same pattern: perfect dragging is approached precisely when essentially all the matter of the model universe is included.

Schmid (Phys. Rev. D 74 (2006) 044031; Phys. Rev. D 79 (2009) 064007; arXiv:1406.4665) developed this into rigorous theorems using cosmological gravitomagnetism — the splitting of the linearized Einstein equations of a perturbed Friedmann–Lemaitre–Robertson–Walker (FLRW) spacetime into gravitoelectric and gravitomagnetic parts analogous to electromagnetism. He proved that for arbitrary linear perturbations of any FLRW universe (with spatial curvature K=+1,−1,0K=+1,-1,0, any equation of state), the axes of local inertial frames are dragged exactly by a weighted average of the cosmic energy currents on the observer's past light cone. This establishes Mach's dragging postulate in cosmological GR — but only at first order in perturbation theory, and it does not invoke or vindicate any Schwarzschild-radius criterion.

Braeck, Grøn and Farup (Universe 3 (2017) 38) pursued the complementary, causal route. The particle horizon of an observer is the radius beyond which no signal could yet have reached her; because gravitational influence propagates at light speed, they argued that the gravitationally effective, or causal, mass is the mass inside the particle horizon evaluated on the past light cone, where each spherical shell is weighted by the density at the epoch when it emitted the light (or gravitational influence) now arriving. Integrating the flat ΛCDM model (the standard cold-dark-matter-plus-cosmological-constant cosmology) along the past light cone, they found the Schwarzschild radius of this causal mass to be about three times the particle-horizon radius. Motivated by the shell results, they conjectured the BC-condition: there is perfect inertial dragging in a universe if the Schwarzschild radius of the causal mass inside the particle horizon equals or exceeds the horizon radius. Since a Foucault pendulum's swinging plane is observed to follow the stars, they read the fulfilled BC-condition as explaining, via near-perfect dragging, why rotation may be relative — not absolute — in our universe. A contrasting case is the rotating Gödel universe, where dragging is not perfect and rotation is absolute.

The authors were explicit about the limits of the argument. The BC-condition is a conjecture extrapolated from rotating-shell theorems, not a result about realistic cosmologies: no proof (or counterexample) establishes the mathematical relation between the causal-mass criterion and the frame-dragging angular velocity in full, nonlinear GR. The paper infers near-perfect dragging from the Schwarzschild-radius comparison but never computes the dragging angular velocity that the ΛCDM causal mass-energy-current distribution actually produces. And the construction presupposes a boundary-condition stance: in GR the usual asymptotically empty Minkowski boundary silently introduces an absolute standard of rotation, and the authors propose — without implementation — replacing it by a mass shell at its own Schwarzschild radius, so that even the empty Milne universe would acquire causal perfect dragging. How such boundary choices are to be implemented and justified globally remains unsettled; Braeck's 2023 sequel (Universe 9 (2023) 120) showed, for axistationary asymptotically flat spacetimes with a relaxed boundary condition, that Einstein's equations determine only differences of rotation rates of zero-angular-momentum observers, supporting the relativity of rotation locally but leaving the cosmological boundary question open.

Problem Statement

Determine, within full (nonlinear) general relativity, whether and why the cosmic causal mass fixes the local inertial frames — making the Schwarzschild-radius heuristic of Braeck, Grøn and Farup (2017) into a theorem, refuting it, or replacing it with a first-principles computation. Concretely, the problem as left open by the source comprises three components. A complete answer resolves at least one of the first two — (1) the BC-conjecture or (2) the first-principles computation; component (3), the boundary-condition question, is required background rather than a stand-alone deliverable: it must be addressed within whichever of the two routes is taken, but resolving it on its own does not complete the task. (1) The BC-conjecture: prove, or give a counterexample to, the statement that perfect inertial dragging holds in a cosmological spacetime if the Schwarzschild radius of the causal mass — the mass inside the particle horizon integrated along the past light cone — is equal to or greater than the particle-horizon radius; i.e., establish the precise mathematical relation between this causal-mass criterion and the frame-dragging angular velocity in realistic cosmologies, not just in the slow-rotation shell limit from which it was extrapolated. (2) The computation: calculate from first principles, in the full nonlinear GR treatment of the flat ΛCDM spacetime and its causal distribution of mass and energy currents, the actual inertial-frame dragging angular velocity (or dragging coefficient) experienced by a local observer, thereby testing the inference of near-perfect dragging currently drawn only from the Schwarzschild-radius comparison. (3) The boundary conditions: settle the status of the proposed replacement of the asymptotically empty Minkowski boundary by, e.g., a mass shell at its own Schwarzschild radius — how such a boundary condition is implemented, what it implies for the notions of relative rotation and perfect dragging in cosmology (including whether it renders even the Milne universe causally Machian), and what physically justifies it. Answers must fix operational, coordinate-invariant definitions of causal mass, particle horizon, and dragging coefficient in the class of cosmological solutions considered; the single overarching objective is a rigorous GR account of how the rotating cosmic matter distribution determines the local compass of inertia in our universe.

References

  1. S. Braeck, Ø. Grøn, I. Farup, 'The Cosmic Causal Mass', Universe 3 (2017) 38, DOI 10.3390/universe3020038, arXiv:1705.04461, https://arxiv.org/abs/1705.04461
  2. S. Braeck, 'Inertial Frame Dragging and Relative Rotation of ZAMOs in Axistationary Asymptotically Flat Spacetimes', Universe 9 (2023) 120, DOI 10.3390/universe9030120, arXiv:2303.04868, https://arxiv.org/abs/2303.04868
  3. D. R. Brill, J. M. Cohen, 'Rotating Masses and Their Effect on Inertial Frames', Physical Review 143 (1966) 1011, DOI 10.1103/PhysRev.143.1011, https://doi.org/10.1103/PhysRev.143.1011
  4. C. Schmid, 'Cosmological gravitomagnetism and Mach's principle', Physical Review D 74 (2006) 044031, DOI 10.1103/PhysRevD.74.044031, https://doi.org/10.1103/PhysRevD.74.044031
  5. C. Schmid, 'Mach's principle: Exact frame-dragging via gravitomagnetism in perturbed Friedmann-Robertson-Walker universes with K = (±1, 0)', Physical Review D 79 (2009) 064007, DOI 10.1103/PhysRevD.79.064007, https://doi.org/10.1103/PhysRevD.79.064007
  6. C. Schmid, 'Exact Dragging of Inertial Axes by Cosmic Energy-Currents on the Past Light-Cone', arXiv:1406.4665 (2014), https://arxiv.org/abs/1406.4665
  7. Ø. Grøn, T. Jemterud, 'An interesting consequence of the general principle of relativity', European Physical Journal Plus 131 (2016) 91, DOI 10.1140/epjp/i2016-16091-9, https://doi.org/10.1140/epjp/i2016-16091-9
  8. V. Christianto, F. Smarandache, 'Four Possible Ways to Model Rotating Universe', SSRN preprint (2024), DOI 10.2139/ssrn.4694002, https://doi.org/10.2139/ssrn.4694002
  9. Ø. Grøn, 'Cosmology', chapter in 'Introduction to Einstein's Theory of Relativity', Undergraduate Texts in Physics, Springer (2020), pp. 311–391, DOI 10.1007/978-3-030-43862-3_12, https://doi.org/10.1007/978-3-030-43862-3_12