Dynamical foundations of the Machian integral formulation of Einstein's equations and the selection of 3+1 spacetime dimensions

Einstein's equations admit an integral, Machian reformulation in which the metric is expressed as a retarded integral over the matter stress-energy tensor, with a kernel given by the Green function of the 'Mach operator' — the second functional variation of the Einstein-Hilbert action with respect to the contravariant metric components. Boris Altshuler showed in 2012 that demanding this integral representation be ghost-free acts as a kinematic selection rule on the dimensionality of highly symmetric backgrounds: on de Sitter and Einstein-static backgrounds the Mach operator carries tensor ghost modes in every dimensionality from five spacetime dimensions upward, while the four-dimensional backgrounds are ghost-free, and a ghost-free condition also constrains Freund-Rubin compactifications. The paper itself emphasizes that these rules are 'just a mathematical observation': the integral formulation is written down by hand without dynamical grounds, Wheeler's question of whether gravitational-wave energy-momentum belongs in the source is unanswered, and no mechanism connects the rules to a dynamical cosmology in which only three spatial dimensions expand. The open problem is to supply those dynamical foundations — or prove them impossible — and thereby convert (or definitively fail to convert) the ghost-avoidance rule into a genuine explanation of why spacetime is (3+1)-dimensional.

Background

Mach's principle, in the formulation Einstein gave it, asserts that spacetime geometry is entirely created by matter: there should be no 'empty' spacetime, and local inertia should be fixed by the global matter distribution. A concrete realization, proposed by Altshuler in 1966 and independently pursued by Lynden-Bell and by Sciama, Waylen and Gilman, is an integral form of Einstein's equations, written for an $N$-dimensional spacetime as $g_{AB}(x)=\kappa N\int G^{ret}_{AB}{}^{PQ}(x,y|g)T_{PQ}(y)\sqrt{-g}\,d^{N}y$. The kernel is a retarded bi-tensor Green function — a two-point kernel, carrying indices at both $x$ and $y$, that inverts a differential operator and vanishes unless $y$ lies in the causal past of $x$ — of a covariant differential operator $E_{AB}{}^{CD}$ (the 'Mach operator'), defined as the second variation of the Einstein-Hilbert action $\int R\sqrt{-g}$ with respect to the contravariant metric components; acting with $E$ on the integral relation reproduces the Einstein equations, while the integral form itself excludes empty and asymptotically flat solutions, making the theory Machian. Writing down this representation 'by hand' raises the question Wheeler put to Altshuler in 1968: why is the energy-momentum of gravitational waves themselves absent from the source on the right-hand side?

Altshuler's 2012 paper analyzes when such an integral representation can be self-consistent. If the homogeneous equation $E_{AB}{}^{CD}u_{CD}=0$ admits 'ghost' modes — transverse-traceless (TT) tensor perturbations (divergence-free, trace-free metric perturbations) with negative eigenvalue, i.e. normalizable negative-energy-squared solutions — the retarded boundary condition underpinning the Green-function representation fails and the integral form is invalidated. On the Einstein-static background — Einstein's 1917 static closed universe, the product $T\times S^{n}$ of a time line and an $n$-sphere of radius $r_{0}$, with $N=1+n$ spacetime dimensions — the TT tensor spectrum gives $\mu^{2}r_{0}^{2}=l(l+n-1)-2-n(n-3)$ for tensor spherical harmonic number $l$; tensor modes exist only for $n\geq 3$ (there are none on the 2-sphere), and at the lowest mode $l=1$ the expression equals $4n-2-n^{2}$, positive only for $n=3$: only the (3+1)-dimensional Einstein universe survives. (The Mach operator differs from the standard Lichnerowicz operator — the operator governing metric perturbations, whose first four terms it shares — by curvature terms that produce exactly this ghost shift; the underlying TT spectral machinery follows Gibbons and Hartnoll's study of gravitational instability in higher dimensions.) On de Sitter backgrounds — the maximally symmetric solutions with positive cosmological constant, written in static coordinates with an $S^{n}$ angular factor so that $N=n+2$ — the normalizability condition $\sqrt{5n^{2}+6n+1}>2l+n+1$ for a negative-energy-squared mode is satisfied at the lowest tensor momentum numbers on every background of five and more spacetime dimensions, while four-dimensional de Sitter space carries no tensor ghosts (its $S^{2}$ supports no tensor modes): again $N=4$ is singled out; on the Freund-Rubin background $AdS_{m}\times S^{n}$ (a product of anti-de Sitter and spherical spaces stabilized by flux) the Breitenlohner-Freedman bound — the lower bound on effective mass below which modes of an anti-de Sitter space become unstable — yields the ghost-free condition $mn+9\geq 5(m+n)$.

The selection of $N=4$ is thus established only as a kinematic fact about operator spectra on fixed, highly symmetric backgrounds. The analysis is restricted to gauge-invariant TT tensor modes (vector modes only partially, scalar gauge-invariant modes not at all), the integral formulation is postulated rather than derived, and independent dynamical proposals for why three spatial dimensions expanded — string gas cosmology (Brandenberger and Vafa; winding modes of strings, which can annihilate only in at most three spatial dimensions, block the growth of extra dimensions), brane relaxation (Karch and Randall; a brane-world scenario in which an initially higher-dimensional universe dynamically relaxes to three large spatial dimensions), and Lorentzian matrix-model simulations (Kim, Nishimura and Tsuchiya; real-time simulations of a matrix regularization of string theory in which a (3+1)-dimensional expanding universe emerges) — neither use nor derive the Machian criterion. The unresolved target is the gap between the spectral selection rule and dynamics.

Problem Statement

The problem, as posed by Altshuler's 2012 paper and left standing by the literature since, is to convert the kinematic ghost-avoidance selection rule of the Machian integral formulation into a dynamical statement about spacetime dimensionality — or to show rigorously that no such conversion is possible. It has two jointly linked components serving this single objective, and both must be settled to resolve it — establishing (1) alone does not complete the task:

(1) Dynamical foundations. Derive the integral representation of Einstein's equations — the metric expressed as a retarded integral over matter through the Green function of the second-variation (Mach) operator — from a specified underlying dynamics (an action principle or more fundamental theory), instead of postulating it by hand. The derivation must in particular settle Wheeler's question: what belongs in the source on the right-hand side, whether the energy-momentum of gravitational waves must be included, and whether that inclusion changes the definition or spectrum of the Mach operator and hence the ghost criteria.

(2) The dimension-selection mechanism. Using those foundations, establish — or exclude — a dynamical cosmological mechanism whereby the ghost-free condition selects a spacetime with three expanding spatial dimensions: for instance, an early-universe epoch in which higher-dimensional Machian backgrounds are invalidated by Mach ghosts while a (3+1)-dimensional branch survives, with the Freund-Rubin dimensionalities evaluated in the same framework rather than remaining 'numerology'.

A negative result is admissible, equally answers the problem, and need not also settle (2): a proof that no consistent dynamical grounds for the integral formulation exist, or a demonstration that the $N=4$ selection is an artifact (for example, of the restriction to transverse-traceless tensor modes on highly symmetric backgrounds, or curable by standard higher-derivative or extra-field modifications that shift the Mach operator's spectrum) cleanly severs the rule from dynamics and retires the Machian route to explaining 3+1 dimensionality.

References

1. Boris L. Altshuler, 'Mach's principle selects four space-time dimensions', Physical Review D 86, 044006 (2012). DOI: 10.1103/PhysRevD.86.044006, https://doi.org/10.1103/PhysRevD.86.044006; preprint (titled 'Mach's Principle selects 4 space-time dimensions'): arXiv:1205.0279, https://arxiv.org/abs/1205.0279
2. B. L. Altshuler, 'Integral form of Einstein equations and covariant formulation of Mach's principle', Zhurnal Eksperimental'noi i Teoreticheskoi Fiziki 51, 1143 (1966); Soviet Physics JETP 24, 766 (1967). No DOI exists; https://arxiv.org/abs/1205.0279
3. D. Lynden-Bell, 'On the Origins of Space-Time and Inertia', Monthly Notices of the Royal Astronomical Society 135, 413-428 (1967). DOI: 10.1093/mnras/135.4.413, https://doi.org/10.1093/mnras/135.4.413
4. D. W. Sciama, P. C. Waylen, R. C. Gilman, 'Generally Covariant Integral Formulation of Einstein's Field Equations', Physical Review 187, 1762-1766 (1969). DOI: 10.1103/PhysRev.187.1762, https://doi.org/10.1103/PhysRev.187.1762
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8. S.-W. Kim, J. Nishimura, A. Tsuchiya, 'Expanding (3+1)-Dimensional Universe from a Lorentzian Matrix Model for Superstring Theory in (9+1) Dimensions', Physical Review Letters 108, 011601 (2012). DOI: 10.1103/PhysRevLett.108.011601; arXiv:1108.1540, https://arxiv.org/abs/1108.1540
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10. P. Ehrenfest, 'Welche Rolle spielt die Dreidimensionalität des Raumes in den Grundgesetzen der Physik?', Annalen der Physik 366, 440-446 (1920). DOI: 10.1002/andp.19203660503, https://doi.org/10.1002/andp.19203660503
11. Eva Deli, 'Evaluation of Mach's Principle in a Universe with Four Spatial Dimensions', viXra:1704.0336 (2017), https://vixra.org/abs/1704.0336 (unrefereed preprint)