Non-Tilted BKS Solutions in Topo-GR

• Non-tilted BKS solutions are cosmological models on closed three-manifolds where a background connection encodes topology in the modified field equations.
• They exhibit unique isotropization and static vacuum properties, allowing FLRW-type dynamics under shear-free conditions and a positive cosmological constant.
• The methodology uses a modified ADM framework with Milnor frames to derive global constraints that prevent recollapse, contrasting with standard GR behaviors.

Non-tilted BKS (Bianchi–Kantowski–Sachs) solutions refer to cosmological models exhibiting spatial homogeneity and admitting a foliation by closed three-manifolds with Thurston geometry, within the framework of a gravitational theory—referred to as topo-GR—that explicitly encodes topology via a background connection. In contrast to the standard General Relativity (GR) context, these solutions feature a fluid flow orthogonal to the homogeneous hypersurfaces (“non-tilted”), and the field equations depend not only on the dynamical metric but also on a reference Ricci tensor determined by the underlying topology. The resulting set of solutions exhibits unique existence, isotropization, and static vacuum properties, which differ crucially from their GR analogues (Vigneron et al., 8 Dec 2025).

1. Metric Ansatz for Non-Tilted BKS Spacetimes

Non-tilted BKS solutions are defined on spacetimes of the form $M\cong\mathbb{R}_t\times\Sigma$, where $\Sigma$ is a closed three-manifold admitting one of the eight Thurston geometries. The metric in Arnowitt–Deser–Misner (ADM) coordinates is

$$\mathrm{d}s^2 = -N^2(t)\,\mathrm{d}t^2 + h_{ij}(t)\,\omega^i \otimes \omega^j,$$

where $\{\omega^i\}$ is a set of left-invariant one-forms on the symmetry group (Bianchi or Kantowski–Sachs), and $h_{ij}(t)$ is diagonal in a Milnor frame. In an orthonormal frame adapted to the foliation, one can write

$$g = -e^0 \otimes e^0 + \delta_{ij}\,e^i\otimes e^j, \quad e^0 = N\,\mathrm{d}t, \quad e^i = e^i{}_k(t)\,\omega^k.$$

This formalism accommodates all topologically distinct, spatially homogeneous cosmologies classified by BKS.

2. Field Equations in the Thurston-Based Theory of Gravity (topo-GR)

Topologically-augmented gravity (topo-GR) augments the Einstein–Hilbert action with a background connection $\bar\nabla$, fixed by topology, whose Ricci tensor $\bar R_{\mu\nu}$ appears in the field equations: $$R_{\mu\nu} - \bar R_{\mu\nu} = \kappa\left( T_{\mu\nu} - \tfrac{1}{2} T g_{\mu\nu} \right) + \Lambda\,g_{\mu\nu},\quad \bar\nabla\text{ fixed}.$$ For perfect fluids, the relevant $3+1$ split yields a modified Hamiltonian constraint,

$$\tfrac{2}{3}\theta^2 - \left(\sigma_{ij}\sigma^{ij} + \tfrac{1}{N^2} \bar R_{ij} h^{ij}\right) + R - \bar R = 2\kappa\rho + 2\Lambda,$$

and modified momentum/evolution equations, with additional terms proportional to $\bar R_{ij}$ and possible “reference tilt” contributions of the form $\bar q_i \propto \bar R_{ij} n^j$ (Vigneron et al., 8 Dec 2025). This structure distinguishes topo-GR from GR, especially in the treatment of the anisotropic and topological source terms.

3. Shear-Free Perfect Fluid Solutions and Friedmann Dynamics

Non-tilted, shear-free solutions are defined by

$\sigma_{ij}=0, \quad q_i=0, \quad \pi_{ij}=0.$

Under these conditions, the anisotropic piece of $\bar R_{ij}$ is balanced by the reference fluid stresses, and the metric assumes the form

$$\mathrm{d}s^2 = -\mathrm{d}t^2 + a^2(t)\,\bar h_{ij}\,\omega^i\omega^j,$$

where $\bar h_{ij}$ is a maximal, time-independent Thurston metric on $\Sigma$. The dynamical evolution reduces universally to a flat FLRW-type Friedmann system: $$3H^2 = \kappa\rho, \qquad \dot H = -\tfrac{1}{2}(3w+1)H^2,$$ with $p=w\rho$. This result, in contrast with GR, holds for all Thurston topologies, not just for maximally-symmetric cases (Vigneron et al., 8 Dec 2025).

4. Static Vacuum Solutions in Thurston Topologies

Imposing $\rho = p = 0$ in the shear-free ansatz yields static, vacuum solutions of the form

$$\mathrm{d}s^2 = -\mathrm{d}t^2 + \bar h_{ij}\,\omega^i\omega^j,$$

for every Thurston geometry, where $\bar h_{ij}$ is the maximal metric associated to the geometry of $\Sigma$. This existence result contrasts with GR, where static LSH (locally spatially homogeneous) vacua are restricted to $E^3$ and $S^1\times S^2$ topologies. In topo-GR, all eight Thurston classes admit such vacua, with the only free parameter being the overall spatial scale.

5. Isotropization with Positive Cosmological Constant and Absence of Recollapse

A key global inequality satisfied by nearly all non-LRS (locally rotationally symmetric) BKS types, except Bianchi II, is

$R - \bar R \leq 0.$

The corresponding Hamiltonian constraint enforces

$$3H^2 = \kappa\rho + \Lambda + \frac{1}{2}\left(\sigma_{ij}\sigma^{ij} + \frac{1}{N^2} \bar R_{ij} h^{ij} \right) - \frac{1}{2}(R - \bar R),$$

so $H^2 \geq \Lambda/3$ if $H>0$ initially. Extending the argument of Wald, for any non-tilted BKS solution with $\Lambda>0$, matter satisfying the weak and strong energy conditions, and initial expansion,

$$H(t)>0 \quad \forall t \ge t_0, \qquad \sigma_{ij}\to0 \text{ as } t\to\infty,$$

and recollapse is precluded. In GR, by contrast, Bianchi IX and KS spacetimes can recollapse or fail to isotropize without additional fine-tuning. This property in topo-GR is robust and does not require extra free parameters relative to GR (Vigneron et al., 8 Dec 2025).

6. Comparison with Standard General Relativity

Several distinctions arise between topo-GR and standard GR in the context of non-tilted BKS spacetimes:

• Shear-Free FLRW-Type Solutions: These exist for all Thurston geometries in topo-GR, though in GR only for maximally-symmetric $E^3$, $S^3$, $H^3$ topologies.
• Static Vacuum Solutions: All topologies admit static vacua in topo-GR, but only $E^3$ and $S^1\times S^2$ do so in GR.
• Isotropization and Recollapse: Topo-GR prevents recollapse and ensures isotropization generically under $\Lambda>0$, whereas in GR Bianchi IX and KS require extensive fine-tuning.
• Dynamical Role of Curvature: The spatial curvature term ${}^3 R$ in GR is replaced by $\bar R_{ij}$ from the topological connection in topo-GR, leading to fundamental differences in the dynamics, particularly in the shear-free and static vacuum sectors.

7. Principal Theorems and Foundational Propositions

The theoretical foundation relies on several results:

Item	Statement	Applicability
Thurston–Hamilton–Perelman Theorem	Every closed 3-manifold is a connected sum of Thurston-geometric pieces	All closed 3-manifolds
Topo-GR Field Equations	$$R_{\mu\nu} - \bar R_{\mu\nu} = \kappa (T_{\mu\nu} - \tfrac{1}{2} T g_{\mu\nu}) + \Lambda g_{\mu\nu}$$	Topo-GR, background $\bar\nabla$ fixed
Shear-Free Existence Proposition	Every Thurston class admits $\sigma_{ij}=0$ FLRW-type solutions with $3H^2=\kappa\rho$	All Thurston geometries
Static Vacuum Proposition	For $\rho=p=0$ in any shear-free ansatz, static vacuum exists: $$\mathrm{d}s^2 = -\mathrm{d}t^2 + \bar h_{ij}\omega^i\omega^j$$	All Thurston geometries
Isotropization & No-Recollapse Theorem	Under $\Lambda>0$, weak & strong energy, $H(t_0)>0$: $H>0$ for all $t$, $\sigma_{ij}\to0$, no recollapse (except Bianchi II)	All non-LRS BKS except BII

These principles underscore the universality of certain dynamical and kinematical features in non-tilted BKS solutions within topo-GR and delineate the modifications induced by the explicit topological dependence in the gravitational sector (Vigneron et al., 8 Dec 2025).