On the geometry of silent and anisotropic big bang singularities

TL;DR

This work develops a geometric framework to analyze silent and anisotropic big bang singularities in general relativity, focusing on the expansion normalised Weingarten map $\mathcal{K}$ and its associated quantities to capture asymptotics without symmetry assumptions. By coupling this framework to Einstein's equations, the authors obtain exponential decay and convergence results for $\mathcal{K}$ and for the expansion-normalised normal derivative, and reveal how the Kasner map and a scalar bounce function $(\check{\gamma}^{1}_{23})^{2}$ organize the dynamics in 3+1 dimensions. The results unify and extend prior analyses (including BKL-type oscillations and quiescent regimes) across vacuum and scalar-field matter, and produce explicit curvature bounds, Hamiltonian constraint implications, and initial-data-on-the-singularity perspectives within a compatible bootstrap scheme. The framework also accommodates non-symmetric settings and connects with companion linear-wave analyses (RinWave) and symmetry-reduced models (Gowdy, Bianchi), highlighting when oscillatory versus quiescent behavior arises and quantifying convergence rates and data on the singularity.

Abstract

This article is the second of two in which we develop a geometric framework for analysing silent and anisotropic big bang singularities. In the present article, we record geometric conclusions obtained by combining the geometric framework with Einstein's equations. The main features of the results are the following: The assumptions do not involve any symmetry requirements and are weak enough to be consistent with most big bang singularities for which the asymptotic geometry is understood. The framework gives a clear picture of the asymptotic geometry. It also reproduces the Kasner map, conjectured in the physics literature to constitute the essence of the asymptotic dynamics for vacuum solutions to Einstein's equations. When combined with Einstein's equations, the framework yields partial improvements of the assumptions concerning, e.g., the expansion normalised Weingarten map $\mathcal{K}$ (one of the central objects of the framework, defined as the Weingarten map of the leaves of the foliation divided by the mean curvature). For example, the expansion normalised normal derivative of $\mathcal{K}$ can, under suitable assumptions concerning the eigenvalues of $\mathcal{K}$, be demonstrated to decay exponentially and $\mathcal{K}$ can be demonstrated to converge exponentially, even though we initially only impose weighted bounds on these quantities. Finally, the framework gives a unified perspective on the existing results. Moreover, in $3+1$-dimensions, the only parameters necessary to interpret the results are the eigenvalues of $\mathcal{K}$ and an additional scalar function determined by the geometry induced on the leaves of the foliation. In the companion article, we obtain conclusions concerning the asymptotic behaviour of solutions to linear systems of wave equations on the backgrounds consistent with the framework.

Figures (6)

• Figure 1: The Kasner circle with the special points $T_{i}$, antipodal points $Q_{i}$ and segments $\mathscr{K}_i$, $i=1,2,3$, indicated. At each of the points $T_{i}$, $Q_{i}$, $i=1,2,3$, two of the eigenvalues coincide. In between these points, the eigenvalues are all distinct. Moreover, on $\mathscr{K}_i$, $l_i$ is the smallest eigenvalue, $i=1,2,3$. We have also indicated that between $Q_{1}$ and $T_{2}$, $l_{1}<l_{3}<l_{2}$ etc. Note also that between $T_{2}$ and $Q_{3}$, $l_{3}<l_{1}<l_{2}$ and that between $Q_{2}$ and $T_{3}$, $l_{2}<l_{1}<l_{3}$.
• Figure 2: The state space of Bianchi type I orthogonal stiff fluid solutions (i.e., orthogonal perfect fluids with equation of state $p=\rho$) is depicted on the left, where $\omega^{2}:=\Omega_{\mathrm{wh}}$, $\omega\geq 0$. Since the Hamiltonian constraint in this case reads $\omega^{2}+\Sigma_{+}^{2}+\Sigma_{-}^{2}=1$, where $\omega\geq 0$, it is sometimes more convenient to represent the state space by the Kasner disc; see the image on the right.
• Figure 3: The gray area in the figure on the right indicates the subset of the Kasner disc in which $\mathcal{K}$ is positive definite. The gray area in the figure on the left indicates the subset of the Kasner disc to which Bianchi type II solutions converge in case $N_{1}\neq 0$. As expected, the complement of this area corresponds to the part of the phase space where $l_{1}\leq 0$ (since $\gamma^{1}_{23}\neq 0$, $\gamma^{2}_{31}=0$ and $\gamma^{3}_{12}=0$). The gray area in the figure in the center indicates the subset of the Kasner disc to which Bianchi type VI${}_{0}$ and VII${}_{0}$ solutions converge in case $N_{2}\neq 0$ and $N_{3}\neq 0$. As expected, the complement of this area corresponds to the part of the phase space where either $l_{2}\leq 0$ or $l_{3}\leq 0$ (since $\gamma^{1}_{23}=0$, $\gamma^{2}_{31}\neq 0$ and $\gamma^{3}_{12}\neq 0$). In the case of Bianchi type VIII and IX, $\gamma^{1}_{23}\neq 0$, $\gamma^{2}_{31}\neq 0$ and $\gamma^{3}_{12}\neq 0$. Moreover, Bianchi type VIII and IX orthogonal stiff fluid solutions converge, as expected, to a point inside the triangle on the right.
• Figure 4: Equating the right hand sides of (\ref{['seq:hUellpmintro']}) with zero yields a map from the Kasner circle to itself, called the Kasner map (the straight line connecting $S$ and $\kappa(S)$ is the projection of a Bianchi type II solution to the Einstein vacuum equations). Given a point $S$ on the circle, $\kappa(S)$ is obtained by taking the nearest corner of the triangle, drawing a straight line from this corner to $S$, and then continuing this straight line to the next intersection with the circle. This next intersection defines $\kappa(S)$.
• Figure 5: The Kasner parabola (representing the Kasner solutions) with the two special points $T1$ and $T2$ indicated (these points correspond to flat Kasner solutions). In the vacuum setting, generic Bianchi type VI${}_{\eta}$ and VII${}_{\eta}$ solutions converge to a point on the Kasner parabola to the right of $T2$.

Theorems & Definitions (163)

• Definition 1
• Remark 2
• Definition 3
• Remark 4
• Remark 5
• Remark 6
• Definition 7
• Definition 8
• Remark 9
• Remark 10