Filtered expansions in general relativity I

TL;DR

This paper develops a formal power series approach to solving vacuum Einstein equations, focusing on one bounce in inhomogeneous spacetimes, using a graded Lie algebra and homological methods.

Contribution

It introduces a novel perturbative framework with a filtration that encodes BKL features, enabling systematic solutions to Einstein's equations.

Findings

Constructed formal solutions with one bounce in inhomogeneous spacetimes

Established an unobstructed perturbative problem using a graded Lie algebra

Implemented a filtration capturing key BKL features

Abstract

This is the first of two papers in which we construct formal power series solutions in external parameters to the vacuum Einstein equations, implementing one bounce for the Belinskii-Khalatnikov-Lifshitz (BKL) proposal for spatially inhomogeneous spacetimes. We use a graded Lie algebra, homological framework. A dedicated filtration encodes key features of the BKL proposal, and we use it to set up an unobstructed perturbative problem.

Tables

Table 1. Table 1: Partial dictionary for comparison with the non-rigorous BKL literature. The correspondence gives the closest analogue. Beware that the Kasner solution is a well-known exact solution to the vacuum Einstein equations, whereas the generalized Kasner spacetime is a vague concept from the BKL literature. An analogy may be made with polydifferential operators with the Gerstenhaber bracket. The Kasner solution is analogous to the constant coefficient Moyal product, whereas the generalized Kasner spacetime is analogous to a generalized non-constant coefficient Moyal product, which is an MC-element in the associated graded of a suitable filtration on polydifferential operators.

this paper	closest analogue in the BKL papers
$\text{MC}\left(ℰ\right)=\{x\in {ℰ}^1\mid [x,x]=0\}$	(possibly singular) Ricci-flat metrics
$ℰ$	–
$F_0{ℰ}^1$	spatial derivatives ignored to leading order
$F_{\alpha }{ℰ}^1$	smallness hierarchy
$F_{\alpha }ℰ$	–
${𝒫}_{\text{free}}=$ Rees algebra for 3-index $F_{\alpha }ℰ$	–
${𝒫}_{\text{bounce}}=$ Rees algebra for 2-index $F_{\alpha }ℰ$	–
$\text{MC}\left({𝒜}_{\text{free}}\right)$, with ${𝒜}_{\text{free}}=\mathrm{Gr}{𝒫}_{\text{free}}$	generalized Kasner spacetimes
$\text{MC}\left({𝒜}_{\text{bounce}}\right)$, with ${𝒜}_{\text{bounce}}=\mathrm{Gr}{𝒫}_{\text{bounce}}$	leading term for one bounce spacetimes
$\text{MC}\left({𝒫}_{\text{free}}\right)$	–
$\text{MC}\left({𝒫}_{\text{bounce}}\right)$	–

Table 2. Table 2: Definition of G α ​ ℰ G subscript 𝐺 𝛼 subscript ℰ 𝐺 G_{\alpha}\mathcal{E}_{G} associated to a conformally orthonormal basis θ 0 , θ 1 , θ 2 , θ 3 subscript 𝜃 0 subscript 𝜃 1 subscript 𝜃 2 subscript 𝜃 3 \theta_{0},\theta_{1},\theta_{2},\theta_{3} . We omit wedges, so θ 0 ​ θ 1 = θ 0 ∧ θ 1 subscript 𝜃 0 subscript 𝜃 1 subscript 𝜃 0 subscript 𝜃 1 \theta_{0}\theta_{1}=\theta_{0}\wedge\theta_{1} . The basis-dependent injection Der ⁡ ( C ∞ ) ↪ CDer ⁡ ( W ) ↪ Der superscript 𝐶 CDer 𝑊 \operatorname{Der}({C^{\infty}})\hookrightarrow\operatorname{CDer}(W) is implicit rt1 . All elements in the table are elements of ℰ ℰ \mathcal{E} via the canonical surjection ℒ ↠ ℰ ↠ ℒ ℰ \mathcal{L}\twoheadrightarrow\mathcal{E} .

Equations

$$F_p\mathcal{E}={⨁}_{q\le p}{⨁}_{p_2,p_3}{G}_{q,p_2,p_3}\mathcal{E}$$

$$F_{p_2{p}_3}\mathcal{E}=F_{p_2}^{\prime }\mathcal{E}\cap F_{p_3}^{\prime \prime }\mathcal{E}{F}_{p_1{p}_2{p}_3}\mathcal{E}=F_{p_1}\mathcal{E}\cap F_{p_2}^{\prime }\mathcal{E}\cap F_{p_3}^{\prime \prime }\mathcal{E}$$

$$\mathcal{P}=\{{\sum }_{p\ge 0}{s}^p{x}_{(p)}\mid x_{(p)}\in F_p\mathcal{E}\}$$

$$s\mathcal{P}\supseteq s^2\mathcal{P}\supseteq s^3\mathcal{P}\supseteq \dots$$

$$\text{MC}(\mathfrak{g})=\{x\in {\mathfrak{g}}^1\mid [x,x]=0\}$$

$$[F_p\mathfrak{g},F_q\mathfrak{g}]\subseteq F_{p+q}\mathfrak{g}$$

$$\mathfrak{p}=\{{\sum }_{p\ge 0}{s}^p{x}_{(p)}\mid x_{(p)}\in F_p\mathfrak{g}\}$$

$$\frac{\{x\in \text{MC}(\mathfrak{p})\mid x\mathrm{mod}s\mathfrak{p}=x_0\}}{\exp (s{\mathfrak{p}}^0[[s]])}$$

$$s\mathfrak{p}\supseteq s^2\mathfrak{p}\supseteq s^3\mathfrak{p}\supseteq \dots$$

$$d=[x_0,-]\in {\mathrm{End}}^1(\mathfrak{a})$$

$$(f,g)\mapsto {\sum }_p\frac{1}{p!}{s}^{2p}{\pi }^{i_1{k}_1}\cdots {\pi }^{i_p{k}_p}({\partial }_{i_1}\cdots {\partial }_{i_p}f)({\partial }_{k_1}\cdots {\partial }_{k_p}g)$$

$$F_{⊲p}\mathfrak{g}=\{g\in {\mathfrak{g}}_{p\mathrm{mod}2}\mid \forall q\ge 0:g(F_{⊲q}X)\subseteq F_{⊲q+p}X\}$$

$$W=W_0\oplus W_1$$

$$\mathcal{E}\to \mathrm{End}(\wedge W)\oplus \mathrm{End}(\wedge W\otimes \Omega )$$

$${\mathcal{E}}_G$$

$$G_{\alpha }\mathcal{E}$$

$$\mathcal{E}$$

$$F_{⊲p}\mathcal{E}={⨁}_{k\ge 0}{⨁}_{p_2,p_3}{G}_{p-2k,p_2,p_3}\mathcal{E}$$

$$W=C^{\infty }{\theta }_0\oplus C^{\infty }{\theta }_1\oplus C^{\infty }{\theta }_2\oplus C^{\infty }{\theta }_3$$

$${\mathcal{P}}_{\text{bounce}}$$

$${\mathcal{P}}_{\text{free}}$$

$${\mathcal{A}}_{\text{bounce}}$$

$${\mathcal{A}}_{\text{free}}$$

$$D_0=\frac{\partial }{\partial t}$$

$$\mathcal{U}$$

$${\mathcal{U}}_G$$

$${\theta }_0{D}_0+{\mathcal{U}}_G^1\subseteq {\mathcal{U}}^1$$

$${\theta }_0{D}_0+{\sum }_{\text{cyclic (i,j,k)}}(-$$

$$-$$

$$+$$

$$-$$

$$+$$

$${\beta }_1^0={\beta }_2^0={\beta }_3^0=0⟹{\gamma }_1^6={\gamma }_2^6={\gamma }_3^6=0$$

Full text

1. 1 Introduction
2. 2 Filtered expansions
3. 3 The BKL filtration, informal motivation
4. 4 The 1-index BKL filtration
5. 5 The 2-index and 3-index BKL filtrations
6. 6 Maurer-Cartan elements in $\mathcal{A}_{\textnormal{free}}$
7. 7 The constraint equations
8. 8 Maurer-Cartan elements in $\mathcal{A}_{\textnormal{bounce}}$
9. 9 Some automorphisms of $\mathcal{P}_{\textnormal{bounce}}$ and $\mathcal{P}_{\textnormal{free}}$
10. A Review of the spectral sequence of a filtered complex

1 Introduction

We formulate and prove theorems about the Belinskii-Khalatnikov-Lifshitz (BKL) proposal for spatially inhomogeneous spacetimes in general relativity bkl . In this paper and a sequel, we construct formal power series solutions in external parameters, giving meaning to ‘one BKL bounce’. Naively a problem with nonzero homological obstruction space, a filtration is used to set up an unobstructed Maurer-Cartan (MC) perturbation problem. See Gerstenhaber Gerstenhaber .

In rt1 the vacuum Einstein equations were formulated as the MC-equation $[x,x]=0$ for an unknown $x\in\mathcal{E}^{1}$ of degree one in a graded Lie algebra (gLa) $\mathcal{E}$. Here we construct formal power series solutions, namely MC-elements in the gLa $\mathcal{E}[[s]]$ where $s$ stands for one or more symbols. These are formal power series whose leading terms are, in BKL terminology that we will not adopt, the so-called

• •

generalized Kasner spacetime

• •

one bounce spacetime

They are rigorous building blocks for the BKL proposal. The full BKL proposal would involve sticking together an infinite sequence of bounces.

This paper is organized around a gLa filtration $(F_{p}\mathcal{E})_{p\geq 0}$ that we call the BKL filtration. It provides a clear framework for the BKL proposal. See also Table 1. The BKL filtration is in all homological degrees, comprehensively organizing the gauge symmetries, the unknown, the MC-equations, and the differential identities.

Recall from rt1 that $\mathcal{E}$ is a graded Lie algebroid over the graded commutative algebra ${\wedge W}$; here $W$ is the ${C^{\infty}}$-module of sections of a real vector bundle with a conformal inner product of signature ${-}{+}{+}{+}$ on a 4-dim manifold. So $\mathcal{E}$ comes with an anchor map $\mathcal{E}\to\operatorname{Der}({\wedge W})$. The BKL filtration is defined once an orthogonal decomposition of $W$ into rank two timelike and spacelike submodules is chosen.

Theorem 1.1 (The 1-index BKL filtration)

Let $W=W_{0}\oplus W_{1}$ be a fiberwise orthogonal decomposition with $W_{0}$, $W_{1}$ having signatures ${-}{+}$ and ${+}{+}$ respectively. Associated to it is a canonical gLa filtration of $\mathcal{E}$, induced by a corresponding $\mathbbm{Z}_{2}$-graded filtration on ${\wedge W}$ via the anchor map. Explicitly, with the summands given in Table 2,

[TABLE]

In the table, it is understood that $W_{0}={C^{\infty}}\theta_{0}\oplus{C^{\infty}}\theta_{1}$ and $W_{1}={C^{\infty}}\theta_{2}\oplus{C^{\infty}}\theta_{3}$ for a conformally orthonormal basis $\theta_{0},\theta_{1},\theta_{2},\theta_{3}$.

To obtain the 2- and 3-index BKL filtrations, choose a conformally orthonormal basis associated to which are three orthogonal decompositions of $W$, hence three 1-index BKL filtrations denoted $F$, $F^{\prime}$, $F^{\prime\prime}$ respectively, see Definition 6. Their intersections

[TABLE]

are the 2- and 3-index filtrations. The 3-index grading $G_{p_{1}p_{2}p_{3}}\mathcal{E}$ gives all these filtrations. The significance of these filtrations is that they guide one in constructing MC-elements for BKL.

We motivate these filtrations by reference to the existing non-rigorous BKL literature, but also independently on algebraic and geometric grounds. Once these filtrations are accepted, there is a logical framework in which to prove theorems. In this introduction we pretend that $p$ is a single index, rather than two or three.

The filtration organizes formal power series expansions. Namely, MC-elements are sought in the Rees algebra

[TABLE]

morally the generating function of the filtration. We discuss such ‘filtered expansions’ generally, for any gLa, in Section 2. So the $p$-th term of a formal power series MC-element is drawn from $F_{p}\mathcal{E}^{1}$. The clumsy notation $x_{(p)}$ indicates that constructing the $x_{(p)}$ in succession is conceptually the wrong approach. Instead, one solves the equations successively modulo the descending chain of ideals

[TABLE]

The leading term of such an expansion is naturally an MC-element $x_{0}\in\textnormal{{{MC}}}(\mathcal{A})$ of the associated graded gLa $\mathcal{A}=\operatorname{Gr}\mathcal{P}=\mathcal{P}/s\mathcal{P}$. It contains more data than the naive leading term $x_{(0)}$. An important point is that the associated differential $[x_{0},-]$ on $\mathcal{A}$ controls perturbation theory on $\mathcal{P}$, in the usual sense of MC perturbation and obstruction theory. So its first homology is the linearized solution space, its second homology is the obstruction space.

Even if the differential $[x_{(0)},-]$ on $\mathcal{E}$ has a nonzero obstruction space, the differential $[x_{0},-]$ on $\mathcal{A}$ may have a vanishing obstruction space. This scenario plays out for BKL, as we show in examples, giving an unobstructed problem.

An informal motivation for the BKL filtration is in Section 3. The 1-index BKL filtration is in Section 4. The 2- and 3-index filtrations, whose Rees algebras we suggestively call $\mathcal{P}_{\textnormal{bounce}}$ and $\mathcal{P}_{\textnormal{free}}$, are in Section 5. We construct MC-elements of the associated gradeds $\mathcal{A}_{\textnormal{bounce}}$ and $\mathcal{A}_{\textnormal{free}}$, that are interesting closed-form solutions, in Sections 6 and 8. This requires solving the so-called constraint equations, which we do in the real analytic class for simplicity, in Section 7. We discuss pertinent gLa automorphisms in Section 9, and review spectral sequences in Appendix A.

The differential on the associated gradeds defines a filtered complex that is studied using two tools: spectral sequences for a filtered complex; and a contraction based on gauge-fixing rt1 for general relativity.

Non-rigorous work on the BKL proposal bkl include berger ; lim ; uggla and references therein. See also the exchange exc ; exc2 . Table 1 provides a partial dictionary for comparison with the existing BKL literature.

The BKL proposal has so far only been rigorously implemented for the non-generic class of spatially homogeneous spacetimes, see bklhom and references therein, where the vacuum Einstein equations reduce to ordinary differential equations. There one-bounce solution are constructed via a fixed point method, and an infinite sequence of bounces were stuck together to get a semiglobal solution. The spatially homogeneous problem is non-trivial since small divisors appear when sticking bounces together. But the gauge freedom, that is the homological nature of general relativity, can be avoided, unlike in the spatially inhomogeneous case that this paper is about.

This work departs conceptually from the existing, non-rigorous BKL literature by consistently employing a homological framework. This does due justice to the gauge theory nature of general relativity. An attempt to construct formal power series can run into obstructions, and the homological framework treats the obstruction space as an entity. The gLa $\mathcal{E}$ contains geometrically degenerate elements, without metric counterpart, as regular elements, including elements with a frame collapsed to timelike lines, which we perturb to construct BKL bounces, in a controlled way dictated by the BKL filtration.

Combining this paper with a no obstructions result in the companion paper rt3 , one obtains the existence of formal power series solutions to the vacuum Einstein equations, interpreted as one BKL bounce.

Theorem (One bounce MC-elements - informal version)

Spatially homogeneous elements in $\textnormal{{{MC}}}(\mathcal{A}_{\textnormal{bounce}})$ have zero homological obstruction space $H^{2}$ in $\mathcal{A}_{\textnormal{free}}$ and $\mathcal{A}_{\textnormal{bounce}}$. This yields MC-elements in $\mathcal{P}_{\textnormal{free}}$ and $\mathcal{P}_{\textnormal{bounce}}$ respectively.

This theorem concerns spatially inhomogeneous perturbations of homogeneous MC-elements; inhomogeneous because the spaces $\mathcal{A}_{\textnormal{free}}$ and $\mathcal{A}_{\textnormal{bounce}}$ involve no homogeneity condition. Starting from homogeneous MC-elements is for simplicity.

We speculate that matching adjacent BKL bounces at the formal perturbative level will define a discrete dynamical system of $L_{\infty}$ maps, describing gravitational scattering across a BKL bounce. In this paper we construct what one would expect to be the leading term of this scattering map.

Theorem (Bounce map - informal version of Theorem 6.1)

A subset of $\textnormal{{{MC}}}(\mathcal{A}_{\textnormal{bounce}})$ has future and past asymptotic limits (see Remark 3) that are in $\textnormal{{{MC}}}(\mathcal{A}_{\textnormal{free}})$. This gives a partially defined map $\mathcal{B}:\textnormal{{{MC}}}_{\textnormal{normal}}(\mathcal{A}_{\textnormal{free}})\nrightarrow\textnormal{{{MC}}}_{\textnormal{normal}}(\mathcal{A}_{\textnormal{free}})$ where ‘normal’ means that elements are in a distinguished coordinate system and frame.

Iterating $\mathcal{B}$ gives a discrete dynamical system. Analyzing this system can give insight into the problem of concatenating several bounces, but it does not amount to studying the full inhomogeneous BKL proposal, because it is only for the leading term.

The following quote from BKL exc2 is prescient:

As to the general solution (the oscillatory regime), it cannot be represented in a closed analytical form, although it admits a very detailed description refexc2 (the same refers even to the simpler case of the oscillatory regime in homogeneous models). The construction of this solution […] is based only on […] the estimation of the terms in the equations which are omitted in the asymptotic limit. It is just these estimates […] which prove the possibility to neglect the particular terms with spatial derivatives in the equations.

It shows that BKL viewed their calculations as providing an approximation to true solutions, and that estimates are called for. We do not see that their estimates are amenable to rigorous mathematics. In this paper we elevate their ‘detailed description’ to a formal power series expansion for one bounce. Natural next steps are:

• •

The construction of true one bounce solutions, either by showing that the formal series converge, or using energy estimates to construct nearby solutions. Energy estimates for the MC-equation in the gLa $\mathcal{E}$ are obtained by gauge-fixing. By construction, the gauges in rt1 yield symmetric hyperbolic equations.

• •

By matching adjacent BKL bounces, first at the formal perturbative level. This will require suitable gauge transformations, and as we have mentioned, ought to result in a discrete dynamical system of $L_{\infty}$ maps.

The BKL claim that one can concatenate infinitely many bounces, though made rigorous for spatially homogeneous spacetimes bklhom , remains extraordinary for spatially inhomogeneous spacetimes. We believe that the validity of this claim will largely be decided at the formal perturbative level. Many potential issues have not been dealt with by BKL, including the global causal structure; small divisors; the necessity of repeated gauge transformations; and repeated localization in space to avoid resonances. The spatially inhomogeneous BKL proposal remains wide open.

2 Filtered expansions

Let $\mathfrak{g}$ be a real gLa $\mathfrak{g}$ with a filtration. The filtration will be used as a comprehensive organizing tool for formal expansions. In favorable cases, a problem with nonzero naive obstruction space is turned into one with zero obstruction space.

Define the Maurer-Cartan set

[TABLE]

which would be a real quadratic variety if $\mathfrak{g}$ were finite-dimensional. In favorable cases, the infinitesimal action of the Lie algebra $\mathfrak{g}^{0}$ can be integrated, tracing out orbits on $\textnormal{{{MC}}}(\mathfrak{g})$, and it is the space of orbits that is the basic object. Rigorous sense can be made in formal perturbation theory about a point of $\textnormal{{{MC}}}(\mathfrak{g})$, effectively by tensoring with an auxiliary nilpotent algebra, giving the deformation functor.

Definition 1 (Filtration, its Rees algebra and its associated graded)

Let $\mathfrak{g}$ be a real gLa. By a filtration we mean a collection $(F_{p}\mathfrak{g})_{p\geq 0}$ where

• •

$F_{p}\mathfrak{g}\subseteq\mathfrak{g}$ is a graded subspace

• •

$F_{p}\mathfrak{g}\subseteq F_{p+1}\mathfrak{g}$

• •

there is a finite $p$ such that $F_{p}\mathfrak{g}=\mathfrak{g}$

• •

$F_{p}\mathfrak{g}$ is consistent with the bracket,

[TABLE]

Associated to the filtration is its Rees algebra. Using a formal parameter $s$, the Rees algebra is the following gLa over $\mathbbm{R}[[s]]$, a subalgebra of $\mathfrak{g}[[s]]$:

[TABLE]

The associated graded is the graded gLa $\mathfrak{a}=\mathfrak{p}/s\mathfrak{p}$.

Condition (2) encodes a wealth of conditions. In particular, for the Lie algebra representation $\mathfrak{g}^{0}\otimes\mathfrak{g}^{i}\to\mathfrak{g}^{i}$, and for the MC-map $\mathfrak{g}^{1}\otimes\mathfrak{g}^{1}\to\mathfrak{g}^{2}$.

Assuming each $F_{p}\mathfrak{g}$ admits a complement in $F_{p+1}\mathfrak{g}$, then non-canonically $\mathfrak{a}\simeq\mathfrak{g}$ as graded vector spaces and $\mathfrak{p}\simeq\mathfrak{g}[[s]]$ as graded $\mathbbm{R}[[s]]$-modules, however not as gLa.

Definition 2 (Filtered expansion, or expansion subordinate to a filtration)

For every $x\in\textnormal{{{MC}}}(\mathfrak{p})$ we say that its leading term is the element $x\bmod s\mathfrak{p}\in\textnormal{{{MC}}}(\mathfrak{a})$. Conversely, to every $x_{0}\in\textnormal{{{MC}}}(\mathfrak{a})$ we associate the moduli space of solutions

[TABLE]

where the denominator is a Lie algebra over $\mathbbm{R}[[s]]$ using Baker-Campbell-Hausdorff.

Remark 1

If $x=\sum_{p}s^{p}x_{(p)}\in\textnormal{{{MC}}}(\mathfrak{p})$ then we expressly do not view $x_{(0)}\in\textnormal{{{MC}}}(F_{0}\mathfrak{g})$ as the leading term, and we contend that constructing the $x_{(p)}$ in succession is the wrong approach. Instead, formal perturbation theory acquires a uniform structure if one solves the equations successively modulo the descending chain of ideals

[TABLE]

That is, we view $\mathfrak{p}$ as the projective limit $\varprojlim_{p}\mathfrak{p}/s^{p}\mathfrak{p}$.

Lemma 1 (The differential controlling formal perturbations)

Given an element $x_{0}\in\textnormal{{{MC}}}(\mathfrak{a})$, then formal perturbations about $x_{0}$ in the sense of (3), organized using the descending chain of ideals (4), are controlled by the differential

[TABLE]

In particular $H^{2}(d)$ is the obstruction space in the sense of Gerstenhaber.

Proof

This is a slightly informal version of a theorem proved in rt1 . ∎

So filtered expansions in $\mathfrak{p}$ are based on some element of $\textnormal{{{MC}}}(\mathfrak{a})$ and are controlled by the associated differential on $\mathfrak{a}$. This differential is lower triangular.

Lemma 2 (Block lower triangularity)

Relative to the grading $\mathfrak{a}=\bigoplus_{p}F_{p}\mathfrak{g}/F_{p-1}\mathfrak{g}$, the differential (5) is a block lower triangular, finite square matrix.

Proof

The $p$-th subdiagonal is determined by $x_{(p)}\bmod F_{p-1}\mathfrak{g}$. ∎

So $d$ is naturally studied via its spectral sequence, reviewed in Appendix A. A particularly intuitive situation occurs if the block diagonal part of $d$ has nonzero second homology, but the subdiagonal entries kill off that homology to give an unobstructed problem. Situations of this kind motivate the abstract setup.

Similar statements hold for 2-index and many-index filtrations.

Example 1 (Generalized Moyal product analogous to generalized Kasner spacetime)

We will make sense of BKL’s generalized Kasner spacetime as an MC-element in an associated graded gLa. An analogous algebraic example is a generalized Moyal product. Let $C^{\infty}$ be the smooth functions on $\mathbbm{R}^{n}$. Let $\mathfrak{g}$ be the gLa of polyderivations k with the Gerstenhaber bracket, so $\mathfrak{g}^{i}\subseteq\operatorname{Hom}_{\mathbbm{R}}((C^{\infty})^{\otimes(i+1)},C^{\infty})$. The elements of $\textnormal{{{MC}}}(\mathfrak{g})$ are the associative maps, for example pointwise multiplication. Let $F_{p}\mathfrak{g}$ be the filtration by the total number of derivatives; $F_{p}\mathfrak{g}\neq\mathfrak{g}$ for all $p$ but we gloss over that. Given $\pi^{ik}=-\pi^{ki}\in C^{\infty}$ the generalized non-constant coefficient Moyal product

[TABLE]

is not in $\textnormal{{{MC}}}(\mathfrak{p})$ but it is in $\textnormal{{{MC}}}(\mathfrak{a})$.

3 The BKL filtration, informal motivation

There are different motivations for the BKL filtration:

• •

Historic motivation: The filtration distills the algebraic content of the BKL proposal. The BKL proposal simplifies the Einstein equations in an ad-hoc way, in particular by dropping terms containing spatial derivatives. What remains are ordinary differential equations along distinguished timelike lines that foliate the spacetime. The simplified equations are explicitly solvable. We make sense of the ad-hoc simplifications by introducing a filtration of $\mathcal{E}$ that we call the BKL filtration. The $\textnormal{{{MC}}}(\mathcal{A})$ equations amount to the simplified equations of BKL.

• •

Geometric motivation: The filtration encodes a controlled degenerate to nondegenerate perturbation. It is natural to try to construct elements in $\textnormal{{{MC}}}(\mathcal{E})$ by perturbing simpler ones. Every element of $\mathcal{E}^{1}$ defines a map $W^{\ast}\to\operatorname{Der}({C^{\infty}})$ of rank four ${C^{\infty}}$-modules, conventionally called a frame. One can try to perturb explicitly solvable MC-elements with a degenerate frame of rank one into ones with a nondegenerate frame of rank four, and the BKL filtration is designed to do that.

• •

Algebraic motivation: The filtration is functorial given an orthogonal direct sum decomposition of $W$. One can appreciate the simplicity and functoriality of the filtration, and how it is constructed using the anchor map $\mathcal{E}\to\operatorname{Der}({\wedge W})$.

• •

The ends justify the means. The BKL filtration enables us to construct new formal power series solutions, in external parameters, naturally interpreted as inhomogeneous one bounce spacetimes.

4 The 1-index BKL filtration

From this point on, we use notation from rt1 . The gLa $\mathcal{E}=\mathcal{L}/\mathcal{I}$ is also a graded Lie algebroid over the graded commutative algebra ${\wedge W}$. Tensor products are over ${C^{\infty}}$. Here $W$, as in rt1 , is a rank four free ${C^{\infty}}$-module with a conformal inner product of signature ${-}{+}{+}{+}$. A conformally orthonormal basis is denoted $\theta_{0},\ldots,\theta_{3}$ as in rt1 .

We define a filtration on $\mathcal{E}$ that is functorially associated to an orthogonal direct sum decomposition of $W$ into two 2-dimensional submodules, one timelike meaning it has signature ${-}{+}$, one spacelike meaning it has signature ${+}{+}$.

The key is the consistency (2) with the bracket, a quadratic system of inequalities. Some filtrations come from a filtered representation space, in fact, a gLa representation $\mathcal{E}\to\operatorname{End}X$ with a filtration of $X$ by graded subspaces induces a gLa filtration of $\mathcal{E}$. We do something like this, except that we use filtrations with compatible $\mathbbm{Z}_{2}$-grading, and our notion of induced filtration also depends on the $\mathbbm{Z}_{2}$-grading.

Definition 3 (Filtration with $\mathbbm{Z}_{2}$-grading)

Let $X$ be a $\mathbbm{Z}$-graded vector space. A filtration with compatible $\mathbbm{Z}_{2}$-grading is data $X_{0}$, $X_{1}$ and $(F_{p}X)_{p\geq 0}$ where:

• •

$X_{0}$, $X_{1}$, $F_{p}X$ are graded subspaces of $X$, with respect to its $\mathbbm{Z}$-grading

• •

$X=X_{0}\oplus X_{1}$ as an internal direct sum

• •

$F_{p}X\subseteq F_{p+1}X$, and $F_{p}X=X$ for some finite $p$

• •

$F_{p}X=F_{\lhd p}X\oplus F_{\lhd p-1}X$ where $F_{\lhd p}X=F_{p}X\cap X_{p\bmod 2}$

It is equivalently determined by a collection $(F_{\lhd p}X)_{p\geq 0}$ with suitable properties. Beware that the $\mathbbm{Z}_{2}$-grading and the original $\mathbbm{Z}$-grading are separate data.

Below $\operatorname{End}(X)$ is the gLa of $\mathbbm{Z}$-graded endomorphisms with the graded commutator. It also inherits a $\mathbbm{Z}_{2}$-grading from $X$.

Lemma 3 (Induced gLa filtration)

Suppose $\mathfrak{g}$ is a gLa that also has a $\mathbbm{Z}_{2}$-grading $\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}$ compatible with the bracket and the $\mathbbm{Z}$-grading. Suppose $X$ is as in Definition 3. Given a gLa representation $\mathfrak{g}\to\operatorname{End}(X)$ that respects the $\mathbbm{Z}_{2}$-gradings, then there is on $\mathfrak{g}$ an induced filtration as in Definition 3 determined by

[TABLE]

Furthermore, $[F_{\lhd p}\mathfrak{g},F_{\lhd q}\mathfrak{g}]\subseteq F_{\lhd p+q}\mathfrak{g}$ so both the $\mathbbm{Z}_{2}$-grading and the induced filtration $F_{p}\mathfrak{g}$ are consistent with the bracket.

Proof

Omitted. The fact that the filtration on $X$ only starts at $p=0$ and exhausts at a finite $p$ is important. ∎

Definition 4 (Orthogonal decomposition of $W$)

This is a decomposition

[TABLE]

with $W_{0}={C^{\infty}}\theta_{0}\oplus{C^{\infty}}\theta_{1}$ and $W_{1}={C^{\infty}}\theta_{2}\oplus{C^{\infty}}\theta_{3}$ for some conformally orthonormal basis. Set $F_{\lhd 0}W=W_{0}$ and $F_{\lhd 1}W=W_{1}$ to obtain a filtration as in Definition 3.

One could define a groupoid of rank 4 vector bundles with both a conformal inner product and an orthogonal decomposition of $W$. The BKL filtration defines a functor out of this groupoid, into, for example, the category of filtered gLa.

Lemma 4 (The 1-index BKL filtration)

For every orthogonal decomposition $W=W_{0}\oplus W_{1}$ there are associated filtrations as in Definition 3 as follows:

• •

On ${\wedge W}$ the smallest such filtration that respects the unital algebra structure and the injection $W\hookrightarrow{\wedge W}$. So $F_{\lhd p}({\wedge W})$ is ${C^{\infty}}$-spanned by any monomial in $\theta_{0},\theta_{1},\theta_{2},\theta_{3}$, such that the total degree in $\theta_{2},\theta_{3}$ is at most $p$ and is even respectively odd depending on whether $p$ is even respectively odd. So $F_{2}({\wedge W})={\wedge W}$.

• •

On ${\wedge W}\otimes\Omega$ the filtration $F_{\lhd p}({\wedge W}\otimes\Omega)=F_{\lhd p}({\wedge W})\otimes\Omega$. Here $\Omega=\wedge^{4}W$ rt1 .

• •

*On $\mathcal{E}$ the filtration induced by the $\mathbbm{Z}_{2}$-grading $\mathcal{E}=\mathcal{E}_{0}\oplus\mathcal{E}_{1}$ coming from $W=W_{0}\oplus W_{1}$, together with Lemma 3 via the natural gLa representation *rt1

[TABLE]

The representation on the first summand is the anchor $\mathcal{E}\to\operatorname{Der}({\wedge W})$. The representation descends from a representation of $\mathcal{L}={\wedge W}\otimes\operatorname{CDer}(W)$, given explicitly by $\omega\delta\mapsto(\omega^{\prime}z\mapsto(\omega\delta(\omega^{\prime}))z+(\omega\omega^{\prime})\delta(z))$ with $z=1$ respectively $z\in\Omega$.

We call $(F_{p}\mathcal{E})_{p\geq 0}$ the BKL filtration associated to the decomposition $W=W_{0}\oplus W_{1}$.

Proof

The $\mathbbm{Z}_{2}$-gradings are natural and are compatible. ∎

One can extend the filtration to a graded Lie algebroid filtration of $\mathcal{E}$ over the filtered graded commutative algebra ${\wedge W}$, with filtered module and anchor maps.

We have defined the BKL filtration via a representation, which makes it look natural. One would expect that this representation should play an important role in using the BKL filtration. Actually, we proceed with a more explicit description.

Lemma 5 (Decompositions of $\mathcal{E}$)

*For every conformally orthonormal basis of $W$, there are graded free ${C^{\infty}}$-submodules111We are abusing notation of rt1 . Here $\mathcal{E}_{G}$ is a new symbol, not immediately related to the output of the gauge fixing algorithm in rt1 , though it can be obtained from it by a limiting procedure.

$G_{\alpha}\mathcal{E}_{G}\subseteq\mathcal{E}$ defined by Table 2, with $\alpha=p_{1}p_{2}p_{3}$. The ranks are given in the table. Set*

[TABLE]

Here ${\wedge W}$-module multiplication by $\theta_{0}$ acts injectively. All direct summands are free graded ${C^{\infty}}$-submodules. The $\mathbbm{Z}_{\geq 0}^{3}$-grading $G_{\alpha}\mathcal{E}$ is invariant under scalings of the conformally orthonormal basis by a positive function.

Proof

The direct sums are well-defined; linear independence is self-evident in $\mathcal{L}$ because it is a product, but not self-evident in $\mathcal{E}$. The $G_{\alpha}\mathcal{E}_{G}$ are not invariant under conformal scalings of the basis, since the injection $\operatorname{Der}({C^{\infty}})\hookrightarrow\operatorname{CDer}(W)$ changes by terms proportional to $\sigma_{0}$. Hence in $G_{211}\mathcal{E}_{G}$ the term $\theta_{2}\theta_{3}\operatorname{Der}({C^{\infty}})$ generates new terms $\theta_{2}\theta_{3}\sigma_{0}$; more details will be in Theorem 9.1. While such terms are not contained in $G_{211}\mathcal{E}_{G}$, they are in $G_{211}\mathcal{E}$. This example generalizes. ∎

The space $\mathcal{E}_{G}$ will be used for calculations, at the moment it plays a minor role.

Lemma 6 (The 1-index BKL filtration,

explicit definition)

The 1-index BKL filtration associated to $W_{0}={C^{\infty}}\theta_{0}\oplus{C^{\infty}}\theta_{1}$ and $W_{1}={C^{\infty}}\theta_{2}\oplus{C^{\infty}}\theta_{3}$ is

[TABLE]

Proof

The $\mathbbm{Z}_{2}$-grading is important because, for example, $\theta_{0}\theta_{3}\sigma_{1}+\theta_{0}\theta_{1}\sigma_{3}\in G_{101}\mathcal{E}$ is mapped to the zero map under (6) and hence would be a candidate for $F_{0}\mathcal{E}$ if it were not for the $\mathbbm{Z}_{2}$-grading, which puts it in $F_{\lhd 1}\mathcal{E}$. One checks that in Table 2, the entries with $p_{1}$ even (odd) are in the even (odd) sector of $\mathcal{E}=\mathcal{E}_{0}\oplus\mathcal{E}_{1}$ for its $\mathbbm{Z}_{2}$-grading associated to $W_{0}\oplus W_{1}$. For example, $A_{\pm}=\theta_{2}\sigma_{3}\pm\theta_{3}\sigma_{2}$ are even and indeed appear in the table with $p_{1}$ even. Note that $A_{+}$ annihilates $\theta_{0}$, $\theta_{1}$ and from $\theta_{2}$, $\theta_{3}$ produces $\theta_{3}\theta_{0}$, $\theta_{2}\theta_{0}$ terms not increasing filtration degree, and using the derivation property we get $A_{+}\in F_{\lhd 0}\mathcal{E}$. By contrast, applying $A_{-}$ to $\theta_{0}\in F_{\lhd 0}({\wedge W})$ gives $\theta_{2}\theta_{3}\notin F_{\lhd 0}({\wedge W})$, which implies $A_{-}\notin F_{\lhd 0}\mathcal{E}$, so it is only in $F_{\lhd 2}\mathcal{E}$. By similar arguments, no linear combination of elements in $G_{2p_{2}p_{3}}\mathcal{E}_{G}$, and in fact $G_{2p_{2}p_{3}}\mathcal{E}$, can be in $F_{\lhd 0}\mathcal{E}$. The ${\wedge W}\otimes\Omega$ part in (6) is used, for example, to make $\theta_{0}\theta_{1}\theta_{2}\theta_{3}\sigma_{0}\notin F_{\lhd 0}\mathcal{E}$. ∎

As an application, note that elements of $F_{0}\mathcal{E}^{1}$ define a frame of rank at most two, hence a degenerate frame, because it contains $\theta_{i}\operatorname{Der}({C^{\infty}})$ with $i=0,1$ but not $i=2,3$. Degenerate frames were part of the motivation in Section 3.

5 The 2-index and 3-index BKL filtrations

Definition 5 (Full orthogonal decomposition of $W$)

This is a decomposition

[TABLE]

for some conformally orthonormal basis, with the understanding that two bases that differ only by a scaling by a positive function define the same decomposition.

Definition 6 (2-index and 3-index BKL filtrations)

Given an orthogonal decomposition of $W$ introduce the following 1-index BKL filtrations:

• •

$(F_{p}\mathcal{E})$ the BKL filtration associated to $W_{0}={C^{\infty}}\theta_{0}\oplus{C^{\infty}}\theta_{1}$, $W_{1}={C^{\infty}}\theta_{2}\oplus{C^{\infty}}\theta_{3}$.

• •

$(F_{p}^{\prime}\mathcal{E})$ the BKL filtration associated to $W_{0}={C^{\infty}}\theta_{0}\oplus{C^{\infty}}\theta_{2}$, $W_{1}={C^{\infty}}\theta_{3}\oplus{C^{\infty}}\theta_{1}$.

• •

$(F_{p}^{\prime\prime}\mathcal{E})$ the BKL filtration associated to $W_{0}={C^{\infty}}\theta_{0}\oplus{C^{\infty}}\theta_{3}$, $W_{1}={C^{\infty}}\theta_{1}\oplus{C^{\infty}}\theta_{2}$.

Then (1) defines the 2-index and 3-index BKL filtrations. They are gLa filtrations, and222Here $\beta=q_{1}q_{2}q_{3}\leq\alpha=p_{1}p_{2}p_{3}$ iff $q_{1}\leq p_{1}$ and $q_{2}\leq p_{2}$ and $q_{3}\leq p_{3}$ $F_{\alpha}\mathcal{E}=\bigoplus_{\beta\leq\alpha}G_{\beta}\mathcal{E}$. The filtration is free over ${C^{\infty}}\mathbbm{1}\oplus{C^{\infty}}\theta_{0}$ with $\theta_{0}^{2}=0$, the exterior algebra in one generator.

Elements of $F_{00}\mathcal{E}^{1}$ and $F_{000}\mathcal{E}^{1}$ define a frame of rank at most one, hence a degenerate frame, because they contain $\theta_{i}\operatorname{Der}({C^{\infty}})$ only when $i=0$. See also Section 3.

Lemma 7 (Rees algebras)

Let $\mathbbm{R}[[s_{1},s_{2},s_{3}]]$ be the $\mathbbm{R}$-algebra of formal power series in the symbols $s_{1},s_{2},s_{3}$. There are gLa

[TABLE]

where $(s_{2},s_{3})$ is the ideal generated by these elements, similar for $(s_{1},s_{2},s_{3})$.

Proof

Omitted. Freeness is defined in the discussion of MC perturbations in rt1 . ∎

6 Maurer-Cartan elements in $\mathcal{A}_{\textnormal{free}}$

Elements of $\textnormal{{{MC}}}(\mathcal{A}_{\textnormal{free}})$ and their differentials are key in filtered expansions on $\mathcal{P}_{\textnormal{free}}$, see Section 2. This section contains logically complete statements for $\textnormal{{{MC}}}(\mathcal{A}_{\textnormal{free}})$. We keep this section compact by postponing some important things:

• •

The solution of certain partial differential equations in three dimensions, called ‘constraints’ in the vernacular of general relativity, is postponed to Section 7.

• •

We write down a partial map from $\textnormal{{{MC}}}(\mathcal{A}_{\textnormal{free}})$ to itself that we call the bounce map. It is astonishing that the constraints are preserved. This can be checked by direct, unrevealing calculation, but the rationale is that this map is a ‘scattering map’ for an element of $\textnormal{{{MC}}}(\mathcal{A}_{\textnormal{bounce}})$ that we postpone to Section 8.

• •

The elements of $\textnormal{{{MC}}}(\mathcal{A}_{\textnormal{free}})$ in this section are essentially general, but to see this one needs certain automorphisms of $\mathcal{A}_{\textnormal{free}}$ that are postponed to Section 9.

• •

Studying the differential on $\mathcal{A}_{\textnormal{free}}$ is postponed. In particular, the obstruction space.

Definition 7 (Foliated base manifold

and orthogonal decomposition)

The base manifold is $\mathbbm{R}\times S$ for a manifold $S\simeq\mathbbm{R}^{3}$. The projection out of the first factor $t:\mathbbm{R}\times S\to\mathbbm{R}$ is called ‘time’, the partial derivative along the first factor is denoted

[TABLE]

We assume $W={C^{\infty}}\theta_{0}\oplus{C^{\infty}}\theta_{1}\oplus{C^{\infty}}\theta_{2}\oplus{C^{\infty}}\theta_{3}$ is a full orthonormal decomposition, see Definition 5. There is canonical injection $\operatorname{Der}({C^{\infty}})\hookrightarrow\operatorname{CDer}(W)$ that produces elements that annihilate the basis elements $\theta_{i}$, and that we use implicitly.

The base manifold carries a foliation by affine lines, suitable for BKL. This is made precise by letting Definition 7 define the objects of a groupoid, see Theorem 9.1.

Cyclic $(i,j,k)$ means that they run over $\{(1,2,3),(2,3,1),(3,1,2)\}$.

Lemma 8 (Master space

and overparametrization of an affine gauge subspace)

Given Definition 7 and given three symbols $s_{1},s_{2},s_{3}$, and abbreviating $\alpha=p_{1}p_{2}p_{3}$ and $s^{\alpha}=s_{1}^{p_{1}}s_{2}^{p_{2}}s_{3}^{p_{3}}$, define the external direct sums

[TABLE]

Then the ${C^{\infty}}$-module ‘master space’ $\mathcal{U}$ is canonically isomorphic to each of $\mathcal{E}$, $\mathcal{A}_{\textnormal{bounce}}$, $\mathcal{A}_{\textnormal{free}}$ and enables identification between any two of them. In this way, $\mathcal{U}$ inherits three different real gLa structures. The ‘affine gauge subspace’

[TABLE]

admits an over-parametrization (surjection) in terms of, with $i=1,2,3$,

• •

$\beta_{i}^{\mu},\,\gamma_{i}^{a},\,\mu_{i}\in C^{\infty}(\mathbbm{R}\times S,\mathbbm{R})$* with $\mu=0,1,2,3$ and $a=0,1,2,3,4,5,6$*

• •

$D_{i}$* a frame of vector fields on $S$ extended to $\operatorname{Der}({C^{\infty}})$ by $D_{i}(t)=0$*

Concretely,

[TABLE]

with the abbreviations $\mathbf{s}_{i}=s_{i}e^{\mu_{i}}$ and $\beta_{i}=\beta_{i}^{0}D_{0}+\beta_{i}^{1}D_{1}+\beta_{i}^{2}D_{2}+\beta_{i}^{3}D_{3}$. The notation in this lemma is used consistently in the following.

Proof

Clear. Use Table 2. ∎

The $\mu_{i}$ will simplify later statements; we often require $D_{0}(\mu_{i})=\gamma_{i}^{0}$. An element for which $D_{0},\beta_{1},\beta_{2},\beta_{3}$ is a frame may be thought of as infinitesimally nondegenerate.

Lemma 9 (Synchronous frame)

If an element of $\theta_{0}D_{0}+\mathcal{U}_{G}^{1}$ is an MC-element of any one of $\mathcal{E}$, $\mathcal{A}_{\textnormal{bounce}}$, $\mathcal{A}_{\textnormal{free}}$ then we have the implication

[TABLE]

Proof

If the $\beta_{i}^{0}$ vanish, then only the $\gamma_{i}^{6}$ appear as coefficients in front of $s_{i}^{2}s_{j}s_{k}\theta_{j}\theta_{k}D_{0}$ in the MC-equations, hence the $\gamma_{i}^{6}$ have to vanish. To check this, note that the bracket of any two of the eleven elements $D_{0},D_{i},\sigma_{0},\sigma_{i},\sigma_{jk}$ does not produce a $D_{0}$. ∎

Lemma 10 (Necessary conditions for $\textnormal{{{MC}}}(\mathcal{A}_{\textnormal{free}})$)

To be in

[TABLE]

requires $D_{0}(\beta_{i}^{0,1,2,3})=D_{0}(\gamma_{i}^{0,1,2,3,4,6})=D_{0}(e^{\mu_{j}+\mu_{k}}\gamma_{i}^{5})=0$ for cyclic $(i,j,k)$.

Proof

Required for the $\mathcal{A}_{\textnormal{free}}^{2}/{\mathcal{A}_{\textnormal{free}}^{2}}_{,G}$ part of the MC-equations. The parametrization in Lemma 8 was designed to get simple equations. ∎

It should be clear from this lemma that the $\gamma_{i}^{5}$ are three odd ducks, that potentially behave badly. Luckily, one can get rid of them using nilpotent automorphisms of333Crucially, these nilpotent automorphisms are induced from automorphisms of $\mathcal{P}_{\textnormal{free}}$ and hence extend to formal perturbation theory in $\mathcal{P}_{\textnormal{free}}$. $\mathcal{A}_{\textnormal{free}}$ that leave the affine gauge subspace invariant. See Section 9, particularly Remark 4. For the same reasons, the additional conditions in the next lemma do not entail a significant restriction of generality.

Below we associate to the frame $D_{i}$ the ‘structure functions’ $c_{jk}^{i}\in C^{\infty}(M,\mathbbm{R})$ by

[TABLE]

Definition 8

Let $\textnormal{{{MC}}}_{\textnormal{pre}}(\mathcal{A}_{\textnormal{free}})$ be the set of tuples

[TABLE]

with $g_{1}^{0},g_{2}^{0},g_{3}^{0},\xi\in C^{\infty}(S,\mathbbm{R})$ subject to the ‘constraint’ equations, for $(i,j,k)$ cyclic,

[TABLE]

with (7) understood. To every such tuple we associate $g_{i}^{1,2,3,4}\in C^{\infty}(S,\mathbbm{R})$ by

[TABLE]

for all cyclic $(i,j,k)$.

Lemma 11 (Sufficient conditions for $\textnormal{{{MC}}}(\mathcal{A}_{\textnormal{free}})$)

An element is in

[TABLE]

iff $\gamma_{i}^{0,1,2,3,4}=g_{i}^{0,1,2,3,4}$ for some element of $\textnormal{{{MC}}}_{\textnormal{pre}}(\mathcal{A}_{\textnormal{free}})$.

Proof

By direct calculation. Here are some remarks. Note that $\beta_{i}=D_{i}$ means $\beta_{i}^{0}=0$ and $\beta_{i}^{j}=\delta_{i}^{j}$ for $i,j=1,2,3$. Since $D_{1},D_{2},D_{3}$ is a frame, some constraint equations amount to the Jacobi identity and are dropped. The MC-equations also imply $D_{j}(g_{k}^{2})-D_{k}(g_{j}^{2})=\textstyle\sum_{p=1}^{3}c_{jk}^{p}g_{p}^{2}$, which modulo the other MC-equations is equivalent to $g_{i}^{2}=D_{i}(\xi)$ by the Poincare lemma, since $S\simeq\mathbbm{R}^{3}$ is simply connected. ∎

Remark 2

In Lemma 11 a special class of MC-elements is given by requiring that $\gamma_{i}^{0,1}$ be constant and that $\gamma_{i}^{2,3,4,5,6}=0$. These elements will be called homogeneous. Here only the structure coefficients $c^{i}_{jk}$ with cyclic $(i,j,k)$ can be nonzero.

Definition 9

Let $\textnormal{{{MC}}}_{\textnormal{normal}}(\mathcal{A}_{\textnormal{free}})\subseteq\textnormal{{{MC}}}_{\textnormal{pre}}(\mathcal{A}_{\textnormal{free}})$ be the elements with

[TABLE]

for some $u\in C^{\infty}(S,\mathbbm{R})$ with $u>0$.

Lemma 12 (Normal form elements in $\textnormal{{{MC}}}(\mathcal{A}_{\textnormal{free}})$)

• (a)

Given an element of $\textnormal{{{MC}}}_{\textnormal{pre}}(\mathcal{A}_{\textnormal{free}})$, then for all $A,B\in C^{\infty}(S,\mathbbm{R})$ with $A,B>0$ and for all $\sigma\in\{-1,+1\}$ we have

[TABLE]

• (b)

For every element of $\textnormal{{{MC}}}_{\textnormal{pre}}(\mathcal{A}_{\textnormal{free}})$ that satisfies $g_{1}^{0}>0$ and $g_{2}^{0},g_{3}^{0}<0$ and $c_{23}^{1}\neq 0$ everywhere on $S$, there is a unique transformation as in (b) that yields an element of $\textnormal{{{MC}}}_{\textnormal{normal}}(\mathcal{A}_{\textnormal{free}})$. This yields a partial map, that we call normalization,

[TABLE]

Proof

Claim (a) by direct calculation; we will see later that these transformations actually come from automorphisms of $\mathcal{A}_{\textnormal{free}}$. For (b), equation (8) implies that the sum of any two $g_{i}^{0}$ is negative, so we must take $A=1/(2g_{1}^{0})$ and we get $u=-(g_{1}^{0}+g_{2}^{0})/g_{1}^{0}>0$, and we must choose $B$, $\sigma$ so that $\sigma AB^{2g_{1}^{0}}c_{23}^{1}=-2$. ∎

Lemma 13

Given an element in $\textnormal{{{MC}}}_{\textnormal{normal}}(\mathcal{A}_{\textnormal{free}})$ with $u\neq\tfrac{1}{2},2$ everywhere then

[TABLE]

Proof

By direct unrevealing calculation, (8) and (9) hold. Alternatively and more rationally, note that the element of $\textnormal{{{MC}}}(\mathcal{A}_{\textnormal{bounce}})$ written out in Theorem 8.1 below is asymptotic to an element of $\textnormal{{{MC}}}(\mathcal{A}_{\textnormal{free}})$ as $t\to-\infty$ and another as $t\to+\infty$, using the module identifications $\mathcal{A}_{\textnormal{bounce}}\simeq\mathcal{U}\simeq\mathcal{A}_{\textnormal{free}}$. ∎

Theorem 6.1 (The inhomogeneous bounce map)

There is a partial map

[TABLE]

that assigns to every $X\in\textnormal{{{MC}}}_{\textnormal{normal}}(\mathcal{A}_{\textnormal{free}})$, with associated $g_{i}^{0,1,2,3,4}$ and $u$, the following element $\mathcal{B}(X)$, in each case provided only that the partial map $\mathcal{N}$ is defined:

• •

If $0<u<1$ and $u\neq\tfrac{1}{2}$ everywhere then set, with ${u}=\tfrac{1}{u}-1$,

[TABLE]

• •

If $u>1$ and $u\neq 2$ everywhere then set, with ${u}=u-1$,

[TABLE]

Proof

By Lemmas 12, 13. Permute $1,2,3\mapsto 3,1,2$ respectively $1,2,3\mapsto 3,2,1$ so that $\frac{1}{2}(1-u)>0$ respectively $\frac{1}{2}(1-\frac{1}{u})>0$ appear first, so that $\mathcal{N}$ has a chance of being defined. We have applied a transformation in Lemma 12 (a) to bring out the map $u\mapsto{u}$ explicitly (this map is well-known from the BKL papers). ∎

If $\mathcal{B}$ is not defined for some element, it may be defined after restricting to an open subset of $S\simeq\mathbbm{R}^{3}$. So iterating $\mathcal{B}$ may require restricting to subsets, again and again.

The conceptual interpretation of $\mathcal{B}$ is to come. As remarked in the proofs of Lemmas 12 and 13, it arises from taking past and future limits of elements in $\textnormal{{{MC}}}(\mathcal{A}_{\textnormal{bounce}})$ and applying automorphisms to bring them into a normal form. These automorphisms can be interpreted as selecting a distinguished coordinate system and frame.

7 The constraint equations

We parametrize the space of real analytic germs of solutions to the constraint equations in Lemma 11, at the origin $0\in\mathbbm{R}^{3}$. We confine ourselves to real analytic germs for simplicity. Every representative of a germ is defined on some open ball, so we get an actual solution by defining $S$ to be such a ball. This proves existence of elements in $\textnormal{{{MC}}}(\mathcal{A}_{\textnormal{free}})$ and, via Theorem 8.1 below, of $\textnormal{{{MC}}}(\mathcal{A}_{\textnormal{bounce}})$.

• •

The coordinates on $\mathbbm{R}^{3}$ are denoted $x^{1},x^{2},x^{3}$, the partial derivatives $\partial_{1},\partial_{2},\partial_{3}$.

• •

We assume

[TABLE]

where $D_{i}=D_{i}^{1}\partial_{1}+D_{i}^{2}\partial_{2}+D_{i}^{3}\partial_{3}$. There is always a local real analytic change of coordinates that maps the origin to itself and brings $D_{1}$, $D_{2}$, $D_{3}$ into this form.

• •

We assume, with $a$ and $u>0$ germs of real analytic functions,

[TABLE]

Lemma 14 (All real analytic germs of elements in $\textnormal{{{MC}}}(\mathcal{A}_{\textnormal{free}})$)

Let $\mathcal{G}$ be the set of all real analytic germs at $0\in\mathbbm{R}^{3}$ of functions

[TABLE]

that satisfy (7), (8), (9), (10), (11), in particular the $g_{i}^{0}$ are given in terms of $(a,u)$. Let $\mathcal{D}$ be the set of 6 real analytic germs at $0\in\mathbbm{R}^{3}$ and 9 real analytic germs at $0\in\{x\in\mathbbm{R}^{3}\mid x^{1}=0\}$, that by an abuse of notation we denote by:

• •

Germs $c_{31}^{i}$ and $c_{12}^{i}$ at $0\in\mathbbm{R}^{3}$.

• •

Germs $D_{2}^{i}$, $D_{3}^{i}$, $a$, $u$, $\xi$ at $0\in\{x\in\mathbbm{R}^{3}\mid x^{1}=0\}$, subject to $D_{2}^{1}(0),\,D_{3}^{1}(0),\,u(0)>0$.

Then the map $\mathcal{G}\to\mathcal{D}$, given by the identity on the first 6 and by restriction to the $x^{1}=0$ hypersurface for the other 9, is bijective.

Proof

We use Cauchy Kowalewski to construct a unique element in $\mathcal{G}$ for a given element in $\mathcal{D}$. Equation (8) is automatic for elements of $\mathcal{G}$ since we assume (11). Given an element of $\mathcal{D}$, equation (7) uniquely determines the restriction of $c_{23}^{i}$ to $x^{1}=0$. By the Jacobi identity for vector fields, we necessarily have in $\mathcal{G}$,

[TABLE]

We apply Cauchy Kowalewski to the following data:

• •

We use $x^{1}=0$ as the initial hypersurface.

• •

The unknown vector is $v=(D_{2}^{i},D_{3}^{i},c_{23}^{i},a,u,\xi)$. Its restriction to $x^{1}=0$ is given by the data in $\mathcal{D}$ or, in the case of $c_{23}^{i}$, forced by the data as explained above.

• •

The system of differential equations consists of equation (9), the equations for $[D_{3},D_{1}]$ and $[D_{1},D_{2}]$ in (7), and equation (12).

So there are 12 scalar unknowns and 12 scalar equations. Using $D_{2}^{i}(0),D_{3}^{i}(0),u(0)>0$ one can bring these 12 equations into the quasilinear form

[TABLE]

with $A$, $A_{2}$, $A_{3}$ real analytic in a neighborhood of $(0,v(0))$. In fact, (9) can be solved for $\partial_{1}a$, $\partial_{1}u$, $\partial_{1}\xi$ which follows from a short calculation. By Cauchy Kowalewski there exists a unique solution germ for these 12 equations. This is not yet the system defining $\mathcal{G}$. To see that the equation for $[D_{2},D_{3}]$ in (7) also holds, observe that it holds along $x^{1}=0$ by choice of $c_{23}^{i}$, and then everywhere by (12). ∎

8 Maurer-Cartan elements in $\mathcal{A}_{\textnormal{bounce}}$

The following theorem has already been referred to. It is the result of a computer calculation. It is somewhat remarkable that one has a closed form solution for elements in $\textnormal{{{MC}}}(\mathcal{A}_{\textnormal{bounce}})$, and we only found it after experimentation with different $\mathcal{E}_{G}$ spaces, and different over-parametrizations as in Lemma 8.

Theorem 8.1

Suppose an element of $\textnormal{{{MC}}}_{\textnormal{normal}}(\mathcal{A}_{\textnormal{free}})$ is given that satisfies $u\neq\tfrac{1}{2},2$ everywhere, with their associated $g_{i}^{0,1,2,3,4}\in C^{\infty}(S,\mathbbm{R})$. Then an element of

[TABLE]

is given, setting $\chi=\tfrac{1}{2}(1+\tanh t)$ and using the over-parametrization Lemma 8, by

[TABLE]

and

[TABLE]

with the abbreviations $\mathbf{A}_{1},\ldots,\mathbf{A}_{7}\in C^{\infty}(S,\mathbbm{R})$ given by

[TABLE]

Proof

By direct computer calculation, using the constraint equations in Definition 8. All denominators are nonzero since $u,\,u-2,\,2u-1\neq 0$ everywhere. ∎

Remark 3

This solution satisfies $\lim_{t\to-\infty}\gamma_{i}^{0,1,2,3,4}=g_{i}^{0,1,2,3,4}$ with exponentially fast convergence, and as $|t|\to\infty$ we have, again exponentially quickly,

[TABLE]

In this sense, one recovers the given element of $\textnormal{{{MC}}}_{\textnormal{normal}}(\mathcal{A}_{\textnormal{free}})$ as the past limit, and another element of $\textnormal{{{MC}}}_{\textnormal{pre}}(\mathcal{A}_{\textnormal{free}})$ as the future limit, see Lemma 13 and Theorem 6.1. This is convergence in a rather weak sense, future work will have to strengthen this in order to construct the speculative ($L_{\infty}$) scattering map mentioned in the introduction.

9 Some automorphisms of $\mathcal{P}_{\textnormal{bounce}}$ and $\mathcal{P}_{\textnormal{free}}$

Automorphisms provide conceptual background for various statements that have been made. From a broader perspective, the BKL problem is sufficiently complicated that fixing a gauge globally is unlikely to succeed, one will have to invoke automorphisms. The normalization operator $\mathcal{N}$ in Theorem 6.1 exemplifies this point of view.

Theorem 9.1 (The $1+3$ groupoid and isomorphisms)

• •

There is a groupoid, the ‘1+3 groupoid’, where an object is as in444The objects are in one-to-one correspondence with manifolds $S\simeq\mathbbm{R}^{3}$.* Definition 7, and a morphism is a map $\Phi:\mathbbm{R}\times S\to\mathbbm{R}\times S^{\prime}$, $(t,x)\mapsto(a(x)t+b(x),\varphi(x))$ together with a bundle isomorphism given on sections by $W^{\prime}\to W$, $\sum_{i}f_{i}\theta_{i}\mapsto\sum_{i}c(f_{i}\circ\Phi)\theta_{i}$, with $a,b,c\in C^{\infty}(S,\mathbbm{R})$ and $a,c>0$ and $\varphi\in\operatorname{Diff}^{\infty}(S,S^{\prime})$.*

• •

A functor from the 1+3 groupoid into the category of real gLa is given by the construction of $\mathcal{E}$. As a special case, morphisms in the 1+3 groupoid with $\varphi=\mathbbm{1}$ are mapped to the isomorphism induced by555Morphisms with $a=c=1$ and $b=0$ and general $\varphi$ are clear. :

[TABLE]

for all $f\in{C^{\infty}}$ and all $X\in\operatorname{Der}({C^{\infty}}(S,\mathbbm{R}))$ extended by $X(t)=0$. As always, the basis-dependent inclusion $\operatorname{Der}({C^{\infty}})\hookrightarrow\operatorname{CDer}(W)$ is implicit.

• •

This functor induces functors into the category of real filtered gLa, by the construction of the 1- or 2- or 3-index BKL filtrations. The corresponding isomorphisms of $\mathcal{E}[[s_{2},s_{3}]]$, $\mathcal{E}[[s_{1},s_{2},s_{3}]]$ restrict to isomorphisms of $\mathcal{P}_{\textnormal{free}}$, $\mathcal{P}_{\textnormal{bounce}}$ and these, in turn, induce isomorphisms of the associated gradeds $\mathcal{A}_{\textnormal{free}}$, $\mathcal{A}_{\textnormal{bounce}}$.

Proof

By direct calculation. ∎

The $1+3$ groupoid in Theorem 9.1 restricts the class of automorphisms, enough so that the BKL filtration can be defined, and then some more to fix a foliation by timelike lines, reflecting the structure (rank 1 frame) of MC-elements in the associated gradeds about which we perturb. Informally, of the original 11 gauge degrees of freedom, only 5 are left in Theorem 9.1, the remaining 6 are still present as nilpotent automorphisms.

Lemma 15 (Some nilpotent automorphisms of $\mathcal{A}_{\textnormal{free}}$)

Every $x\in\bigoplus_{\alpha\neq(0,0,0)}G_{\alpha}\mathcal{A}_{\textnormal{free}}^{0}$ generates a nilpotent automorphism $\exp([x,-])\in\operatorname{Aut}(\mathcal{A}_{\textnormal{free}})$ that need not preserve the $\alpha$-grading. Every such automorphism is induced by an $\mathbbm{R}[[s_{1},s_{2},s_{3}]]$-linear automorphism of $\mathcal{P}_{\textnormal{free}}$. In particular, for all $f\in{C^{\infty}}=C^{\infty}(\mathbbm{R}\times S,\mathbbm{R})$ we have on $\mathcal{A}_{\textnormal{free}}^{1}$, with formulas only given for some elements of $\mathcal{A}_{\textnormal{free}}^{1}$, for all cyclic $(i,j,k)$:

• (i)

The ${C^{\infty}}$-linear automorphism $\exp([fs_{j}s_{k}\sigma_{i},-])$ acts as:

[TABLE]

• (ii)

The ${C^{\infty}}$-linear automorphism $\exp([fs_{j}s_{k}\sigma_{jk},-])$ acts as:

[TABLE]

These automorphisms map a given element of $\theta_{0}D_{0}+{\mathcal{A}_{\textnormal{free}}^{1}}_{,G}$ back to this space if and only if $f$ satisfies, respectively:

• (i)

$D_{0}(f)-(\gamma_{j}^{0}+\gamma_{k}^{0})f=0$. Equivalently, $D_{0}(e^{-\mu_{j}-\mu_{k}}f)=0$ if $D_{0}(\mu_{i})=\gamma_{i}^{0}$ for all $i$.

• (ii)

$D_{0}(f)=0$.

Proof

A preimage of $x$ under the surjection $\mathcal{P}_{\textnormal{free}}^{0}\to\mathcal{A}_{\textnormal{free}}^{0}$ generates a (formal power series) automorphism of $\mathcal{P}_{\textnormal{free}}$ by exponentiation that induces the nilpotent automorphism generated by $x$. The elements $x\in\mathcal{A}_{\textnormal{free}}^{0}$ in (i), (ii) satisfy $\exp([x,-])=\mathbbm{1}+[x,-]$ when restricted to $\mathcal{A}_{\textnormal{free}}^{1}$ since $G_{p_{1}p_{2}p_{3}}\mathcal{A}_{\textnormal{free}}^{1}=0$ if two of $p_{1}p_{2}p_{3}$ are $\geq 2$. We have not written out how these nilpotent automorphisms act on all elements of $\mathcal{A}_{\textnormal{free}}^{1}$, for example what $s_{i}^{2}s_{j}s_{k}$ terms they generate, but such terms are in ${\mathcal{A}_{\textnormal{free}}^{1}}_{,G}$ anyway. ∎

Remark 4

By Lemma 15, every MC-element as in Lemma 10 can be brought, via a nilpotent automorphism of $\mathcal{A}_{\textnormal{free}}$, into the more specific form in Lemma 11, provided the $\smash{\gamma_{j}^{0}-\gamma_{k}^{0}}$ do not vanish, which informally means that the MC-element has three preferred spatial directions. Use (i) to set the $\beta_{i}^{0}$ to zero which yields a synchronous frame and makes the $\gamma_{i}^{6}$ zero by Lemma 9; use (ii) to set the $\gamma_{i}^{5}$ to zero, informally to align the MC-element with the orthogonal decomposition of $W$ that defines the filtration; finally choose $\mu_{i}$ and $D_{i}$ appropriately.

Lemma 16 (Automorphism giving the transformation in Lemma 12)

The transformation (a) in Lemma 12 is given by composing the following automorphisms of $\mathcal{A}_{\textnormal{free}}$, induced by automorphisms of $\mathcal{P}_{\textnormal{free}}$, in this specific order:

An automorphism with $a=c=A$ and $b=\log B$ as in Theorem 9.1.

• 2)

The nilpotent $\exp([f_{1}s_{2}s_{3}\sigma_{1},-])\circ\exp([f_{2}s_{3}s_{1}\sigma_{2},-])\circ\exp([f_{3}s_{1}s_{2}\sigma_{3},-])$ with

[TABLE]

• 3)

Reflection by $\sigma\mathbbm{1}$ (with $\sigma$ as in Lemma 12) of the basis elements $\theta_{1},\theta_{2},\theta_{3}$.

Proof

We use the over-parametrization in Lemma 8 even if intermediate results lie outside its range. Step 1) uses $a=c$ to keep $\theta_{0}D_{0}$ in place. The choice $a=c=A$ brings the $\gamma_{i}^{0}$ into the desired form, and $b=\log B$ brings the $\beta_{i}$ into the desired form. In Step 2) the $f_{i}$ are uniquely determined by requiring that they remove any $s_{j}s_{k}\theta_{i}D_{0}$ terms that we may have inadvertently introduced in Step 1), so now $\beta_{i}^{0}=0$. Any $s_{j}s_{k}\theta_{i}\sigma_{0}$ that we may have inadvertently introduced in Step 1) only appear in the combination $s_{j}s_{k}(\theta_{0}\sigma_{i}+\theta_{i}\sigma_{0})$, so after Step 2) we are back in the range of Lemma 8. The $\gamma_{i}^{5}$ are zero because no $s_{j}s_{k}(\theta_{k}\sigma_{j}+\theta_{j}\sigma_{k})$ was introduced. The $\gamma_{i}^{6}$ are zero by Lemma 9. Note that Step 2) did not change $\gamma_{i}^{0}$ or $\beta_{i}$. Step 3) is clear. ∎

Appendix A Review of the spectral sequence of a filtered complex

We review spectral sequences in a concrete form, suited for our application. Suppose we have a complex in the most mundane sense, namely a vector space $V$ and a $d\in\operatorname{End}(V)$ with $d^{2}=0$. Its homology is $H(d)=\ker d/\operatorname{image}d$.

Suppose a bounded non-increasing filtration $V_{\geq i}$ is fixed that respects the differential in the sense that $dV_{\geq i}\subseteq V_{\geq i}$. For concreteness, suppose the filtration comes from a grading, so $V_{\geq i}=V_{i}\oplus V_{i+1}\oplus\ldots\oplus V_{P}$ for some grading $V=V_{0}\oplus V_{1}\oplus\ldots\oplus V_{P}$. Then $d$ is a lower triangular matrix with entries $d_{ij}\in\operatorname{Hom}(V_{j},V_{i})$. For every $p>0$ let $d_{ip}^{+}$ be the $p\times p$ submatrix of $d$ with upper left corner $d_{ii}$. For example

[TABLE]

Likewise, let $d_{ip}^{-}$ be the $p\times p$ submatrix with lower right corner $d_{ii}$.

Lemma 17 (The spectral sequence)

Set $Z_{i0}=V_{i}$ and $B_{i0}=0$ and for $p>0$ set

[TABLE]

Then $B_{ip}\subseteq Z_{ip}\subseteq V_{i}$. Define $\bullet_{ip}=Z_{ip}/B_{ip}$. A well-defined $D_{i+p,i}\in\operatorname{Hom}(\bullet_{ip},\bullet_{i+p,p})$ is given by $D_{ii}=d_{ii}$ if $p=0$, and if $p>0$ induced by the map $Z_{ip}\to\bullet_{i+p,p}$ given by

[TABLE]

with any filler $\ast$ as in the definition of $Z_{ip}$. For every fixed $p$, the vector space $\bigoplus_{i}\bullet_{ip}$ together with the maps $(D_{i+p,i})_{i}$ (combined in a block matrix with entries only along the $p$-th subdiagonal) is a complex. The homology at $\bullet_{ip}$ is equal to $\bullet_{i,p+1}$.

Proof

By working out the ramifications of $d^{2}=0$. ∎

Even though one iteratively takes homology, each $\bullet_{ip}$ is a concrete space, namely a subquotient of $V_{i}$. This is because, a subquotient of a subquotient is a subquotient.

Theorem A.1 (Main fact)

Let $H(d)_{\geq i}\subseteq H(d)$ be all elements that have a representative in $V_{\geq i}$. Then the associated graded, $\operatorname{Gr}H(d)$, is isomorphic, as a graded vector space, to the last page $\bigoplus_{i}\bullet_{i,P+1}$ of the spectral sequence.

Proof

The map $Z_{i,P+1}\to H(d)_{\geq i}/H(d)_{\geq i+1}$ defined by $x\mapsto(\begin{smallmatrix}x\\ \ast\end{smallmatrix})$, with $\ast$ any filler as in the definition of $Z_{i,P+1}$, is independent of $\ast$, surjective, and has kernel $B_{i,P+1}$. ∎

The definition of $\bullet_{ip}$ given above makes the proof of Theorem A.1 straightforward. In practice, it is important to know the differentials on the first few pages.

Lemma 18 (The first few pages)

Choose $h_{i}\in\operatorname{End}(V_{i})$ such that $d_{ii}h_{i}d_{ii}=d_{ii}$. Then:

• •

$D_{ii}=d_{ii}$.

• •

$D_{i+1,i}$* is induced by $d_{i+1,i}$.*

• •

$D_{i+2,i}$* is induced by $d_{i+2,i}-d_{i+2,i+1}h_{i+1}d_{i+1,i}$.*

Proof

Omitted. ∎

If $V$ carries an additional homological grading, the construction is compatible with it. This is useful in practice, but irrelevant for the logic of spectral sequences.