Cosmological dynamics and anisotropic shear

Let $c>0$ be a fixed conversion constant in metres per second. Let $I\subseteq \mathbb{R}$ be a nonempty open interval of physical time in seconds, and let ${\mathbb{T}}_{2\pi }^3={\mathbb{R}}^3\mathrm{/}(2\pi \mathbb{Z}{)}^3$ have dimensionless angular coordinates $x^1,x^2,x^3$. Its quotient charts are local lifts to ${\mathbb{R}}^3$; chart changes are translations by elements of $(2\pi \mathbb{Z}{)}^3$. Give $M=I\times {\mathbb{T}}_{2\pi }^3$ the product smooth structure, so $M$ is a smooth four-manifold without boundary. Here a $C^2$ time-oriented Lorentzian metric datum means a symmetric covariant $2$-tensor field $g$ with $C^2$ components in smooth charts, whose value at each point has inertia $(3,1,0)$, together with a smooth vector field $N$ satisfying $g(N,N)<0$ everywhere. This is the explicit $C^2$ variant of the pointwise signature and time-orientation conditions in Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model; that supplier's metric regularity itself is smooth. Choose positive $C^2$ functions $a_i:I\to (0,\mathrm{\infty })$, $i=1,2,3$, with values in metres, and set

$g=-c^{2}d{t}^2+{\sum }_{i=1}^3{a}_i(t{)}^2(d{x}^i{)}^2,N:={\mathrm{\partial }}_t\mathrm{.}$

In every lifted product chart, $g$ has matrix $\mathrm{diag}(-c^2,a_1^2,a_2^2,a_3^2)$, so it is symmetric, $C^2$, and has inertia $(3,1,0)$; $N$ is smooth and $g(N,N)=-c^2<0$. Thus $(M,g,N)$ is a $C^2$ time-oriented Lorentzian metric datum in the stated sense. This diagonal, spatially homogeneous metric datum is called a diagonal Bianchi I datum here; homogeneity means its coefficients are independent of the torus coordinates, and no isotropy is required. Define the directional expansion rates, their mean, the spatial volume scale and the squared directional shear by

$H_i:=\frac{{\dot{a}}_i}{a_i},H:=\frac{1}{3}{\sum }_{i=1}^3{H}_i,V:=a_1{a}_2{a}_3,{\sigma }^2:=\frac{1}{2}{\sum }_{i=1}^3(H_i-H{)}^2\mathrm{.}$

Here ${\dot{a}}_i=d{a}_i\mathrm{/}dt$; $H_i$ and $H$ have units ${\mathrm{s}}^{-1}$, $V$ has units ${\mathrm{m}}^3$, and ${\sigma }^2$ has units ${\mathrm{s}}^{-2}$. The symbol ${\sigma }^2$ denotes a nonnegative scalar defined by this formula, not a tensor norm on a spacetime frame. This is a geometric ansatz and makes no field-equation or observational claim. The product and quotient constructions carry the inherited axiom of countable choice ${\mathrm{A}\mathrm{C}}_{\omega }$ and use no stronger choice principle.

For comparison with smooth Lorentzian geometry, use the curvature convention $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z$ and the component order $R({\mathrm{\partial }}_c,{\mathrm{\partial }}_d){\mathrm{\partial }}_b=R^a{}_{bcd}{\mathrm{\partial }}_a$. The local connection and Ricci components for this $C^2$ metric are defined separately.

The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s

Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model

The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

For a diagonal Bianchi I datum, use local quotient coordinates $x^0=t,x^1,x^2,x^3$, with indices $a,b,c,d\in \{0,1,2,3\}$, and let $(g_{ab})$ and $(g^{ab})$ be the displayed metric matrix and its inverse. At this $C^2$ regularity, define

${\mathrm{\Gamma }}^a{}_{bc}:=\frac{1}{2}{g}^{ad}({\mathrm{\partial }}_b{g}_{dc}+{\mathrm{\partial }}_c{g}_{bd}-{\mathrm{\partial }}_d{g}_{bc}),$

$R^a{}_{bcd}:={\mathrm{\partial }}_c{\mathrm{\Gamma }}^a{}_{db}-{\mathrm{\partial }}_d{\mathrm{\Gamma }}^a{}_{cb}+{\sum }_{e=0}^3({\mathrm{\Gamma }}^a{}_{ce}{\mathrm{\Gamma }}^e{}_{db}-{\mathrm{\Gamma }}^a{}_{de}{\mathrm{\Gamma }}^e{}_{cb}),$

${\mathrm{Ric}}_{bd}:={\sum }_{a=0}^3{R}^a{}_{bad},\mathcal{R}:={\sum }_{b,d=0}^3{g}^{bd}{\mathrm{Ric}}_{bd},G_{ab}:={\mathrm{Ric}}_{ab}-\frac{1}{2}\mathcal{R}{g}_{ab},G^a{}_b:={\sum }_{c=0}^3{g}^{ac}{G}_{cb}\mathrm{.}$

Repeated upper and lower indices in the first two displays are summed over $\{0,1,2,3\}$. The first formula gives coefficients symmetric in $b,c$ and, by expansion,

${\mathrm{\Gamma }}^d{}_{ca}{g}_{db}+{\mathrm{\Gamma }}^d{}_{cb}{g}_{ad}=\frac{1}{2}({\mathrm{\partial }}_c{g}_{ab}+{\mathrm{\partial }}_a{g}_{cb}-{\mathrm{\partial }}_b{g}_{ca})+\frac{1}{2}({\mathrm{\partial }}_c{g}_{ab}+{\mathrm{\partial }}_b{g}_{ac}-{\mathrm{\partial }}_a{g}_{cb})={\mathrm{\partial }}_c{g}_{ab}\mathrm{.}$

Thus these coefficients define a torsion-free, metric-compatible connection in every local quotient chart. It is unique: if ${\mathrm{\Gamma }}^{\mathrm{\prime }}$ is another such connection, the lowered difference $C_{abc}:=g_{ad}({\mathrm{\Gamma }}^{\mathrm{\prime }d}{}_{bc}-{\mathrm{\Gamma }}^d{}_{bc})$ is symmetric in its last two indices by torsion freedom and skew in its first two by metric compatibility. Hence

$C_{abc}=C_{acb}=-C_{cab}=-C_{cba}=C_{bca}=C_{bac}=-C_{abc},$

so $C=0$. The curvature sign and index order agree with the smooth convention cited in the references below. Since $g$ is $C^2$, the coefficients $\mathrm{\Gamma }$ are $C^1$ and the curvature and Ricci components are continuous. On overlaps, two local lifts differ by a constant translation: the Jacobian is the identity and all second derivatives of the transition map vanish. The metric arrays and their derivatives therefore agree, so the displayed connection and curvature arrays patch consistently over the quotient atlas. This is a local $C^2$ extension for this fixed atlas; no general low-regularity tensoriality theorem is being assumed. It inherits only ${\mathrm{A}\mathrm{C}}_{\omega }$ from the torus and product definitions.

Diagonal Bianchi I model

The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s

The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors

The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)

For a diagonal Bianchi I datum, write $H_{\mathrm{\Sigma }}:=H_1+H_2+H_3=3H$. In every local quotient chart, the only nonzero Christoffel symbols are ${\mathrm{\Gamma }}^0{}_{ii}=a_i^2{H}_i\mathrm{/}{c}^2$ and ${\mathrm{\Gamma }}^i{}_{0i}={\mathrm{\Gamma }}^i{}_{i0}=H_i$. The possibly nonzero lowered curvature components are exactly those obtained from the following values by the displayed sign reversals:

$R_{0i0i}=R_{i0i0}=-a_i^2({\dot{H}}_i+H_i^2),R_{0ii0}=R_{i00i}=a_i^2({\dot{H}}_i+H_i^2),$

$R_{ijij}=R_{jiji}=\frac{a_i^2{a}_j^2}{c^2}{H}_i{H}_j,R_{ijji}=R_{jiij}=-\frac{a_i^2{a}_j^2}{c^2}{H}_i{H}_j(1\le i<j\le 3)\mathrm{.}$

The Ricci components are

${\mathrm{Ric}}_{00}=-{\sum }_{i=1}^3({\dot{H}}_i+H_i^2),{\mathrm{Ric}}_{ii}=\frac{a_i^2}{c^2}({\dot{H}}_i+H_i{H}_{\mathrm{\Sigma }}),$

and every off-diagonal Ricci component is zero.

Diagonal Bianchi I model

C2 curvature for the diagonal Bianchi I quotient metric

Let $a_i(t)=A_i(t\mathrm{/}{t}_0{)}^{p_i}$ on a nonempty open interval $I\subseteq (0,\mathrm{\infty })$, where $A_i>0$ are lengths in metres, $t_0>0$ is a time in seconds, and $p_i\in \mathbb{R}$ are dimensionless exponents. These $p_i$ are called Kasner exponents for this power-law branch; define their sums $P:={\sum }_{i=1}^3{p}_i$ and $Q:={\sum }_{i=1}^3{p}_i^2$. For a nonstatic datum, meaning that not all $p_i$ vanish, the geometric vacuum equation $\mathrm{Ric}=0$ in the $C^2$ curvature definition of this page holds exactly when $P=Q=1$. If all $p_i=0$, then the full curvature tensor vanishes, so the metric is flat in the curvature sense. These are exact statements about the specified finite-dimensional metrics; no singularity or generic cosmological conclusion follows.

Diagonal Bianchi I model

C2 curvature for the diagonal Bianchi I quotient metric

Curvature components of the diagonal Bianchi I metric

For a diagonal Bianchi I datum, let $\mathrm{\Lambda }$ be a constant in ${\mathrm{m}}^{-2}$ and let $Q_i:I\to \mathbb{R}$ be continuous functions in ${\mathrm{m}}^{-2}$. Suppose the spatial diagonal equations

$G^i{}_i+\mathrm{\Lambda }=Q_i(t)(i=1,2,3;\text{ no sum over }i)$

hold, where $G^i{}_i$ is defined by the local $C^2$ curvature convention on this page. With ${\mathrm{\Delta }}_{ij}:=H_i-H_j$ and $H_{\mathrm{\Sigma }}:=3H$, subtracting the equations gives

${\dot{\mathrm{\Delta }}}_{ij}+3H{\mathrm{\Delta }}_{ij}=c^2(Q_i-Q_j)\mathrm{.}$

If the diagonal source is isotropic, meaning $Q_1=Q_2=Q_3$, then $V{\mathrm{\Delta }}_{ij}$ is constant for each pair, and there is a constant $C\ge 0$ with units ${\mathrm{m}}^6{\mathrm{s}}^{-2}$ such that ${\sigma }^2=C{V}^{-2}$ throughout $I$. For unequal diagonal sources the right-hand side remains; this proposition makes no unsourced $V^{-2}$ claim in that case.

Diagonal Bianchi I model

C2 curvature for the diagonal Bianchi I quotient metric

Curvature components of the diagonal Bianchi I metric