Diagonal Bianchi I model

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 (${\mathrm{A}\mathrm{C}}_{\omega }$)