Riemann Curvature of the FLRW Metric

The Friedmann--Lemaître--Robertson--Walker geometry models a homogeneous, isotropic universe. In comoving coordinates $(w,r,\theta,\phi)$ and units $c=1$,

$$ ds^2=-dw^2+a(w)^2\left( \frac{dr^2}{1-Kr^2}+r^2d\theta^2+r^2\sin^2\theta\,d\phi^2 \right). $$

The scale factor $a(w)$ is left as an arbitrary symbolic function and $K$ is the constant spatial-curvature parameter.

Metric data

Write $W(r)=(1-Kr^2)^{-1}$. The inverse metric is entered explicitly so that selected connection and curvature components stay responsive.

declare symbol w, r, θ, φ, K: MathValue

def x : Vector MathValue := [| w, r, θ, φ |]
def a : MathValue := function (w)

def W (r: MathValue) : MathValue := 1 / `(1 - K * r^2)

def g_i_j : Matrix MathValue :=
  [| [| -1, 0, 0, 0 |]
   , [| 0, a^2 * W r, 0, 0 |]
   , [| 0, 0, a^2 * r^2, 0 |]
   , [| 0, 0, 0, a^2 * r^2 * (sin θ)^2 |]
   |]_i_j

def g~i~j : Matrix MathValue :=
  [| [| -1, 0, 0, 0 |]
   , [| 0, 1 / (a^2 * W r), 0, 0 |]
   , [| 0, 0, 1 / (a^2 * r^2), 0 |]
   , [| 0, 0, 0, 1 / (a^2 * r^2 * (sin θ)^2) |]
   |]~i~j

W r
$(-K r^{2} + 1)^{-1}$
g_#_#
$\begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & (-K r^{2} + 1)^{-1} a^{2} & 0 & 0 \\ 0 & 0 & a^{2} r^{2} & 0 \\ 0 & 0 & 0 & \sin(θ)^{2} a^{2} r^{2} \\ \end{pmatrix}_{\#\#}^{\;\;}$
g~#~#
$\begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & (-K r^{2} + 1) a^{-2} & 0 & 0 \\ 0 & 0 & a^{-2} r^{-2} & 0 \\ 0 & 0 & 0 & \sin(θ)^{-2} a^{-2} r^{-2} \\ \end{pmatrix}_{\;\;}^{\#\#}$

Expansion enters the connection

Time derivatives of the spatial metric generate Christoffel symbols proportional to $a'(w)$. In particular, radial expansion appears in both $\Gamma^w{}_{rr}$ and $\Gamma^r{}_{wr}$.

def Γ_i_j_k : Tensor MathValue :=
  (1 / 2) * (∂/∂ g_i_k x~j + ∂/∂ g_i_j x~k - ∂/∂ g_j_k x~i)

def Γ~i_j_k : Tensor MathValue := withSymbols [m]
  g~i~m . Γ_m_j_k

Γ~1_2_2
$(-K r^{2} + 1)^{-1} a \frac{\partial a}{\partial 1}$
Γ~2_1_2
$a^{-1} \frac{\partial a}{\partial 1}$

Riemann tensor and contractions

Components mixing time and space measure the acceleration of the scale factor; purely spatial components combine $(a')^2$ with $K$. The complete contractions are

$$ \operatorname{Ric}_{ij}=R^m{}_{imj},\qquad \mathcal R=g^{ij}\operatorname{Ric}_{ij}. $$

def R~i_j_k_l : Tensor MathValue := withSymbols [m]
  ∂/∂ Γ~i_j_l x~k - ∂/∂ Γ~i_j_k x~l
    + Γ~m_j_l . Γ~i_m_k - Γ~m_j_k . Γ~i_m_l

def Ric_i_j : Matrix MathValue := withSymbols [m]
  sum (contract R~m_i_m_j)

def scalarCurvature : MathValue := withSymbols [i, j]
  expandAll' (g~i~j . Ric_i_j)

R~1_2_1_2
$(-K r^{2} + 1)^{-1} a \frac{\partial^2 a}{\partial 1^2}$
R~2_3_2_3
$\frac{\partial a}{\partial 1}^{2} r^{2} + K r^{2}$

Scalar curvature and interpretation

With this convention, the expected scalar is

$$ \mathcal R =6\left(\frac{a''(w)}{a(w)}+ \frac{a'(w)^2+K}{a(w)^2}\right) =\frac{6\bigl(a''a+(a')^2+K\bigr)}{a^2}. $$

The selected Riemann components expose the two geometric ingredients: accelerated expansion and curvature of spatial slices. Evaluating scalarCurvature asks the CAS to form and simplify the full contraction for an arbitrary function $a$ and can take several minutes, so it is deliberately a definition rather than an automatic output cell.

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