riemann-curvature-tensor-of-FLRW-metric.egi

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

-- Parameters
def x : Vector MathValue := [| w, r, θ, φ |]

-- Scale factor function a(w)
def a := function (w)

assertEqual "x" x [| w, r, θ, φ |]

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

assertEqual "W r = 1/(1-Kr^2)"
  (W r)
  (1 / `(1 - K * r^2))

-- Metric tensor
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 |]
   |]

def g~i~j := M.inverse g_#_#

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

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

-- Riemann curvature
def R~i_j_k_l := 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

-- Ricci curvature
def Ric_i_j := withSymbols [m]
  sum (contract R~m_i_m_j)

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

-- Expected scalar curvature of the FLRW metric (in units c=1):
--   R = 6 (a''(w) a + (a'(w))^2 + K) / a^2
-- The full computation is expensive (>5min on this machine); evaluating
-- `scalarCurvature` is left as the file's final value, not asserted, to
-- keep `cabal test` runtime reasonable. To verify the expected form
-- explicitly, run: `cabal run egison -- -t <this-file>` and compare.