Tensor calculus

alkahest.tensor computes curvature from a metric on a coordinate chart and offers an abstract-index layer with Einstein summation.

import alkahest as ak
from alkahest import sin, cos

p = ak.ExprPool()
t, r, th, ph, M, a = (p.symbol(s) for s in ("t", "r", "theta", "phi", "M", "a"))

# Kerr, Boyer–Lindquist
sigma = r**2 + a**2 * cos(th)**2
delta = r**2 - 2*M*r + a**2
gtt = -(1 - 2*M*r/sigma)
gtp = -2*M*r*a*sin(th)**2/sigma
gpp = (r**2 + a**2 + 2*M*r*a**2*sin(th)**2/sigma) * sin(th)**2
g = [[gtt, 0, 0, gtp], [0, sigma/delta, 0, 0], [0, 0, sigma, 0], [gtp, 0, 0, gpp]]

m = ak.Metric(g, [t, r, th, ph])
m.is_ricci_flat()      # True — proved, not sampled
m.timings()            # seconds per stage: convert, inverse, christoffel, riemann, ricci

#The metric layer

Metric(g, coords) takes an n x n symmetric list of lists (or a Matrix) and n distinct coordinate symbols. Entries may contain parameters, sin/cos/tan/…, sinh/cosh/…, exp, log, radicals and undefined functions of the coordinates — the FLRW scale factor is just p.func("a", [t]), and its derivatives appear as Derivative(a(t), t).

Sign conventions are MTW / Carroll / Wald: R^a_bcd = ∂_c Γ^a_db − ∂_d Γ^a_cb + Γ^a_ce Γ^e_db − Γ^a_de Γ^e_cb, so the round 2-sphere of radius ρ has R = 2/ρ² and Schwarzschild has K = 48 M²/r⁶.

#Why the zeros are exact

Every component lives in one exact field: a quotient of FLINT integer polynomials over the coordinates, the parameters and one generator per transcendental atom. The relations sin² = 1 − cos², cosh² − sinh² = 1, i² = −1 and (√b)² = b are applied after every operation, and each denominator is multiplied by its conjugate until it is free of sin, radicals and i. Partial derivatives act on the generators by their own rules (∂ sin v = cos v · ∂v, ∂ √b = √b · ∂b / 2b, ∂ a(t) = Derivative(a(t), t)), which respect the relations. The resulting normal form is canonical: two components are equal exactly when their forms are identical, so a component that vanishes is the literal 0 — no sampling, no floats. That is what lets is_ricci_flat() prove the Kerr metric is a vacuum solution, which the cancel + simplify_trig route could not.

Results are printed factored over ℤ, with (cos θ − 1)(cos θ + 1) folded back to −sin² θ (so Γ^φ_θφ reads cos θ · sin θ⁻¹).

Only independent components are computed (Γ_abc for b ≤ c, R_abcd for a < b, c < d, (a,b) ≤ (c,d), R_ab for a ≤ b); the rest are filled in by symmetry. Each stage is cached.

#Abstract indices

a, b, c, d = ak.tensor_indices("a b c d")
Ric, g = m.ricci(), m.as_tensor()
R = Ric[a, -a]                         # raise one slot with g^ab, contract: the Ricci scalar
G = Ric[-a, -b] - g[-a, -b] * R / 2    # same components as m.einstein()
m.riemann()[c, -a, -c, -b]             # R^c_acb = R_ab
Rl = m.riemann_lower()[-a, -b, -c, -d]
Rl.antisymmetrize(b, c, d).is_zero()   # first Bianchi identity: True

-a is a lower index and +a (or a) an upper one. Indexing a Tensor in the other variance than it is stored raises or lowers that slot with the metric; a repeated index — once up, once down — is summed. IndexedTensor supports +, -, * (outer product plus contraction), scalar * and /, symmetrize, antisymmetrize, is_zero(), components and tensor(). Components are kept in the metric's canonical form.

#Refusals

TensorError with a stable code:

An active Budget is checked once per component; a trip raises BudgetExceededError.