Curvature 2-form ( $R^{a}_{b}$ )

The curvature two-forms are

R^{a}_{b} = d ω^{a}_{b} + ω^{a}_{c} \land ω^{c}_{b} = \frac{1}{2} R^{a}_{b c d} θ^{c} \land θ^{d} .

Cartan curvature forms for a static spherical metric ( $g = - V d t^{2} + V^{- 1} d r^{2} + r^{2} d Ω^{2}$ )

In the exterior orthonormal coframe

E^{0} = V d t

E^{1} = d r / V

E^{2} = r d θ

E^{3} = r sin θ d ϕ

, use

Ω^{a}_{b} = \frac{1}{2} R^{a}_{b c d} E^{c} \land E^{d}

. The independent mixed-index curvature 2-forms are

Ω^{0}_{1} = - V^{''} E^{0} \land E^{1} /2

Ω^{0}_{A} = - V^{'} E^{0} \land E^{A} / (2 r)

Ω^{1}_{A} = - V^{'} E^{1} \land E^{A} / (2 r)

for

A = 2, 3

, and

Ω^{2}_{3} = (1 - V) E^{2} \land E^{3} / r^{2}

. The orthonormal Ricci tensor is diagonal with entries

V^{''} /2 + V^{'} / r

- V^{''} /2 - V^{'} / r

, and twice

(1 - V - r V^{'}) / r^{2}

V = 1 - 2 M / r + r^{2} / l^{2}

The curvature operator is diagonal in an orthonormal coframe

{e^{a}}

when every curvature 2-form satisfies

R^{ab} = K_{ab} e^{a} \land e^{b}

. Its matrix on the corresponding basis of 2-forms is then diagonal.

R^{ab}

is proportional to

e^{a} \land e^{b}

for every pair, then a curvature component can be nonzero only when its two index pairs describe the same coordinate two-plane. The contraction

R_{b d} = R^{a}_{ba d}

therefore vanishes for

b \neq = d

, so the Ricci tensor is diagonal in the same frame.

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