{e_{a}}

satisfying

g (e_{a}, e_{b}) = η_{ab}

. The metric determines it only up to a point-dependent Lorentz transformation, with orientation and time orientation optionally restricting the component of the Lorentz group. The orthonormal coframe in spacetime is defined by

θ^{a} (e_{b}) = δ_{b}^{a},

and the metric is

g = η_{ab} θ^{a} \otimes θ^{b} .

b

Solution

The connection 1-forms are defined by

\nabla e_{b} = ω^{a}_{b} \otimes e_{a}

ω_{ab} = - ω_{ba}

, while vanishing torsion gives Cartan's first structure equation

d θ^{a} + ω^{a}_{b} \land θ^{b} = 0.

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},

which displays their relation to the Riemann curvature tensor.

Choose

θ^{0} = f d t, θ^{1} = \frac{d r}{f}, θ^{2} = r d ϑ, θ^{3} = r sin ϑ d φ .

Cartan's first structure equation gives the independent nonzero connection 1-forms

ω^{0}_{1} = \frac{f ^{'}}{2 f} θ^{0}, ω^{2}_{1} = \frac{f}{r} θ^{2}, ω^{3}_{1} = \frac{f}{r} θ^{3}, ω^{3}_{2} = \frac{c o t ϑ}{r} θ^{3},

with the remaining forms obtained by metric antisymmetry. Cartan's second structure equation then gives

R^{0}_{1} R^{0}_{2} R^{1}_{2} R^{2}_{3} = - \frac{f ^{''}}{2} θ^{0} \land θ^{1}, = - \frac{f ^{'}}{2 r} θ^{0} \land θ^{2}, = - \frac{f ^{'}}{2 r} θ^{1} \land θ^{2}, = \frac{1 - f}{r ^{2}} θ^{2} \land θ^{3} . R^{0}_{3} R^{1}_{3} = - \frac{f ^{'}}{2 r} θ^{0} \land θ^{3}, = - \frac{f ^{'}}{2 r} θ^{1} \land θ^{3},

These signs follow the curvature convention stated in Question 1.

k = e_{0} + e_{2}

, which has a nonzero angular component. The curvature forms give

R_{ab} k^{a} k^{b} = R_{00} + R_{22} = \frac{1}{2} f^{''} - \frac{f - 1}{r ^{2}} .

R_{ab} k^{a} k^{b} \geq 0

. Therefore

f^{''} \geq \frac{2 ( f - 1 )}{r ^{2}} .

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