Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 309 / 4 / d

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- Solution d

Solution

Substituting the forms from part (b) into Cartan's second structure equation gives the independent mixed-index curvature 2-forms

Θ^{0}_{1} = \frac{2 m}{r ^{3}} e^{0} \land e^{1}, Θ^{0}_{2} = - \frac{m}{r ^{3}} e^{0} \land e^{2}, Θ^{0}_{3} = - \frac{m}{r ^{3}} e^{0} \land e^{3},

Θ^{1}_{2} = - \frac{m}{r ^{3}} e^{1} \land e^{2}, Θ^{1}_{3} = - \frac{m}{r ^{3}} e^{1} \land e^{3}, Θ^{2}_{3} = \frac{2 m}{r ^{3}} e^{2} \land e^{3} .

Every form is proportional to its corresponding basis 2-form, so the curvature operator is diagonal. The diagonal Ricci tensor components are contractions of these sectional curvature coefficients. In each case the coefficients cancel in the pattern

2 - 1 - 1 = 0

, with the Lorentzian sign included when the time direction is contracted. Hence

R_{μν} = 0 .

This is consistent with Schwarzschild spacetime being a vacuum solution away from

r = 0

.

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