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The curvature operator is diagonal in an orthonormal coframe $\{e^a\}$ when every curvature 2-form satisfies ${\mathcal{R}}^{ab}=K_{ab}{e}^a\wedge e^b$. Its matrix on the corresponding basis of 2-forms is then diagonal.

Substituting the forms from part (b) into Cartan's second structure equation gives the independent mixed-index curvature 2-formsEvery 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. HenceThis is consistent with Schwarzschild spacetime being a vacuum solution away from $r=0$.

The curvature 2-forms show that orthonormal curvature components grow as $m/r^3$. The geodesic deviation equation converts them into the relative acceleration of nearby parts of an observer. As $r\to 0$, the Schwarzschild tidal force stretches radial separations and compresses transverse separations with unbounded magnitude. An extended body therefore undergoes destructive tidal deformation before reaching the Schwarzschild singularity.

The connection 1-forms are defined by $\nabla e_b={\omega }^a{}_b\otimes e_a$. Metric compatibility gives ${\omega }_{ab}=-{\omega }_{ba}$, while vanishing torsion gives Cartan's first structure equationThe curvature 2-forms arewhich displays their relation to the Riemann curvature tensor.