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

Let $v=\sqrt{2m/r}$. The proposed coframe giveswhich expands to the stated metric. Hence it is the orthonormal coframe in Painlevé–Gullstrand coordinates with signature $(-,+,+,+)$.

SetDirect exterior differentiation givesSolving Cartan's first structure equation and imposing ${\omega }_{\mu \nu }=-{\omega }_{\nu \mu }$ gives the independent mixed-index connection 1-formsThe remaining forms follow from Lorentz-signature antisymmetry: ${\omega }^1{}_0={\omega }^0{}_1$, ${\omega }^2{}_0={\omega }^0{}_2$, ${\omega }^3{}_0={\omega }^0{}_3$, and ${\omega }^j{}_i=-{\omega }^i{}_j$ for spatial indices.

WriteFor a diagonal curvature operator, ${\Theta }^{\mu \nu }$ contains only $f^{\mu }\wedge f^{\nu }$. Consequently $R^{\mu }{}_{\nu \rho \sigma }$ can be nonzero only when the plane indexed by $\rho ,\sigma$ is the same as the plane indexed by $\mu ,\nu$. In the contractionthis condition cannot hold for $\nu \ne \sigma$. Thereforewhich is the statement that a diagonal curvature operator implies diagonal Ricci tensor.

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.