Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 38C (General Relativity)

Contract the defining formula in its first and third indices. In four dimensions,Thus

The Weyl tensor has the same algebraic symmetries as the Riemann curvature tensor: it is antisymmetric within each index pair, symmetric under exchanging the pairs, and obeys the first Bianchi identity. A contraction within an antisymmetric pair vanishes against the symmetric metric. Every contraction between the two pairs can be converted by the pair symmetries to the one just calculated, possibly with a sign. Hence every contraction vanishes, proving the Trace-free property of the Weyl tensor.

WriteThe Levi-Civita connection ofisThe assumed first-derivative condition makes ${\Gamma }_{\alpha \beta }^{\rho }(p)=0$. Differentiating once more gives, at $p$,Its contractions areSubstitution into the defining expression for $C_{\alpha \beta \gamma \delta }$ cancels every Hessian term, leavingSince $p$ was arbitrary, this is the Weyl tensor of a conformally flat metric.

The Schwarzschild exterior is vacuum, soand consequently $C_{\alpha \beta \gamma \delta }=R_{\alpha \beta \gamma \delta }$. PutThe relevant Christoffel symbols areWith the conventionone findsSince $f^{\prime \prime }=-4M/r^3$,This is the leading Radial Weyl curvature of Schwarzschild spacetime. The opposite convention for the Riemann tensor reverses the sign but not the physical tidal magnitude.

This curvature component produces radial tidal acceleration: nearby freely falling particles separated radially are stretched relative to one another, while transverse separations are compressed. At the Earth, after removing the Earth's own field, the Sun and especially the Moon produce the familiar ocean tides known since antiquity.

There is no conflict with the equivalence principle. A freely falling local frame can remove the uniform gravitational acceleration at one event by setting the connection to zero there. It cannot remove the Riemann curvature tensor, which controls relative acceleration through geodesic deviation and is detectable across an extended body such as the Earth. Tides measure this spatial variation of gravity rather than gravity at a single point.