Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 52 / 4

A spacetime is a smooth four-dimensional manifold equipped with a smooth Lorentzian metric, conventionally of metric signature $(-,+,+,+)$, and a choice of time orientation. The usual manifold assumptions include the Hausdorff space and second-countable space conditions. A physical model also specifies matter fields and requires the Einstein field equations and matter equations.

Diffeomorphism invariance of general relativity means that relabelling events by a smooth invertible map, while transforming the metric tensor and every matter field together, preserves the form of the equations. Passively, a coordinate change gives new components for the same geometric fields. Actively, pulling all fields back by a diffeomorphism gives another representative of the same physical geometry, subject to any prescribed boundary conditions or boundary symmetries.

The geometry and matter together carry the physical content; coordinate labels do not. A diffeomorphism need not be an isometry of a fixed metric, and unrelated metrics are not automatically physically equivalent.

Use the connection convention ${\nabla }_{\alpha }{Y}^{\rho }={\partial }_{\alpha }{Y}^{\rho }+{\Gamma }^{\rho }{}_{\mu \alpha }{Y}^{\mu }$, with the derivative index last. This is consistent with the printed curvature formula and the final formula in part (iv). The paper's $T_{\mu \nu }{}^{\rho }={\Gamma }^{\rho }{}_{\mu \nu }-{\Gamma }^{\rho }{}_{\nu \mu }$ is then the negative of the geometric torsion tensor defined by $\mathcal{T}(X,Y)={\nabla }_{X}Y-{\nabla }_{Y}X-[X,Y]$. Keep the paper's component convention throughout this question.

Set $A_{\mu \nu \rho }=g_{\rho \lambda }({\Gamma }^{\lambda }{}_{\mu \nu }-S^{\lambda }{}_{\mu \nu })$. The difference of affine connections is a tensor. Since the Levi-Civita connection $S$ is symmetric and metric compatible, expanding the nonmetricity tensor givesHere $T_{\mu \nu \rho }=g_{\rho \lambda }{T}_{\mu \nu }{}^{\lambda }$ is antisymmetric in its first two slots, while $N_{\mu \nu \rho }$ is symmetric in its first two slots. Cyclically permuting these identities and eliminating the other components givesRaise the last slot to obtain the connection decompositionwhereThe PDF is missing the factor $1/2$ in its definition of $K$. With the printed $K$, the decomposition would instead contain $-K/2$. In these formulas $T^{\rho }{}_{\mu \nu }$ raises the first slot of the fully lowered tensor; it must not be confused with $T_{\mu \nu }{}^{\rho }$, which raises the last slot.

The Levi-Civita connection is the unique affine connection that has both vanishing torsion tensor and metric compatibility. The affine connection decomposition shows exactly how an arbitrary affine connection departs from it: the contorsion tensor contributes the metric-compatible torsion correction, and the disformation tensor contributes the nonmetricity correction. The torsion and nonmetricity tensor, together with the metric tensor, determine the difference uniquely.

In the paper's convention the actual contorsion correction is $-K$, with the corrected half factor from part (i), and the disformation correction is $W$. Both corrections are tensors, although the separate connection coefficients $\Gamma$ and $S$ are not tensors. If both torsion and nonmetricity vanish, then $\boxed{\Gamma =S}$. Vanishing nonmetricity alone permits torsion, and vanishing torsion alone permits nonmetricity.

Let ${\gamma }_X(s)$ be the affinely parametrized geodesic of the affine connection with ${\gamma }_X(0)=p$ and ${\dot{\gamma }}_X(0)=X$. For fixed $t$, the curve $\eta (u)={\gamma }_X(tu)$ has initial velocity $tX$. The geodesic equation is homogeneous of degree two in the velocity, so its left-hand side for $\eta$ is $t^2$ times the left-hand side for ${\gamma }_X$, and is zero. Uniqueness of the geodesic initial value problem gives $\eta ={\gamma }_{tX}$.

By the definition of the affine exponential map, rescaling the initial velocity rescales affine parameter:At $t=0$ both sides equal $p$. The statement is made where the geodesics exist; without geodesic completeness, the affine exponential map is defined only on a suitable neighbourhood of zero in $T_{p}M$, not necessarily all of $T_{p}M$. An affine parameter distance need not be a metric length.

Choose a basis of $T_{p}M$ and use the affine exponential map to define affine normal coordinates. Its differential at zero is the identity, so the inverse function theorem gives a chart near $p$. By part (iii), radial geodesics have coordinates $x^{\mu }(s)=s{X}^{\mu }$. Their geodesic equations at $p$ imply ${\Gamma }^{\lambda }{}_{\mu \nu }(p)X^{\mu }{X}^{\nu }=0$ for every $X$. The polarization identity therefore gives ${\Gamma }^{\lambda }{}_{(\mu \nu )}(p)=0$.

The disformation tensor $W_{\mu \nu }{}^{\lambda }$ is symmetric in $\mu ,\nu$, whereas $T_{\mu \nu }{}^{\lambda }$ is antisymmetric. Also $T_{\nu }{}^{\lambda }{}_{\mu }=-T^{\lambda }{}_{\nu \mu }$. Taking the symmetric part of the corrected affine connection decomposition consequently gives the normal-coordinate identityThus the final identity in the PDF is correct after the half-factor repair in part (i). In general ${\Gamma }^{\lambda }{}_{\mu \nu }(p)=\frac{1}{2}{T}_{\mu \nu }{}^{\lambda }(p)$ in the paper's convention, rather than zero. Neither $S$ nor the first derivatives of the metric tensor need vanish in these affine normal coordinates. With the Levi-Civita connection, both torsion and nonmetricity vanish and the usual vanishing of the Christoffel symbols is recovered.