Past exam of the general relativity course of the University of Oxford / 2022 / 2

Choose normal coordinates ${\xi }^{\alpha }$ centered at an event $p$. At $p$, the metric tensor in these coordinates is the constant Minkowski metric ${\eta }_{\alpha \beta }=\mathrm{diag}(-1,1,1,1)$, with ${\xi }^0$ measured in length units, and its first partial derivatives vanish. In arbitrary coordinates $x^{\mu }$, the chain rule for the line element givessoThis construction is pointwise. At nearby events, the metric tensor in the chosen normal coordinates is $g_{\alpha \beta }^{(\xi )}(\xi )={\eta }_{\alpha \beta }+O(|\xi {|}^2)$, with the quadratic corrections determined by the Riemann curvature tensor. Replacing $g^{(\xi )}$ by one constant Minkowski metric throughout an open region would make that region flat, so it cannot describe general curved spacetime exactly.

For a contravariant vector field $A^{\lambda }$, the covariant derivative isThe Levi-Civita connection is the unique torsion-free connection satisfying metric compatibility. Its Christoffel symbols in a coordinate basis areThere is generally no symmetry interchanging the upper index with a lower one. The Christoffel symbols themselves are not the components of a tensor.

An ordinary partial derivative of the components of a vector field includes the apparent change caused by the varying coordinate basis. The connection term corrects for that change, so ${\nabla }_{\nu }{A}^{\lambda }$ obeys the tensor component transformation law. It gives coordinate-independent ways to express parallel transport, geodesics and stress-energy conservation in curved spacetime.

Each lower index contributes a minus connection term to the covariant derivative of the metric tensor:Substitute the Christoffel symbols of the Levi-Civita connection and use the inverse metric identity $g_{\rho \nu }{g}^{\rho \sigma }={\delta }_{\nu }^{\sigma }$:Adding cancels the first and third terms and leaves exactly ${\partial }_{\lambda }{g}_{\mu \nu }$. HenceThis is metric compatibility: the Levi-Civita connection preserves the inner product under parallel transport.

The metric tensor in a basis ${\mathbf{e}}_a$ is $g_{ab}={\mathbf{e}}_a\cdot {\mathbf{e}}_b$. Differentiating this inner product with the product rule and using the assumed basis-derivative relation yieldsThus the coefficients satisfy metric compatibility. For a coordinate basis with zero torsion tensor, symmetrizing this relation in $a,c$, adding the corresponding two equations and subtracting the third givesThey are therefore the Christoffel symbols of the Levi-Civita connection. Metric compatibility by itself does not establish zero torsion tensor; that also uses the coordinate basis and its commuting coordinate derivatives.

For an embedded curved surface, the stated ambient basis-derivative relation needs a qualification: the Gauss formula is ${\partial }_c{\mathbf{e}}_a={\Gamma }_{ac}^b{\mathbf{e}}_b+h_{ac}\mathbf{n}$, where $\mathbf{n}$ is a unit normal and $h_{ac}$ is the second fundamental form. The normal term has zero inner product with each tangent basis vector, so the displayed identity for ${\partial }_c{g}_{ab}$ still holds. For a concrete counterexample to dropping that term, on a two-sphere one has ${\partial }_{\theta }{\mathbf{e}}_{\theta }=-R\mathbf{n}$, whereas every ${\Gamma }_{\theta \theta }^b$ is zero. Thus the intrinsic connection describes the tangential projection of the ambient derivative.

In standard spherical coordinates, parametrize the two-sphere byThe coordinate basis transforms from the fixed Cartesian basis according to ${\mathbf{e}}_a=(\partial x^i/\partial q^a){\mathbf{e}}_i$. ThereforeTheir inner products are ${\mathbf{e}}_{\theta }^2=R^2$, ${\mathbf{e}}_{\theta }\cdot {\mathbf{e}}_{\varphi }=0$ and ${\mathbf{e}}_{\varphi }^2=R^2{\sin }^2\theta$. Thus the induced metric isThe source's intermediate display places an extra $R^2$ outside entries already labelled $g_{\theta \theta },\dots$. Interpreting those entries as the actual metric tensor components would double-count the factor: for example, $R=2$ gives $g_{\theta \theta }=4$ directly from ${\mathbf{e}}_{\theta }^2$, not $16$. Its final matrix is the correct induced metric; the entries of its intermediate matrix must instead denote dimensionless coefficients.

The only nonconstant entry of the metric tensor is $g_{\varphi \varphi }=R^2{\sin }^2\theta$, whose partial derivative is ${\partial }_{\theta }{g}_{\varphi \varphi }=2R^2\sin \theta \cos \theta$. The nonzero Christoffel symbols areAlong a fixed latitude with $0<\theta <\pi$, parallel transport givesSet $X=V^{\theta }$ and $Y=\sin \theta V^{\varphi }$. Since $\theta$ is fixed, this linear ordinary differential equation becomesIt is a rotation at constant rate $-\cos \theta$. Its general solution, also valid at the equator where the rate is zero, isThe initial data give $A=1$ and $B=0$. At $\theta =\pi /3$, a complete circuit has $\varphi \cos \theta =\pi$, soThe physical squared length uses the metric tensor, rather than the Euclidean sum of coordinate components:It is constant and equals $R^2$ for these data. More generally, metric compatibility gives $dg(V,V)/d\varphi =({\nabla }_{{\partial }_{\varphi }}g)(V,V)+2g({\nabla }_{{\partial }_{\varphi }}V,V)=0$. The final change of direction is holonomy, exemplified by parallel transport around a latitude of the unit sphere; parallel transport preserves length without requiring the direction to return unchanged.