Covariant Derivative and the Metric Connection

Derivation

Covariant Derivative and the Metric Connection

D-386 Home PU-403 Threads symmetry · fields Depends on Geodesic Equation from Extremal Proper Time

Statement

On a manifold carrying a metric \(g_{\mu\nu}\), demanding that differentiation of tensor fields again produce a tensor forces the introduction of a connection \(\Gamma^{\lambda}{}_{\mu\nu}\) with a fixed inhomogeneous transformation law. Adding the two conditions of vanishing torsion (\(\Gamma^{\lambda}{}_{\mu\nu}=\Gamma^{\lambda}{}_{\nu\mu}\)) and metric compatibility (\(\nabla_{\rho}g_{\mu\nu}=0\)) fixes it uniquely as the Levi-Civita connection \(\Gamma^{\lambda}{}_{\mu\nu}=\tfrac12 g^{\lambda\rho}\!\left(\partial_{\mu}g_{\rho\nu}+\partial_{\nu}g_{\rho\mu}-\partial_{\rho}g_{\mu\nu}\right)\), and the associated covariant derivative \(\nabla_{\mu}\) is the unique metric-preserving, torsion-free derivative operator.

Why it matters

The partial derivative \(\partial_{\mu}\) of a tensor is not a tensor: the second-derivative terms in a coordinate change spoil the transformation law. Without a repair, no equation containing derivatives could be written coordinate-independently, and the whole apparatus of curved-space physics, from geodesics to the Einstein equations, would be ill-defined.

The covariant derivative supplies that repair, and metric compatibility ties differentiation to the geometry that measures lengths and angles. The Christoffel symbols it produces are the same objects that appear in the geodesic equation obtained from extremising proper time, so the two derivations describe one and the same connection from two directions.

Assumptions

A smooth, invertible metric \(g_{\mu\nu}\) exists.Without \(g^{\mu\nu}\) the index can never be raised in the final contraction, so no unique connection is singled out; only an affine connection with extra free data remains. The connection is torsion-free, \(\Gamma^{\lambda}{}_{\mu\nu}=\Gamma^{\lambda}{}_{\nu\mu}\).If dropped, the antisymmetric part \(T^{\lambda}{}_{\mu\nu}=\Gamma^{\lambda}{}_{\mu\nu}-\Gamma^{\lambda}{}_{\nu\mu}\) survives as an independent tensor and the Christoffel formula acquires torsion terms (Einstein–Cartan theory). The connection is metric-compatible, \(\nabla_{\rho}g_{\mu\nu}=0\).If dropped, lengths of parallel-transported vectors drift (non-metricity \(Q_{\rho\mu\nu}=\nabla_{\rho}g_{\mu\nu}\neq0\)); index raising and covariant differentiation no longer commute. The coordinate patch is differentiable with a well-defined Jacobian.At coordinate singularities (e.g. \(r=0\) in polar coordinates) individual \(\Gamma\) blow up even in flat space, and the construction must be reinterpreted in a regular chart.

Derivation

\[ \partial'_{\nu}V'^{\mu}=\frac{\partial x^{\beta}}{\partial x^{\prime\nu}}\!\left[\frac{\partial x^{\prime\mu}}{\partial x^{\alpha}}\,\partial_{\beta}V^{\alpha}+\frac{\partial^{2}x^{\prime\mu}}{\partial x^{\beta}\partial x^{\alpha}}\,V^{\alpha}\right] \]

Transform a contravariant field \(V'^{\mu}=\frac{\partial x^{\prime\mu}}{\partial x^{\alpha}}V^{\alpha}\) and differentiate by the product rule. The first bracketed term is tensorial; the second, carrying \(\partial^{2}x^{\prime}\), is not — so \(\partial_{\nu}V^{\mu}\) is not a tensor. B

\[ \nabla_{\mu}V^{\nu}\equiv\partial_{\mu}V^{\nu}+\Gamma^{\nu}{}_{\mu\lambda}V^{\lambda},\qquad \Gamma^{\prime\mu}{}_{\nu\rho}=\frac{\partial x^{\prime\mu}}{\partial x^{\alpha}}\frac{\partial x^{\beta}}{\partial x^{\prime\nu}}\frac{\partial x^{\gamma}}{\partial x^{\prime\rho}}\Gamma^{\alpha}{}_{\beta\gamma}-\frac{\partial x^{\beta}}{\partial x^{\prime\nu}}\frac{\partial x^{\gamma}}{\partial x^{\prime\rho}}\frac{\partial^{2}x^{\prime\mu}}{\partial x^{\beta}\partial x^{\gamma}} \]

Add a correction \(\Gamma^{\nu}{}_{\mu\lambda}V^{\lambda}\) whose inhomogeneous transformation is chosen precisely to cancel the offending \(\partial^{2}x^{\prime}\) term from Step 1. This defines what a connection must be; the extra piece is not a tensor, but \(\nabla_{\mu}V^{\nu}\) is. C

\[ \nabla_{\mu}\omega_{\nu}=\partial_{\mu}\omega_{\nu}-\Gamma^{\lambda}{}_{\mu\nu}\omega_{\lambda},\qquad \nabla_{\rho}T^{\mu}{}_{\nu}=\partial_{\rho}T^{\mu}{}_{\nu}+\Gamma^{\mu}{}_{\rho\lambda}T^{\lambda}{}_{\nu}-\Gamma^{\lambda}{}_{\rho\nu}T^{\mu}{}_{\lambda} \]

Require \(\nabla_{\mu}\) to obey the Leibniz rule and to reduce to \(\partial_{\mu}\) on scalars. Applying it to the scalar \(\omega_{\nu}V^{\nu}\) forces the sign flip for lower indices; each index contributes one \(\Gamma\). B

\[ \Gamma^{\lambda}{}_{\mu\nu}=\Gamma^{\lambda}{}_{\nu\mu} \]

Impose the torsion-free condition. Acting on a scalar \(f\), \((\nabla_{\mu}\nabla_{\nu}-\nabla_{\nu}\nabla_{\mu})f=-\big(\Gamma^{\lambda}{}_{\mu\nu}-\Gamma^{\lambda}{}_{\nu\mu}\big)\partial_{\lambda}f\); demanding this vanish (so mixed second derivatives commute) fixes the lower pair as symmetric. B

\[ \nabla_{\rho}g_{\mu\nu}=\partial_{\rho}g_{\mu\nu}-\Gamma^{\lambda}{}_{\rho\mu}g_{\lambda\nu}-\Gamma^{\lambda}{}_{\rho\nu}g_{\mu\lambda}=0 \]

Impose metric compatibility using the rule from Step 3 with two lower indices. This is one tensor equation; rearranged, \(\partial_{\rho}g_{\mu\nu}=\Gamma^{\lambda}{}_{\rho\mu}g_{\lambda\nu}+\Gamma^{\lambda}{}_{\rho\nu}g_{\mu\lambda}\). B

\[ \partial_{\mu}g_{\nu\rho}=\Gamma^{\lambda}{}_{\mu\nu}g_{\lambda\rho}+\Gamma^{\lambda}{}_{\mu\rho}g_{\nu\lambda},\qquad \partial_{\nu}g_{\rho\mu}=\Gamma^{\lambda}{}_{\nu\rho}g_{\lambda\mu}+\Gamma^{\lambda}{}_{\nu\mu}g_{\rho\lambda} \]

Write the Step 5 relation twice more under the cyclic permutation \(\rho\to\mu\to\nu\to\rho\). Three equations now contain every \(\Gamma\) contracted once with the metric. A

\[ \partial_{\mu}g_{\nu\rho}+\partial_{\nu}g_{\rho\mu}-\partial_{\rho}g_{\mu\nu}=2\,\Gamma^{\lambda}{}_{\mu\nu}\,g_{\lambda\rho} \]

Add the two Step 6 equations and subtract the Step 5 equation. Using the symmetry of \(g\) and the torsion-free symmetry \(\Gamma^{\lambda}{}_{\mu\nu}=\Gamma^{\lambda}{}_{\nu\mu}\) from Step 4, four of the six \(\Gamma\)-terms cancel pairwise and the remaining two combine. C

\[ \Gamma^{\sigma}{}_{\mu\nu}=\tfrac12\,g^{\sigma\rho}\!\left(\partial_{\mu}g_{\rho\nu}+\partial_{\nu}g_{\rho\mu}-\partial_{\rho}g_{\mu\nu}\right) \]

Contract Step 7 with \(\tfrac12 g^{\sigma\rho}\), using \(g^{\sigma\rho}g_{\lambda\rho}=\delta^{\sigma}_{\lambda}\) to free the upper index. Invertibility of \(g\) (Assumption 1) guarantees this step and hence uniqueness. B

Result

\[ \boxed{\;\Gamma^{\lambda}{}_{\mu\nu}=\tfrac12\,g^{\lambda\rho}\!\left(\partial_{\mu}g_{\rho\nu}+\partial_{\nu}g_{\rho\mu}-\partial_{\rho}g_{\mu\nu}\right)\;}\qquad \nabla_{\mu}V^{\nu}=\partial_{\mu}V^{\nu}+\Gamma^{\nu}{}_{\mu\lambda}V^{\lambda} \]

Reading. Among all connections, exactly one is simultaneously torsion-free and metric-compatible, and its coefficients are built purely from first derivatives of the metric. The covariant derivative it defines turns \(\partial_{\mu}\) into a genuine tensor operation: the \(\Gamma\)-terms are the "fictitious force" bookkeeping that accounts for how the coordinate basis vectors themselves twist and stretch from point to point.

Units check. With coordinates \(x^{\mu}\) of dimension length, \(g_{\mu\nu}\) is dimensionless for like-index pairs, so \(\partial g\) carries \([\text{length}]^{-1}\); \(g^{\lambda\rho}\) is dimensionless and the product gives \([\Gamma]=[\text{length}]^{-1}\). Then \(\Gamma^{\nu}{}_{\mu\lambda}V^{\lambda}\) has the units of \([V]/[\text{length}]\), matching \(\partial_{\mu}V^{\nu}\); both terms of \(\nabla_{\mu}V^{\nu}\) are dimensionally consistent.

Limiting cases

Cartesian / inertial coordinates: \(g_{\mu\nu}=\text{const}\Rightarrow\partial g=0\Rightarrow\Gamma^{\lambda}{}_{\mu\nu}=0\) and \(\nabla_{\mu}\to\partial_{\mu}\). The covariant derivative reduces to the ordinary one.
Curvilinear coordinates in flat space: \(\Gamma\neq0\) but the Riemann tensor \(R^{\rho}{}_{\sigma\mu\nu}\) built from \(\Gamma\) still vanishes — nonzero connection does not imply curvature.
Locally inertial frame at a point \(p\): one can always choose coordinates with \(g_{\mu\nu}(p)=\eta_{\mu\nu}\) and \(\Gamma^{\lambda}{}_{\mu\nu}(p)=0\) (though \(\partial\Gamma\neq0\)), the mathematical statement of the equivalence principle.
Along a geodesic: \(\nabla_{u}u^{\mu}=0\) with \(u^{\mu}=dx^{\mu}/d\tau\) reproduces the geodesic equation \(\ddot{x}^{\mu}+\Gamma^{\mu}{}_{\alpha\beta}\dot{x}^{\alpha}\dot{x}^{\beta}=0\) obtained from extremal proper time.

Breaks when

The metric is degenerate or non-invertible (\(\det g=0\)), as on a null hypersurface or at a horizon in singular coordinates: \(g^{\lambda\rho}\) does not exist, so the contraction in Step 8 fails and no unique metric connection can be formed.
Torsion is physically present (Einstein–Cartan gravity with spin sources): the antisymmetric part \(\Gamma^{\lambda}{}_{[\mu\nu]}\neq0\) survives, Step 4 is invalid, and the Christoffel formula gains explicit torsion contributions \(-\tfrac12(T^{\lambda}{}_{\mu\nu}+T_{\mu}{}^{\lambda}{}_{\nu}+T_{\nu}{}^{\lambda}{}_{\mu})\).
Non-metricity is allowed (metric-affine / Weyl geometry): dropping metric compatibility leaves \(\nabla_{\rho}g_{\mu\nu}=Q_{\rho\mu\nu}\neq0\), Step 5 no longer vanishes, and the connection is under-determined by the metric alone.
At a genuine curvature singularity (e.g. \(r=0\) in Schwarzschild): components of \(\Gamma\) and its derivatives diverge in every chart and the tangent-space construction ceases to exist.

Failure modes

Treating \(\Gamma^{\lambda}{}_{\mu\nu}\) as a tensor. It is not — its transformation carries the inhomogeneous \(\partial^{2}x^{\prime}\) term. One cannot conclude "\(\Gamma=0\) in one frame \(\Rightarrow\Gamma=0\) in all frames."
Wrong sign on lower indices. Writing \(\nabla_{\mu}\omega_{\nu}=\partial_{\mu}\omega_{\nu}+\Gamma\,\omega\) instead of \(-\Gamma\,\omega\); the sign is fixed by demanding \(\nabla(\omega_{\nu}V^{\nu})=\partial(\omega_{\nu}V^{\nu})\).
Index-order confusion in the covariant derivative. Placing the differentiation index in the wrong \(\Gamma\) slot; for the torsion-free connection the lower pair is symmetric, but the contracted index must still match the tensor slot it corrects.
Forgetting the factor \(\tfrac12\) or the inverse metric. Reading off \(\Gamma\) directly from \(\partial g\) without the \(\tfrac12 g^{\lambda\rho}\) contraction.
Assuming \(\Gamma\neq0\) means curved space. Curvature lives in \(R^{\rho}{}_{\sigma\mu\nu}=\partial_{\mu}\Gamma^{\rho}{}_{\nu\sigma}-\partial_{\nu}\Gamma^{\rho}{}_{\mu\sigma}+\Gamma\Gamma-\Gamma\Gamma\), not in \(\Gamma\) itself.
Mismatching the sign convention for the third metric-derivative term. The combination is \((+,+,-)\): \(\partial_{\mu}g_{\rho\nu}+\partial_{\nu}g_{\rho\mu}-\partial_{\rho}g_{\mu\nu}\); permuting these carelessly gives a symmetric-in-wrong-slots object.

Discussion

The construction shows that "how to differentiate" is not extra geometric data once a metric is fixed: metric compatibility plus torsion-freedom are exactly enough constraints to solve for all \(D^2(D+1)/2\) independent connection coefficients in \(D\) dimensions. This is the fundamental theorem of Riemannian geometry, and it is what lets general relativity be a theory of the metric alone — the connection, and hence gravity's "force," is a derived quantity.

Physically, \(\Gamma^{\lambda}{}_{\mu\nu}\) encodes how the coordinate basis vectors \(\mathbf{e}_{\mu}\) rotate and rescale as one moves: \(\partial_{\nu}\mathbf{e}_{\mu}=\Gamma^{\lambda}{}_{\nu\mu}\mathbf{e}_{\lambda}\). The covariant derivative subtracts this basis-drift so that \(\nabla_{\mu}V^{\nu}\) measures the genuine change of the vector, not the artefact of a moving frame. This is why in an inertial (freely falling) frame the \(\Gamma\) vanish at a point: locally, gravity is transformed away.

The same symbols govern parallel transport, \(\nabla_{u}V=0\), which defines what it means for a vector to stay "the same" along a curve. Because the transport depends on the path, the failure of two transports to agree is curvature — the commutator \([\nabla_{\mu},\nabla_{\nu}]V^{\rho}=R^{\rho}{}_{\sigma\mu\nu}V^{\sigma}\). Thus one object, the metric connection, unifies differentiation, geodesic motion, and curvature.

At a deeper level the metric connection is the unique \(GL\)-connection compatible with the reduction of the frame bundle to the orthogonal (Lorentz) subgroup: metric compatibility is the statement that the connection one-form takes values in \(\mathfrak{so}(1,D-1)\) when expressed in an orthonormal frame, and torsion-freedom is the vanishing of the soldering-form's exterior covariant derivative, \(d e^{a}+\omega^{a}{}_{b}\wedge e^{b}=0\). Solving this Cartan structure equation for the spin connection \(\omega^{a}{}_{b}\) is the frame-bundle counterpart of Steps 5–8, and reproduces the Christoffel symbols upon pulling back to a coordinate basis.

Common misconceptions. A nonzero Christoffel symbol does not signal curvature (polar coordinates on a flat plane already have \(\Gamma\neq0\)); the covariant derivative of the metric is identically zero by construction, so "\(\nabla g\)" never appears as a source term; and lowering an index inside a covariant derivative is legal precisely because \(\nabla_{\rho}g_{\mu\nu}=0\) lets \(g\) pass through \(\nabla\).

Worked examples

Polar coordinates on the flat plane: \(\;ds^{2}=dr^{2}+r^{2}d\theta^{2}\)

Read off \(g_{rr}=1,\;g_{\theta\theta}=r^{2}\) (off-diagonals zero), so \(g^{rr}=1,\;g^{\theta\theta}=1/r^{2}\). Only \(g_{\theta\theta}\) depends on a coordinate (\(r\)). A

\[ \Gamma^{r}{}_{\theta\theta}=\tfrac12 g^{rr}\big(2\partial_{\theta}g_{r\theta}-\partial_{r}g_{\theta\theta}\big)=\tfrac12(1)\big(0-2r\big)=-r \]

Apply Step 8 with \(\sigma=r,\ \mu=\nu=\theta\); the only surviving metric derivative is \(\partial_{r}g_{\theta\theta}=2r\). A

\[ \Gamma^{\theta}{}_{r\theta}=\Gamma^{\theta}{}_{\theta r}=\tfrac12 g^{\theta\theta}\,\partial_{r}g_{\theta\theta}=\tfrac12\frac{1}{r^{2}}(2r)=\frac{1}{r} \]

Now \(\sigma=\theta,\ \mu=r,\ \nu=\theta\); symmetry in the lower pair gives the second equal component. All other \(\Gamma\) vanish. A

\[ \Gamma^{r}{}_{\theta\theta}=-r,\qquad \Gamma^{\theta}{}_{r\theta}=\Gamma^{\theta}{}_{\theta r}=\frac{1}{r}\quad(\text{at }r=2:\ -2\ \text{and}\ 0.5\ \mathrm{m^{-1}}) \]

Reading. The plane is flat (one checks \(R^{\rho}{}_{\sigma\mu\nu}=0\)), yet the connection is nonzero purely because the polar basis vectors turn and lengthen. The geodesic equation with these \(\Gamma\) reproduces straight lines written in polar form.

Round 2-sphere of radius \(a\): \(\;ds^{2}=a^{2}\,d\theta^{2}+a^{2}\sin^{2}\!\theta\,d\phi^{2}\)

Read off \(g_{\theta\theta}=a^{2},\;g_{\phi\phi}=a^{2}\sin^{2}\theta\), hence \(g^{\theta\theta}=1/a^{2},\;g^{\phi\phi}=1/(a^{2}\sin^{2}\theta)\). Only \(g_{\phi\phi}\) depends on \(\theta\). B

\[ \Gamma^{\theta}{}_{\phi\phi}=\tfrac12 g^{\theta\theta}\big(-\partial_{\theta}g_{\phi\phi}\big)=\tfrac12\frac{1}{a^{2}}\big(-a^{2}\cdot 2\sin\theta\cos\theta\big)=-\sin\theta\cos\theta \]

Step 8 with \(\sigma=\theta,\ \mu=\nu=\phi\): the two \(+\) terms carry \(\partial_{\phi}g_{\theta\phi}=0\), leaving only \(-\partial_{\theta}g_{\phi\phi}\). Note \(a\) cancels. B

\[ \Gamma^{\phi}{}_{\theta\phi}=\Gamma^{\phi}{}_{\phi\theta}=\tfrac12 g^{\phi\phi}\,\partial_{\theta}g_{\phi\phi}=\tfrac12\frac{1}{a^{2}\sin^{2}\theta}\big(a^{2}\cdot2\sin\theta\cos\theta\big)=\cot\theta \]

Now \(\sigma=\phi,\ \mu=\theta,\ \nu=\phi\); the surviving derivative is \(\partial_{\theta}g_{\phi\phi}\). All others vanish. B

\[ \Gamma^{\theta}{}_{\phi\phi}=-\sin\theta\cos\theta,\qquad \Gamma^{\phi}{}_{\theta\phi}=\cot\theta\quad\left(\text{at }\theta=\tfrac{\pi}{3}:\ -\tfrac{\sqrt3}{4}\approx-0.433,\ \tfrac{1}{\sqrt3}\approx0.577\right) \]

Reading. Unlike the plane, these feed a nonzero Riemann tensor and Gaussian curvature \(K=1/a^{2}\). The radius \(a\) drops out of the connection because \(\Gamma\) scales as (derivative of metric)/(metric); curvature reintroduces \(a\) through the second derivatives.

Problems

(A) Show that in Cartesian coordinates on flat 3-space, \(ds^{2}=dx^{2}+dy^{2}+dz^{2}\), every Christoffel symbol vanishes, and state what \(\nabla_{\mu}\) becomes.
Solution
Here \(g_{\mu\nu}=\delta_{\mu\nu}=\text{const}\), so \(\partial_{\rho}g_{\mu\nu}=0\) for all indices. Then \(\Gamma^{\lambda}{}_{\mu\nu}=\tfrac12 g^{\lambda\rho}(0+0-0)=0\). With \(\Gamma=0\), \(\nabla_{\mu}V^{\nu}=\partial_{\mu}V^{\nu}\): the covariant derivative reduces to the ordinary partial derivative, which is why elementary vector calculus needs no connection.
(B) For 3D cylindrical coordinates, \(ds^{2}=dr^{2}+r^{2}d\phi^{2}+dz^{2}\), find all nonzero Christoffel symbols.
Solution
\(g_{rr}=1,\ g_{\phi\phi}=r^{2},\ g_{zz}=1\); inverses \(g^{rr}=1,\ g^{\phi\phi}=1/r^{2},\ g^{zz}=1\). Only \(g_{\phi\phi}\) varies (with \(r\)). Exactly as in the polar example: \(\Gamma^{r}{}_{\phi\phi}=\tfrac12 g^{rr}(-\partial_{r}g_{\phi\phi})=\tfrac12(-2r)=-r\), and \(\Gamma^{\phi}{}_{r\phi}=\Gamma^{\phi}{}_{\phi r}=\tfrac12 g^{\phi\phi}\partial_{r}g_{\phi\phi}=\tfrac12\frac{1}{r^{2}}(2r)=\frac1r\). All symbols involving \(z\) vanish because \(g_{zz}\) is constant and \(g\) is diagonal. So the nonzero ones are \(\Gamma^{r}{}_{\phi\phi}=-r\) and \(\Gamma^{\phi}{}_{r\phi}=1/r\).
(B) Using only \(\nabla_{\rho}g_{\mu\nu}=0\) and the Leibniz rule, prove that \(\nabla_{\rho}g^{\mu\nu}=0\).
Solution
Since \(g^{\mu\nu}g_{\nu\sigma}=\delta^{\mu}_{\sigma}\) is constant, \(\nabla_{\rho}\delta^{\mu}_{\sigma}=0\). By the Leibniz rule, \(0=\nabla_{\rho}(g^{\mu\nu}g_{\nu\sigma})=(\nabla_{\rho}g^{\mu\nu})g_{\nu\sigma}+g^{\mu\nu}(\nabla_{\rho}g_{\nu\sigma})\). The second term vanishes by metric compatibility, leaving \((\nabla_{\rho}g^{\mu\nu})g_{\nu\sigma}=0\). Contract with \(g^{\sigma\alpha}\): \(\nabla_{\rho}g^{\mu\nu}\,\delta^{\alpha}_{\nu}=\nabla_{\rho}g^{\mu\alpha}=0\). Hence the inverse metric is also covariantly constant, which is what allows indices to be raised inside a covariant derivative.
(C) Prove the contracted-connection identity \(\Gamma^{\lambda}{}_{\mu\lambda}=\partial_{\mu}\ln\sqrt{|g|}\), where \(g=\det g_{\mu\nu}\).
Solution
From Step 8, \(\Gamma^{\lambda}{}_{\mu\lambda}=\tfrac12 g^{\lambda\rho}\big(\partial_{\mu}g_{\rho\lambda}+\partial_{\lambda}g_{\rho\mu}-\partial_{\rho}g_{\mu\lambda}\big)\). The last two terms cancel: relabel \(\lambda\leftrightarrow\rho\) in \(g^{\lambda\rho}\partial_{\lambda}g_{\rho\mu}\) and use \(g^{\lambda\rho}=g^{\rho\lambda}\) to get \(g^{\lambda\rho}\partial_{\rho}g_{\lambda\mu}=g^{\lambda\rho}\partial_{\rho}g_{\mu\lambda}\), identical to the third term. Thus \(\Gamma^{\lambda}{}_{\mu\lambda}=\tfrac12 g^{\lambda\rho}\partial_{\mu}g_{\rho\lambda}\). By Jacobi's formula \(\partial_{\mu}\ln|\det g|=g^{\lambda\rho}\partial_{\mu}g_{\rho\lambda}=\mathrm{tr}(g^{-1}\partial_{\mu}g)\), so \(\Gamma^{\lambda}{}_{\mu\lambda}=\tfrac12\partial_{\mu}\ln|g|=\partial_{\mu}\ln\sqrt{|g|}\). This is the identity behind the covariant divergence \(\nabla_{\mu}V^{\mu}=\frac{1}{\sqrt{|g|}}\partial_{\mu}(\sqrt{|g|}\,V^{\mu})\).
(C) For the spatially flat expanding universe, \(ds^{2}=-dt^{2}+a(t)^{2}\big(dx^{2}+dy^{2}+dz^{2}\big)\), compute \(\Gamma^{t}{}_{xx}\) and \(\Gamma^{x}{}_{tx}\), and evaluate them for \(a(t)=e^{Ht}\) with \(H=2.2\times10^{-18}\ \mathrm{s^{-1}}\) at the present epoch \(a=1\).
Solution
Nonzero metric components: \(g_{tt}=-1,\ g_{xx}=a^{2}\), inverses \(g^{tt}=-1,\ g^{xx}=1/a^{2}\). For \(\Gamma^{t}{}_{xx}\): \(\Gamma^{t}{}_{xx}=\tfrac12 g^{tt}\big(2\partial_{x}g_{tx}-\partial_{t}g_{xx}\big)=\tfrac12(-1)\big(0-2a\dot a\big)=a\dot a\). For \(\Gamma^{x}{}_{tx}\): \(\Gamma^{x}{}_{tx}=\tfrac12 g^{xx}\big(\partial_{t}g_{xx}+\partial_{x}g_{xt}-\partial_{x}g_{tx}\big)=\tfrac12\frac{1}{a^{2}}(2a\dot a)=\frac{\dot a}{a}\). With \(a=e^{Ht}\), \(\dot a=Ha\), so \(\Gamma^{x}{}_{tx}=H\) and \(\Gamma^{t}{}_{xx}=a\dot a=Ha^{2}\). At \(a=1\): \(\Gamma^{x}{}_{tx}=H=2.2\times10^{-18}\ \mathrm{s^{-1}}\) and \(\Gamma^{t}{}_{xx}=H=2.2\times10^{-18}\ \mathrm{s^{-1}}\) (in units where \(x\) is measured so that \(a\) is dimensionless). The factor \(\dot a/a=H\) is exactly the Hubble rate, and these symbols are what turn \(\nabla_{u}u=0\) into the cosmological redshift of freely moving particles.