Connection Coefficients and Coordinate Change

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Connection coefficients can vanish at a point because they describe a connection relative to a changing coordinate frame, not a tensor field. Under a nonlinear coordinate change they acquire a term containing second derivatives of the map. That term permits the symmetric part of the coefficients to be canceled at one point, while tensorial data—most notably torsion—and derivatives that encode curvature cannot generally be removed.

Required background. Tensor Fields and Natural Operations supplies coordinate bases and tensor transformation laws; Pseudo-Riemannian Metrics and Signature supplies metric components without assuming that every connection is Levi-Civita.

Helpful background. Calculus and Linear Algebra for Relativity supplies Jacobians, second derivatives, and matrix changes of basis.

The inhomogeneous transformation law

Let $x^\mu$ and $x^{\mu'}$ be overlapping coordinate systems, with

\nabla_{\partial_\mu}\partial_\nu =\Gamma^\rho{}_{\mu\nu}\partial_\rho.

Write each primed basis vector in the unprimed basis and use the Leibniz rule. One obtains

\begin{aligned} \Gamma^{\rho'}{}_{\mu'\nu'} &= \frac{\partial x^{\rho'}}{\partial x^\rho} \frac{\partial x^\mu}{\partial x^{\mu'}} \frac{\partial x^\nu}{\partial x^{\nu'}} \Gamma^\rho{}_{\mu\nu}\\ &\quad+ \frac{\partial x^{\rho'}}{\partial x^\rho} \frac{\partial^2x^\rho} {\partial x^{\mu'}\partial x^{\nu'}}. \end{aligned}

The first line is the transformation law of a $(1,2)$ tensor. The second, inhomogeneous line is present for nonlinear changes of coordinates and proves that the coefficients themselves are not tensor components. Substitution into

\nabla_{\mu'}V^{\rho'} =\partial_{\mu'}V^{\rho'} +\Gamma^{\rho'}{}_{\mu'\nu'}V^{\nu'}

cancels the second derivatives arising from $\partial_{\mu'}V^{\rho'}$ . The result is tensorial.

This derivation and cancellation are given in Tong 2019, §3.2.1, pp. 101–104, PDF.

Flat space in polar coordinates

Start with the Euclidean plane in Cartesian coordinates, where $\Gamma^i{}_{jk}=0$ . Let

x=r\cos\phi, \qquad y=r\sin\phi.

The inhomogeneous term alone then produces

\Gamma^r{}_{\phi\phi}=-r, \qquad \Gamma^\phi{}_{r\phi} =\Gamma^\phi{}_{\phi r}=\frac1r.

These coefficients are nonzero although the underlying connection is flat. They record that $\partial_r$ and $\partial_\phi$ change direction and length from point to point. Conversely, returning to Cartesian coordinates makes them vanish throughout this flat patch. A nonzero coefficient is therefore not evidence of curvature, just as a vanishing coefficient at one point is not evidence of flatness.

One can independently recover the same coefficients from the metric $g_{rr}=1$ , $g_{\phi\phi}=r^2$ using the Levi-Civita formula. Agreement between the coordinate transformation and metric calculation checks both the sign $-r$ and the factor $1/r$ .

What coordinates can remove

At a chosen point $p$ , the second derivatives of a coordinate transformation are symmetric in $\mu'$ and $\nu'$ . They can therefore cancel $\Gamma^\rho{}_{(\mu\nu)}(p)$ , but not an antisymmetric part. In a coordinate basis,

T^\rho{}_{\mu\nu} =2\Gamma^\rho{}_{[\mu\nu]}.

Torsion is tensorial, so nonzero torsion cannot be transformed away. For a torsion-free connection, coordinates can be chosen with $\Gamma^\rho{}_{\mu\nu}(p)=0$ . For the Levi-Civita connection, exponential normal coordinates provide one such choice. Vanishing coefficients at $p$ alone does not make an arbitrary chart an exponential normal chart: the latter additionally represents geodesics through $p$ by radial straight lines.

Even then, the statement concerns only one point. Derivatives $\partial_\sigma\Gamma^\rho{}_{\mu\nu}(p)$ generally remain, and their antisymmetrized combinations contribute to curvature. Parallel transport around finite or infinitesimal loops can therefore be nontrivial despite vanishing coefficients at the base point.

Frames, neighborhoods, and failure cases

A noncoordinate frame has its own connection coefficients, but its commutator $[e_a,e_b]=C^c{}_{ab}e_c$ also enters torsion:

T^a{}_{bc} =\Gamma^a{}_{bc}-\Gamma^a{}_{cb}-C^a{}_{bc}.

Thus the simple identification of torsion with twice the antisymmetric coefficient part applies only to coordinate frames. Parallel-propagating a frame along a selected curve makes the coefficients of the derivative in the curve’s tangent direction zero. This says nothing about derivatives in other directions, and that frame need not extend to a coordinate basis on a neighborhood.

The ability to erase coefficients is local and basis dependent. A single chart cannot make a curved Levi-Civita connection vanish throughout an open set. If it could, the curvature components built from its derivatives and quadratic terms would vanish there. This is why “zero at a point,” “zero along a curve,” and “zero on an open region” are sharply different claims.

There is also a regularity requirement hidden in every transformation law: the coordinate map must be a diffeomorphism on the overlap. Polar coordinates do not form a valid chart at $r=0$ , so the divergence of $1/r$ coefficients there cannot be used to diagnose a singular connection. Away from the origin, their finite nonzero values are exactly the inhomogeneous contribution of the nonlinear Cartesian-to-polar map.

More generally, prescribing a coordinate system’s first derivatives at $p$ chooses its basis, while prescribing symmetric second derivatives adjusts the symmetric coefficient part. Mixed partial derivatives force that adjustment to be symmetric in the two lower coordinate indices. This jet-level argument explains both results at once: every torsion-free connection admits coordinates with all coefficients zero at $p$ , and a torsionful connection does not. A general noncoordinate frame has more freedom, but then its commutator coefficients carry the torsion information that a coordinate frame could not erase.

Exercises

Use the Levi-Civita formula for $ds^2=dr^2+r^2d\phi^2$ to compute all nonzero connection coefficients. Then evaluate them at $r=R>0$ and explain why their nonzero values do not contradict the existence of Cartesian coordinates with zero coefficients.

Solution

The inverse metric has $g^{rr}=1$ and $g^{\phi\phi}=r^{-2}$ . Since $\partial_rg_{\phi\phi}=2r$ ,

\Gamma^r{}_{\phi\phi} =-\frac12\partial_rg_{\phi\phi}=-r, \qquad \Gamma^\phi{}_{r\phi} =\Gamma^\phi{}_{\phi r} =\frac12g^{\phi\phi}\partial_rg_{\phi\phi}=\frac1r.

At $r=R$ these are $-R$ and $1/R$ . The polar-to-Cartesian map is nonlinear, so its second derivatives supply exactly these inhomogeneous terms. The connection is the same flat connection in both charts; only its coefficients change.

References

Tong, David. General Relativity. University of Cambridge lecture notes, 2019, §3.2.1, pp. 101–104. Official course PDF.