Affine connection for axial symmetry

Difficulty:   ★★★★☆   graduate

Suppose you have an axially symmetric vector field. Can we define an affine connection which keeps the vectors “parallel”, under rotation about the axis? For example, we wish the vectors illustrated below to get parallel-transported around the circle:

an axially symmetric vector field
We seek an affine connection which declares vectors at a given radius “parallel”, for any vector field with circular symmetry / cylindrical symmetry / rotational symmetry.

Take Minkowski spacetime in cylindrical coordinates (t,r,\phi,z), with metric -dt^2+dr^2+r^2d\phi^2+dz^2, and consider a vector field \mathbf u whose components are independent of \phi:

    \[u^\mu = (A{\scriptstyle(t,r,z)},B{\scriptstyle(t,r,z)},C{\scriptstyle(t,r,z)},D{\scriptstyle(t,r,z)}).\]

The covariant derivative in the tangential direction \partial_\phi has components:

    \[(\nabla_{\partial_\phi}\mathbf u)^\alpha = \Big(0,-rC{\scriptstyle(t,r,z)},\frac{B{\scriptstyle(t,r,z)}}{r},0\Big).\]

We want this to vanish, but first a quick recap (Lee  §4, 5). Recall a connection is defined by \nabla_{\partial_\mu}\partial_\nu = \Gamma_{\mu\nu}^\alpha \partial_\alpha, in terms of our coordinate vector frame (\partial_t,\partial_r,\partial_\phi,\partial_z). This extends to a covariant derivative of arbitrary vectors and tensors, also denoted “\nabla”. The derivative of \mathbf u above assumed the Levi–Civita connection, which is inherited from the metric: it is the unique symmetric, metric-compatible connection. In that case the set of \Gamma are called Christoffel symbols, but in general they are called connection coefficients.

The offending Christoffel symbols which prevent our vector field from being parallel-transported are: \Gamma_{\phi\phi}^r = -r and \Gamma_{\phi r}^\phi = 1/r. But we are free to simply define a new connection for which these vanish: \tilde\Gamma_{\phi\phi}^r := 0 =: \tilde\Gamma_{\phi r}^\phi! Given a frame, any set of smooth functions \tilde\Gamma yields a valid connection (Lee, Lemma 4.10). It is natural to hold on to the other Christoffel symbols, to accord whatever respect remains for the metric. In fact only one is nonzero, \Gamma_{r\phi}^\phi = 1/r. To set this to zero would essentially deny the increase in circumference with the radius. Incidentally, even with keeping \Gamma_{r\phi}^\phi, the new connection is flat, meaning its associated Riemann curvature tensor vanishes.

The new connection may be expressed as the Levi–Civita one with a bilinear correction:

    \[\tilde\nabla_{\mathbf v}\mathbf u = \nabla_{\mathbf v}\mathbf u - \big( \frac{1}{r}\partial_\phi\otimes d\phi\otimes dr - r\partial_r\otimes d\phi\otimes d\phi \big) (\mathbf v,\mathbf u),\]

where \mathbf v and \mathbf u are arbitrary vectors, to be substituted into the 2nd and 3rd slots respectively of the (1,2)-tensor in parentheses. This is much simpler than it looks, as the terms just pick out r and \phi-components, and return basis vectors. Equivalently, the correction may be written -\langle d\phi,\mathbf v\rangle \big( r^{-1}\langle dr,\mathbf u\rangle \partial_\phi - r\langle d\phi,\mathbf u\rangle \partial_r \big), where the angle brackets mean contraction of a 1-form and vector. Notice from here and the two “offending” Christoffel symbols mentioned earlier, that only (the component of) the derivative in the \phi–direction is affected.

These expressions obscure some beautiful symmetry. Let’s raise one index and lower another, in the correction term:

    \[\tilde\nabla_{\mathbf v}\mathbf u = \nabla_{\mathbf v}\mathbf u - \frac{1}{r}\langle d\phi,\mathbf v\rangle \cdot (2\partial_r\wedge\partial_\phi)\lrcorner\mathbf u^\flat.\]

Here 2\partial_r\wedge\partial_\phi := \partial_r\otimes \partial_\phi - \partial_\phi\otimes\partial_r is a wedge product, \mathbf u^\flat is just the 1-form with components u_\mu, and “\boldsymbol\lrcorner” is a contraction. The correction’s components are simply -r^{-1}v^\phi\cdot(0,-u_\phi,u_r,0). This is a vector, even though some lowered indices appear in the expression. The correction is just a rotation in the r\phi–plane! From inspection of the diagram, this is unsurprising.

This is analogous to Fermi–Walker transport. Given a worldline, this corrects the (Levi–Civita connection) time-derivative \nabla_{\mathbf u} by a rotation in the plane spanned by \mathbf u and the 4-acceleration vector \nabla_{\mathbf u}\mathbf u. Under Fermi–Walker transport, orthonormal frames stay orthonormal over time, and their orientation agrees with gyroscopes. For both our connection and the Fermi-Walker derivative, there is a preferred differentiation direction, along which a rotation is added to the Levi-Civita derivative.

I previously wrote about a connection for a spherically symmetric vector field. This has been a good learning experience about connections other than Levi-Civita. Many of us completed general relativity courses in which the curvature quantities were merely formulae with no intuitive understanding. However questions from mathematicians like: “Which connection are you using?” prompted me to learn more. (At least I have never been asked which differential structure I am using, nor which point-set topology, which is fortunate for all involved. 🙂 ) There are various physically-motivated connections defined in research paper  §2. I intend to apply this to the rotating disc, and to an observer field in Schwarzschild spacetime. Also, I accidentally came across Rothman+ 2001  about parallel transport in Schwarzschild spacetime, with numerous followup papers by various authors. All of this struck me again with a sense of fascination about curvature: how rich and deep it is.

🡐 Spherical symmetry connection | Curvature | ⸻ 🠦

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