Suppose you have a spherically symmetric vector field, as in the diagram. Can we find an affine connection which transports the vectors into one-another? That is, a geometry in which they are all “parallel”?
Take Schwarzschild spacetime, in the usual coordinates . The coordinate basis vectors are , , , and . I will write these as , so for for example, this is the vector with components . Recall a connection is defined by:
where the are the connection coefficients, also called Christoffel symbols in the specific case of the Levi-Civita connection. (Recall the Levi-Civita connection is the one inherited from the metric: it is the unique symmetric and metric-compatible connection.) For each pair , this definition is interpreted as the derivative of the field, in the direction .
Now consider an arbitrary vector field of the form:
We would not expect the sought-for parallel transport to work for vectors with components in the or -directions — at least, not without imposing extra choices. In particular, the “hairy ball theorem” states no smooth, non-vanishing vector field along the 2-sphere exists: that is, within its 2-dimensional tangent bundle. For Schwarzschild spacetime, we move around a 2-sphere of constant and , by taking “directional derivatives” along the -plane. As expected, does not vanish, even in these directions:
The offending Christoffel symbols turn out to be and . These arise from and . These quantify how the radial coordinate vector changes as you move around on a sphere.
One option is to simply define new connection coefficients for which these vanish: and , and keep the remaining Christoffel symbols, in order to remain as close as possible to the metric connection. This procedure is justified, because given a frame field, any choice of smooth functions yields a valid connection (Lee 2018 , Introduction to Riemannian manifolds, Lemma 4.10). We can also write this new connection as the usual (Levi-Civita) covariant derivative plus a bilinear correction:
The parenthetical term is a (1,2)-tensor we interpret as accepting the vectors in the last two slots ( in the second slot, and into the last), returning another vector. The correction term may also be written , where the angle brackets mean contraction of a 1-form and vector in this case. Intuitively, the parenthetical term just above is also a projection, returning only the angular part of the differentiation direction . This is the blue arrow in the original diagram. For large , the basis vectors and grow very large, but the red vectors must adjust only by the angle rotated through, hence the multiplier. returns the radial component .
As a check, as required. The new connection is not symmetric, because and remain non-vanishing. Hence the connection has “torsion”. I won’t write out its Riemann and Ricci tensors, but the scalar curvature is ! At face value this violates the Einstein field equations, for which the Ricci tensor (and hence the scalar curvature) always vanish in a vacuum, however Einstein’s equations use the Levi-Civita connection. Curiously, the value is precisely the scalar curvature for a 2-sphere.
We can also construct a symmetric connection for which additionally . In the (somewhat) index-free expression:
where is the symmetric product, and analogously for . This connection has Ricci tensor equal to the metric in the and components, apart from a scalar factor , and vanishing elsewhere. Its scalar curvature is .
Hence we have constructed connections which parallel transport our spherically symmetric vector field around a sphere, and deviate as little as possible from the Levi-Civita connection. Neither of the new connections are “metric-compatible”, for instance . Hence . The same holds for .
If you find some formulae here do not work for you, compare your convention for the connection coefficient index order, or try swapping and in the correction terms. I had problems myself, so undertook a painstaking review of my own conventions, and wrote a new page describing them. Finally, beware of coordinate basis vectors! The “vectors” and actually depend on all four coordinates, which is related to the so-called “second fundamental confusion of calculus”! In case of ambiguity, perhaps some should be replaced with and , or scalar multiples thereof. I avoided this technicality in the interests of readability. This concern only applies to coordinate systems in which the metric is non-diagonal.