# Affine and metric geodesics

In D'Inverno's "Introducing Einstein's Relativity", an affine geodesic is defined as a privileged curve along which the tangent vector is propagated parallel to itself. Choosing an affine parameter, the affine geodesic equation reduces to

$$\frac{d^2x^a}{ds^2}+\Gamma^a_{bc}\frac{dx^b}{ds}\frac{dx^c}{ds}=0\tag{1}$$

Later, a metric is introduced, and a timelike metric geodesic is defined as a privileged curve that makes the interval $$s=\int_{P_1}^{P_2}\sqrt{g_{ab}\frac{dx^a}{du}\frac{dx^b}{du}}du\tag{2}$$ stationary under small variations. From Euler-Lagrange equations, and choosing a parameter $$u$$ which is linearly related to the interval $$s$$, the equation for a metric geodesic becomes

$$\frac{d^2x^a}{ds^2}+\frac 12 g^{ad}\left(\partial_cg_{bd}+\partial_bg_{cd}-\partial_dg_{bc}\right)\frac{dx^b}{ds}\frac{dx^c}{ds}=0.\tag{3}$$

Then, it states that once we have a manifold endowed with both an affine connection $$\Gamma^a_{bc}$$ and a metric, there are two classes of geodesics: affine ones and metric ones, which are different in general. However, if we take the metric connection

$$\Gamma^a_{bc}\equiv\frac 12 g^{ad}\left(\partial_cg_{bd}+\partial_bg_{cd}-\partial_dg_{bc}\right),\tag{4}$$

they will coincide. This definition leads to the identity $$\nabla_a g_{bc}=0\tag{5}$$ and, as $$(4)$$ is symmetric, the torsion is $$0$$.

Conversely, having a symmetric connection and requiring $$(5)$$ leads to the metric connection $$(4)$$.

My questions are:

1. Why are we asking that both classes of geodesics coincide? What does it mean? Is there any physical motivation or is it general in differential geometry?
2. Is it more natural to require this coincidence or that $$\nabla_a g_{bc}=0$$ and null torsion?
3. In my lessons I was told that non-zero torsion has to be taken into account in quantum gravity theories . Without the goal of fully understanding why this is so, as it is still far from my knowledge, does it mean that metric geodesics and affine geodesics don't coincide in these theories?
• IMO it's just an axiom. The Levi-Civita connection is the most natural one, but as always, the reason for using it is that the results match the observations. I don't see much point trying to motivate it with arguments like this. Mar 14, 2021 at 14:48

You're correct that if one works with a general affine connection $$\bar{\nabla}$$ that isn't the Levi-Civita connection, the two notions do not coincide: i.e. the autoparallel curves satisfying the "geodesic equation" $$\bar{\nabla}_{\tau} X^{\mu} = 0 \iff \ddot{x}^{\mu} + \bar{\Gamma}^{\mu}_{\rho \sigma}\dot{x}^{\rho} \dot{x}^{\sigma} = 0$$ are not the same as the "geodesics" obtained by extermising your equation (2), which always leads to the Levi-Civita connection as you observed. I'm not sure there are very satisfying answers to all your questions 1 to 3, but there are of course arguments for why one wants a Levi-Civita connection in general$$^{1}$$: e.g. we want parallel transport to preserve the inner product (if $$\nabla g \neq 0$$ this isn't the case). Moreover, extremising (2) to obtain a geodesic only seems to make sense if the metric is constant with respect to the connection (i.e. we have vanishing non-metricity).

1. See above. Though I'm not sure of any physical arguments why we'd need the two types of geodesics (the autoparallel curve and extremised action) to be the same.

2. Requiring these to be the same doesn't actually pin down the Levi-Civita connection. If a general affine connection is given by $$\bar{\Gamma}^{\lambda}_{\mu \nu} = \Gamma^{\lambda}_{\mu \nu} + K^{\lambda}{}_{\mu \nu}$$ with $$\Gamma$$ the Levi-Civita connection, then for the geodesics obtained from (2) to coincide with autoparallel curves we only need that $$K^{\lambda}{}_{\mu \nu}$$ is antisymmetric over $$\nu \mu$$. In other words, torsion as defined by $$K^{\lambda}{}_{[\mu \nu]}$$ doesn't actually affect geodesic motion derived from autoparallel curves, see  for more details. So we really need other reasons for wanting $$\nabla g=0$$ and a symmetric connection.

3. See answer 2, this isn't actually the case: the geodesic equations can be the same. See this question for another quick proof: geodesics for two connections differing by torsion. What they probably meant in the lesson is that the effects of torsion become important when considering spinors because torsion couples to spin (but I won't go into the details).

$$^{1}$$This question has been asked a few times here too, e.g. What is the motivation from Physics for the Levi-Civita connection on GR?, Why can we assume torsion is zero in GR?, but of course Einstein-Cartan theory metric-affine gravity are both alternatives, so we're not required to choose this connection.

 Also https://arxiv.org/abs/gr-qc/9402012 might be useful for general reading.

1.It means that both are consistent and all the arguments concerning the affine space, that do not utilize distance via the metric, are applicable. He is showing that a lot can be done without a metric and that the metric adds distance.

2.I don't know. The theorem on page 85 says that the covariant derivative of the metric being 0 is a necessary and sufficient condition for the connection to be a metric connection. It seems natural to me that there should be a coincidence between the two geodesic equations. It would be weird if the geodesics couldn't match up. Because the arguments regarding the affine connection should all equally apply to the metric space.

Why are we asking that both classes of geodesics coincide? What does it mean?

Affine geodesics and geodesics are essentially the same thing. It turns out that geodesics can be affinely reparametrised without affecting the property that it is a geodesic. It's for this reason they are called affine geodesics, to emphasis this property. Personally, I prefer the term geodesic without the affine qualifier.

There is also a term affine connection which is simply a linear connection on a vector bundle. There are also connections on affine bundles which may be more justly called affine connections. In fact, connections are defined in all generality on fibre bundles of which the tangent bundle is an example. In fact, the tangent bundle is a vector bundle and hence we require a linear or affine connection. Personally, I prefer the term linear connection so as to reserve the term affine connection for a connection on an affine bundle.

In general, we cannot differentiate vector fields on a manifold without a connection. Hence a connection is extra structure that is required when we want to differentiate. Each connection has its own notion of what a geodesic is, hence by choosing different connections there are many different classes of geodesics. Here, choosing an affine or linear connection will give you the class of linear or affine geodesics. In this sense, the term affine geodesic is different to the sense I allued to in the first paragraph, and is what probably what d'Inervo is referring to.

The question then is how to choose the right connection. After all, nature chooses a single connection in GR and not all of them. It turns out that when the manifold is pseudo-Riemannian, of which Lorentzian manifolds in GR are a particular case, there is a canonical (that is unique) connection called the Levi-Civita connection. It's this connection that is used in GR. It's also termed a metric connection and it's class of geodesics are called metric geodesics.

Thus metric geodesics are a special case of linear or affine geodesics. They aren't the same.

• This doesn't address the questions OP has asked. You're probably confusing affine geodesics with the affine connection. Mar 14, 2021 at 19:35
• @Eletie: No, I'm not. The affine connection is also referred to as the linear connection which is a general connection on a vector bundle with a linear connector. There are also connections on affine bundles which can also be termed affine connections. The affine connection that you're thinkingbof is probably the canonical Levi-Civita connection on a pseudo-Riemannian manifold. Mar 14, 2021 at 23:53
• @Eletie: I know that, which you can see from my updated answer. Like I said, I prefer the term affine connection for connections on affine bundles and vector connections for connections on vector bundles - it causes less confusion. Mar 15, 2021 at 0:18
• OP's question is about the autoparallel curves of a general affine connection not coinciding with the geodesic equation obtained from extremising the action (2). I.e. two types of "geodesics" which go by differing names in the literature but are distinct for a general affine connection. I see you've edited your answer, but unfortunately this is still missing the point of the question and just giving definitions that OP appears to already understand. See my answer for an explanation of these differing concepts. Mar 15, 2021 at 0:19
• @Eletie: I've said that geodesics for general affine connections don't coincide for those of a metric connection as they're different connections. What exactly is your point? Mar 15, 2021 at 0:22