The geodesic equation in general relativity is famously invariant under affine reparametrization, i.e., under the reparametrization $\tau \to a\tau + b$ where $\tau $ is the proper time. This can be read off directly from the geodesic equation
\begin{align*}
\frac{d^2 x^\lambda}{d\tau^2}+\Gamma^{\lambda}_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}&=0\text{ (where }d\tau^2\equiv g_{\mu\nu}dx^\mu dx^\nu\text{)}\\
\end{align*}
One can reinterpret the affine reparametrization as a transformation $x^\lambda\to ax^\lambda$ where $a$ is a constant. One also observes that the geodesic equation is invariant under the scale transformation $x^\lambda \to a x^\lambda$. Let's just consider the transformation $x^\lambda \to a x^\lambda$. Now, under this transformation, $d\tau \to a d\tau$. The first term in the geodesic equation picks up a factor of $1/a$ ($a$ from the numerator and $a^2$ from the denominator). The second term in the geodesic equation also picks up a factor of $1/a$ due to the one partial derivative in each of the terms in the Christoffel symbols. So, the overall factor of $1/a$ cancels out. Thus, the geodesic equation is invariant under this scale transformation. This is a somewhat weird invariance in my estimation for the following reasons:
- It is not generically an isometry, because the metric is not necessarily invariant under this transformation.
- Consequently, the action $S=-m\int d\tau$ is not generically invariant under this transformation. So, we have an equation of motion invariant under a transformation without the action being invariant under the same transformation.
So, I am not sure how to think about this invariance. Is it an artifact but not a real symmetry (like the diffeomorphism invariance of GR which is not generically an isometry)? But if so, how exactly? Because unlike in the case of diffeomorphisms, we are not actually transforming the metric (or anything else) precisely in such a way that it cancels out the factors arising from the transformation of coordinates.
It is also somewhat curious that the geodesic equation is scale-invariant even in the presence of matter when the Einstein-Hilbert action is not scale-invariant. Of course, that is not a contradiction but still, it seems somewhat interesting and I would be interested in finding out if it associates to some nice physical implication.
I should emphasize the point that I made in the comments:
The geodesic equation is invariant under π₯βππ₯ not only when I consider π₯βππ₯ as a diffeomorphism (which would be unsurprising) but also when I consider it as a "real" transformation, i.e., I don't covariantly change the metric accordingly. The Christoffel symbols do need to change here, but not as they would change under a diffeomorphism. They only pick up an overall factor of 1/π from the partial derivatives picking up a factor of 1/π as a direct consequence of π₯βππ₯.
And yet, this is not an isometry as I pointed out earlier.
To put it very simply, I don't know what to do with an invariance that is coming neither from an isometry nor from a diffeomorphism.