Invariance in general relativity, university in problems question

Question

From Problem #5 here,

Free falling particles' worldlines in General Relativity are geodesics of the spacetime, i.e the curves $x^\mu(\lambda)$ with tangent vector $u^\mu=dx^\mu/d\lambda$, such that covariant derivative of the latter along the curve equals to zero: $$u^\mu\nabla_\mu u^\nu=0.$$

In a (pseudo-)Euclidean space the geodesics are straight lines. Obtain the general equation of geodesics in terms of the connection coefficients. Show that the quantity $u^\mu u_\mu$ is conserved along the geodesic.

Is $u^{\mu}u_{\mu}$ not invariant in general relativity? Is this not enough to say that it is constant along a geodesic? Am I understanding something wrong?

Answer 1

"Invariant" in this case means "invariant under Lorentz transformations". This means that it is observed to be the same by different observers passing through one particular spacetime point; two different observers with differing velocities will have their observable quantities related by a Lorentz transformation.

However, this doesn't imply that the quantity $u^\mu u_\mu$ is the same at different points along the geodesic. As an (imperfect) analogy, two people using Euclidean coordinates rotated relative to each other will agree that the electric field at a particular point has a particular magnitude; but that doesn't imply that the electric field has a constant magnitude everywhere in space.

There can be invariants along geodesics, if the spacetime has enough symmetry (look up "Killing vector" and "conserved quantity"), but these are always expressed as contractions of $u^\mu$ with another vector (or sometimes tensor) rather than a contraction of $u^\mu$ with itself.