Symmetry of geodesic equations under the transformation of Christoffel symbols

Question

I am not asking for a solution of the following problem that appears in my assignment. However, I don't understand the question and I would like someone to explain the what the question actually is asking AND also hints for a solution as in where to look for or what to think in solving this,

Express the geodesic equation as a differential equation for
$x^{mu}(tau) .$ What is the most general transformation of the
Christoffel symbols that will leave these equations invariant?

(The question assumes metric compatibility and that the connection torsionfree.) The Christoffel can change if there's a transformation $xrightarrowbar{x}$ and $g_{munu}rightarrowbar{g}_{munu}$. Also it can change if $taurightarrow{tau^prime}$, via,
$$frac{d^2 x^mu}{dtau^2}+Gamma^mu_{alphabeta}(x(tau))frac{dx^alpha}{dtau}frac{dx^beta}{dtau}=0tag{1}$$
What transformations are the question asking for (may be both! Or maybe a different kind of transformation for Christoffel symbols altogeher)? How to approach the question?

Qmechanic · Answer

Let $nabla^{LC}$ denote the Levi-Civita (LC) connection on a pseudo-Riemannian manifold $(M,g)$. OP's eq. (1) is the affine geodesic equation $nabla^{LC}_{dot{gamma}}dot{gamma}=0$, which is independent of the local $x$-coordinate system for $M$, but depends on the world-line parametrization $tau$: Eq. (1) holds when the parameter $lambda$ is affinely related to the arc length $s=atau+b$ of the geodesic.
For more information, see my related Phys.SE answer here.

Symmetry of geodesic equations under the transformation of Christoffel symbols

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