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For an affine parameter, the geodesic equation isVary the energy action with endpoint-fixed variations. Its Euler-Lagrange equation isMultiplying by $g^{da}/2$ and symmetrizing the velocity indices determines the Levi-Civita connection:This determines the torsion-free metric connection; an antisymmetric-in-$b,c$ addition could not be detected by the geodesic energy variation alone.

For the displayed metric on $t>0$, the nonzero coefficients are ${\Gamma }_{xt}^x={\Gamma }_{tx}^x=-1/t$ and ${\Gamma }_{xx}^t={\Gamma }_{tt}^t=-1/t$. ConsequentlyFor a timelike curve choose proper time so $({\dot{x}}^2-{\dot{t}}^2)/t^2=-1$. The first equation integrates to $\dot{x}=p{t}^2$, and then ${\dot{t}}^2=t^2+p^2{t}^4$. If $p\ne 0$, dividing and integrating givesThe timelike condition is ${\alpha }^2-4\beta =4p^{-2}>0$, with the appropriate branch in the coordinate patch. Conversely these curves with that condition obey the first integrals and admit a timelike affine parametrization.

There is an omitted family in the printed claim: if $p=0$, then $x=x_c$ is a timelike vertical geodesic, with $t=t_{\ast }{e}^{\pm \tau }$ in proper time. No fixed finite $\alpha ,\beta$ can represent all points of such a vertical curve by the displayed quadratic. The quadratic therefore describes the nonvertical timelike geodesics; the vertical family must be included for completeness.

Together the two families describe the timelike geodesics of an inverse-time conformal metric.