but can “chase” light arbitrarily closely—fixes a Weyl structure, an equivalence class of conformal metrics and affine connections such that $$\begin{aligned}
    \nabla_a g_{bc} = A_a g_{bc} \qquad
    \text{for some vector field $A$.}
    \label{a1}\end{aligned}$$

As Einstein first noted, though (see [@Einstein]), unless $A_a$ is of the form $\partial_a\varphi$ for some scalar $\varphi$, eqn.(\[a1\]) implies that lengths change under parallel transport. This would lead to a “second clock effect,” a dependence of the rate of a clock on its history. The observation that this is not the case in our Universe suggests a further condition:

1.  [**The absence of a “second clock effect”**]{} implies that the vector $A$ can be eliminated by a Weyl transformation, picking out a unique representative of the conformal class of metrics. This requirement can be replaced by a number of others: for instance, that neighboring clocks remain synchronized [@EPS] or that matter waves follow geodesics in the short wavelength limit [@Audretsch].

Together, these observations imply that motion in a gravitational field can be described as geodesic motion in a Lorentzian spacetime, with a metric providing a full description of the field. To go beyond this kinematic setting and obtain the dynamics of general relativity, more is needed. 