Albeit useful in defining post-Newtonian expansions, the parameter is not fundamental in characterizing a gravitational field in Einstein’s theory. Indeed, the geodesic equation and the Einstein field equation (or equivalently, the Einstein–Hilbert action [1]) are written in terms of the Ricci scalar, the Ricci tensor, and the Riemann tensor, all of which measure the curvature of the field and not its potential. As a result, when we consider deviations from general relativity that arise by adding terms linearly to the Einstein–Hilbert action, the critical strength of the gravitational field beyond which these additional terms become important is typically given in terms of the spacetime curvature.

For example, in the presence of a cosmological constant, the metric of a spherically-symmetric object becomes

and the Newtonian approximation becomes invalid when In this case, a gravitational field is “weak” if the spacetime curvature is smaller than 1/6, independent of the value of the parameter . In the opposite extreme, if there are additional terms in the action of the gravitational field beyond the Einstein–Hilbert term, such as then the general relativistic predictions become inaccurate at strong gravitational fields defined by the condition even if the parameter is much smaller than unity. Note that, in Equation (8), is an appropriate constant with units of (length)Similar considerations lead to a condition on curvature when we add to the Einstein–Hilbert action terms that invoke additional scalar, vector, and tensor fields. In all these cases, a strong gravitational field is characterized not by a large gravitational potential (i.e., a high value of the parameter ) but rather by a large curvature

Because the condition that the curvature needs to satisfy in order for a gravitational field to be considered “strong” depends on the particular deviation from general relativity under study (cf. Equations [7] and [9]) I will not normalize the parameter to any particular energy density but rather leave it, hereafter, as a dimensional quantity.This is an appropriate parameter with which to measure the strength of a gravitational field in a geometric theory of gravity, such as general relativity, because the curvature is the lowest order quantity of the gravitational field that cannot be set to zero by a coordinate transformation. Moreover, because the curvature measures energy density, a limit on curvature will correspond to an energy scale beyond which additional gravitational degrees of freedom may become important.

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