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 ) 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 becomes2 and I have set the Ricci scalar to (I use this here as an order of magnitude estimate and do not consider the fact that, if the distance is larger than the radius of the object, then the Ricci scalar in general relativity vanishes).
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
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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