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 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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