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3.1 When is a gravitational field strong?

Looking at the Schwarzschild spacetime, it is natural to measure the “strength” of the gravitational field at a distance r away from an object of mass M by the parameter
ε ≡ GM---, (5 ) rc2
which is proportional to the Newtonian gravitational potential and is also directly related to the redshift. Infinitesimal gravitational fields correspond to the limit ε → 0, which leads to the Minkowski spacetime of special relativity. Weak gravitational fields correspond to ε ≪ 1, which leads to Newtonian gravity. Finally, the strongest gravitational fields accessible to an observer are characterized by ε → 1, at which point the black-hole horizon of an object of mass M is approached. (Note that formally the horizon of a Schwarzschild black hole occurs at ε = 2; I drop here the factor of 2, as I am mostly interested in dimensional arguments.)

Albeit useful in defining post-Newtonian expansions, the parameter ε 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 [1View Equation]) 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

( 2) ( 2)− 1 ds2 = − 1 − 2GM---− Λr-- dt2 + 1 − 2GM---− Λr-- dr2 + r2(dθ2 + sin2 θdφ2) (6 ) rc2 3 rc2 3
and the Newtonian approximation becomes invalid when
GM 1 -----≪ -Λ . (7 ) c2r3 6
In this case, a gravitational field is “weak” if the spacetime curvature is smaller than 1/6Λ, independent of the value of the parameter ε. In the opposite extreme, if there are additional terms in the action of the gravitational field beyond the Einstein–Hilbert term, such as
4 ∫ --c--- 4 √ --- 2 S = 16 πG d x − g(R + αR ) , (8 )
then the general relativistic predictions become inaccurate at strong gravitational fields defined by the condition
GM 1 -----≫ --, (9 ) c2r3 α
even if the parameter ε is much smaller than unity. Note that, in Equation (8View Equation), α is an appropriate constant with units of (length)2 and I have set the Ricci scalar to R ∼ GM ∕r3c2 (I use this here as an order of magnitude estimate and do not consider the fact that, if the distance r 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 ε) but rather by a large curvature

GM--- ξ ≡ r3c2 . (10 )
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 [7View Equation] and [9View Equation]) I will not normalize the parameter ξ 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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