Extended gravity theories and the Einstein--Hilbert action

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Extended graviry theories 27 1 where 4 (1 + k'p)/G~ and m2 f (4) = LY~*/G~ = (GD@ - 1)*/(4aG~). O'Hanlon introduced his action to produce a 'fifth force' Yukawa type interaction in the quasi- Newtonian gravitational potential, so it is not surprising that it also appears in the weak-field limit of fourth-order gravity [15]. This equivalence applies to any gravitational Lagrangian that is a function of the Ricci scalar. where ' denotes differentiation with respect to 9. The field equation for 'p again gives 9 = R, provided F" # 0, which substituted into the metric field equations gives the standard field equations for the F(R) Lagrangian. In O'Hanlon's notation, we have Go@ F'('p) Gom2 f (4) = 'pF' - F. (9) 3. Higher-order gravity as scalar-tensor gravity We can extend this equivalence to sixth- or arbitrarily high-order gravity theories derived from Lagrangians that are functions not only of R but also OR or O"R, where U is the d'Alembertian. Such terms can also be generated by quantum corrections to general relativity and these theories have recently been studied in the context of inflationary cosmology [16,17]. Notice that apparently different Lagrangians F(O'R) can yield identical field equations if they differ only by a total divergence, as this can be written as a boundary term which cannot contribute to the Euler-Lagrange equations. For instance, OiR alone is a total divergence and can be ignored, while a term O"ROmR can be integrated by parts to give RGmt"R. Thus I can take any polynomial function F(0'R) to be linear in its highest-order derivative (O"R), and that this ierm is multiplied only by a function of the Ricci scalar, F,(R). Its field equations contain (2n + 4)-order derivatives of the metric tensor. Any Lagrangian containing a term 0"R multiplied by derivatives of the Ricci scalar, or any other fields, corresponds to a still higher-order gravity theory as it can be integmted by p m to give terms at least of order U""+' R. Consider then the action for (2n + 4)-order gravity, with a polynomial F(O'R) = Fo(0'"" R) + F, (R)O"R The classical field equations are [I61 I = ~XGDT'" + -g""(F - OR) + (g"gvK - g""g")O;~, 2
212 D W ad where I have used We introduce the new variables 'pi = a. 'pI, . . . 'pn and write the function F = F(OiR) as F(rpi), selecting a new action whose field equations will require 'pi = O'R, with OoR R.t Thus we choose the dynamically equivalent action and the (OX field equations are where n j=O Fjt(OjR - rpj) = 0 Then this does indeed require O'R = 'pi, subject now to the condition that the matrix Fjk is non-degenerate (i.e. det& # 0). and it can be verified that the metric field equations are indeed the same as in the original higher-order gravity theory. The Lagrangian, which still contains derivatives of R, can be reduced to a Lagrangian linear in the Ricci scalar by integration by parts. For instance aF dDxfiRD- arpi Since I am only concerned in this section with actions yielding equivalent Euler-Lagrange field equations I will continue to disregard boundary terms. That is, I am assuming that to obtain the variation of the action to first-order, I can neglect the contribution due to the variation of the fields' derivatives on the boundary, aM. There is nothing lost by this, as the correct form of the boundary term required in higher-order gravity theories to avoid this requirement is not in general known [lS]; a fact I will return to later. Thus the action can be brought to t Notice that these fields have !he non-standard dimensions of (len5h)-2"+'), We could re-define lhe fields to give them more usual dimensions of ras for a scalar field, but for brevity of notation I will stick to 'pi as defined here.
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Extended gravity theories and the Einstein--Hilbert action

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