Derivation of field equations In metric
f(
R) gravity, one arrives at the field equations by varying the action with respect to the
metric and not treating the
connection \Gamma^\mu_{\alpha\beta} independently. For completeness we will now briefly mention the basic steps of the variation of the action. The main steps are the same as in the case of the variation of the
Einstein–Hilbert action (see the article for more details) but there are also some important differences. The variation of the determinant is as always: \delta \sqrt{-g}= -\frac{1}{2} \sqrt{-g} g_{\mu\nu} \delta g^{\mu\nu} The
Ricci scalar is defined as R = g^{\mu\nu} R_{\mu\nu}. Therefore, its variation with respect to the inverse metric g^{\mu\nu} is given by \begin{align} \delta R &= R_{\mu\nu} \delta g^{\mu\nu} + g^{\mu\nu} \delta R_{\mu\nu}\\ &= R_{\mu\nu} \delta g^{\mu\nu} + g^{\mu\nu} \left (\nabla_\rho \delta \Gamma^\rho_{\nu\mu} - \nabla_\nu \delta \Gamma^\rho_{\rho\mu} \right ) \end{align} For the second step see the article about the
Einstein–Hilbert action. Since \delta\Gamma^\lambda_{\mu\nu} is the difference of two connections, it should transform as a tensor. Therefore, it can be written as \delta \Gamma^\lambda_{\mu\nu}=\frac{1}{2}g^{\lambda a}\left(\nabla_\mu\delta g_{a\nu}+\nabla_\nu\delta g_{a\mu}-\nabla_a\delta g_{\mu\nu} \right). Substituting into the equation above: \delta R= R_{\mu\nu} \delta g^{\mu\nu}+g_{\mu\nu}\Box \delta g^{\mu\nu}-\nabla_\mu \nabla_\nu \delta g^{\mu\nu} where \nabla_\mu is the
covariant derivative and \square = g^{\mu\nu}\nabla_\mu\nabla_\nu is the
d'Alembert operator. Denoting F(R) = \frac{df}{d R}, the variation in the action reads: \begin{align} \delta S[g]&= \int \frac{1}{2\kappa} \left(\delta f(R) \sqrt{-g}+f(R) \delta \sqrt{-g} \right)\, \mathrm{d}^4x \\ &= \int \frac{1}{2\kappa} \left(F(R) \delta R \sqrt{-g}-\frac{1}{2} \sqrt{-g} g_{\mu\nu} \delta g^{\mu\nu} f(R)\right) \, \mathrm{d}^4x \\ &= \int \frac{1}{2\kappa} \sqrt{-g}\left(F(R)(R_{\mu\nu} \delta g^{\mu\nu}+g_{\mu\nu}\Box \delta g^{\mu\nu}-\nabla_\mu \nabla_\nu \delta g^{\mu\nu}) -\frac{1}{2} g_{\mu\nu} \delta g^{\mu\nu} f(R) \right)\, \mathrm{d}^4x \end{align} Doing
integration by parts on the second and third terms (and neglected the boundary contributions), we get: \delta S[g]= \int \frac{1}{2\kappa} \sqrt{-g}\delta g^{\mu\nu} \left(F(R)R_{\mu\nu}-\frac{1}{2}g_{\mu\nu} f(R)+[g_{\mu\nu}\Box -\nabla_\mu \nabla_\nu]F(R) \right)\, \mathrm{d}^4x. By demanding that the action remains invariant under variations of the metric, \frac{\delta S}{\delta g^{\mu\nu}}=0, one obtains the field equations: F(R)R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}f(R)+\left[ g_{\mu\nu} \Box-\nabla_\mu \nabla_\nu \right]F(R) = \kappa T_{\mu\nu}, where T_{\mu\nu} is the
energy–momentum tensor defined as T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g} \mathcal L_\mathrm{m})}{\delta g^{\mu\nu}}, where \mathcal L_m is the matter Lagrangian.
Generalized Friedmann equations Assuming a
Robertson–Walker metric with scale factor a(t) we can find the generalized
Friedmann equations to be (in units where \kappa = 1): 3F H^{2} = \rho_+\rho_+\frac{1}{2}(FR-f)-3H{\dot F} -2F\dot{H} = \rho_+\frac{4}{3}\rho_+\ddot{F}-H\dot{F}, where H = \frac{\dot{a}}{a} is the
Hubble parameter, the dot is the derivative with respect to the cosmic time , and the terms m and rad represent the matter and radiation densities respectively; these satisfy the
continuity equations: \dot{\rho}_+3H\rho_=0; \dot{\rho}_+4H\rho_=0.
Modified gravitational constant An interesting feature of these theories is the fact that the
gravitational constant is time and scale dependent. To see this, add a small scalar perturbation to the metric (in the
Newtonian gauge): \mathrm{d}s^2 = -(1+2\Phi)\mathrm{d}t^2 +\alpha^2 (1-2\Psi)\delta_{ij}\mathrm{d}x^i \mathrm{d}x^j where and are the Newtonian potentials and use the field equations to first order. After some lengthy calculations, one can define a
Poisson equation in the Fourier space and attribute the extra terms that appear on the right-hand side to an effective gravitational constant eff. Doing so, we get the gravitational potential (valid on sub-
horizon scales ): \Phi = -4\pi G_\mathrm{eff} \frac{a^2}{k^2} \delta\rho_\mathrm{m} where m is a perturbation in the matter density, is the Fourier scale and eff is: G_\mathrm{eff}=\frac{1}{8\pi F}\frac{1+4\frac{k^2}{a^2R}m}{1+3\frac{k^2}{a^2R}m}, with m\equiv\frac{RF_{,R}}{F}.
Massive gravitational waves This class of theories when linearized exhibits three polarization modes for the
gravitational waves, of which two correspond to the massless
graviton (helicities ±2) and the third (scalar) is coming from the fact that if we take into account a conformal transformation, the fourth order theory () becomes
general relativity plus a
scalar field. To see this, identify \Phi \to f'(R) \quad \textrm{and} \quad \frac{dV}{d\Phi}\to\frac{2f(R)-R f'(R)}{3}, and use the field equations above to get \Box \Phi=\frac{\mathrm{d}V}{\mathrm{d}\Phi} Working to first order of perturbation theory: g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu} \Phi=\Phi_0+\delta \Phi and after some tedious algebra, one can solve for the metric perturbation, which corresponds to the gravitational waves. A particular frequency component, for a wave propagating in the -direction, may be written as h_{\mu\nu}(t,z;\omega)=A^{+}(\omega)\exp(-i\omega(t-z))e^{+}_{\mu\nu}+A^{\times}(\omega)\exp(-i\omega(t-z))e^{\times}_{\mu\nu} +h_f(v_\mathrm{g} t-z;\omega) \eta_{\mu\nu} where h_f\equiv \frac{\delta \Phi}{\Phi_0}, and g() = d/d is the
group velocity of a
wave packet centred on wave-vector . The first two terms correspond to the usual
transverse polarizations from general relativity, while the third corresponds to the new massive polarization mode of () theories. This mode is a mixture of massless transverse breathing mode (but not traceless) and massive longitudinal scalar mode. The transverse and traceless modes (also known as tensor modes) propagate at the
speed of light, but the massive scalar mode moves at a speed G f(R) = \alpha R^2 (also known as pure R^2 model), the third polarization mode is a pure breathing mode and propagate with the speed of light through the spacetime. == Equivalent formalism ==