Английская Википедия:F(R) gravity

Шаблон:Var(Шаблон:Var) is a type of modified gravity theory which generalizes Einstein's general relativity. Шаблон:Var(Шаблон:Var) gravity is actually a family of theories, each one defined by a different function, Шаблон:Var, of the Ricci scalar, Шаблон:Var. The simplest case is just the function being equal to the scalar; this is general relativity. As a consequence of introducing an arbitrary function, there may be freedom to explain the accelerated expansion and structure formation of the Universe without adding unknown forms of dark energy or dark matter. Some functional forms may be inspired by corrections arising from a quantum theory of gravity. Шаблон:Var(Шаблон:Var) gravity was first proposed in 1970 by Hans Adolph Buchdahl^[1] (although Шаблон:Var was used rather than Шаблон:Var for the name of the arbitrary function). It has become an active field of research following work by Starobinsky on cosmic inflation.^[2] A wide range of phenomena can be produced from this theory by adopting different functions; however, many functional forms can now be ruled out on observational grounds, or because of pathological theoretical problems.

Introduction

In Шаблон:Var(Шаблон:Var) gravity, one seeks to generalize the Lagrangian of the Einstein–Hilbert action: <math display="block">S[g]= \int {1 \over 2\kappa} R \sqrt{-g} \, \mathrm{d}^4x </math> to <math display="block">S[g]= \int {1 \over 2\kappa} f(R) \sqrt{-g} \, \mathrm{d}^4x </math> where <math>\kappa=\tfrac{8\pi G}{c^4}, g = \det g_{\mu\nu}</math> is the determinant of the metric tensor, and <math>f(R)</math> is some function of the Ricci scalar.^[3]

There are two ways to track the effect of changing <math>R</math> to <math>f(R)</math>, i.e., to obtain the theory field equations. The first is to use metric formalism and the second is to use the Palatini formalism.^[3] While the two formalisms lead to the same field equations for General Relativity, i.e., when <math>f(R)=R</math>, the field equations may differ when <math>f(R) \neq R</math>.

Metric Шаблон:Var(Шаблон:Var) gravity

Derivation of field equations

In metric Шаблон:Var(Шаблон:Var) gravity, one arrives at the field equations by varying the action with respect to the metric and not treating the connection <math>\Gamma^\mu_{\alpha\beta}</math> 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: <math display="block">\delta \sqrt{-g}= -\frac{1}{2} \sqrt{-g} g_{\mu\nu} \delta g^{\mu\nu}</math>

The Ricci scalar is defined as <math display="block"> R = g^{\mu\nu} R_{\mu\nu}.</math>

Therefore, its variation with respect to the inverse metric <math>g^{\mu\nu}</math> is given by

<math display="block">\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}</math>

For the second step see the article about the Einstein–Hilbert action. Since <math>\delta\Gamma^\lambda_{\mu\nu}</math>is the difference of two connections, it should transform as a tensor. Therefore, it can be written as <math display="block">\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).</math>

Substituting into the equation above: <math display="block">\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}</math>

where <math>\nabla_\mu</math>is the covariant derivative and <math>\square = g^{\mu\nu}\nabla_\mu\nabla_\nu</math> is the d'Alembert operator.

Denoting <math>F(R) = \frac{df}{d R}</math>, the variation in the action reads: <math display="block">\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}</math>

Doing integration by parts on the second and third terms (and neglected the boundary contributions), we get: <math display="block">\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.</math>

By demanding that the action remains invariant under variations of the metric, <math>\frac{\delta S}{\delta g^{\mu\nu}}=0</math>, one obtains the field equations: <math display="block">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},</math> where <math>T_{\mu\nu}</math>is the energy–momentum tensor defined as <math display="block">T_{\mu\nu}=-\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g} \mathcal L_\mathrm{m})}{\delta g^{\mu\nu}},</math> where <math>\mathcal L_m</math>is the matter Lagrangian.

The generalized Friedmann equations

Assuming a Robertson–Walker metric with scale factor <math>a(t)</math> we can find the generalized Friedmann equations to be (in units where <math>\kappa = 1</math>): <math display="block"> 3F H^{2} = \rho_Шаблон:\rm m+\rho_Шаблон:\rm rad+\frac{1}{2}(FR-f)-3H{\dot F}</math> <math display="block">-2F\dot{H} = \rho_Шаблон:\rm m+\frac{4}{3}\rho_Шаблон:\rm rad+\ddot{F}-H\dot{F},</math> where <math display="block">H = \frac{\dot{a}}{a}</math> is the Hubble parameter, the dot is the derivative with respect to the cosmic time Шаблон:Var, and the terms Шаблон:Var_m and Шаблон:Var_rad represent the matter and radiation densities respectively; these satisfy the continuity equations: <math display="block"> \dot{\rho}_Шаблон:\rm m+3H\rho_Шаблон:\rm m=0;</math> <math display="block"> \dot{\rho}_Шаблон:\rm rad+4H\rho_Шаблон:\rm rad=0.</math>

Modified Newton's constant

An interesting feature of these theories is the fact that the gravitational constant is time and scale dependent.^[4] To see this, add a small scalar perturbation to the metric (in the Newtonian gauge): <math display="block">\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</math> where Шаблон:Varserif and Шаблон:Varserif 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 Шаблон:Var_eff. Doing so, we get the gravitational potential (valid on sub-horizon scales Шаблон:Nowrap): <math display="block">\Phi = -4\pi G_\mathrm{eff} \frac{a^2}{k^2} \delta\rho_\mathrm{m} </math> where Шаблон:Var Шаблон:Var_m is a perturbation in the matter density, Шаблон:Var is the Fourier scale and Шаблон:Var_eff is: <math display="block"> 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},</math> with <math display="block"> m\equiv\frac{RF_{,R}}{F}.</math>

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 Шаблон:Var(Шаблон:Var) becomes general relativity plus a scalar field. To see this, identify <math display="block"> \Phi \to f'(R) \quad \textrm{and} \quad \frac{dV}{d\Phi}\to\frac{2f(R)-R f'(R)}{3},</math> and use the field equations above to get <math display="block">\Box \Phi=\frac{\mathrm{d}V}{\mathrm{d}\Phi}</math>

Working to first order of perturbation theory: <math display="block"> g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu} </math> <math display="block"> \Phi=\Phi_0+\delta \Phi</math> 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 Шаблон:Var-direction, may be written as <math display="block">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} </math> where <math display="block"> h_f\equiv \frac{\delta \Phi}{\Phi_0},</math>

and Шаблон:Var_g(Шаблон:Var) = dШаблон:Var/dШаблон:Var is the group velocity of a wave packet Шаблон:Var_{Шаблон:Var} centred on wave-vector Шаблон:Var. The first two terms correspond to the usual transverse polarizations from general relativity, while the third corresponds to the new massive polarization mode of Шаблон:Var(Шаблон:Var) theories. This mode is a mixture of massless transverse breathing mode (but not traceless) and massive longitudinal scalar mode. ^[5] ^[6] 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 Шаблон:Var_G < 1 (in units where Шаблон:Var = 1), this mode is dispersive. However, in Шаблон:Var(Шаблон:Var) gravity metric formalism, for the model <math> f(R) = \alpha R^2 </math> (also known as pure <math> R^2 </math> model), the third polarization mode is a pure breathing mode and propagate with the speed of light through the spacetime. ^[7]

Equivalent formalism

Under certain additional conditions^[8] we can simplify the analysis of Шаблон:Var(Шаблон:Var) theories by introducing an auxiliary field Шаблон:Varserif. Assuming <math>f(R) \neq 0</math> for all Шаблон:Var, let Шаблон:Var(Шаблон:Var) be the Legendre transformation of Шаблон:Var(Шаблон:Var) so that <math>\Phi = f'(R)</math> and <math>R=V'(\Phi)</math>. Then, one obtains the O'Hanlon (1972) action: <math display="block">S = \int d^4x \sqrt{-g} \left[ \frac{1}{2\kappa}\left(\Phi R - V(\Phi)\right) + \mathcal{L}_{\text{m}}\right].</math>

We have the Euler–Lagrange equations <math display="block">V'(\Phi)=R</math> <math display="block">\Phi \left( R_{\mu\nu} - \frac{1}{2}g_{\mu\nu} R \right) + \left(g_{\mu\nu}\Box -\nabla_\mu \nabla_\nu \right) \Phi + \frac{1}{2} g_{\mu\nu}V(\Phi) = \kappa T_{\mu\nu}</math>

Eliminating Шаблон:Varserif, we obtain exactly the same equations as before. However, the equations are only second order in the derivatives, instead of fourth order.

We are currently working with the Jordan frame. By performing a conformal rescaling <math display="block">\tilde{g}_{\mu\nu}=\Phi g_{\mu\nu},</math> we transform to the Einstein frame: <math display="block">R = \Phi \left[ \tilde{R} + \frac{3\tilde{\Box} \Phi}{\Phi} -\frac{9}{2}\left(\frac{\tilde{\nabla} \Phi}{\Phi}\right)^2 \right]</math> <math display="block">S = \int d^4x \sqrt{-\tilde{g}}\frac{1}{2\kappa}\left[ \tilde{R} - \frac{3}{2}\left( \frac{\tilde{\nabla}\Phi}{\Phi} \right)^2 - \frac{V(\Phi)}{\Phi^2} \right]</math> after integrating by parts.

Defining <math>\tilde{\Phi} = \sqrt{3} \ln{\Phi}</math>, and substituting <math display="block">S = \int \mathrm{d}^4x \sqrt{-\tilde{g}}\frac{1}{2\kappa}\left[ \tilde{R} - \frac{1}{2}\left(\tilde{\nabla}\tilde{\Phi}\right)^2 - \tilde{V}(\tilde{\Phi}) \right]</math> <math display="block">\tilde{V}(\tilde{\Phi}) = e^{-\frac{2}{\sqrt{3}} \tilde{\Phi}} V \left (e^{\tilde{\Phi}/\sqrt{3}} \right ).</math>

This is general relativity coupled to a real scalar field: using Шаблон:Var(Шаблон:Var) theories to describe the accelerating universe is practically equivalent to using quintessence. (At least, equivalent up to the caveat that we have not yet specified matter couplings, so (for example) Шаблон:Var(Шаблон:Var) gravity in which matter is minimally coupled to the metric (i.e., in Jordan frame) is equivalent to a quintessence theory in which the scalar field mediates a fifth force with gravitational strength.)

Palatini Шаблон:Var(Шаблон:Var) gravity

In Palatini Шаблон:Var(Шаблон:Var) gravity, one treats the metric and connection independently and varies the action with respect to each of them separately. The matter Lagrangian is assumed to be independent of the connection. These theories have been shown to be equivalent to Brans–Dicke theory with Шаблон:Nowrap.^[9]^[10] Due to the structure of the theory, however, Palatini Шаблон:Var(Шаблон:Var) theories appear to be in conflict with the Standard Model,^[9]^[11] may violate Solar system experiments,^[10] and seem to create unwanted singularities.^[12]

Metric-affine Шаблон:Var(Шаблон:Var) gravity

In metric-affine Шаблон:Var(Шаблон:Var) gravity, one generalizes things even further, treating both the metric and connection independently, and assuming the matter Lagrangian depends on the connection as well.

Observational tests

As there are many potential forms of Шаблон:Var(Шаблон:Var) gravity, it is difficult to find generic tests. Additionally, since deviations away from General Relativity can be made arbitrarily small in some cases, it is impossible to conclusively exclude some modifications. Some progress can be made, without assuming a concrete form for the function Шаблон:Var(Шаблон:Var) by Taylor expanding <math display="block">f(R) = a_0 + a_1 R + a_2 R^2 + \cdots</math>

The first term is like the cosmological constant and must be small. The next coefficient Шаблон:Var₁ can be set to one as in general relativity. For metric Шаблон:Var(Шаблон:Var) gravity (as opposed to Palatini or metric-affine Шаблон:Var(Шаблон:Var) gravity), the quadratic term is best constrained by fifth force measurements, since it leads to a Yukawa correction to the gravitational potential. The best current bounds are Шаблон:Nowrap or equivalently Шаблон:Nowrap^[13]^[14]

The parameterized post-Newtonian formalism is designed to be able to constrain generic modified theories of gravity. However, Шаблон:Var(Шаблон:Var) gravity shares many of the same values as General Relativity, and is therefore indistinguishable using these tests.^[15] In particular light deflection is unchanged, so Шаблон:Var(Шаблон:Var) gravity, like General Relativity, is entirely consistent with the bounds from Cassini tracking.^[13]

Starobinsky gravity

Шаблон:Main Starobinsky gravity has the following form <math display="block"> f(R) = R + \frac{R^2}{6M^2}</math> where <math>M</math> has the dimensions of mass.^[16]

Starobinsky gravity provides a mechanism for the cosmic inflation, just after the Big Bang when <math>R</math> was still large. However, it is not suited to describe the present universe acceleration since at present <math>R</math> is very small.^[17]^[18]^[19] This implies that the quadratic term in <math>f(R) = R + \frac{R^2}{6M^2}</math> is negligible, i.e., one tends to <math>f(R) = R </math> which is General Relativity with a null cosmological constant.

Gogoi-Goswami gravity

Gogoi-Goswami gravity has the following form <math display="block"> f(R) = R - \frac{\alpha}{\pi} R_c \cot^{-1} \left( \frac{R_c^2}{R^2} \right) - \beta R_c \left[ 1 - \exp\left( - \frac{R}{R_c} \right) \right] </math> where <math> \alpha </math> and <math> \beta </math> are two dimensionless positive constants and <math> R_c </math> is a characteristic curvature constant. ^[20]

Tensorial generalization

Шаблон:Var(Шаблон:Var) gravity as presented in the previous sections is a scalar modification of general relativity. More generally, we can have a <math display="block">\int \mathrm{d}^Dx \sqrt{-g}\, f(R, R^{\mu\nu}R_{\mu\nu}, R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma})</math> coupling involving invariants of the Ricci tensor and the Weyl tensor. Special cases are Шаблон:Var(Шаблон:Var) gravity, conformal gravity, Gauss–Bonnet gravity and Lovelock gravity. Notice that with any nontrivial tensorial dependence, we typically have additional massive spin-2 degrees of freedom, in addition to the massless graviton and a massive scalar. An exception is Gauss–Bonnet gravity where the fourth order terms for the spin-2 components cancel out.

References

External links

Шаблон:Theories of gravitation

[1] Шаблон:Cite journal

[2] Шаблон:Cite journal

[DE_textbook_Amendola-Tsujikawa-3] 3,0 ^3,1 L. Amendola and S. Tsujikawa (2013) “Dark Energy, Theory and Observations” Cambridge University Press

[4] Шаблон:Cite journal

[5] Шаблон:Cite journal

[6] Шаблон:Cite journal

[7] Шаблон:Cite journal

[8] Шаблон:Cite journal

[flanagan04-9] 9,0 ^9,1 Шаблон:Cite journal

[olmo05-10] 10,0 ^10,1 Шаблон:Cite journal

[11] Шаблон:Cite journal

[12] Шаблон:Cite journal

[Berry-13] 13,0 ^13,1 Шаблон:Cite journal

[14] Шаблон:Cite journal

[15] Шаблон:Cite journal

[16] Шаблон:Cite journal

[NASA_Shape-17] Шаблон:Cite web

[Fermi_Flat-18] Шаблон:Cite web

[19] Шаблон:Cite journal

[20] Шаблон:Cite journal

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Английская Википедия:F(R) gravity

Содержание