f(R) gravity
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() is a type of modified gravity theory which generalizes Einstein's
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. () gravity is actually a family of theories, each one defined by a different function, , of the
Ricci scalar In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometr ...
, . 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 In physical cosmology, structure formation is the formation of galaxies, galaxy clusters and larger structures from small early density fluctuations. The universe, as is now known from observations of the cosmic microwave background radiation, beg ...
of the Universe without adding unknown forms of
dark energy In physical cosmology and astronomy, dark energy is an unknown form of energy that affects the universe on the largest scales. The first observational evidence for its existence came from measurements of supernovas, which showed that the univer ...
or
dark matter Dark matter is a hypothetical form of matter thought to account for approximately 85% of the matter in the universe. Dark matter is called "dark" because it does not appear to interact with the electromagnetic field, which means it does not a ...
. Some functional forms may be inspired by corrections arising from a
quantum theory of gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...
. () gravity was first proposed in 1970 by Hans Adolph Buchdahl (although was used rather than for the name of the arbitrary function). It has become an active field of research following work by Starobinsky on cosmic inflation. 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 () gravity, one seeks to generalize the
Lagrangian Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
of the
Einstein–Hilbert action The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the ac ...
: S \int R \sqrt \, \mathrm^4x to S \int f(R) \sqrt \, \mathrm^4x where \kappa=\tfrac, g = \det g_ is the determinant of the metric tensor, and f(R) is some function of the
Ricci scalar In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometr ...
.L. Amendola and S. Tsujikawa (2013) “Dark Energy, Theory and Observations”
Cambridge University Press
There are two ways to track the effect of changing R to f(R), i.e., to obtain the theory
field equation In theoretical physics and applied mathematics, a field equation is a partial differential equation which determines the dynamics of a physical field, specifically the time evolution and spatial distribution of the field. The solutions to the equ ...
s. The first is to use metric formalism and the second is to use the
Palatini formalism Palatini may refer to: * Attilio Palatini (1889–1949), Italian mathematician * (1855–1914), Italian politician * Palatini identity * Palatini variation * Latin plural of Palatine * Palatini (Roman military) The ''palatini'' ( Latin for "pala ...
. While the two formalisms lead to the same field equations for General Relativity, i.e., when f(R)=R, the field equations may differ when f(R) \neq R.


Metric () gravity


Derivation of field equations

In metric () gravity, one arrives at the field equations by varying the action with respect to the
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
and not treating the connection \Gamma^\mu_ 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 The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the ac ...
(see the article for more details) but there are also some important differences. The variation of the determinant is as always: \delta \sqrt= -\frac \sqrt g_ \delta g^ The
Ricci scalar In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometr ...
is defined as R = g^ R_. Therefore, its variation with respect to the inverse metric g^ is given by \begin \delta R &= R_ \delta g^ + g^ \delta R_\\ &= R_ \delta g^ + g^ \left (\nabla_\rho \delta \Gamma^\rho_ - \nabla_\nu \delta \Gamma^\rho_ \right ) \end For the second step see the article about the
Einstein–Hilbert action The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the ac ...
. Since \delta\Gamma^\lambda_is the difference of two connections, it should transform as a tensor. Therefore, it can be written as \delta \Gamma^\lambda_=\fracg^\left(\nabla_\mu\delta g_+\nabla_\nu\delta g_-\nabla_a\delta g_ \right). Substituting into the equation above: \delta R= R_ \delta g^+g_\Box \delta g^-\nabla_\mu \nabla_\nu \delta g^ where \nabla_\muis the covariant derivative and \square = g^\nabla_\mu\nabla_\nu is the
d'Alembert operator In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of Mi ...
. Denoting F(R) = \frac, the variation in the action reads: \begin \delta S = \int \frac \left(\delta f(R) \sqrt+f(R) \delta \sqrt \right)\, \mathrm^4x \\ &= \int \frac \left(F(R) \delta R \sqrt-\frac \sqrt g_ \delta g^ f(R)\right) \, \mathrm^4x \\ &= \int \frac \sqrt\left(F(R)(R_ \delta g^+g_\Box \delta g^-\nabla_\mu \nabla_\nu \delta g^) -\frac g_ \delta g^ f(R) \right)\, \mathrm^4x \end Doing
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
on the second and third terms (and neglected the boundary contributions), we get: \delta S \int \frac \sqrt\delta g^ \left(F(R)R_-\fracg_ f(R)+ _\Box -\nabla_\mu \nabla_\nu(R) \right)\, \mathrm^4x. By demanding that the action remains invariant under variations of the metric, \frac=0, one obtains the field equations: F(R)R_-\fracf(R)g_+\left g_ \Box-\nabla_\mu \nabla_\nu \right(R) = \kappa T_, where T_is the
energy–momentum tensor Energy–momentum may refer to: * Four-momentum * Stress–energy tensor * Energy–momentum relation {{dab ...
defined as T_=-\frac\frac, where \mathcal L_mis the matter Lagrangian.


The generalized Friedmann equations

Assuming a Robertson–Walker metric with scale factor a(t) we can find the generalized
Friedmann equations The Friedmann equations are a set of equations in physical cosmology that govern the expansion of space in homogeneous and isotropic models of the universe within the context of general relativity. They were first derived by Alexander Friedmann ...
to be (in units where \kappa = 1): 3F H^ = \rho_+\rho_+\frac(FR-f)-3H -2F\dot = \rho_+\frac\rho_+\ddot-H\dot, where H = \frac is the
Hubble parameter Hubble's law, also known as the Hubble–Lemaître law, is the observation in physical cosmology that galaxies are moving away from Earth at speeds proportional to their distance. In other words, the farther they are, the faster they are moving a ...
, 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 equation A continuity equation or transport equation is an equation that describes the transport of some quantity. It is particularly simple and powerful when applied to a conserved quantity, but it can be generalized to apply to any extensive quantity. S ...
s: \dot_+3H\rho_=0; \dot_+4H\rho_=0.


Modified Newton's constant

An interesting feature of these theories is the fact that the
gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...
is time and scale dependent. To see this, add a small scalar perturbation to the metric (in the
Newtonian gauge In general relativity, the Newtonian gauge is a perturbed form of the Friedmann–Lemaître–Robertson–Walker line element. The gauge freedom of general relativity is used to eliminate two scalar degrees of freedom of the metric, so that it can ...
): \mathrms^2 = -(1+2\Phi)\mathrmt^2 +\alpha^2 (1-2\Psi)\delta_\mathrmx^i \mathrmx^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 Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...
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 The horizon is the apparent line that separates the surface of a celestial body from its sky when viewed from the perspective of an observer on or near the surface of the relevant body. This line divides all viewing directions based on whether i ...
scales ): \Phi = -4\pi G_\mathrm \frac \delta\rho_\mathrm where m is a perturbation in the matter density, is the Fourier scale and eff is: G_\mathrm=\frac\frac, with m\equiv\frac.


Massive gravitational waves

This class of theories when linearized exhibits three polarization modes for the
gravitational wave Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside in 1 ...
s, of which two correspond to the massless
graviton In theories of quantum gravity, the graviton is the hypothetical quantum of gravity, an elementary particle that mediates the force of gravitational interaction. There is no complete quantum field theory of gravitons due to an outstanding mathem ...
(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 General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
plus a
scalar field In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
. To see this, identify \Phi \to f'(R) \quad \textrm \quad \frac\to\frac, and use the field equations above to get \Box \Phi=\frac Working to first order of perturbation theory: g_=\eta_+h_ \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_(t,z;\omega)=A^(\omega)\exp(-i\omega(t-z))e^_+A^(\omega)\exp(-i\omega(t-z))e^_ +h_f(v_\mathrm t-z;\omega) \eta_ where h_f\equiv \frac, and g() = d/d is the group velocity of a
wave packet In physics, a wave packet (or wave train) is a short "burst" or "envelope" of localized wave action that travels as a unit. A wave packet can be analyzed into, or can be synthesized from, an infinite set of component sinusoidal waves of diff ...
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 The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
, but the massive scalar mode moves at a speed G < 1 (in units where  = 1), this mode is dispersive. However, in () gravity metric formalism, for the model 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

Under certain additional conditions we can simplify the analysis of () theories by introducing an
auxiliary field In physics, and especially quantum field theory, an auxiliary field is one whose equations of motion admit a single solution. Therefore, the Lagrangian describing such a field A contains an algebraic quadratic term and an arbitrary linear term, wh ...
. Assuming f''(R) \neq 0 for all , let () be the
Legendre transformation In mathematics, the Legendre transformation (or Legendre transform), named after Adrien-Marie Legendre, is an involutive transformation on real-valued convex functions of one real variable. In physical problems, it is used to convert functions ...
of () so that \Phi = f'(R) and R=V'(\Phi). Then, one obtains the O'Hanlon (1972) action: S = \int d^4x \sqrt \left \frac\left(\Phi R - V(\Phi)\right) + \mathcal_\right We have the Euler–Lagrange equations V'(\Phi)=R \Phi \left( R_ - \fracg_ R \right) + \left(g_\Box -\nabla_\mu \nabla_\nu \right) \Phi + \frac g_V(\Phi) = \kappa T_ Eliminating , 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 \tilde_=\Phi g_, we transform to the Einstein frame: R = \Phi \left \tilde + \frac -\frac\left(\frac\right)^2 \right/math> S = \int d^4x \sqrt\frac\left \tilde - \frac\left( \frac \right)^2 - \frac \right/math> after integrating by parts. Defining \tilde = \sqrt \ln, and substituting S = \int \mathrm^4x \sqrt\frac\left \tilde - \frac\left(\tilde\tilde\right)^2 - \tilde(\tilde) \right/math> \tilde(\tilde) = e^ V \left (e^ \right ). This is general relativity coupled to a real scalar field: using () 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) () 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 () gravity

In Palatini () 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 . Due to the structure of the theory, however, Palatini () theories appear to be in conflict with the Standard Model, may violate Solar system experiments, and seem to create unwanted singularities.


Metric-affine () gravity

In metric-affine () 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 () 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 () by Taylor expanding f(R) = a_0 + a_1 R + a_2 R^2 + \cdots The first term is like the
cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field eq ...
and must be small. The next coefficient 1 can be set to one as in general relativity. For metric () gravity (as opposed to Palatini or metric-affine () gravity), the quadratic term is best constrained by
fifth force In physics, there are four observed fundamental interactions (also known as fundamental forces) that form the basis of all known interactions in nature: gravitational, electromagnetic, strong nuclear, and weak nuclear forces. Some speculative the ...
measurements, since it leads to a Yukawa correction to the gravitational potential. The best current bounds are or equivalently The
parameterized post-Newtonian formalism In physics, precisely in the study of the theory of general relativity and many alternatives to it, the post-Newtonian formalism is a calculational tool that expresses Einstein's (nonlinear) equations of gravity in terms of the lowest-order devi ...
is designed to be able to constrain generic modified theories of gravity. However, () gravity shares many of the same values as General Relativity, and is therefore indistinguishable using these tests. In particular light deflection is unchanged, so () gravity, like General Relativity, is entirely consistent with the bounds from Cassini tracking.


Starobinsky gravity

Starobinsky gravity has the following form f(R) = R + \frac where M has the dimensions of mass. Starobinsky gravity provides a mechanism for the cosmic
inflation In economics, inflation is an increase in the general price level of goods and services in an economy. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a reductio ...
, just after the
Big Bang The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...
when R was still large. However, it is not suited to describe the present universe acceleration since at present R is very small. This implies that the quadratic term in f(R) = R + \frac is negligible, i.e., one tends to f(R) = R which is General Relativity with a null
cosmological constant In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: ), alternatively called Einstein's cosmological constant, is the constant coefficient of a term that Albert Einstein temporarily added to his field eq ...
.


Gogoi-Goswami gravity

Gogoi-Goswami gravity has the following form f(R) = R - \frac R_c \cot^ \left( \frac \right) - \beta R_c \left 1 - \exp\left( - \frac \right) \right where \alpha and \beta are two dimensionless positive constants and R_c is a characteristic curvature constant.


Tensorial generalization

() gravity as presented in the previous sections is a scalar modification of general relativity. More generally, we can have a \int \mathrm^Dx \sqrt\, f(R, R^R_, R^R_) coupling involving invariants of the
Ricci tensor In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measur ...
and the
Weyl tensor In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. Like the Riemann curvature tensor, the Weyl tensor expresses the tidal f ...
. Special cases are () gravity,
conformal gravity Conformal gravity refers to gravity theories that are invariant under conformal transformations in the Riemannian geometry sense; more accurately, they are invariant under Weyl transformations g_\rightarrow\Omega^2(x)g_ where g_ is the metric te ...
,
Gauss–Bonnet gravity In general relativity, Gauss–Bonnet gravity, also referred to as Einstein–Gauss–Bonnet gravity, is a modification of the Einstein–Hilbert action to include the Gauss–Bonnet term (named after Carl Friedrich Gauss and Pierre Ossian Bonne ...
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.


See also

* Extended theories of gravity *
Gauss–Bonnet gravity In general relativity, Gauss–Bonnet gravity, also referred to as Einstein–Gauss–Bonnet gravity, is a modification of the Einstein–Hilbert action to include the Gauss–Bonnet term (named after Carl Friedrich Gauss and Pierre Ossian Bonne ...
* Lovelock gravity


References


Further reading

* See Chapter 29 in the textbook on "Particles and Quantum Fields" by Kleinert, H. (2016)
World Scientific (Singapore, 2016)
(also availabl
online
* * * *Salvatore Capozziello and Mariafelicia De Laurentis, (2015) "F(R) theories of gravitation". Scholarpedia
doi:10.4249/scholarpedia.31422
*Kalvakota, Vaibhav R., (2021) "Investigating f(R)" gravity and cosmologies". Mathematical physics preprint archive, https://web.ma.utexas.edu/mp_arc/c/21/21-38.pdf


External links


''f''(''R'') gravity on arxiv.orgExtended Theories of Gravity
{{theories of gravitation Theories of gravity