Alternatives to general relativity

Alternatives to general relativity are physical theories that attempt to describe the phenomenon of gravitation in competition with Einstein's theory of general relativity. There have been many different attempts at constructing an ideal theory of gravity.^[1]

These attempts can be split into four broad categories based on their scope. In this article, straightforward alternatives to general relativity are discussed, which do not involve quantum mechanics or force unification. Other theories which do attempt to construct a theory using the principles of quantum mechanics are known as theories of quantized gravity. Thirdly, there are theories which attempt to explain gravity and other forces at the same time; these are known as classical unified field theories. Finally, the most ambitious theories attempt to both put gravity in quantum mechanical terms and unify forces; these are called theories of everything.

None of these alternatives to general relativity have gained wide acceptance. General relativity has withstood many tests,^[2][3] remaining consistent with all observations so far. In contrast, many of the early alternatives have been definitively disproven. However, some of the alternative theories of gravity are supported by a minority of physicists, and the topic remains the subject of intense study in theoretical physics.

History of gravitational theory through general relativity

Main article: History of gravitational theory

At the time it was published in the 17th century, Isaac Newton's theory of gravity was the most accurate theory of gravity. Since then, a number of alternatives were proposed. The theories which predate the formulation of general relativity in 1915 are discussed in history of gravitational theory.

General relativity

Main articles: General relativity and History of general relativity

This theory^[4][5] is what we now call "general relativity" (included here for comparison). Discarding the Minkowski metric entirely, Einstein gets:

${\displaystyle \delta \int ds=0\,}$ 

${\displaystyle {ds}^{2}=g_{\mu \nu }\,dx^{\mu }\,dx^{\nu }\,}$ 

${\displaystyle R_{\mu \nu }={\frac {8\pi G}{c^{4}}}\left(T_{\mu \nu }-{\frac {1}{2}}g_{\mu \nu }T\right)\,}$ 

which can also be written

${\displaystyle T^{\mu \nu }={c^{4} \over 8\pi G}\left(R^{\mu \nu }-{\frac {1}{2}}g^{\mu \nu }R\right)\,.}$ 

Five days before Einstein presented the last equation above, Hilbert had submitted a paper containing an almost identical equation. See General relativity priority dispute. Hilbert was the first to correctly state the Einstein–Hilbert action for general relativity, which is:

${\displaystyle S={c^{4} \over 16\pi G}\int R{\sqrt {-g}}\ d^{4}x+S_{m}\,}$ 

where ${\displaystyle G\,}$  is Newton's gravitational constant, ${\displaystyle R=R_{\mu }^{~\mu }\,}$  is the Ricci curvature of space, ${\displaystyle g=\det(g_{\mu \nu })\,}$  and ${\displaystyle S_{m}\,}$  is the action due to mass.

General relativity is a tensor theory, the equations all contain tensors. Nordström's theories, on the other hand, are scalar theories because the gravitational field is a scalar. Other proposed alternatives include scalar–tensor theories that contain a scalar field in addition to the tensors of general relativity, and other variants containing vector fields as well have been developed recently.

Motivations

After general relativity, attempts were made either to improve on theories developed before general relativity, or to improve general relativity itself. Many different strategies were attempted, for example the addition of spin to general relativity, combining a general relativity-like metric with a spacetime that is static with respect to the expansion of the universe, getting extra freedom by adding another parameter. At least one theory was motivated by the desire to develop an alternative to general relativity that is free of singularities.

Experimental tests improved along with the theories. Many of the different strategies that were developed soon after general relativity were abandoned, and there was a push to develop more general forms of the theories that survived, so that a theory would be ready when any test showed a disagreement with general relativity.

By the 1980s, the increasing accuracy of experimental tests had all confirmed general relativity; no competitors were left except for those that included general relativity as a special case. Further, shortly after that, theorists switched to string theory which was starting to look promising, but has since lost popularity. In the mid-1980s a few experiments were suggesting that gravity was being modified by the addition of a fifth force (or, in one case, of a fifth, sixth and seventh force) acting in the range of a few meters. Subsequent experiments eliminated these.

Motivations for the more recent alternative theories are almost all cosmological, associated with or replacing such constructs as "inflation", "dark matter" and "dark energy". Investigation of the Pioneer anomaly has caused renewed public interest in alternatives to general relativity.

Notation in this article

Main articles: Mathematics of general relativity and Ricci calculus

${\displaystyle c\;}$  is the speed of light, ${\displaystyle G\;}$  is the gravitational constant. "Geometric variables" are not used.

Latin indices go from 1 to 3, Greek indices go from 0 to 3. The Einstein summation convention is used.

${\displaystyle \eta _{\mu \nu }\;}$  is the Minkowski metric. ${\displaystyle g_{\mu \nu }\;}$  is a tensor, usually the metric tensor. These have signature (−,+,+,+).

Partial differentiation is written ${\displaystyle \partial _{\mu }\varphi \;}$  or ${\displaystyle \varphi _{,\mu }\;}$ . Covariant differentiation is written ${\displaystyle \nabla _{\mu }\varphi \;}$  or ${\displaystyle \varphi _{;\mu }\;}$ .

Classification of theories

Theories of gravity can be classified, loosely, into several categories. Most of the theories described here have:

• an 'action' (see the principle of least action, a variational principle based on the concept of action)
• a Lagrangian density
• a metric

If a theory has a Lagrangian density for gravity, say ${\displaystyle L\,}$ , then the gravitational part of the action ${\displaystyle S\,}$  is the integral of that:

${\displaystyle S=\int L{\sqrt {-g}}\,\mathrm {d} ^{4}x}$ .

In this equation it is usual, though not essential, to have ${\displaystyle g=-1\,}$  at spatial infinity when using Cartesian coordinates. For example, the Einstein–Hilbert action uses

${\displaystyle L\,\propto \,R}$ 

where R is the scalar curvature, a measure of the curvature of space.

Almost every theory described in this article has an action. It is the most efficient known way to guarantee that the necessary conservation laws of energy, momentum and angular momentum are incorporated automatically; although it is easy to construct an action where those conservation laws are violated. Canonical methods provide another way to construct systems that have the required conservation laws, but this approach is more cumbersome to implement.^[6] The original 1983 version of MOND did not have an action.

A few theories have an action but not a Lagrangian density. A good example is Whitehead,^[7] the action there is termed non-local.

A theory of gravity is a "metric theory" if and only if it can be given a mathematical representation in which two conditions hold:
Condition 1: There exists a symmetric metric tensor ${\displaystyle g_{\mu \nu }\,}$  of signature (−, +, +, +), which governs proper-length and proper-time measurements in the usual manner of special and general relativity:

${\displaystyle {d\tau }^{2}=-g_{\mu \nu }\,dx^{\mu }\,dx^{\nu }\,}$ 

where there is a summation over indices ${\displaystyle \mu }$  and ${\displaystyle \nu }$ .
Condition 2: Stressed matter and fields being acted upon by gravity respond in accordance with the equation:

${\displaystyle 0=\nabla _{\nu }T^{\mu \nu }={T^{\mu \nu }}_{,\nu }+\Gamma _{\sigma \nu }^{\mu }T^{\sigma \nu }+\Gamma _{\sigma \nu }^{\nu }T^{\mu \sigma }\,}$ 

where ${\displaystyle T^{\mu \nu }\,}$  is the stress–energy tensor for all matter and non-gravitational fields, and where ${\displaystyle \nabla _{\nu }}$  is the covariant derivative with respect to the metric and ${\displaystyle \Gamma _{\sigma \nu }^{\alpha }\,}$  is the Christoffel symbol. The stress–energy tensor should also satisfy an energy condition.

Metric theories include (from simplest to most complex):

• Scalar field theories (includes conformally flat theories & Stratified theories with conformally flat space slices)
  • Bergman
  • Coleman
  • Einstein (1912)
  • Einstein–Fokker theory
  • Lee–Lightman–Ni
  • Littlewood
  • Ni
  • Nordström's theory of gravitation (first metric theory of gravity to be developed)
  • Page–Tupper
  • Papapetrou
  • Rosen (1971)
  • Whitrow–Morduch
  • Yilmaz theory of gravitation (attempted to eliminate event horizons from the theory.)
• Quasilinear theories (includes Linear fixed gauge)
  • Bollini–Giambiagi–Tiomno
  • Deser–Laurent
  • Whitehead's theory of gravity (intended to use only retarded potentials)
• Tensor theories
  • Einstein's general relativity
  • Fourth-order gravity (allows the Lagrangian to depend on second-order contractions of the Riemann curvature tensor)
  • f(R) gravity (allows the Lagrangian to depend on higher powers of the Ricci scalar)
  • Gauss–Bonnet gravity
  • Lovelock theory of gravity (allows the Lagrangian to depend on higher-order contractions of the Riemann curvature tensor)
  • Infinite derivative gravity
• Scalar–tensor theories
  • Bekenstein
  • Bergmann–Wagoner
  • Brans–Dicke theory (the most well-known alternative to general relativity, intended to be better at applying Mach's principle)
  • Jordan
  • Nordtvedt
  • Thiry
  • Chameleon
  • Pressuron
• Vector–tensor theories
  • Hellings–Nordtvedt
  • Will–Nordtvedt
• Bimetric theories
  • Lightman–Lee
  • Rastall
  • Rosen (1975)
• Other metric theories

(see section Modern theories below)

Non-metric theories include

• Belinfante–Swihart
• Einstein–Cartan theory (intended to handle spin-orbital angular momentum interchange)
• Kustaanheimo (1967)
• Teleparallelism
• Gauge theory gravity

A word here about Mach's principle is appropriate because a few of these theories rely on Mach's principle (e.g. Whitehead^[7]), and many mention it in passing (e.g. Einstein–Grossmann,^[8] Brans–Dicke^[9]). Mach's principle can be thought of a half-way-house between Newton and Einstein. It goes this way:^[10]

• Newton: Absolute space and time.
• Mach: The reference frame comes from the distribution of matter in the universe.
• Einstein: There is no reference frame.

Theories from 1917 to the 1980s

This section includes alternatives to general relativity published after general relativity but before the observations of galaxy rotation that led to the hypothesis of "dark matter". Those considered here include (see Will^[11][12] Lang^[13][14]):

Theories from 1917 to the 1980s.

Publication year(s)	Author(s)	Theory name	Theory type
1922^[7]	Alfred North Whitehead	Whitehead's theory of gravitation	Quasilinear
1922,^[15] 1923^[16]	Élie Cartan	Einstein–Cartan theory	Non-metric
1939^[17]	Markus Fierz, Wolfgang Pauli
1943^[18]	George David Birkhoff
1948^[19]	Edward Arthur Milne	Kinematic Relativity
1948^[20]	Yves Thiry
1954^[21]^[22]	Achilles Papapetrou		Scalar field
1953^[23]	Dudley E. Littlewood		Scalar field
1955^[24]	Pascual Jordan
1956^[25]	Otto Bergmann		Scalar field
1957^[26]^[27]	Frederik Belinfante, James C. Swihart
1958,^[28] 1973^[29]	Huseyin Yilmaz	Yilmaz theory of gravitation
1961^[9]	Carl H. Brans, Robert H. Dicke	Brans–Dicke theory	Scalar–tensor
1960,^[30] 1965^[31]	Gerald James Whitrow, G. E. Morduch		Scalar field
1966^[32]	Paul Kustaanheimo [de]
1967^[33]	Paul Kustaanheimo, V. S. Nuotio
1968^[34]	Stanley Deser, B. E. Laurent		Quasilinear
1968^[35]	C. Page, B. O. J. Tupper		Scalar field
1968^[36]	Peter Bergmann		Scalar–tensor
1970^[37]	C. G. Bollini, J. J. Giambiagi, J. Tiomno		Quasilinear
1970^[38]	Kenneth Nordtvedt
1970^[39]	Robert V. Wagoner		Scalar–tensor
1971^[40]	Nathan Rosen		Scalar field
1975^[41]	Nathan Rosen		Bimetric
1972,^[12] 1973^[42]	Ni Wei-tou		Scalar field
1972^[43]	Clifford Martin Will, Kenneth Nordtvedt		Vector–tensor
1973^[44]	Ronald Hellings, Kenneth Nordtvedt		Vector–tensor
1973^[45]	Alan Lightman, David L. Lee		Scalar field
1974^[46]	David L. Lee, Alan Lightman, Ni Wei-tou
1977^[47]	Jacob Bekenstein		Scalar–tensor
1978^[48]	B. M. Barker		Scalar–tensor
1979^[49]	P. Rastall		Bimetric

These theories are presented here without a cosmological constant or added scalar or vector potential unless specifically noted, for the simple reason that the need for one or both of these was not recognized before the supernova observations by the Supernova Cosmology Project and High-Z Supernova Search Team. How to add a cosmological constant or quintessence to a theory is discussed under Modern Theories (see also Einstein–Hilbert action).

Scalar field theories

See also: Scalar theories of gravitation

The scalar field theories of Nordström^[50][51] have already been discussed. Those of Littlewood,^[23] Bergman,^[25] Yilmaz,^[28] Whitrow and Morduch^[30][31] and Page and Tupper^[35] follow the general formula give by Page and Tupper.

According to Page and Tupper,^[35] who discuss all these except Nordström,^[51] the general scalar field theory comes from the principle of least action:

${\displaystyle \delta \int f\left({\tfrac {\varphi }{c^{2}}}\right)\,ds=0}$ 

where the scalar field is,

${\displaystyle \varphi ={\frac {GM}{r}}}$ 

and c may or may not depend on ${\displaystyle \varphi }$ .

In Nordström,^[50]

${\displaystyle f(\varphi /c^{2})=\exp(-\varphi /c^{2}),\qquad c=c_{\infty }}$ 

In Littlewood^[23] and Bergmann,^[25]

${\displaystyle f\left({\frac {\varphi }{c^{2}}}\right)=\exp \left(-{\frac {\varphi }{c^{2}}}-{\frac {(c/\varphi ^{2})^{2}}{2}}\right)\qquad c=c_{\infty }\,}$ 

In Whitrow and Morduch,^[30]

${\displaystyle f\left({\frac {\varphi }{c^{2}}}\right)=1,\qquad c^{2}=c_{\infty }^{2}-2\varphi \,}$ 

In Whitrow and Morduch,^[31]

${\displaystyle f\left({\frac {\varphi }{c^{2}}}\right)=\exp \left(-{\frac {\varphi }{c^{2}}}\right),\qquad c^{2}=c_{\infty }^{2}-2\varphi \,}$ 

In Page and Tupper,^[35]

${\displaystyle f\left({\frac {\varphi }{c^{2}}}\right)={\frac {\varphi }{c^{2}}}+\alpha \left({\frac {\varphi }{c^{2}}}\right)^{2},\qquad {\frac {c_{\infty }^{2}}{c^{2}}}=1+4\left({\frac {\varphi }{c_{\infty }^{2}}}\right)+(15+2\alpha )\left({\frac {\varphi }{c_{\infty }^{2}}}\right)^{2}}$ 

Page and Tupper^[35] matches Yilmaz's theory^[28] to second order when ${\displaystyle \alpha =-7/2}$ .

The gravitational deflection of light has to be zero when c is constant. Given that variable c and zero deflection of light are both in conflict with experiment, the prospect for a successful scalar theory of gravity looks very unlikely. Further, if the parameters of a scalar theory are adjusted so that the deflection of light is correct then the gravitational redshift is likely to be wrong.

Ni^[12] summarized some theories and also created two more. In the first, a pre-existing special relativity space-time and universal time coordinate acts with matter and non-gravitational fields to generate a scalar field. This scalar field acts together with all the rest to generate the metric.

The action is:

${\displaystyle S={1 \over 16\pi G}\int d^{4}x{\sqrt {-g}}L_{\varphi }+S_{m}}$ 

${\displaystyle L_{\varphi }=\varphi R-2g^{\mu \nu }\,\partial _{\mu }\varphi \,\partial _{\nu }\varphi }$ 

Misner et al.^[52] gives this without the ${\displaystyle \varphi R}$  term. ${\displaystyle S_{m}}$  is the matter action.

${\displaystyle \Box \varphi =4\pi T^{\mu \nu }\left[\eta _{\mu \nu }e^{-2\varphi }+\left(e^{2\varphi }+e^{-2\varphi }\right)\,\partial _{\mu }t\,\partial _{\nu }t\right]}$ 

t is the universal time coordinate. This theory is self-consistent and complete. But the motion of the solar system through the universe leads to serious disagreement with experiment.

In the second theory of Ni^[12] there are two arbitrary functions ${\displaystyle f(\varphi )}$  and ${\displaystyle k(\varphi )}$  that are related to the metric by:

${\displaystyle ds^{2}=e^{-2f(\varphi )}dt^{2}-e^{2f(\varphi )}\left[dx^{2}+dy^{2}+dz^{2}\right]}$ 

${\displaystyle \eta ^{\mu \nu }\partial _{\mu }\partial _{\nu }\varphi =4\pi \rho ^{*}k(\varphi )}$ 

Ni^[12] quotes Rosen^[40] as having two scalar fields ${\displaystyle \varphi }$  and ${\displaystyle \psi }$  that are related to the metric by:

${\displaystyle ds^{2}=\varphi ^{2}\,dt^{2}-\psi ^{2}\left[dx^{2}+dy^{2}+dz^{2}\right]}$ 

In Papapetrou^[21] the gravitational part of the Lagrangian is:

${\displaystyle L_{\varphi }=e^{\varphi }\left({\tfrac {1}{2}}e^{-\varphi }\,\partial _{\alpha }\varphi \,\partial _{\alpha }\varphi +{\tfrac {3}{2}}e^{\varphi }\,\partial _{0}\varphi \,\partial _{0}\varphi \right)}$ 

In Papapetrou^[22] there is a second scalar field ${\displaystyle \chi }$ . The gravitational part of the Lagrangian is now:

${\displaystyle L_{\varphi }=e^{{\frac {1}{2}}(3\varphi +\chi )}\left(-{\tfrac {1}{2}}e^{-\varphi }\,\partial _{\alpha }\varphi \,\partial _{\alpha }\varphi -e^{-\varphi }\,\partial _{\alpha }\varphi \,\partial _{\chi }\varphi +{\tfrac {3}{2}}e^{-\chi }\,\partial _{0}\varphi \,\partial _{0}\varphi \right)\,}$ 

Bimetric theories

See also: Bimetric theory

Bimetric theories contain both the normal tensor metric and the Minkowski metric (or a metric of constant curvature), and may contain other scalar or vector fields.

Rosen^[53] (1975) bimetric theory The action is:

${\displaystyle S={1 \over 64\pi G}\int d^{4}x\,{\sqrt {-\eta }}\eta ^{\mu \nu }g^{\alpha \beta }g^{\gamma \delta }(g_{\alpha \gamma |\mu }g_{\alpha \delta |\nu }-\textstyle {\frac {1}{2}}g_{\alpha \beta |\mu }g_{\gamma \delta |\nu })+S_{m}}$ 

${\displaystyle \Box _{\eta }g_{\mu \nu }-g^{\alpha \beta }\eta ^{\gamma \delta }g_{\mu \alpha |\gamma }g_{\nu \beta |\delta }=-16\pi G{\sqrt {g/\eta }}(T_{\mu \nu }-\textstyle {\frac {1}{2}}g_{\mu \nu }T)\,}$ 

Lightman–Lee^[45] developed a metric theory based on the non-metric theory of Belinfante and Swihart.^[26][27] The result is known as BSLL theory. Given a tensor field ${\displaystyle B_{\mu \nu }\,}$ , ${\displaystyle B=B_{\mu \nu }\eta ^{\mu \nu }\,}$ , and two constants ${\displaystyle a\,}$  and ${\displaystyle f\,}$  the action is:

${\displaystyle S={1 \over 16\pi G}\int d^{4}x{\sqrt {-\eta }}(aB^{\mu \nu |\alpha }B_{\mu \nu |\alpha }+fB_{,\alpha }B^{,\alpha })+S_{m}}$ 

and the stress–energy tensor comes from:

${\displaystyle a\Box _{\eta }B^{\mu \nu }+f\eta ^{\mu \nu }\Box _{\eta }B=-4\pi G{\sqrt {g/\eta }}\,T^{\alpha \beta }\left({\frac {\partial g_{\alpha \beta }}{\partial B_{\mu }\nu }}\right)}$ 

In Rastall,^[49] the metric is an algebraic function of the Minkowski metric and a Vector field.^[54] The Action is:

${\displaystyle S={1 \over 16\pi G}\int d^{4}x\,{\sqrt {-g}}F(N)K^{\mu ;\nu }K_{\mu ;\nu }+S_{m}}$ 

where

${\displaystyle F(N)=-{\frac {N}{2+N}}}$  and ${\displaystyle N=g^{\mu \nu }K_{\mu }K_{\nu }\;}$ 

(see Will^[11] for the field equation for ${\displaystyle T^{\mu \nu }\;}$  and ${\displaystyle K_{\mu }\;}$ ).

Quasilinear theories

In Whitehead,^[7] the physical metric ${\displaystyle g\;}$  is constructed (by Synge) algebraically from the Minkowski metric ${\displaystyle \eta \;}$  and matter variables, so it doesn't even have a scalar field. The construction is:

${\displaystyle g_{\mu \nu }(x^{\alpha })=\eta _{\mu \nu }-2\int _{\Sigma ^{-}}{y_{\mu }^{-}y_{\nu }^{-} \over (w^{-})^{3}}\left[{\sqrt {-g}}\rho u^{\alpha }\,d\Sigma _{\alpha }\right]^{-}}$ 

where the superscript (−) indicates quantities evaluated along the past ${\displaystyle \eta \;}$  light cone of the field point ${\displaystyle x^{\alpha }\;}$  and

${\displaystyle {\begin{aligned}(y^{\mu })^{-}&=x^{\mu }-(x^{\mu })^{-},\qquad (y^{\mu })^{-}(y_{\mu })^{-}=0,\\[5pt]w^{-}&=(y^{\mu })^{-}(u_{\mu })^{-},\qquad (u_{\mu })={\frac {dx^{\mu }}{d\sigma }},\\[5pt]d\sigma ^{2}&=\eta _{\mu \nu }\,dx^{\mu }\,dx^{\nu }\end{aligned}}}$ 

Nevertheless, the metric construction (from a non-metric theory) using the "length contraction" ansatz is criticised.^[55]

Deser and Laurent^[34] and Bollini–Giambiagi–Tiomno^[37] are Linear Fixed Gauge theories. Taking an approach from quantum field theory, combine a Minkowski spacetime with the gauge invariant action of a spin-two tensor field (i.e. graviton) ${\displaystyle h_{\mu \nu }\;}$  to define

${\displaystyle g_{\mu \nu }=\eta _{\mu \nu }+h_{\mu \nu }\;}$ 

The action is:

${\displaystyle S={1 \over 16\pi G}\int d^{4}x{\sqrt {-\eta }}\left[2h_{|\nu }^{\mu \nu }h_{\mu \lambda }^{|\lambda }-2h_{|\nu }^{\mu \nu }h_{\lambda |\mu }^{\lambda }+h_{\nu |\mu }^{\nu }h_{\lambda }^{\lambda |\mu }-h^{\mu \nu |\lambda }h_{\mu \nu |\lambda }\right]+S_{m}\;}$ 

The Bianchi identity associated with this partial gauge invariance is wrong. Linear Fixed Gauge theories seek to remedy this by breaking the gauge invariance of the gravitational action through the introduction of auxiliary gravitational fields that couple to ${\displaystyle h_{\mu \nu }\;}$ .

A cosmological constant can be introduced into a quasilinear theory by the simple expedient of changing the Minkowski background to a de Sitter or anti-de Sitter spacetime, as suggested by G. Temple in 1923. Temple's suggestions on how to do this were criticized by C. B. Rayner in 1955.^[56]

Tensor theories

Einstein's general relativity is the simplest plausible theory of gravity that can be based on just one symmetric tensor field (the metric tensor). Others include: Starobinsky (R+R^2) gravity, Gauss–Bonnet gravity, f(R) gravity, and Lovelock theory of gravity.

Starobinsky

See also: Starobinsky inflation

Starobinsky gravity, proposed by Alexei Starobinsky has the Lagrangian

${\displaystyle {\mathcal {L}}={\sqrt {-g}}\left[R+{\frac {R^{2}}{6M^{2}}}\right]}$ 

and has been used to explain inflation, in the form of Starobinsky inflation. Here ${\displaystyle M}$  is a constant.

Gauss–Bonnet

Gauss–Bonnet gravity has the action

${\displaystyle {\mathcal {L}}={\sqrt {-g}}\left[R+R^{2}-4R^{\mu \nu }R_{\mu \nu }+R^{\mu \nu \rho \sigma }R_{\mu \nu \rho \sigma }\right].}$ 

where the coefficients of the extra terms are chosen so that the action reduces to general relativity in 4 spacetime dimensions and the extra terms are only non-trivial when more dimensions are introduced.

Stelle's 4th derivative gravity

Stelle's 4th derivative gravity, which is a generalization of Gauss–Bonnet gravity, has the action

${\displaystyle {\mathcal {L}}={\sqrt {-g}}\left[R+f_{1}R^{2}+f_{2}R^{\mu \nu }R_{\mu \nu }+f_{3}R^{\mu \nu \rho \sigma }R_{\mu \nu \rho \sigma }\right].}$ 

f(R)

f(R) gravity has the action

${\displaystyle {\mathcal {L}}={\sqrt {-g}}f(R)}$ 

and is a family of theories, each defined by a different function of the Ricci scalar. Starobinsky gravity is actually an ${\displaystyle f(R)}$  theory.

Infinite derivative gravity

Infinite derivative gravity is a covariant theory of gravity, quadratic in curvature, torsion free and parity invariant,^[57]

${\displaystyle {\mathcal {L}}={\sqrt {-g}}\left[M_{p}^{2}R+Rf_{1}\left({\frac {\Box }{M_{s}^{2}}}\right)R+R^{\mu \nu }f_{2}\left({\frac {\Box }{M_{s}^{2}}}\right)R_{\mu \nu }+R^{\mu \nu \rho \sigma }f_{3}\left({\frac {\Box }{M_{s}^{2}}}\right)R_{\mu \nu \rho \sigma }\right].}$ 

and

${\displaystyle 2f_{1}\left({\frac {\Box }{M_{s}^{2}}}\right)+f_{2}\left({\frac {\Box }{M_{s}^{2}}}\right)+2f_{3}\left({\frac {\Box }{M_{s}^{2}}}\right)=0,}$ 

in order to make sure that only massless spin −2 and spin −0 components propagate in the graviton propagator around Minkowski background. The action becomes non-local beyond the scale ${\displaystyle M_{s}}$ , and recovers to general relativity in the infrared, for energies below the non-local scale ${\displaystyle M_{s}}$ . In the ultraviolet regime, at distances and time scales below non-local scale, ${\displaystyle M_{s}^{-1}}$ , the gravitational interaction weakens enough to resolve point-like singularity, which means Schwarzschild's singularity can be potentially resolved in infinite derivative theories of gravity.

Lovelock

Lovelock gravity has the action

${\displaystyle {\mathcal {L}}={\sqrt {-g}}\ (\alpha _{0}+\alpha _{1}R+\alpha _{2}\left(R^{2}+R_{\alpha \beta \mu \nu }R^{\alpha \beta \mu \nu }-4R_{\mu \nu }R^{\mu \nu }\right)+\alpha _{3}{\mathcal {O}}(R^{3})),}$ 

and can be thought of as a generalization of general relativity.

Scalar–tensor theories

See also: Scalar–tensor theory, Brans–Dicke theory, Dilaton, Chameleon particle, Pressuron, and Horndeski's theory

These all contain at least one free parameter, as opposed to general relativity which has no free parameters.

Although not normally considered a Scalar–Tensor theory of gravity, the 5 by 5 metric of Kaluza–Klein reduces to a 4 by 4 metric and a single scalar. So if the 5th element is treated as a scalar gravitational field instead of an electromagnetic field then Kaluza–Klein can be considered the progenitor of Scalar–Tensor theories of gravity. This was recognized by Thiry.^[20]

Scalar–Tensor theories include Thiry,^[20] Jordan,^[24] Brans and Dicke,^[9] Bergman,^[36] Nordtveldt (1970), Wagoner,^[39] Bekenstein^[47] and Barker.^[48]

The action ${\displaystyle S\;}$  is based on the integral of the Lagrangian ${\displaystyle L_{\varphi }\;}$ .

${\displaystyle S={1 \over 16\pi G}\int d^{4}x{\sqrt {-g}}L_{\varphi }+S_{m}\;}$ 

${\displaystyle L_{\varphi }=\varphi R-{\omega (\varphi ) \over \varphi }g^{\mu \nu }\,\partial _{\mu }\varphi \,\partial _{\nu }\varphi +2\varphi \lambda (\varphi )\;}$ 

${\displaystyle S_{m}=\int d^{4}x\,{\sqrt {g}}\,G_{N}L_{m}\;}$ 

${\displaystyle T^{\mu \nu }\ {\stackrel {\mathrm {def} }{=}}\ {2 \over {\sqrt {g}}}{\delta S_{m} \over \delta g_{\mu \nu }}}$ 

where ${\displaystyle \omega (\varphi )\;}$  is a different dimensionless function for each different scalar–tensor theory. The function ${\displaystyle \lambda (\varphi )\;}$  plays the same role as the cosmological constant in general relativity. ${\displaystyle G_{N}\;}$  is a dimensionless normalization constant that fixes the present-day value of ${\displaystyle G\;}$ . An arbitrary potential can be added for the scalar.

The full version is retained in Bergman^[36] and Wagoner.^[39] Special cases are:

Nordtvedt,^[38] ${\displaystyle \lambda =0\;}$ 

Since ${\displaystyle \lambda }$  was thought to be zero at the time anyway, this would not have been considered a significant difference. The role of the cosmological constant in more modern work is discussed under Cosmological constant.

Brans–Dicke,^[9] ${\displaystyle \omega \;}$  is constant

Bekenstein^[47] variable mass theory Starting with parameters ${\displaystyle r\;}$  and ${\displaystyle q\;}$ , found from a cosmological solution, ${\displaystyle \varphi =[1-qf(\varphi )]f(\varphi )^{-r}\;}$  determines function ${\displaystyle f\;}$  then

${\displaystyle \omega (\varphi )=-\textstyle {\frac {3}{2}}-\textstyle {\frac {1}{4}}f(\varphi )[(1-6q)qf(\varphi )-1][r+(1-r)qf(\varphi )]^{-2}\;}$ 

Barker^[48] constant G theory

${\displaystyle \omega (\varphi )={\frac {4-3\varphi }{2\varphi -2}}}$ 

Adjustment of ${\displaystyle \omega (\varphi )\;}$  allows Scalar Tensor Theories to tend to general relativity in the limit of ${\displaystyle \omega \rightarrow \infty \;}$  in the current epoch. However, there could be significant differences from general relativity in the early universe.

So long as general relativity is confirmed by experiment, general Scalar–Tensor theories (including Brans–Dicke^[9]) can never be ruled out entirely, but as experiments continue to confirm general relativity more precisely and the parameters have to be fine-tuned so that the predictions more closely match those of general relativity.

The above examples are particular cases of Horndeski's theory,^[58][59] the most general Lagrangian constructed out of the metric tensor and a scalar field leading to second order equations of motion in 4-dimensional space. Viable theories beyond Horndeski (with higher order equations of motion) have been shown to exist.^[60][61][62]

Vector–tensor theories

Before we start, Will (2001) has said: "Many alternative metric theories developed during the 1970s and 1980s could be viewed as "straw-man" theories, invented to prove that such theories exist or to illustrate particular properties. Few of these could be regarded as well-motivated theories from the point of view, say, of field theory or particle physics. Examples are the vector–tensor theories studied by Will, Nordtvedt and Hellings."

Hellings and Nordtvedt^[44] and Will and Nordtvedt^[43] are both vector–tensor theories. In addition to the metric tensor there is a timelike vector field ${\displaystyle K_{\mu }.}$  The gravitational action is:

${\displaystyle S={\frac {1}{16\pi G}}\int d^{4}x{\sqrt {-g}}\left[R+\omega K_{\mu }K^{\mu }R+\eta K^{\mu }K^{\nu }R_{\mu \nu }-\epsilon F_{\mu \nu }F^{\mu \nu }+\tau K_{\mu ;\nu }K^{\mu ;\nu }\right]+S_{m}}$ 

where ${\displaystyle \omega ,\eta ,\epsilon ,\tau }$  are constants and

${\displaystyle F_{\mu \nu }=K_{\nu ;\mu }-K_{\mu ;\nu }.}$  (See Will^[11] for the field equations for ${\displaystyle T^{\mu \nu }}$  and ${\displaystyle K_{\mu }.}$ )

Will and Nordtvedt^[43] is a special case where

${\displaystyle \omega =\eta =\epsilon =0;\quad \tau =1}$ 

Hellings and Nordtvedt^[44] is a special case where

${\displaystyle \tau =0;\quad \epsilon =1;\quad \eta =-2\omega }$ 

These vector–tensor theories are semi-conservative, which means that they satisfy the laws of conservation of momentum and angular momentum but can have preferred frame effects. When ${\displaystyle \omega =\eta =\epsilon =\tau =0}$  they reduce to general relativity so, so long as general relativity is confirmed by experiment, general vector–tensor theories can never be ruled out.

Other metric theories

Others metric theories have been proposed; that of Bekenstein^[63] is discussed under Modern Theories.

Non-metric theories

See also: Einstein–Cartan theory and Cartan connection

Cartan's theory is particularly interesting both because it is a non-metric theory and because it is so old. The status of Cartan's theory is uncertain. Will^[11] claims that all non-metric theories are eliminated by Einstein's Equivalence Principle. Will (2001) tempers that by explaining experimental criteria for testing non-metric theories against Einstein's Equivalence Principle. Misner et al.^[52] claims that Cartan's theory is the only non-metric theory to survive all experimental tests up to that date and Turyshev^[64] lists Cartan's theory among the few that have survived all experimental tests up to that date. The following is a quick sketch of Cartan's theory as restated by Trautman.^[65]

Cartan^[15][16] suggested a simple generalization of Einstein's theory of gravitation. He proposed a model of space time with a metric tensor and a linear "connection" compatible with the metric but not necessarily symmetric. The torsion tensor of the connection is related to the density of intrinsic angular momentum. Independently of Cartan, similar ideas were put forward by Sciama, by Kibble in the years 1958 to 1966, culminating in a 1976 review by Hehl et al.

The original description is in terms of differential forms, but for the present article that is replaced by the more familiar language of tensors (risking loss of accuracy). As in general relativity, the Lagrangian is made up of a massless and a mass part. The Lagrangian for the massless part is:

${\displaystyle {\begin{aligned}L&={1 \over 32\pi G}\Omega _{\nu }^{\mu }g^{\nu \xi }x^{\eta }x^{\zeta }\varepsilon _{\xi \mu \eta \zeta }\\[5pt]\Omega _{\nu }^{\mu }&=d\omega _{\nu }^{\mu }+\omega _{\xi }^{\eta }\\[5pt]\nabla x^{\mu }&=-\omega _{\nu }^{\mu }x^{\nu }\end{aligned}}}$ 

The ${\displaystyle \omega _{\nu }^{\mu }\;}$  is the linear connection. ${\displaystyle \varepsilon _{\xi \mu \eta \zeta }\;}$  is the completely antisymmetric pseudo-tensor (Levi-Civita symbol) with ${\displaystyle \varepsilon _{0123}={\sqrt {-g}}\;}$ , and ${\displaystyle g^{\nu \xi }\,}$  is the metric tensor as usual. By assuming that the linear connection is metric, it is possible to remove the unwanted freedom inherent in the non-metric theory. The stress–energy tensor is calculated from:

${\displaystyle T^{\mu \nu }={1 \over 16\pi G}(g^{\mu \nu }\eta _{\eta }^{\xi }-g^{\xi \mu }\eta _{\eta }^{\nu }-g^{\xi \nu }\eta _{\eta }^{\mu })\Omega _{\xi }^{\eta }\;}$ 

The space curvature is not Riemannian, but on a Riemannian space-time the Lagrangian would reduce to the Lagrangian of general relativity.

Some equations of the non-metric theory of Belinfante and Swihart^[26][27] have already been discussed in the section on bimetric theories.

A distinctively non-metric theory is given by gauge theory gravity, which replaces the metric in its field equations with a pair of gauge fields in flat spacetime. On the one hand, the theory is quite conservative because it is substantially equivalent to Einstein–Cartan theory (or general relativity in the limit of vanishing spin), differing mostly in the nature of its global solutions. On the other hand, it is radical because it replaces differential geometry with geometric algebra.

Modern theories 1980s to present

This section includes alternatives to general relativity published after the observations of galaxy rotation that led to the hypothesis of "dark matter". There is no known reliable list of comparison of these theories. Those considered here include: Bekenstein,^[63] Moffat,^[66] Moffat,^[67] Moffat.^[68][69] These theories are presented with a cosmological constant or added scalar or vector potential.

Motivations

Motivations for the more recent alternatives to general relativity are almost all cosmological, associated with or replacing such constructs as "inflation", "dark matter" and "dark energy". The basic idea is that gravity agrees with general relativity at the present epoch but may have been quite different in the early universe.

In the 1980s, there was a slowly dawning realisation in the physics world that there were several problems inherent in the then-current big-bang scenario, including the horizon problem and the observation that at early times when quarks were first forming there was not enough space on the universe to contain even one quark. Inflation theory was developed to overcome these difficulties. Another alternative was constructing an alternative to general relativity in which the speed of light was higher in the early universe. The discovery of unexpected rotation curves for galaxies took everyone by surprise. Could there be more mass in the universe than we are aware of, or is the theory of gravity itself wrong? The consensus now is that the missing mass is "cold dark matter", but that consensus was only reached after trying alternatives to general relativity, and some physicists still believe that alternative models of gravity may hold the answer.

In the 1990s, supernova surveys discovered the accelerated expansion of the universe, now usually attributed to dark energy. This led to the rapid reinstatement of Einstein's cosmological constant, and quintessence arrived as an alternative to the cosmological constant. At least one new alternative to general relativity attempted to explain the supernova surveys' results in a completely different way. The measurement of the speed of gravity with the gravitational wave event GW170817 ruled out many alternative theories of gravity as explanations for the accelerated expansion.^[70][71][72] Another observation that sparked recent interest in alternatives to General Relativity is the Pioneer anomaly. It was quickly discovered that alternatives to general relativity could explain this anomaly. This is now believed to be accounted for by non-uniform thermal radiation.

Cosmological constant and quintessence

See also: Cosmological constant, Einstein–Hilbert action, and Quintessence (physics)

The cosmological constant ${\displaystyle \Lambda \;}$  is a very old idea, going back to Einstein in 1917.^[5] The success of the Friedmann model of the universe in which ${\displaystyle \Lambda =0\;}$  led to the general acceptance that it is zero, but the use of a non-zero value came back with a vengeance when data from supernovae indicated that the expansion of the universe is accelerating

First, let's see how it influences the equations of Newtonian gravity and General Relativity. In Newtonian gravity, the addition of the cosmological constant changes the Newton–Poisson equation from:

${\displaystyle \nabla ^{2}\varphi =4\pi \rho \ G;}$ 

to

${\displaystyle \nabla ^{2}\varphi +{\frac {1}{2}}\Lambda c^{2}=4\pi \rho \ G;}$ 

In general relativity, it changes the Einstein–Hilbert action from

${\displaystyle S={1 \over 16\pi G}\int R{\sqrt {-g}}\,d^{4}x\,+S_{m}\;}$ 

to

${\displaystyle S={1 \over 16\pi G}\int (R-2\Lambda ){\sqrt {-g}}\,d^{4}x\,+S_{m}\;}$ 

which changes the field equation

${\displaystyle T^{\mu \nu }={1 \over 8\pi G}\left(R^{\mu \nu }-{\frac {1}{2}}g^{\mu \nu }R\right)\;}$ 

to

${\displaystyle T^{\mu \nu }={1 \over 8\pi G}\left(R^{\mu \nu }-{\frac {1}{2}}g^{\mu \nu }R+g^{\mu \nu }\Lambda \right)\;}$ 

In alternative theories of gravity, a cosmological constant can be added to the action in exactly the same way.

The cosmological constant is not the only way to get an accelerated expansion of the universe in alternatives to general relativity. We've already seen how the scalar potential ${\displaystyle \lambda (\varphi )\;}$  can be added to scalar tensor theories. This can also be done in every alternative the general relativity that contains a scalar field ${\displaystyle \varphi \;}$  by adding the term ${\displaystyle \lambda (\varphi )\;}$  inside the Lagrangian for the gravitational part of the action, the ${\displaystyle L_{\varphi }\;}$  part of

${\displaystyle S={1 \over 16\pi G}\int d^{4}x\,{\sqrt {-g}}\,L_{\varphi }+S_{m}\;}$ 

Because ${\displaystyle \lambda (\varphi )\;}$  is an arbitrary function of the scalar field, it can be set to give an acceleration that is large in the early universe and small at the present epoch. This is known as quintessence.

A similar method can be used in alternatives to general relativity that use vector fields, including Rastall^[49] and vector–tensor theories. A term proportional to

${\displaystyle K^{\mu }K^{\nu }g_{\mu \nu }\;}$ 

is added to the Lagrangian for the gravitational part of the action.

Farnes' theories

In December 2018, the astrophysicist Jamie Farnes from the University of Oxford proposed a dark fluid theory, related to notions of gravitationally repulsive negative masses that were presented earlier by Albert Einstein. The theory may help to better understand the considerable amounts of unknown dark matter and dark energy in the universe.^[73]

The theory relies on the concept of negative mass and reintroduces Fred Hoyle's creation tensor in order to allow matter creation for only negative mass particles. In this way, the negative mass particles surround galaxies and apply a pressure onto them, thereby resembling dark matter. As these hypothesised particles mutually repel one another, they push apart the Universe, thereby resembling dark energy. The creation of matter allows the density of the exotic negative mass particles to remain constant as a function of time, and so appears like a cosmological constant. Einstein's field equations are modified to:

${\displaystyle R_{\mu \nu }-{\frac {1}{2}}Rg_{\mu \nu }={\frac {8\pi G}{c^{4}}}\left(T_{\mu \nu }^{+}+T_{\mu \nu }^{-}+C_{\mu \nu }\right)}$ 

According to Occam's razor, Farnes' theory is a simpler alternative to the conventional LambdaCDM model, as both dark energy and dark matter (two hypotheses) are solved using a single negative mass fluid (one hypothesis). The theory will be directly testable using the world's largest radio telescope, the Square Kilometre Array which should come online in 2022.^[74]

Relativistic MOND

Main articles: Modified Newtonian dynamics, Tensor–vector–scalar gravity, and Extended theories of gravity

The original theory of MOND by Milgrom was developed in 1983 as an alternative to "dark matter". Departures from Newton's law of gravitation are governed by an acceleration scale, not a distance scale. MOND successfully explains the Tully–Fisher observation that the luminosity of a galaxy should scale as the fourth power of the rotation speed. It also explains why the rotation discrepancy in dwarf galaxies is particularly large.

There were several problems with MOND in the beginning.

1. It did not include relativistic effects
2. It violated the conservation of energy, momentum and angular momentum
3. It was inconsistent in that it gives different galactic orbits for gas and for stars
4. It did not state how to calculate gravitational lensing from galaxy clusters.

By 1984, problems 2 and 3 had been solved by introducing a Lagrangian (AQUAL). A relativistic version of this based on scalar–tensor theory was rejected because it allowed waves in the scalar field to propagate faster than light. The Lagrangian of the non-relativistic form is:

${\displaystyle L=-{a_{0}^{2} \over 8\pi G}f\left\lbrack {\frac {|\nabla \varphi |^{2}}{a_{0}^{2}}}\right\rbrack -\rho \varphi }$ 

The relativistic version of this has:

${\displaystyle L=-{a_{0}^{2} \over 8\pi G}{\tilde {f}}\left(\ell _{0}^{2}g^{\mu \nu }\,\partial _{\mu }\varphi \,\partial _{\nu }\varphi \right)}$ 

with a nonstandard mass action. Here ${\displaystyle f}$  and ${\displaystyle {\tilde {f}}}$  are arbitrary functions selected to give Newtonian and MOND behaviour in the correct limits, and ${\displaystyle l_{0}=c^{2}/a_{0}\;}$  is the MOND length scale. By 1988, a second scalar field (PCC) fixed problems with the earlier scalar–tensor version but is in conflict with the perihelion precession of Mercury and gravitational lensing by galaxies and clusters. By 1997, MOND had been successfully incorporated in a stratified relativistic theory [Sanders], but as this is a preferred frame theory it has problems of its own. Bekenstein^[63] introduced a tensor–vector–scalar model (TeVeS). This has two scalar fields ${\displaystyle \varphi }$  and ${\displaystyle \sigma \;}$  and vector field ${\displaystyle U_{\alpha }}$ . The action is split into parts for gravity, scalars, vector and mass.

${\displaystyle S=S_{g}+S_{s}+S_{v}+S_{m}}$ 

The gravity part is the same as in general relativity.

${\displaystyle {\begin{aligned}S_{s}&=-{\frac {1}{2}}\int \left[\sigma ^{2}h^{\alpha \beta }\varphi _{,\alpha }\varphi _{,\beta }+{\frac {1}{2}}G\ell _{0}^{-2}\sigma ^{4}F(kG\sigma ^{2})\right]{\sqrt {-g}}\,d^{4}x\\[5pt]S_{v}&=-{\frac {K}{32\pi G}}\int \left[g^{\alpha \beta }g^{\mu \nu }U_{[\alpha ,\mu ]}U_{[\beta ,\nu ]}-{\frac {2\lambda }{K}}\left(g^{\mu \nu }U_{\mu }U_{\nu }+1\right)\right]{\sqrt {-g}}\,d^{4}x\\[5pt]S_{m}&=\int L\left({\tilde {g}}_{\mu \nu },f^{\alpha },f_{|\mu }^{\alpha },\ldots \right){\sqrt {-g}}\,d^{4}x\end{aligned}}}$ 

where

${\displaystyle h^{\alpha \beta }=g^{\alpha \beta }-U^{\alpha }U^{\beta }}$ 

${\displaystyle {\tilde {g}}^{\alpha \beta }=e^{2\varphi }g^{\alpha \beta }+2U^{\alpha }U^{\beta }\sinh(2\varphi )}$ 

${\displaystyle k,K}$  are constants, square brackets in indices ${\displaystyle U_{[\alpha ,\mu ]}}$  represent anti-symmetrization, ${\displaystyle \lambda }$  is a Lagrange multiplier (calculated elsewhere), and L is a Lagrangian translated from flat spacetime onto the metric ${\displaystyle {\tilde {g}}^{\alpha \beta }}$ . Note that G need not equal the observed gravitational constant ${\displaystyle G_{Newton}}$ . F is an arbitrary function, and

${\displaystyle F(\mu )={\frac {3}{4}}{\mu ^{2}(\mu -2)^{2} \over 1-\mu }}$ 

is given as an example with the right asymptotic behaviour; note how it becomes undefined when ${\displaystyle \mu =1}$ 

The Parametric post-Newtonian parameters of this theory are calculated in,^[75] which shows that all its parameters are equal to general relativity's, except for

${\displaystyle {\begin{aligned}\alpha _{1}&={\frac {4G}{K}}\left((2K-1)e^{-4\varphi _{0}}-e^{4\varphi _{0}}+8\right)-8\\[5pt]\alpha _{2}&={\frac {6G}{2-K}}-{\frac {2G(K+4)e^{4\varphi _{0}}}{(2-K)^{2}}}-1\end{aligned}}}$ 

both of which expressed in geometric units where ${\displaystyle c=G_{Newtonian}=1}$ ; so

${\displaystyle G^{-1}={\frac {2}{2-K}}+{\frac {k}{4\pi }}.}$ 

Moffat's theories

J. W. Moffat^[66] developed a non-symmetric gravitation theory. This is not a metric theory. It was first claimed that it does not contain a black hole horizon, but Burko and Ori^[76] have found that nonsymmetric gravitational theory can contain black holes. Later, Moffat claimed that it has also been applied to explain rotation curves of galaxies without invoking "dark matter". Damour, Deser & MaCarthy^[77] have criticised nonsymmetric gravitational theory, saying that it has unacceptable asymptotic behaviour.

The mathematics is not difficult but is intertwined so the following is only a brief sketch. Starting with a non-symmetric tensor ${\displaystyle g_{\mu \nu }\;}$ , the Lagrangian density is split into

${\displaystyle L=L_{R}+L_{M}\;}$ 

where ${\displaystyle L_{M}\;}$  is the same as for matter in general relativity.

${\displaystyle L_{R}={\sqrt {-g}}\left[R(W)-2\lambda -{\frac {1}{4}}\mu ^{2}g^{\mu \nu }g_{[\mu \nu ]}\right]-{\frac {1}{6}}g^{\mu \nu }W_{\mu }W_{\nu }\;}$ 

where ${\displaystyle R(W)\;}$  is a curvature term analogous to but not equal to the Ricci curvature in general relativity, ${\displaystyle \lambda \;}$  and ${\displaystyle \mu ^{2}\;}$  are cosmological constants, ${\displaystyle g_{[\nu \mu ]}\;}$  is the antisymmetric part of ${\displaystyle g_{\nu \mu }\;}$ . ${\displaystyle W_{\mu }\;}$  is a connection, and is a bit difficult to explain because it's defined recursively. However, ${\displaystyle W_{\mu }\approx -2g_{[\mu \nu ]}^{,\nu }\;}$ 

Haugan and Kauffmann^[78] used polarization measurements of the light emitted by galaxies to impose sharp constraints on the magnitude of some of nonsymmetric gravitational theory's parameters. They also used Hughes-Drever experiments to constrain the remaining degrees of freedom. Their constraint is eight orders of magnitude sharper than previous estimates.

Moffat's^[68] metric-skew-tensor-gravity (MSTG) theory is able to predict rotation curves for galaxies without either dark matter or MOND, and claims that it can also explain gravitational lensing of galaxy clusters without dark matter. It has variable ${\displaystyle G\;}$ , increasing to a final constant value about a million years after the big bang.

The theory seems to contain an asymmetric tensor ${\displaystyle A_{\mu \nu }\;}$  field and a source current ${\displaystyle J_{\mu }\;}$  vector. The action is split into:

${\displaystyle S=S_{G}+S_{F}+S_{FM}+S_{M}\;}$ 

Both the gravity and mass terms match those of general relativity with cosmological constant. The skew field action and the skew field matter coupling are:

${\displaystyle S_{F}=\int d^{4}x\,{\sqrt {-g}}\left({\frac {1}{12}}F_{\mu \nu \rho }F^{\mu \nu \rho }-{\frac {1}{4}}\mu ^{2}A_{\mu \nu }A^{\mu \nu }\right)\;}$ 

${\displaystyle S_{FM}=\int d^{4}x\,\epsilon ^{\alpha \beta \mu \nu }A_{\alpha \beta }\partial _{\mu }J_{\nu }\;}$ 

where

${\displaystyle F_{\mu \nu \rho }=\partial _{\mu }A_{\nu \rho }+\partial _{\rho }A_{\mu \nu }}$ 

and ${\displaystyle \epsilon ^{\alpha \beta \mu \nu }\;}$  is the Levi-Civita symbol. The skew field coupling is a Pauli coupling and is gauge invariant for any source current. The source current looks like a matter fermion field associated with baryon and lepton number.

Scalar–tensor–vector gravity

Main article: Scalar–tensor–vector gravity

Moffat's Scalar–tensor–vector gravity^[69] contains a tensor, vector and three scalar fields. But the equations are quite straightforward. The action is split into: ${\displaystyle S=S_{G}+S_{K}+S_{S}+S_{M}}$  with terms for gravity, vector field ${\displaystyle K_{\mu },}$  scalar fields ${\displaystyle G,\omega ,\mu }$  and mass. ${\displaystyle S_{G}}$  is the standard gravity term with the exception that ${\displaystyle G}$  is moved inside the integral.

${\displaystyle S_{K}=-\int d^{4}x\,{\sqrt {-g}}\omega \left({\frac {1}{4}}B_{\mu \nu }B^{\mu \nu }+V(K)\right),\qquad {\text{where }}\quad B_{\mu \nu }=\partial _{\mu }K_{\nu }-\partial _{\nu }K_{\mu }.}$ 

${\displaystyle S_{S}=-\int d^{4}x\,{\sqrt {-g}}{\frac {1}{G^{3}}}\left({\frac {1}{2}}g^{\mu \nu }\,\nabla _{\mu }G\,\nabla _{\nu }G-V(G)\right)+{\frac {1}{G}}\left({\frac {1}{2}}g^{\mu \nu }\,\nabla _{\mu }\omega \,\nabla _{\nu }\omega -V(\omega )\right)+{1 \over \mu ^{2}G}\left({\frac {1}{2}}g^{\mu \nu }\,\nabla _{\mu }\mu \,\nabla _{\nu }\mu -V(\mu )\right).}$ 

The potential function for the vector field is chosen to be:

${\displaystyle V(K)=-{\frac {1}{2}}\mu ^{2}\varphi ^{\mu }\varphi _{\mu }-{\frac {1}{4}}g\left(\varphi ^{\mu }\varphi _{\mu }\right)^{2}}$ 

where ${\displaystyle g}$  is a coupling constant. The functions assumed for the scalar potentials are not stated.

Infinite derivative gravity

Main article: Infinite derivative gravity

In order to remove ghosts in the modified propagator, as well as to obtain asymptotic freedom, Biswas, Mazumdar and Siegel (2005) considered a string-inspired infinite set of higher derivative terms

${\displaystyle S=\int \mathrm {d} ^{4}x{\sqrt {-g}}\left({\frac {R}{2}}+RF(\Box )R\right)}$ 

where ${\displaystyle F(\Box )}$  is the exponential of an entire function of the D'Alembertian operator.^[79][80] This avoids a black hole singularity near the origin, while recovering the 1/r fall of the general relativity potential at large distances.^[81] Lousto and Mazzitelli (1997) found an exact solution to this theories representing a gravitational shock-wave.^[82]

General relativity self-interaction (GRSI)

The General Relativity Self-interaction or GRSI model^[83] is an attempt to explain astrophysical and cosmological observations without dark matter, dark energy by adding self-interaction terms when calculating the gravitational effects in general relativity, analogous to the self-interaction terms in quantum chromodynamics.^[84] Additionally, the model explains the Tully-Fisher relation,^[85] the radial acceleration relation,^[86] observations that are currently challenging to understand within Lambda-CDM.

Testing of alternatives to general relativity

Main article: Tests of general relativity

Any putative alternative to general relativity would need to meet a variety of tests for it to become accepted. For in-depth coverage of these tests, see Misner et al.^[52] Ch.39, Will^[11] Table 2.1, and Ni.^[12] Most such tests can be categorized as in the following subsections.

Self-consistency

Self-consistency among non-metric theories includes eliminating theories allowing tachyons, ghost poles and higher order poles, and those that have problems with behaviour at infinity. Among metric theories, self-consistency is best illustrated by describing several theories that fail this test. The classic example is the spin-two field theory of Fierz and Pauli;^[17] the field equations imply that gravitating bodies move in straight lines, whereas the equations of motion insist that gravity deflects bodies away from straight line motion. Yilmaz (1971)^[29] contains a tensor gravitational field used to construct a metric; it is mathematically inconsistent because the functional dependence of the metric on the tensor field is not well defined.

Completeness

To be complete, a theory of gravity must be capable of analysing the outcome of every experiment of interest. It must therefore mesh with electromagnetism and all other physics. For instance, any theory that cannot predict from first principles the movement of planets or the behaviour of atomic clocks is incomplete.

Many early theories are incomplete in that it is unclear whether the density ${\displaystyle \rho }$  used by the theory should be calculated from the stress–energy tensor ${\displaystyle T}$  as ${\displaystyle \rho =T_{\mu \nu }u^{\mu }u^{\nu }}$  or as ${\displaystyle \rho =T_{\mu \nu }\delta ^{\mu \nu }}$ , where ${\displaystyle u}$  is the four-velocity, and ${\displaystyle \delta }$  is the Kronecker delta. The theories of Thirry (1948) and Jordan^[24] are incomplete unless Jordan's parameter ${\displaystyle \eta \;}$  is set to -1, in which case they match the theory of Brans–Dicke^[9] and so are worthy of further consideration. Milne^[19] is incomplete because it makes no gravitational red-shift prediction. The theories of Whitrow and Morduch,^[30][31] Kustaanheimo^[32] and Kustaanheimo and Nuotio^[33] are either incomplete or inconsistent. The incorporation of Maxwell's equations is incomplete unless it is assumed that they are imposed on the flat background space-time, and when that is done they are inconsistent, because they predict zero gravitational redshift when the wave version of light (Maxwell theory) is used, and nonzero redshift when the particle version (photon) is used. Another more obvious example is Newtonian gravity with Maxwell's equations; light as photons is deflected by gravitational fields (by half that of general relativity) but light as waves is not.

Classical tests

Main article: Tests of general relativity

There are three "classical" tests (dating back to the 1910s or earlier) of the ability of gravity theories to handle relativistic effects; they are gravitational redshift, gravitational lensing (generally tested around the Sun), and anomalous perihelion advance of the planets. Each theory should reproduce the observed results in these areas, which have to date always aligned with the predictions of general relativity. In 1964, Irwin I. Shapiro found a fourth test, called the Shapiro delay. It is usually regarded as a "classical" test as well.

Agreement with Newtonian mechanics and special relativity

As an example of disagreement with Newtonian experiments, Birkhoff^[18] theory predicts relativistic effects fairly reliably but demands that sound waves travel at the speed of light. This was the consequence of an assumption made to simplify handling the collision of masses.^{[citation needed]}

The Einstein equivalence principle

Main article: Equivalence principle

Einstein's Equivalence Principle has three components. The first is the uniqueness of free fall, also known as the Weak Equivalence Principle. This is satisfied if inertial mass is equal to gravitational mass. η is a parameter used to test the maximum allowable violation of the Weak Equivalence Principle. The first tests of the Weak Equivalence Principle were done by Eötvös before 1900 and limited η to less than 5×10⁻⁹. Modern tests have reduced that to less than 5×10⁻¹³. The second is Lorentz invariance. In the absence of gravitational effects the speed of light is constant. The test parameter for this is δ. The first tests of Lorentz invariance were done by Michelson and Morley before 1890 and limited δ to less than 5×10⁻³. Modern tests have reduced this to less than 1×10⁻²¹. The third is local position invariance, which includes spatial and temporal invariance. The outcome of any local non-gravitational experiment is independent of where and when it is performed. Spatial local position invariance is tested using gravitational redshift measurements. The test parameter for this is α. Upper limits on this found by Pound and Rebka in 1960 limited α to less than 0.1. Modern tests have reduced this to less than 1×10⁻⁴.^[2]

Schiff's conjecture states that any complete, self-consistent theory of gravity that embodies the Weak Equivalence Principle necessarily embodies Einstein's Equivalence Principle. This is likely to be true if the theory has full energy conservation. Metric theories satisfy the Einstein Equivalence Principle. Extremely few non-metric theories satisfy this. For example, the non-metric theory of Belinfante & Swihart^[26][27] is eliminated by the THεμ formalism for testing Einstein's Equivalence Principle. Gauge theory gravity is a notable exception, where the strong equivalence principle is essentially the minimal coupling of the gauge covariant derivative.

Parametric post-Newtonian formalism

Main article: Parameterized post-Newtonian formalism

See also Tests of general relativity, Misner et al.^[52] and Will^[11] for more information.

Work on developing a standardized rather than ad hoc set of tests for evaluating alternative gravitation models began with Eddington in 1922 and resulted in a standard set of Parametric post-Newtonian numbers in Nordtvedt and Will^[87] and Will and Nordtvedt.^[43] Each parameter measures a different aspect of how much a theory departs from Newtonian gravity. Because we are talking about deviation from Newtonian theory here, these only measure weak-field effects. The effects of strong gravitational fields are examined later.

These ten are: ${\displaystyle \gamma ,\beta ,\eta ,\alpha _{1},\alpha _{2},\alpha _{3},\zeta _{1},\zeta _{2},\zeta _{3},\zeta _{4}.}$ 

• ${\displaystyle \gamma }$  is a measure of space curvature, being zero for Newtonian gravity and one for general relativity.
• ${\displaystyle \beta }$  is a measure of nonlinearity in the addition of gravitational fields, one for general relativity.
• ${\displaystyle \eta }$  is a check for preferred location effects.
• ${\displaystyle \alpha _{1},\alpha _{2},\alpha _{3}}$  measure the extent and nature of "preferred-frame effects". Any theory of gravity in which at least one of the three is nonzero is called a preferred-frame theory.
• ${\displaystyle \zeta _{1},\zeta _{2},\zeta _{3},\zeta _{4},\alpha _{3}}$  measure the extent and nature of breakdowns in global conservation laws. A theory of gravity possesses 4 conservation laws for energy-momentum and 6 for angular momentum only if all five are zero.

Strong gravity and gravitational waves

Main article: Tests of general relativity

Parametric post-Newtonian is only a measure of weak field effects. Strong gravity effects can be seen in compact objects such as white dwarfs, neutron stars, and black holes. Experimental tests such as the stability of white dwarfs, spin-down rate of pulsars, orbits of binary pulsars and the existence of a black hole horizon can be used as tests of alternative to general relativity. General relativity predicts that gravitational waves travel at the speed of light. Many alternatives to general relativity say that gravitational waves travel faster than light, possibly breaking causality. After the multi-messaging detection of the GW170817 coalescence of neutron stars, where light and gravitational waves were measured to travel at the same speed with an error of 1/10¹⁵, many of those modified theories of gravity were excluded.

Cosmological tests

Useful cosmological scale tests are just beginning to become available.^[2]: 88 Given the limited astronomical data and the complexity of the theories, comparisons involve complex parameters. For example, Reyes et al.,^[88] analyzed 70,205 luminous red galaxies with a cross-correlation involving galaxy velocity estimates and gravitational potentials estimated from lensing and yet results are still tentative.^[1]: 164

For those theories that aim to replace dark matter, observations like the galaxy rotation curve, the Tully–Fisher relation, the faster rotation rate of dwarf galaxies, and the gravitational lensing due to galactic clusters act as constraints. For those theories that aim to replace inflation, the size of ripples in the spectrum of the cosmic microwave background radiation is the strictest test. For those theories that incorporate or aim to replace dark energy, the supernova brightness results and the age of the universe can be used as tests. Another test is the flatness of the universe. With general relativity, the combination of baryonic matter, dark matter and dark energy add up to make the universe exactly flat.

Results of testing theories

Parametric post-Newtonian parameters for a range of theories

(See Will^[11] and Ni^[12] for more details. Misner et al.^[52] gives a table for translating parameters from the notation of Ni to that of Will)

General Relativity is now more than 100 years old, during which one alternative theory of gravity after another has failed to agree with ever more accurate observations. One illustrative example is Parameterized post-Newtonian formalism. The following table lists Parametric post-Newtonian values for a large number of theories. If the value in a cell matches that in the column heading then the full formula is too complicated to include here.

	${\displaystyle \gamma }$	${\displaystyle \beta }$	${\displaystyle \xi }$	${\displaystyle \alpha _{1}}$	${\displaystyle \alpha _{2}}$	${\displaystyle \alpha _{3}}$	${\displaystyle \zeta _{1}}$	${\displaystyle \zeta _{2}}$	${\displaystyle \zeta _{3}}$	${\displaystyle \zeta _{4}}$
Einstein general relativity^[4]	1	1	0	0	0	0	0	0	0	0
Scalar–tensor theories
Bergmann,^[36] Wagoner^[39]	${\displaystyle \textstyle {\frac {1+\omega }{2+\omega }}}$	${\displaystyle \beta }$	0	0	0	0	0	0	0	0
Nordtvedt,^[38] Bekenstein^[47]	${\displaystyle \textstyle {\frac {1+\omega }{2+\omega }}}$	${\displaystyle \beta }$	0	0	0	0	0	0	0	0
Brans–Dicke^[9]	${\displaystyle \textstyle {\frac {1+\omega }{2+\omega }}}$	1	0	0	0	0	0	0	0	0
Vector–tensor theories
Hellings–Nordtvedt^[44]	${\displaystyle \gamma }$	${\displaystyle \beta }$	0	${\displaystyle \alpha _{1}}$	${\displaystyle \alpha _{2}}$	0	0	0	0	0
Will–Nordtvedt^[43]	1	1	0	0	${\displaystyle \alpha _{2}}$	0	0	0	0	0
Bimetric theories
Rosen^[41]	1	1	0	0	${\displaystyle c_{0}/c_{1}-1}$	0	0	0	0	0
Rastall^[49]	1	1	0	0	${\displaystyle \alpha _{2}}$	0	0	0	0	0
Lightman–Lee^[45]	${\displaystyle \gamma }$	${\displaystyle \beta }$	0	January 01, 1970 Latest articles December 27, 2022 Stade Joseph Marien December 27, 2022 Stade Marcel-Deflandre January 07, 2023 Stade Municipal de Berrechid December 27, 2022 Stade Municipal (Akonolinga) December 27, 2022 Stade Municipal (Ouagadougou) January 06, 2023 Stade Municipal (Sokodé) January 06, 2023 Stade Port Sudan December 27, 2022 Stade Perruc January 07, 2023 Stade Taïeb El Mhiri January 12, 2023 Stade de Moutendé Most Read December 30, 2022 Parker Jointless January 06, 2023 Parker Pen Company December 31, 2022 Parker v Clark December 26, 2022 Parking pawl December 28, 2022 Parkins Mills, Virginia alternatives, general, relativity, physical, theories, that, attempt, describe, phenomenon, gravitation, competition, with, einstein, theory, general, relativity, there, have, been, many, different, attempts, constructing, ideal, theory, gravity, these, attemp. Alternatives to general relativity are physical theories that attempt to describe the phenomenon of gravitation in competition with Einstein s theory of general relativity There have been many different attempts at constructing an ideal theory of gravity 1 These attempts can be split into four broad categories based on their scope In this article straightforward alternatives to general relativity are discussed which do not involve quantum mechanics or force unification Other theories which do attempt to construct a theory using the principles of quantum mechanics are known as theories of quantized gravity Thirdly there are theories which attempt to explain gravity and other forces at the same time these are known as classical unified field theories Finally the most ambitious theories attempt to both put gravity in quantum mechanical terms and unify forces these are called theories of everything None of these alternatives to general relativity have gained wide acceptance General relativity has withstood many tests 2 3 remaining consistent with all observations so far In contrast many of the early alternatives have been definitively disproven However some of the alternative theories of gravity are supported by a minority of physicists and the topic remains the subject of intense study in theoretical physics Contents 1 History of gravitational theory through general relativity 1 1 General relativity 2 Motivations 3 Notation in this article 4 Classification of theories 5 Theories from 1917 to the 1980s 5 1 Scalar field theories 5 2 Bimetric theories 5 3 Quasilinear theories 5 4 Tensor theories 5 4 1 Starobinsky 5 4 2 Gauss Bonnet 5 4 3 Stelle s 4th derivative gravity 5 4 4 f R 5 4 5 Infinite derivative gravity 5 4 6 Lovelock 5 5 Scalar tensor theories 5 6 Vector tensor theories 5 7 Other metric theories 5 8 Non metric theories 6 Modern theories 1980s to present 6 1 Motivations 6 2 Cosmological constant and quintessence 6 3 Farnes theories 6 4 Relativistic MOND 6 5 Moffat s theories 6 5 1 Scalar tensor vector gravity 6 6 Infinite derivative gravity 6 7 General relativity self interaction GRSI 7 Testing of alternatives to general relativity 7 1 Self consistency 7 2 Completeness 7 3 Classical tests 7 4 Agreement with Newtonian mechanics and special relativity 7 5 The Einstein equivalence principle 7 6 Parametric post Newtonian formalism 7 7 Strong gravity and gravitational waves 7 8 Cosmological tests 8 Results of testing theories 8 1 Parametric post Newtonian parameters for a range of theories 8 2 Theories that fail other tests 9 References 10 External linksHistory of gravitational theory through general relativity editMain article History of gravitational theory At the time it was published in the 17th century Isaac Newton s theory of gravity was the most accurate theory of gravity Since then a number of alternatives were proposed The theories which predate the formulation of general relativity in 1915 are discussed in history of gravitational theory General relativity edit Main articles General relativity and History of general relativity This theory 4 5 is what we now call general relativity included here for comparison Discarding the Minkowski metric entirely Einstein gets d d s 0 displaystyle delta int ds 0 nbsp d s 2 g m n d x m d x n displaystyle ds 2 g mu nu dx mu dx nu nbsp R m n 8 p G c 4 T m n 1 2 g m n T displaystyle R mu nu frac 8 pi G c 4 left T mu nu frac 1 2 g mu nu T right nbsp which can also be written T m n c 4 8 p G R m n 1 2 g m n R displaystyle T mu nu c 4 over 8 pi G left R mu nu frac 1 2 g mu nu R right nbsp Five days before Einstein presented the last equation above Hilbert had submitted a paper containing an almost identical equation See General relativity priority dispute Hilbert was the first to correctly state the Einstein Hilbert action for general relativity which is S c 4 16 p G R g d 4 x S m displaystyle S c 4 over 16 pi G int R sqrt g d 4 x S m nbsp where G displaystyle G nbsp is Newton s gravitational constant R R m m displaystyle R R mu mu nbsp is the Ricci curvature of space g det g m n displaystyle g det g mu nu nbsp and S m displaystyle S m nbsp is the action due to mass General relativity is a tensor theory the equations all contain tensors Nordstrom s theories on the other hand are scalar theories because the gravitational field is a scalar Other proposed alternatives include scalar tensor theories that contain a scalar field in addition to the tensors of general relativity and other variants containing vector fields as well have been developed recently Motivations editAfter general relativity attempts were made either to improve on theories developed before general relativity or to improve general relativity itself Many different strategies were attempted for example the addition of spin to general relativity combining a general relativity like metric with a spacetime that is static with respect to the expansion of the universe getting extra freedom by adding another parameter At least one theory was motivated by the desire to develop an alternative to general relativity that is free of singularities Experimental tests improved along with the theories Many of the different strategies that were developed soon after general relativity were abandoned and there was a push to develop more general forms of the theories that survived so that a theory would be ready when any test showed a disagreement with general relativity By the 1980s the increasing accuracy of experimental tests had all confirmed general relativity no competitors were left except for those that included general relativity as a special case Further shortly after that theorists switched to string theory which was starting to look promising but has since lost popularity In the mid 1980s a few experiments were suggesting that gravity was being modified by the addition of a fifth force or in one case of a fifth sixth and seventh force acting in the range of a few meters Subsequent experiments eliminated these Motivations for the more recent alternative theories are almost all cosmological associated with or replacing such constructs as inflation dark matter and dark energy Investigation of the Pioneer anomaly has caused renewed public interest in alternatives to general relativity Notation in this article editMain articles Mathematics of general relativity and Ricci calculus c displaystyle c nbsp is the speed of light G displaystyle G nbsp is the gravitational constant Geometric variables are not used Latin indices go from 1 to 3 Greek indices go from 0 to 3 The Einstein summation convention is used h m n displaystyle eta mu nu nbsp is the Minkowski metric g m n displaystyle g mu nu nbsp is a tensor usually the metric tensor These have signature Partial differentiation is written m f displaystyle partial mu varphi nbsp or f m displaystyle varphi mu nbsp Covariant differentiation is written m f displaystyle nabla mu varphi nbsp or f m displaystyle varphi mu nbsp Classification of theories editTheories of gravity can be classified loosely into several categories Most of the theories described here have an action see the principle of least action a variational principle based on the concept of action a Lagrangian density a metric If a theory has a Lagrangian density for gravity say L displaystyle L nbsp then the gravitational part of the action S displaystyle S nbsp is the integral of that S L g d 4 x displaystyle S int L sqrt g mathrm d 4 x nbsp In this equation it is usual though not essential to have g 1 displaystyle g 1 nbsp at spatial infinity when using Cartesian coordinates For example the Einstein Hilbert action uses L R displaystyle L propto R nbsp where R is the scalar curvature a measure of the curvature of space Almost every theory described in this article has an action It is the most efficient known way to guarantee that the necessary conservation laws of energy momentum and angular momentum are incorporated automatically although it is easy to construct an action where those conservation laws are violated Canonical methods provide another way to construct systems that have the required conservation laws but this approach is more cumbersome to implement 6 The original 1983 version of MOND did not have an action A few theories have an action but not a Lagrangian density A good example is Whitehead 7 the action there is termed non local A theory of gravity is a metric theory if and only if it can be given a mathematical representation in which two conditions hold Condition 1 There exists a symmetric metric tensor g m n displaystyle g mu nu nbsp of signature which governs proper length and proper time measurements in the usual manner of special and general relativity d t 2 g m n d x m d x n displaystyle d tau 2 g mu nu dx mu dx nu nbsp where there is a summation over indices m displaystyle mu nbsp and n displaystyle nu nbsp Condition 2 Stressed matter and fields being acted upon by gravity respond in accordance with the equation 0 n T m n T m n n G s n m T s n G s n n T m s displaystyle 0 nabla nu T mu nu T mu nu nu Gamma sigma nu mu T sigma nu Gamma sigma nu nu T mu sigma nbsp where T m n displaystyle T mu nu nbsp is the stress energy tensor for all matter and non gravitational fields and where n displaystyle nabla nu nbsp is the covariant derivative with respect to the metric and G s n a displaystyle Gamma sigma nu alpha nbsp is the Christoffel symbol The stress energy tensor should also satisfy an energy condition Metric theories include from simplest to most complex Scalar field theories includes conformally flat theories amp Stratified theories with conformally flat space slices Bergman Coleman Einstein 1912 Einstein Fokker theory Lee Lightman Ni Littlewood Ni Nordstrom s theory of gravitation first metric theory of gravity to be developed Page Tupper Papapetrou Rosen 1971 Whitrow Morduch Yilmaz theory of gravitation attempted to eliminate event horizons from the theory Quasilinear theories includes Linear fixed gauge Bollini Giambiagi Tiomno Deser Laurent Whitehead s theory of gravity intended to use only retarded potentials Tensor theories Einstein s general relativity Fourth order gravity allows the Lagrangian to depend on second order contractions of the Riemann curvature tensor f R gravity allows the Lagrangian to depend on higher powers of the Ricci scalar Gauss Bonnet gravity Lovelock theory of gravity allows the Lagrangian to depend on higher order contractions of the Riemann curvature tensor Infinite derivative gravity Scalar tensor theories Bekenstein Bergmann Wagoner Brans Dicke theory the most well known alternative to general relativity intended to be better at applying Mach s principle Jordan Nordtvedt Thiry Chameleon Pressuron Vector tensor theories Hellings Nordtvedt Will Nordtvedt Bimetric theories Lightman Lee Rastall Rosen 1975 Other metric theories see section Modern theories below Non metric theories include Belinfante Swihart Einstein Cartan theory intended to handle spin orbital angular momentum interchange Kustaanheimo 1967 Teleparallelism Gauge theory gravity A word here about Mach s principle is appropriate because a few of these theories rely on Mach s principle e g Whitehead 7 and many mention it in passing e g Einstein Grossmann 8 Brans Dicke 9 Mach s principle can be thought of a half way house between Newton and Einstein It goes this way 10 Newton Absolute space and time Mach The reference frame comes from the distribution of matter in the universe Einstein There is no reference frame Theories from 1917 to the 1980s editThis section includes alternatives to general relativity published after general relativity but before the observations of galaxy rotation that led to the hypothesis of dark matter Those considered here include see Will 11 12 Lang 13 14 Theories from 1917 to the 1980s Publication year s Author s Theory name Theory type 1922 7 Alfred North Whitehead Whitehead s theory of gravitation Quasilinear 1922 15 1923 16 Elie Cartan Einstein Cartan theory Non metric 1939 17 Markus Fierz Wolfgang Pauli 1943 18 George David Birkhoff 1948 19 Edward Arthur Milne Kinematic Relativity 1948 20 Yves Thiry 1954 21 22 Achilles Papapetrou Scalar field 1953 23 Dudley E Littlewood Scalar field 1955 24 Pascual Jordan 1956 25 Otto Bergmann Scalar field 1957 26 27 Frederik Belinfante James C Swihart 1958 28 1973 29 Huseyin Yilmaz Yilmaz theory of gravitation 1961 9 Carl H Brans Robert H Dicke Brans Dicke theory Scalar tensor 1960 30 1965 31 Gerald James Whitrow G E Morduch Scalar field 1966 32 Paul Kustaanheimo de 1967 33 Paul Kustaanheimo V S Nuotio 1968 34 Stanley Deser B E Laurent Quasilinear 1968 35 C Page B O J Tupper Scalar field 1968 36 Peter Bergmann Scalar tensor 1970 37 C G Bollini J J Giambiagi J Tiomno Quasilinear 1970 38 Kenneth Nordtvedt 1970 39 Robert V Wagoner Scalar tensor 1971 40 Nathan Rosen Scalar field 1975 41 Nathan Rosen Bimetric 1972 12 1973 42 Ni Wei tou Scalar field 1972 43 Clifford Martin Will Kenneth Nordtvedt Vector tensor 1973 44 Ronald Hellings Kenneth Nordtvedt Vector tensor 1973 45 Alan Lightman David L Lee Scalar field 1974 46 David L Lee Alan Lightman Ni Wei tou 1977 47 Jacob Bekenstein Scalar tensor 1978 48 B M Barker Scalar tensor 1979 49 P Rastall Bimetric These theories are presented here without a cosmological constant or added scalar or vector potential unless specifically noted for the simple reason that the need for one or both of these was not recognized before the supernova observations by the Supernova Cosmology Project and High Z Supernova Search Team How to add a cosmological constant or quintessence to a theory is discussed under Modern Theories see also Einstein Hilbert action Scalar field theories edit See also Scalar theories of gravitation The scalar field theories of Nordstrom 50 51 have already been discussed Those of Littlewood 23 Bergman 25 Yilmaz 28 Whitrow and Morduch 30 31 and Page and Tupper 35 follow the general formula give by Page and Tupper According to Page and Tupper 35 who discuss all these except Nordstrom 51 the general scalar field theory comes from the principle of least action d f f c 2 d s 0 displaystyle delta int f left tfrac varphi c 2 right ds 0 nbsp where the scalar field is f G M r displaystyle varphi frac GM r nbsp and c may or may not depend on f displaystyle varphi nbsp In Nordstrom 50 f f c 2 exp f c 2 c c displaystyle f varphi c 2 exp varphi c 2 qquad c c infty nbsp In Littlewood 23 and Bergmann 25 f f c 2 exp f c 2 c f 2 2 2 c c displaystyle f left frac varphi c 2 right exp left frac varphi c 2 frac c varphi 2 2 2 right qquad c c infty nbsp In Whitrow and Morduch 30 f f c 2 1 c 2 c 2 2 f displaystyle f left frac varphi c 2 right 1 qquad c 2 c infty 2 2 varphi nbsp In Whitrow and Morduch 31 f f c 2 exp f c 2 c 2 c 2 2 f displaystyle f left frac varphi c 2 right exp left frac varphi c 2 right qquad c 2 c infty 2 2 varphi nbsp In Page and Tupper 35 f f c 2 f c 2 a f c 2 2 c 2 c 2 1 4 f c 2 15 2 a f c 2 2 displaystyle f left frac varphi c 2 right frac varphi c 2 alpha left frac varphi c 2 right 2 qquad frac c infty 2 c 2 1 4 left frac varphi c infty 2 right 15 2 alpha left frac varphi c infty 2 right 2 nbsp Page and Tupper 35 matches Yilmaz s theory 28 to second order when a 7 2 displaystyle alpha 7 2 nbsp The gravitational deflection of light has to be zero when c is constant Given that variable c and zero deflection of light are both in conflict with experiment the prospect for a successful scalar theory of gravity looks very unlikely Further if the parameters of a scalar theory are adjusted so that the deflection of light is correct then the gravitational redshift is likely to be wrong Ni 12 summarized some theories and also created two more In the first a pre existing special relativity space time and universal time coordinate acts with matter and non gravitational fields to generate a scalar field This scalar field acts together with all the rest to generate the metric The action is S 1 16 p G d 4 x g L f S m displaystyle S 1 over 16 pi G int d 4 x sqrt g L varphi S m nbsp L f f R 2 g m n m f n f displaystyle L varphi varphi R 2g mu nu partial mu varphi partial nu varphi nbsp Misner et al 52 gives this without the f R displaystyle varphi R nbsp term S m displaystyle S m nbsp is the matter action f 4 p T m n h m n e 2 f e 2 f e 2 f m t n t displaystyle Box varphi 4 pi T mu nu left eta mu nu e 2 varphi left e 2 varphi e 2 varphi right partial mu t partial nu t right nbsp t is the universal time coordinate This theory is self consistent and complete But the motion of the solar system through the universe leads to serious disagreement with experiment In the second theory of Ni 12 there are two arbitrary functions f f displaystyle f varphi nbsp and k f displaystyle k varphi nbsp that are related to the metric by d s 2 e 2 f f d t 2 e 2 f f d x 2 d y 2 d z 2 displaystyle ds 2 e 2f varphi dt 2 e 2f varphi left dx 2 dy 2 dz 2 right nbsp h m n m n f 4 p r k f displaystyle eta mu nu partial mu partial nu varphi 4 pi rho k varphi nbsp Ni 12 quotes Rosen 40 as having two scalar fields f displaystyle varphi nbsp and ps displaystyle psi nbsp that are related to the metric by d s 2 f 2 d t 2 ps 2 d x 2 d y 2 d z 2 displaystyle ds 2 varphi 2 dt 2 psi 2 left dx 2 dy 2 dz 2 right nbsp In Papapetrou 21 the gravitational part of the Lagrangian is L f e f 1 2 e f a f a f 3 2 e f 0 f 0 f displaystyle L varphi e varphi left tfrac 1 2 e varphi partial alpha varphi partial alpha varphi tfrac 3 2 e varphi partial 0 varphi partial 0 varphi right nbsp In Papapetrou 22 there is a second scalar field x displaystyle chi nbsp The gravitational part of the Lagrangian is now L f e 1 2 3 f x 1 2 e f a f a f e f a f x f 3 2 e x 0 f 0 f displaystyle L varphi e frac 1 2 3 varphi chi left tfrac 1 2 e varphi partial alpha varphi partial alpha varphi e varphi partial alpha varphi partial chi varphi tfrac 3 2 e chi partial 0 varphi partial 0 varphi right nbsp Bimetric theories edit See also Bimetric theory Bimetric theories contain both the normal tensor metric and the Minkowski metric or a metric of constant curvature and may contain other scalar or vector fields Rosen 53 1975 bimetric theory The action is S 1 64 p G d 4 x h h m n g a b g g d g a g m g a d n 1 2 g a b m g g d n S m displaystyle S 1 over 64 pi G int d 4 x sqrt eta eta mu nu g alpha beta g gamma delta g alpha gamma mu g alpha delta nu textstyle frac 1 2 g alpha beta mu g gamma delta nu S m nbsp h g m n g a b h g d g m a g g n b d 16 p G g h T m n 1 2 g m n T displaystyle Box eta g mu nu g alpha beta eta gamma delta g mu alpha gamma g nu beta delta 16 pi G sqrt g eta T mu nu textstyle frac 1 2 g mu nu T nbsp Lightman Lee 45 developed a metric theory based on the non metric theory of Belinfante and Swihart 26 27 The result is known as BSLL theory Given a tensor field B m n displaystyle B mu nu nbsp B B m n h m n displaystyle B B mu nu eta mu nu nbsp and two constants a displaystyle a nbsp and f displaystyle f nbsp the action is S 1 16 p G d 4 x h a B m n a B m n a f B a B a S m displaystyle S 1 over 16 pi G int d 4 x sqrt eta aB mu nu alpha B mu nu alpha fB alpha B alpha S m nbsp and the stress energy tensor comes from a h B m n f h m n h B 4 p G g h T a b g a b B m n displaystyle a Box eta B mu nu f eta mu nu Box eta B 4 pi G sqrt g eta T alpha beta left frac partial g alpha beta partial B mu nu right nbsp In Rastall 49 the metric is an algebraic function of the Minkowski metric and a Vector field 54 The Action is S 1 16 p G d 4 x g F N K m n K m n S m displaystyle S 1 over 16 pi G int d 4 x sqrt g F N K mu nu K mu nu S m nbsp where F N N 2 N displaystyle F N frac N 2 N nbsp and N g m n K m K n displaystyle N g mu nu K mu K nu nbsp see Will 11 for the field equation for T m n displaystyle T mu nu nbsp and K m displaystyle K mu nbsp Quasilinear theories edit In Whitehead 7 the physical metric g displaystyle g nbsp is constructed by Synge algebraically from the Minkowski metric h displaystyle eta nbsp and matter variables so it doesn t even have a scalar field The construction is g m n x a h m n 2 S y m y n w 3 g r u a d S a displaystyle g mu nu x alpha eta mu nu 2 int Sigma y mu y nu over w 3 left sqrt g rho u alpha d Sigma alpha right nbsp where the superscript indicates quantities evaluated along the past h displaystyle eta nbsp light cone of the field point x a displaystyle x alpha nbsp and y m x m x m y m y m 0 w y m u m u m d x m d s d s 2 h m n d x m d x n displaystyle begin aligned y mu amp x mu x mu qquad y mu y mu 0 5pt w amp y mu u mu qquad u mu frac dx mu d sigma 5pt d sigma 2 amp eta mu nu dx mu dx nu end aligned nbsp Nevertheless the metric construction from a non metric theory using the length contraction ansatz is criticised 55 Deser and Laurent 34 and Bollini Giambiagi Tiomno 37 are Linear Fixed Gauge theories Taking an approach from quantum field theory combine a Minkowski spacetime with the gauge invariant action of a spin two tensor field i e graviton h m n displaystyle h mu nu nbsp to define g m n h m n h m n displaystyle g mu nu eta mu nu h mu nu nbsp The action is S 1 16 p G d 4 x h 2 h n m n h m l l 2 h n m n h l m l h n m n h l l m h m n l h m n l S m displaystyle S 1 over 16 pi G int d 4 x sqrt eta left 2h nu mu nu h mu lambda lambda 2h nu mu nu h lambda mu lambda h nu mu nu h lambda lambda mu h mu nu lambda h mu nu lambda right S m nbsp The Bianchi identity associated with this partial gauge invariance is wrong Linear Fixed Gauge theories seek to remedy this by breaking the gauge invariance of the gravitational action through the introduction of auxiliary gravitational fields that couple to h m n displaystyle h mu nu nbsp A cosmological constant can be introduced into a quasilinear theory by the simple expedient of changing the Minkowski background to a de Sitter or anti de Sitter spacetime as suggested by G Temple in 1923 Temple s suggestions on how to do this were criticized by C B Rayner in 1955 56 Tensor theories edit Einstein s general relativity is the simplest plausible theory of gravity that can be based on just one symmetric tensor field the metric tensor Others include Starobinsky R R 2 gravity Gauss Bonnet gravity f R gravity and Lovelock theory of gravity Starobinsky edit See also Starobinsky inflation Starobinsky gravity proposed by Alexei Starobinsky has the Lagrangian L g R R 2 6 M 2 displaystyle mathcal L sqrt g left R frac R 2 6M 2 right nbsp and has been used to explain inflation in the form of Starobinsky inflation Here M displaystyle M nbsp is a constant Gauss Bonnet edit Gauss Bonnet gravity has the action L g R R 2 4 R m n R m n R m n r s R m n r s displaystyle mathcal L sqrt g left R R 2 4R mu nu R mu nu R mu nu rho sigma R mu nu rho sigma right nbsp where the coefficients of the extra terms are chosen so that the action reduces to general relativity in 4 spacetime dimensions and the extra terms are only non trivial when more dimensions are introduced Stelle s 4th derivative gravity edit Stelle s 4th derivative gravity which is a generalization of Gauss Bonnet gravity has the action L g R f 1 R 2 f 2 R m n R m n f 3 R m n r s R m n r s displaystyle mathcal L sqrt g left R f 1 R 2 f 2 R mu nu R mu nu f 3 R mu nu rho sigma R mu nu rho sigma right nbsp f R edit f R gravity has the action L g f R displaystyle mathcal L sqrt g f R nbsp and is a family of theories each defined by a different function of the Ricci scalar Starobinsky gravity is actually an f R displaystyle f R nbsp theory Infinite derivative gravity edit Infinite derivative gravity is a covariant theory of gravity quadratic in curvature torsion free and parity invariant 57 L g M p 2 R R f 1 M s 2 R R m n f 2 M s 2 R m n R m n r s f 3 M s 2 R m n r s displaystyle mathcal L sqrt g left M p 2 R Rf 1 left frac Box M s 2 right R R mu nu f 2 left frac Box M s 2 right R mu nu R mu nu rho sigma f 3 left frac Box M s 2 right R mu nu rho sigma right nbsp and 2 f 1 M s 2 f 2 M s 2 2 f 3 M s 2 0 displaystyle 2f 1 left frac Box M s 2 right f 2 left frac Box M s 2 right 2f 3 left frac Box M s 2 right 0 nbsp in order to make sure that only massless spin 2 and spin 0 components propagate in the graviton propagator around Minkowski background The action becomes non local beyond the scale M s displaystyle M s nbsp and recovers to general relativity in the infrared for energies below the non local scale M s displaystyle M s nbsp In the ultraviolet regime at distances and time scales below non local scale M s 1 displaystyle M s 1 nbsp the gravitational interaction weakens enough to resolve point like singularity which means Schwarzschild s singularity can be potentially resolved in infinite derivative theories of gravity Lovelock edit Lovelock gravity has the action L g a 0 a 1 R a 2 R 2 R a b m n R a b m n 4 R m n R m n a 3 O R 3 displaystyle mathcal L sqrt g alpha 0 alpha 1 R alpha 2 left R 2 R alpha beta mu nu R alpha beta mu nu 4R mu nu R mu nu right alpha 3 mathcal O R 3 nbsp and can be thought of as a generalization of general relativity Scalar tensor theories edit See also Scalar tensor theory Brans Dicke theory Dilaton Chameleon particle Pressuron and Horndeski s theory These all contain at least one free parameter as opposed to general relativity which has no free parameters Although not normally considered a Scalar Tensor theory of gravity the 5 by 5 metric of Kaluza Klein reduces to a 4 by 4 metric and a single scalar So if the 5th element is treated as a scalar gravitational field instead of an electromagnetic field then Kaluza Klein can be considered the progenitor of Scalar Tensor theories of gravity This was recognized by Thiry 20 Scalar Tensor theories include Thiry 20 Jordan 24 Brans and Dicke 9 Bergman 36 Nordtveldt 1970 Wagoner 39 Bekenstein 47 and Barker 48 The action S displaystyle S nbsp is based on the integral of the Lagrangian L f displaystyle L varphi nbsp S 1 16 p G d 4 x g L f S m displaystyle S 1 over 16 pi G int d 4 x sqrt g L varphi S m nbsp L f f R w f f g m n m f n f 2 f l f displaystyle L varphi varphi R omega varphi over varphi g mu nu partial mu varphi partial nu varphi 2 varphi lambda varphi nbsp S m d 4 x g G N L m displaystyle S m int d 4 x sqrt g G N L m nbsp T m n d e f 2 g d S m d g m n displaystyle T mu nu stackrel mathrm def 2 over sqrt g delta S m over delta g mu nu nbsp where w f displaystyle omega varphi nbsp is a different dimensionless function for each different scalar tensor theory The function l f displaystyle lambda varphi nbsp plays the same role as the cosmological constant in general relativity G N displaystyle G N nbsp is a dimensionless normalization constant that fixes the present day value of G displaystyle G nbsp An arbitrary potential can be added for the scalar The full version is retained in Bergman 36 and Wagoner 39 Special cases are Nordtvedt 38 l 0 displaystyle lambda 0 nbsp Since l displaystyle lambda nbsp was thought to be zero at the time anyway this would not have been considered a significant difference The role of the cosmological constant in more modern work is discussed under Cosmological constant Brans Dicke 9 w displaystyle omega nbsp is constantBekenstein 47 variable mass theory Starting with parameters r displaystyle r nbsp and q displaystyle q nbsp found from a cosmological solution f 1 q f f f f r displaystyle varphi 1 qf varphi f varphi r nbsp determines function f displaystyle f nbsp then w f 3 2 1 4 f f 1 6 q q f f 1 r 1 r q f f 2 displaystyle omega varphi textstyle frac 3 2 textstyle frac 1 4 f varphi 1 6q qf varphi 1 r 1 r qf varphi 2 nbsp Barker 48 constant G theory w f 4 3 f 2 f 2 displaystyle omega varphi frac 4 3 varphi 2 varphi 2 nbsp Adjustment of w f displaystyle omega varphi nbsp allows Scalar Tensor Theories to tend to general relativity in the limit of w displaystyle omega rightarrow infty nbsp in the current epoch However there could be significant differences from general relativity in the early universe So long as general relativity is confirmed by experiment general Scalar Tensor theories including Brans Dicke 9 can never be ruled out entirely but as experiments continue to confirm general relativity more precisely and the parameters have to be fine tuned so that the predictions more closely match those of general relativity The above examples are particular cases of Horndeski s theory 58 59 the most general Lagrangian constructed out of the metric tensor and a scalar field leading to second order equations of motion in 4 dimensional space Viable theories beyond Horndeski with higher order equations of motion have been shown to exist 60 61 62 Vector tensor theories edit Before we start Will 2001 has said Many alternative metric theories developed during the 1970s and 1980s could be viewed as straw man theories invented to prove that such theories exist or to illustrate particular properties Few of these could be regarded as well motivated theories from the point of view say of field theory or particle physics Examples are the vector tensor theories studied by Will Nordtvedt and Hellings Hellings and Nordtvedt 44 and Will and Nordtvedt 43 are both vector tensor theories In addition to the metric tensor there is a timelike vector field K m displaystyle K mu nbsp The gravitational action is S 1 16 p G d 4 x g R w K m K m R h K m K n R m n ϵ F m n F m n t K m n K m n S m displaystyle S frac 1 16 pi G int d 4 x sqrt g left R omega K mu K mu R eta K mu K nu R mu nu epsilon F mu nu F mu nu tau K mu nu K mu nu right S m nbsp where w h ϵ t displaystyle omega eta epsilon tau nbsp are constants and F m n K n m K m n displaystyle F mu nu K nu mu K mu nu nbsp See Will 11 for the field equations for T m n displaystyle T mu nu nbsp and K m displaystyle K mu nbsp Will and Nordtvedt 43 is a special case where w h ϵ 0 t 1 displaystyle omega eta epsilon 0 quad tau 1 nbsp Hellings and Nordtvedt 44 is a special case where t 0 ϵ 1 h 2 w displaystyle tau 0 quad epsilon 1 quad eta 2 omega nbsp These vector tensor theories are semi conservative which means that they satisfy the laws of conservation of momentum and angular momentum but can have preferred frame effects When w h ϵ t 0 displaystyle omega eta epsilon tau 0 nbsp they reduce to general relativity so so long as general relativity is confirmed by experiment general vector tensor theories can never be ruled out Other metric theories edit Others metric theories have been proposed that of Bekenstein 63 is discussed under Modern Theories Non metric theories edit See also Einstein Cartan theory and Cartan connection Cartan s theory is particularly interesting both because it is a non metric theory and because it is so old The status of Cartan s theory is uncertain Will 11 claims that all non metric theories are eliminated by Einstein s Equivalence Principle Will 2001 tempers that by explaining experimental criteria for testing non metric theories against Einstein s Equivalence Principle Misner et al 52 claims that Cartan s theory is the only non metric theory to survive all experimental tests up to that date and Turyshev 64 lists Cartan s theory among the few that have survived all experimental tests up to that date The following is a quick sketch of Cartan s theory as restated by Trautman 65 Cartan 15 16 suggested a simple generalization of Einstein s theory of gravitation He proposed a model of space time with a metric tensor and a linear connection compatible with the metric but not necessarily symmetric The torsion tensor of the connection is related to the density of intrinsic angular momentum Independently of Cartan similar ideas were put forward by Sciama by Kibble in the years 1958 to 1966 culminating in a 1976 review by Hehl et al The original description is in terms of differential forms but for the present article that is replaced by the more familiar language of tensors risking loss of accuracy As in general relativity the Lagrangian is made up of a massless and a mass part The Lagrangian for the massless part is L 1 32 p G W n m g n 3 x h x z e 3 m h z W n m d w n m w 3 h x m w n m x n displaystyle begin aligned L amp 1 over 32 pi G Omega nu mu g nu xi x eta x zeta varepsilon xi mu eta zeta 5pt Omega nu mu amp d omega nu mu omega xi eta 5pt nabla x mu amp omega nu mu x nu end aligned nbsp The w n m displaystyle omega nu mu nbsp is the linear connection e 3 m h z displaystyle varepsilon xi mu eta zeta nbsp is the completely antisymmetric pseudo tensor Levi Civita symbol with e 0123 g displaystyle varepsilon 0123 sqrt g nbsp and g n 3 displaystyle g nu xi nbsp is the metric tensor as usual By assuming that the linear connection is metric it is possible to remove the unwanted freedom inherent in the non metric theory The stress energy tensor is calculated from T m n 1 16 p G g m n h h 3 g 3 m h h n g 3 n h h m W 3 h displaystyle T mu nu 1 over 16 pi G g mu nu eta eta xi g xi mu eta eta nu g xi nu eta eta mu Omega xi eta nbsp The space curvature is not Riemannian but on a Riemannian space time the Lagrangian would reduce to the Lagrangian of general relativity Some equations of the non metric theory of Belinfante and Swihart 26 27 have already been discussed in the section on bimetric theories A distinctively non metric theory is given by gauge theory gravity which replaces the metric in its field equations with a pair of gauge fields in flat spacetime On the one hand the theory is quite conservative because it is substantially equivalent to Einstein Cartan theory or general relativity in the limit of vanishing spin differing mostly in the nature of its global solutions On the other hand it is radical because it replaces differential geometry with geometric algebra Modern theories 1980s to present editThis section includes alternatives to general relativity published after the observations of galaxy rotation that led to the hypothesis of dark matter There is no known reliable list of comparison of these theories Those considered here include Bekenstein 63 Moffat 66 Moffat 67 Moffat 68 69 These theories are presented with a cosmological constant or added scalar or vector potential Motivations edit Motivations for the more recent alternatives to general relativity are almost all cosmological associated with or replacing such constructs as inflation dark matter and dark energy The basic idea is that gravity agrees with general relativity at the present epoch but may have been quite different in the early universe In the 1980s there was a slowly dawning realisation in the physics world that there were several problems inherent in the then current big bang scenario including the horizon problem and the observation that at early times when quarks were first forming there was not enough space on the universe to contain even one quark Inflation theory was developed to overcome these difficulties Another alternative was constructing an alternative to general relativity in which the speed of light was higher in the early universe The discovery of unexpected rotation curves for galaxies took everyone by surprise Could there be more mass in the universe than we are aware of or is the theory of gravity itself wrong The consensus now is that the missing mass is cold dark matter but that consensus was only reached after trying alternatives to general relativity and some physicists still believe that alternative models of gravity may hold the answer In the 1990s supernova surveys discovered the accelerated expansion of the universe now usually attributed to dark energy This led to the rapid reinstatement of Einstein s cosmological constant and quintessence arrived as an alternative to the cosmological constant At least one new alternative to general relativity attempted to explain the supernova surveys results in a completely different way The measurement of the speed of gravity with the gravitational wave event GW170817 ruled out many alternative theories of gravity as explanations for the accelerated expansion 70 71 72 Another observation that sparked recent interest in alternatives to General Relativity is the Pioneer anomaly It was quickly discovered that alternatives to general relativity could explain this anomaly This is now believed to be accounted for by non uniform thermal radiation Cosmological constant and quintessence edit See also Cosmological constant Einstein Hilbert action and Quintessence physics The cosmological constant L displaystyle Lambda nbsp is a very old idea going back to Einstein in 1917 5 The success of the Friedmann model of the universe in which L 0 displaystyle Lambda 0 nbsp led to the general acceptance that it is zero but the use of a non zero value came back with a vengeance when data from supernovae indicated that the expansion of the universe is acceleratingFirst let s see how it influences the equations of Newtonian gravity and General Relativity In Newtonian gravity the addition of the cosmological constant changes the Newton Poisson equation from 2 f 4 p r G displaystyle nabla 2 varphi 4 pi rho G nbsp to 2 f 1 2 L c 2 4 p r G displaystyle nabla 2 varphi frac 1 2 Lambda c 2 4 pi rho G nbsp In general relativity it changes the Einstein Hilbert action from S 1 16 p G R g d 4 x S m displaystyle S 1 over 16 pi G int R sqrt g d 4 x S m nbsp to S 1 16 p G R 2 L g d 4 x S m displaystyle S 1 over 16 pi G int R 2 Lambda sqrt g d 4 x S m nbsp which changes the field equation T m n 1 8 p G R m n 1 2 g m n R displaystyle T mu nu 1 over 8 pi G left R mu nu frac 1 2 g mu nu R right nbsp to T m n 1 8 p G R m n 1 2 g m n R g m n L displaystyle T mu nu 1 over 8 pi G left R mu nu frac 1 2 g mu nu R g mu nu Lambda right nbsp In alternative theories of gravity a cosmological constant can be added to the action in exactly the same way The cosmological constant is not the only way to get an accelerated expansion of the universe in alternatives to general relativity We ve already seen how the scalar potential l f displaystyle lambda varphi nbsp can be added to scalar tensor theories This can also be done in every alternative the general relativity that contains a scalar field f displaystyle varphi nbsp by adding the term l f displaystyle lambda varphi nbsp inside the Lagrangian for the gravitational part of the action the L f displaystyle L varphi nbsp part of S 1 16 p G d 4 x g L f S m displaystyle S 1 over 16 pi G int d 4 x sqrt g L varphi S m nbsp Because l f displaystyle lambda varphi nbsp is an arbitrary function of the scalar field it can be set to give an acceleration that is large in the early universe and small at the present epoch This is known as quintessence A similar method can be used in alternatives to general relativity that use vector fields including Rastall 49 and vector tensor theories A term proportional to K m K n g m n displaystyle K mu K nu g mu nu nbsp is added to the Lagrangian for the gravitational part of the action Farnes theories edit In December 2018 the astrophysicist Jamie Farnes from the University of Oxford proposed a dark fluid theory related to notions of gravitationally repulsive negative masses that were presented earlier by Albert Einstein The theory may help to better understand the considerable amounts of unknown dark matter and dark energy in the universe 73 The theory relies on the concept of negative mass and reintroduces Fred Hoyle s creation tensor in order to allow matter creation for only negative mass particles In this way the negative mass particles surround galaxies and apply a pressure onto them thereby resembling dark matter As these hypothesised particles mutually repel one another they push apart the Universe thereby resembling dark energy The creation of matter allows the density of the exotic negative mass particles to remain constant as a function of time and so appears like a cosmological constant Einstein s field equations are modified to R m n 1 2 R g m n 8 p G c 4 T m n T m n C m n displaystyle R mu nu frac 1 2 Rg mu nu frac 8 pi G c 4 left T mu nu T mu nu C mu nu right nbsp According to Occam s razor Farnes theory is a simpler alternative to the conventional LambdaCDM model as both dark energy and dark matter two hypotheses are solved using a single negative mass fluid one hypothesis The theory will be directly testable using the world s largest radio telescope the Square Kilometre Array which should come online in 2022 74 Relativistic MOND edit Main articles Modified Newtonian dynamics Tensor vector scalar gravity and Extended theories of gravity The original theory of MOND by Milgrom was developed in 1983 as an alternative to dark matter Departures from Newton s law of gravitation are governed by an acceleration scale not a distance scale MOND successfully explains the Tully Fisher observation that the luminosity of a galaxy should scale as the fourth power of the rotation speed It also explains why the rotation discrepancy in dwarf galaxies is particularly large There were several problems with MOND in the beginning It did not include relativistic effects It violated the conservation of energy momentum and angular momentum It was inconsistent in that it gives different galactic orbits for gas and for stars It did not state how to calculate gravitational lensing from galaxy clusters By 1984 problems 2 and 3 had been solved by introducing a Lagrangian AQUAL A relativistic version of this based on scalar tensor theory was rejected because it allowed waves in the scalar field to propagate faster than light The Lagrangian of the non relativistic form is L a 0 2 8 p G f f 2 a 0 2 r f displaystyle L a 0 2 over 8 pi G f left lbrack frac nabla varphi 2 a 0 2 right rbrack rho varphi nbsp The relativistic version of this has L a 0 2 8 p G f ℓ 0 2 g m n m f n f displaystyle L a 0 2 over 8 pi G tilde f left ell 0 2 g mu nu partial mu varphi partial nu varphi right nbsp with a nonstandard mass action Here f displaystyle f nbsp and f displaystyle tilde f nbsp are arbitrary functions selected to give Newtonian and MOND behaviour in the correct limits and l 0 c 2 a 0 displaystyle l 0 c 2 a 0 nbsp is the MOND length scale By 1988 a second scalar field PCC fixed problems with the earlier scalar tensor version but is in conflict with the perihelion precession of Mercury and gravitational lensing by galaxies and clusters By 1997 MOND had been successfully incorporated in a stratified relativistic theory Sanders but as this is a preferred frame theory it has problems of its own Bekenstein 63 introduced a tensor vector scalar model TeVeS This has two scalar fields f displaystyle varphi nbsp and s displaystyle sigma nbsp and vector field U a displaystyle U alpha nbsp The action is split into parts for gravity scalars vector and mass S S g S s S v S m displaystyle S S g S s S v S m nbsp The gravity part is the same as in general relativity S s 1 2 s 2 h a b f a f b 1 2 G ℓ 0 2 s 4 F k G s 2 g d 4 x S v K 32 p G g a b g m n U a m U b n 2 l K g m n U m U n 1 g d 4 x S m L g m n f a f m a g d 4 x displaystyle begin aligned S s amp frac 1 2 int left sigma 2 h alpha beta varphi alpha varphi beta frac 1 2 G ell 0 2 sigma 4 F kG sigma 2 right sqrt g d 4 x 5pt S v amp frac K 32 pi G int left g alpha beta g mu nu U alpha mu U beta nu frac 2 lambda K left g mu nu U mu U nu 1 right right sqrt g d 4 x 5pt S m amp int L left tilde g mu nu f alpha f mu alpha ldots right sqrt g d 4 x end aligned nbsp where h a b g a b U a U b displaystyle h alpha beta g alpha beta U alpha U beta nbsp g a b e 2 f g a b 2 U a U b sinh 2 f displaystyle tilde g alpha beta e 2 varphi g alpha beta 2U alpha U beta sinh 2 varphi nbsp k K displaystyle k K nbsp are constants square brackets in indices U a m displaystyle U alpha mu nbsp represent anti symmetrization l displaystyle lambda nbsp is a Lagrange multiplier calculated elsewhere and L is a Lagrangian translated from flat spacetime onto the metric g a b displaystyle tilde g alpha beta nbsp Note that G need not equal the observed gravitational constant G N e w t o n displaystyle G Newton nbsp F is an arbitrary function and F m 3 4 m 2 m 2 2 1 m displaystyle F mu frac 3 4 mu 2 mu 2 2 over 1 mu nbsp is given as an example with the right asymptotic behaviour note how it becomes undefined when m 1 displaystyle mu 1 nbsp The Parametric post Newtonian parameters of this theory are calculated in 75 which shows that all its parameters are equal to general relativity s except for a 1 4 G K 2 K 1 e 4 f 0 e 4 f 0 8 8 a 2 6 G 2 K 2 G K 4 e 4 f 0 2 K 2 1 displaystyle begin aligned alpha 1 amp frac 4G K left 2K 1 e 4 varphi 0 e 4 varphi 0 8 right 8 5pt alpha 2 amp frac 6G 2 K frac 2G K 4 e 4 varphi 0 2 K 2 1 end aligned nbsp both of which expressed in geometric units where c G N e w t o n i a n 1 displaystyle c G Newtonian 1 nbsp so G 1 2 2 K k 4 p displaystyle G 1 frac 2 2 K frac k 4 pi nbsp Moffat s theories edit J W Moffat 66 developed a non symmetric gravitation theory This is not a metric theory It was first claimed that it does not contain a black hole horizon but Burko and Ori 76 have found that nonsymmetric gravitational theory can contain black holes Later Moffat claimed that it has also been applied to explain rotation curves of galaxies without invoking dark matter Damour Deser amp MaCarthy 77 have criticised nonsymmetric gravitational theory saying that it has unacceptable asymptotic behaviour The mathematics is not difficult but is intertwined so the following is only a brief sketch Starting with a non symmetric tensor g m n displaystyle g mu nu nbsp the Lagrangian density is split into L L R L M displaystyle L L R L M nbsp where L M displaystyle L M nbsp is the same as for matter in general relativity L R g R W 2 l 1 4 m 2 g m n g m n 1 6 g m n W m W n displaystyle L R sqrt g left R W 2 lambda frac 1 4 mu 2 g mu nu g mu nu right frac 1 6 g mu nu W mu W nu nbsp where R W displaystyle R W nbsp is a curvature term analogous to but not equal to the Ricci curvature in general relativity l displaystyle lambda nbsp and m 2 displaystyle mu 2 nbsp are cosmological constants g n m displaystyle g nu mu nbsp is the antisymmetric part of g n m displaystyle g nu mu nbsp W m displaystyle W mu nbsp is a connection and is a bit difficult to explain because it s defined recursively However W m 2 g m n n displaystyle W mu approx 2g mu nu nu nbsp Haugan and Kauffmann 78 used polarization measurements of the light emitted by galaxies to impose sharp constraints on the magnitude of some of nonsymmetric gravitational theory s parameters They also used Hughes Drever experiments to constrain the remaining degrees of freedom Their constraint is eight orders of magnitude sharper than previous estimates Moffat s 68 metric skew tensor gravity MSTG theory is able to predict rotation curves for galaxies without either dark matter or MOND and claims that it can also explain gravitational lensing of galaxy clusters without dark matter It has variable G displaystyle G nbsp increasing to a final constant value about a million years after the big bang The theory seems to contain an asymmetric tensor A m n displaystyle A mu nu nbsp field and a source current J m displaystyle J mu nbsp vector The action is split into S S G S F S F M S M displaystyle S S G S F S FM S M nbsp Both the gravity and mass terms match those of general relativity with cosmological constant The skew field action and the skew field matter coupling are S F d 4 x g 1 12 F m n r F m n r 1 4 m 2 A m n A m n displaystyle S F int d 4 x sqrt g left frac 1 12 F mu nu rho F mu nu rho frac 1 4 mu 2 A mu nu A mu nu right nbsp S F M d 4 x ϵ a b m n A a b m J n displaystyle S FM int d 4 x epsilon alpha beta mu nu A alpha beta partial mu J nu nbsp where F m n r m A n r r A m n displaystyle F mu nu rho partial mu A nu rho partial rho A mu nu nbsp and ϵ a b m n displaystyle epsilon alpha beta mu nu nbsp is the Levi Civita symbol The skew field coupling is a Pauli coupling and is gauge invariant for any source current The source current looks like a matter fermion field associated with baryon and lepton number Scalar tensor vector gravity edit Main article Scalar tensor vector gravity Moffat s Scalar tensor vector gravity 69 contains a tensor vector and three scalar fields But the equations are quite straightforward The action is split into S S G S K S S S M displaystyle S S G S K S S S M nbsp with terms for gravity vector field K m displaystyle K mu nbsp scalar fields G w m displaystyle G omega mu nbsp and mass S G displaystyle S G nbsp is the standard gravity term with the exception that G displaystyle G nbsp is moved inside the integral S K d 4 x g w 1 4 B m n B m n V K where B m n m K n n K m displaystyle S K int d 4 x sqrt g omega left frac 1 4 B mu nu B mu nu V K right qquad text where quad B mu nu partial mu K nu partial nu K mu nbsp S S d 4 x g 1 G 3 1 2 g m n m G n G V G 1 G 1 2 g m n m w n w V w 1 m 2 G 1 2 g m n m m n m V m displaystyle S S int d 4 x sqrt g frac 1 G 3 left frac 1 2 g mu nu nabla mu G nabla nu G V G right frac 1 G left frac 1 2 g mu nu nabla mu omega nabla nu omega V omega right 1 over mu 2 G left frac 1 2 g mu nu nabla mu mu nabla nu mu V mu right nbsp The potential function for the vector field is chosen to be V K 1 2 m 2 f m f m 1 4 g f m f m 2 displaystyle V K frac 1 2 mu 2 varphi mu varphi mu frac 1 4 g left varphi mu varphi mu right 2 nbsp where g displaystyle g nbsp is a coupling constant The functions assumed for the scalar potentials are not stated Infinite derivative gravity edit Main article Infinite derivative gravity In order to remove ghosts in the modified propagator as well as to obtain asymptotic freedom Biswas Mazumdar and Siegel 2005 considered a string inspired infinite set of higher derivative terms S d 4 x g R 2 R F R displaystyle S int mathrm d 4 x sqrt g left frac R 2 RF Box R right nbsp where F displaystyle F Box nbsp is the exponential of an entire function of the D Alembertian operator 79 80 This avoids a black hole singularity near the origin while recovering the 1 r fall of the general relativity potential at large distances 81 Lousto and Mazzitelli 1997 found an exact solution to this theories representing a gravitational shock wave 82 General relativity self interaction GRSI edit The General Relativity Self interaction or GRSI model 83 is an attempt to explain astrophysical and cosmological observations without dark matter dark energy by adding self interaction terms when calculating the gravitational effects in general relativity analogous to the self interaction terms in quantum chromodynamics 84 Additionally the model explains the Tully Fisher relation 85 the radial acceleration relation 86 observations that are currently challenging to understand within Lambda CDM Testing of alternatives to general relativity editMain article Tests of general relativity Any putative alternative to general relativity would need to meet a variety of tests for it to become accepted For in depth coverage of these tests see Misner et al 52 Ch 39 Will 11 Table 2 1 and Ni 12 Most such tests can be categorized as in the following subsections Self consistency edit Self consistency among non metric theories includes eliminating theories allowing tachyons ghost poles and higher order poles and those that have problems with behaviour at infinity Among metric theories self consistency is best illustrated by describing several theories that fail this test The classic example is the spin two field theory of Fierz and Pauli 17 the field equations imply that gravitating bodies move in straight lines whereas the equations of motion insist that gravity deflects bodies away from straight line motion Yilmaz 1971 29 contains a tensor gravitational field used to construct a metric it is mathematically inconsistent because the functional dependence of the metric on the tensor field is not well defined Completeness edit To be complete a theory of gravity must be capable of analysing the outcome of every experiment of interest It must therefore mesh with electromagnetism and all other physics For instance any theory that cannot predict from first principles the movement of planets or the behaviour of atomic clocks is incomplete Many early theories are incomplete in that it is unclear whether the density r displaystyle rho nbsp used by the theory should be calculated from the stress energy tensor T displaystyle T nbsp as r T m n u m u n displaystyle rho T mu nu u mu u nu nbsp or as r T m n d m n displaystyle rho T mu nu delta mu nu nbsp where u displaystyle u nbsp is the four velocity and d displaystyle delta nbsp is the Kronecker delta The theories of Thirry 1948 and Jordan 24 are incomplete unless Jordan s parameter h displaystyle eta nbsp is set to 1 in which case they match the theory of Brans Dicke 9 and so are worthy of further consideration Milne 19 is incomplete because it makes no gravitational red shift prediction The theories of Whitrow and Morduch 30 31 Kustaanheimo 32 and Kustaanheimo and Nuotio 33 are either incomplete or inconsistent The incorporation of Maxwell s equations is incomplete unless it is assumed that they are imposed on the flat background space time and when that is done they are inconsistent because they predict zero gravitational redshift when the wave version of light Maxwell theory is used and nonzero redshift when the particle version photon is used Another more obvious example is Newtonian gravity with Maxwell s equations light as photons is deflected by gravitational fields by half that of general relativity but light as waves is not Classical tests edit Main article Tests of general relativity There are three classical tests dating back to the 1910s or earlier of the ability of gravity theories to handle relativistic effects they are gravitational redshift gravitational lensing generally tested around the Sun and anomalous perihelion advance of the planets Each theory should reproduce the observed results in these areas which have to date always aligned with the predictions of general relativity In 1964 Irwin I Shapiro found a fourth test called the Shapiro delay It is usually regarded as a classical test as well Agreement with Newtonian mechanics and special relativity edit As an example of disagreement with Newtonian experiments Birkhoff 18 theory predicts relativistic effects fairly reliably but demands that sound waves travel at the speed of light This was the consequence of an assumption made to simplify handling the collision of masses citation needed The Einstein equivalence principle edit Main article Equivalence principle Einstein s Equivalence Principle has three components The first is the uniqueness of free fall also known as the Weak Equivalence Principle This is satisfied if inertial mass is equal to gravitational mass h is a parameter used to test the maximum allowable violation of the Weak Equivalence Principle The first tests of the Weak Equivalence Principle were done by Eotvos before 1900 and limited h to less than 5 10 9 Modern tests have reduced that to less than 5 10 13 The second is Lorentz invariance In the absence of gravitational effects the speed of light is constant The test parameter for this is d The first tests of Lorentz invariance were done by Michelson and Morley before 1890 and limited d to less than 5 10 3 Modern tests have reduced this to less than 1 10 21 The third is local position invariance which includes spatial and temporal invariance The outcome of any local non gravitational experiment is independent of where and when it is performed Spatial local position invariance is tested using gravitational redshift measurements The test parameter for this is a Upper limits on this found by Pound and Rebka in 1960 limited a to less than 0 1 Modern tests have reduced this to less than 1 10 4 2 Schiff s conjecture states that any complete self consistent theory of gravity that embodies the Weak Equivalence Principle necessarily embodies Einstein s Equivalence Principle This is likely to be true if the theory has full energy conservation Metric theories satisfy the Einstein Equivalence Principle Extremely few non metric theories satisfy this For example the non metric theory of Belinfante amp Swihart 26 27 is eliminated by the THem formalism for testing Einstein s Equivalence Principle Gauge theory gravity is a notable exception where the strong equivalence principle is essentially the minimal coupling of the gauge covariant derivative Parametric post Newtonian formalism edit Main article Parameterized post Newtonian formalism See also Tests of general relativity Misner et al 52 and Will 11 for more information Work on developing a standardized rather than ad hoc set of tests for evaluating alternative gravitation models began with Eddington in 1922 and resulted in a standard set of Parametric post Newtonian numbers in Nordtvedt and Will 87 and Will and Nordtvedt 43 Each parameter measures a different aspect of how much a theory departs from Newtonian gravity Because we are talking about deviation from Newtonian theory here these only measure weak field effects The effects of strong gravitational fields are examined later These ten are g b h a 1 a 2 a 3 z 1 z 2 z 3 z 4 displaystyle gamma beta eta alpha 1 alpha 2 alpha 3 zeta 1 zeta 2 zeta 3 zeta 4 nbsp g displaystyle gamma nbsp is a measure of space curvature being zero for Newtonian gravity and one for general relativity b displaystyle beta nbsp is a measure of nonlinearity in the addition of gravitational fields one for general relativity h displaystyle eta nbsp is a check for preferred location effects a 1 a 2 a 3 displaystyle alpha 1 alpha 2 alpha 3 nbsp measure the extent and nature of preferred frame effects Any theory of gravity in which at least one of the three is nonzero is called a preferred frame theory z 1 z 2 z 3 z 4 a 3 displaystyle zeta 1 zeta 2 zeta 3 zeta 4 alpha 3 nbsp measure the extent and nature of breakdowns in global conservation laws A theory of gravity possesses 4 conservation laws for energy momentum and 6 for angular momentum only if all five are zero Strong gravity and gravitational waves edit Main article Tests of general relativity Parametric post Newtonian is only a measure of weak field effects Strong gravity effects can be seen in compact objects such as white dwarfs neutron stars and black holes Experimental tests such as the stability of white dwarfs spin down rate of pulsars orbits of binary pulsars and the existence of a black hole horizon can be used as tests of alternative to general relativity General relativity predicts that gravitational waves travel at the speed of light Many alternatives to general relativity say that gravitational waves travel faster than light possibly breaking causality After the multi messaging detection of the GW170817 coalescence of neutron stars where light and gravitational waves were measured to travel at the same speed with an error of 1 1015 many of those modified theories of gravity were excluded Cosmological tests edit Useful cosmological scale tests are just beginning to become available 2 88 Given the limited astronomical data and the complexity of the theories comparisons involve complex parameters For example Reyes et al 88 analyzed 70 205 luminous red galaxies with a cross correlation involving galaxy velocity estimates and gravitational potentials estimated from lensing and yet results are still tentative 1 164 For those theories that aim to replace dark matter observations like the galaxy rotation curve the Tully Fisher relation the faster rotation rate of dwarf galaxies and the gravitational lensing due to galactic clusters act as constraints For those theories that aim to replace inflation the size of ripples in the spectrum of the cosmic microwave background radiation is the strictest test For those theories that incorporate or aim to replace dark energy the supernova brightness results and the age of the universe can be used as tests Another test is the flatness of the universe With general relativity the combination of baryonic matter dark matter and dark energy add up to make the universe exactly flat Results of testing theories editParametric post Newtonian parameters for a range of theories edit See Will 11 and Ni 12 for more details Misner et al 52 gives a table for translating parameters from the notation of Ni to that of Will General Relativity is now more than 100 years old during which one alternative theory of gravity after another has failed to agree with ever more accurate observations One illustrative example is Parameterized post Newtonian formalism The following table lists Parametric post Newtonian values for a large number of theories If the value in a cell matches that in the column heading then the full formula is too complicated to include here g displaystyle gamma nbsp b displaystyle beta nbsp 3 displaystyle xi nbsp a 1 displaystyle alpha 1 nbsp a 2 displaystyle alpha 2 nbsp a 3 displaystyle alpha 3 nbsp z 1 displaystyle zeta 1 nbsp z 2 displaystyle zeta 2 nbsp z 3 displaystyle zeta 3 nbsp z 4 displaystyle zeta 4 nbsp Einstein general relativity 4 1 1 0 0 0 0 0 0 0 0 Scalar tensor theories Bergmann 36 Wagoner 39 1 w 2 w displaystyle textstyle frac 1 omega 2 omega nbsp b displaystyle beta nbsp 0 0 0 0 0 0 0 0 Nordtvedt 38 Bekenstein 47 1 w 2 w displaystyle textstyle frac 1 omega 2 omega nbsp b displaystyle beta nbsp 0 0 0 0 0 0 0 0 Brans Dicke 9 1 w 2 w displaystyle textstyle frac 1 omega 2 omega nbsp 1 0 0 0 0 0 0 0 0 Vector tensor theories Hellings Nordtvedt 44 g displaystyle gamma nbsp b displaystyle beta nbsp 0 a 1 displaystyle alpha 1 nbsp a 2 displaystyle alpha 2 nbsp 0 0 0 0 0 Will Nordtvedt 43 1 1 0 0 a 2 displaystyle alpha 2 nbsp 0 0 0 0 0 Bimetric theories Rosen 41 1 1 0 0 c 0 c 1 1 displaystyle c 0 c 1 1 nbsp 0 0 0 0 0 Rastall 49 1 1 0 0 a 2 displaystyle alpha 2 nbsp 0 0 0 0 0 Lightman Lee 45 g displaystyle gamma nbsp b displaystyle beta nbsp 0 span, wikipedia, wiki, book, books, library, article , read, download, free, free download, mp3, video, mp4, 3gp, jpg, jpeg, gif, png, picture, music, song, movie, book, game, games. © Copyright 2021, All Rights Reserved  |   www.wiki3.en-us.nina.az Rss Feed SiteMap Ping DMCA Contact Us HomePage Closed Call: