Alternatives to general relativity

From Wikipedia, the free encyclopedia

Alternatives to general relativity are physical theories that attempt to describe the phenomenon of gravitation in competition to Einstein's theory of general relativity.

There have been many different attempts at constructing an ideal theory of gravity. These attempts can be split into four broad categories:

• Straightforward alternatives to general relativity (GR), such as the Cartan, Brans–Dicke and Rosen bimetric theories.
• Those that attempt to construct a quantized gravity theory such as loop quantum gravity.
• Those that attempt to unify gravity and other forces such as Kaluza–Klein.
• Those that attempt to do several at once, such as M-theory.

This article deals only with straightforward alternatives to GR. For quantized gravity theories, see the article quantum gravity. For the unification of gravity and other forces, see the article classical unified field theories.

Motivations

Motivations for developing new theories of gravity have changed over the years, with the first one to explain planetary orbits (Newton) and more complicated orbits (e.g. Lagrange). Then came unsuccessful attempts to combine gravity and either wave or corpuscular theories of gravity. The whole landscape of physics was changed with the discovery of Lorentz transformations, and this led to attempts to reconcile it with gravity. At the same time, experimental physicists started testing the foundations of gravity and relativity – Lorentz invariance, the gravitational deflection of light, the Eötvös experiment. These considerations led to and past the development of general relativity.


After that, motivations differ. Two major concerns were the development of quantum theory and the discovery of the strong and weak nuclear forces. Attempts to quantize and unify gravity are outside the scope of this article, and so far none has been completely successful.


After general relativity (GR), attempts were made either to improve on theories developed before GR, or to improve GR itself. Many different strategies were attempted, for example the addition of spin to GR, combining a GR-like metric with a space-time 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 GR that is completely free from singularities.
Experimental tests improved along with the theories. Many of the different strategies that were developed soon after GR were abandoned, and there was a push to develop more general forms of the theories that survived, so that a theory would be ready the moment any test showed a disagreement with GR.


By the 1980s, the increasing accuracy of experimental tests had all led to confirmation of GR, no competitors were left except for those that included GR 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 on the scale of 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

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

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 only 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. 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 (1922), 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 GR
  • 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 theorem gravity
• Scalar-tensor theories
  • Bekenstein
  • Bergmann-Wagoner
  • Brans–Dicke theory (the most well-known alternative to GR, 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 (1922)), and many mention it in passing (e.g. Einstein–Grossmann (1913), Brans–Dicke (1961)). Mach's principle can be thought of a half-way-house between Newton and Einstein. It goes this way:

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

So far, all the experimental evidence points to Mach's principle being wrong, but it has not entirely been ruled out.

Early theories, 1686 to 1916

Newton (1686)

In Newton's (1686) theory (rewritten using more modern mathematics) the density of mass ${\displaystyle \rho \,}$ generates a scalar field, the gravitational potential ${\displaystyle \phi \,}$ in joules per kilogram, by

${\displaystyle {\partial ^{2}\phi \over \partial x^{j}\partial x^{j}}=4\pi G\rho \,.}$

Using the Nabla operator ${\displaystyle \nabla }$ for the gradient and divergence (partial derivatives), this can be conveniently written as:

${\displaystyle \nabla ^{2}\phi =4\pi G\rho \,.}$

This scalar field governs the motion of a free-falling particle by:

${\displaystyle {d^{2}x^{j} \over dt^{2}}=-{\partial \phi \over \partial x^{j}\,}.}$

At distance, r, from an isolated mass, M, the scalar field is

${\displaystyle \phi =-GM/r\,.}$

The theory of Newton, and Lagrange's improvement on the calculation (applying the variational principle), completely fails to take into account relativistic effects of course, and so can be rejected as a viable theory of gravity. Even so, Newton's theory is thought to be exactly correct in the limit of weak gravitational fields and low speeds and all other theories of gravity need to reproduce Newton's theory in the appropriate limits.

Mechanical explanations (1650–1900)

To explain Newton's theory, some mechanical explanations of gravitation (incl. Le Sage's theory) were created between 1650 and 1900, but they were overthrown because most of them lead to an unacceptable amount of drag, which is not observed. Other models are violating the energy conservation law and are incompatible with modern thermodynamics.

Electrostatic models (1870–1900)

At the end of the 19th century, many tried to combine Newton's force law with the established laws of electrodynamics, like those of Weber, Carl Friedrich Gauss, Bernhard Riemann and James Clerk Maxwell. Those models were used to explain the perihelion advance of Mercury. In 1890, Lévy succeeded in doing so by combining the laws of Weber and Riemann, whereby the speed of gravity is equal to the speed of light in his theory. And in another attempt, Paul Gerber (1898) even succeeded in deriving the correct formula for the Perihelion shift (which was identical to that formula later used by Einstein). However, because the basic laws of Weber and others were wrong (for example, Weber's law was superseded by Maxwell's theory), those hypothesis were rejected. In 1900, Hendrik Lorentz tried to explain gravity on the basis of his Lorentz ether theory and the Maxwell equations. He assumed, like Ottaviano Fabrizio Mossotti and Johann Karl Friedrich Zöllner, that the attraction of opposite charged particles is stronger than the repulsion of equal charged particles. The resulting net force is exactly what is known as universal gravitation, in which the speed of gravity is that of light. But Lorentz calculated that the value for the perihelion advance of Mercury was much too low.

Lorentz-invariant models (1905–1910)

Based on the principle of relativity, Henri Poincaré (1905, 1906), Hermann Minkowski (1908), and Arnold Sommerfeld (1910) tried to modify Newton's theory and to establish a Lorentz invariant gravitational law, in which the speed of gravity is that of light. However, as in Lorentz's model, the value for the perihelion advance of Mercury was much too low.

Einstein (1908, 1912)

Einstein's two part publication in 1912 (and before in 1908) is really only important for historical reasons. By then he knew of the gravitational redshift and the deflection of light. He had realized that Lorentz transformations are not generally applicable, but retained them. The theory states that the speed of light is constant in free space but varies in the presence of matter. The theory was only expected to hold when the source of the gravitational field is stationary. It includes the principle of least action:

${\displaystyle \delta \int d\tau =0\,}$

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

where ${\displaystyle \eta _{\mu \nu }\,}$ is the Minkowski metric, and there is a summation from 1 to 4 over indices ${\displaystyle \mu \,}$ and ${\displaystyle \nu \,}$.
Einstein and Grossmann (1913) includes Riemannian geometry and tensor calculus.

${\displaystyle \delta \int d\tau =0\,}$

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

The equations of electrodynamics exactly match those of GR. The equation

${\displaystyle T^{\mu \nu }=\rho {dx^{\mu } \over d\tau }{dx^{\nu } \over d\tau }\,}$

is not in GR. It expresses the stress–energy tensor as a function of the matter density.

Abraham (1912)

While this was going on, Abraham was developing an alternative model of gravity in which the speed of light depends on the gravitational field strength and so is variable almost everywhere. Abraham's 1914 review of gravitation models is said to be excellent, but his own model was poor.

Nordström (1912)

The first approach of Nordström (1912) was to retain the Minkowski metric and a constant value of ${\displaystyle c\,}$ but to let mass depend on the gravitational field strength ${\displaystyle \phi \,}$. Allowing this field strength to satisfy

${\displaystyle \Box \phi =\rho \,}$

where ${\displaystyle \rho \,}$ is rest mass energy and ${\displaystyle \Box \,}$ is the d'Alembertian,

${\displaystyle m=m_{0}\exp(\phi /c^{2})\,}$

and

${\displaystyle -{\partial \phi \over \partial x^{\mu }}={\dot {u}}_{\mu }+{u_{\mu } \over c^{2}{\dot {\phi }}}\,}$

where ${\displaystyle u\,}$ is the four-velocity and the dot is a differential with respect to time.

The second approach of Nordström (1913) is remembered as the first logically consistent relativistic field theory of gravitation ever formulated. From (note, notation of Pais (1982) not Nordström):

${\displaystyle \delta \int \psi d\tau =0\,}$

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

where ${\displaystyle \psi \,}$ is a scalar field,

${\displaystyle -{\partial T^{\mu \nu } \over \partial x^{\nu }}=T{1 \over \psi }{\partial \psi \over \partial x_{\mu }}\,}$

This theory is Lorentz invariant, satisfies the conservation laws, correctly reduces to the Newtonian limit and satisfies the weak equivalence principle.

Einstein and Fokker (1914)

This theory is Einstein's first treatment of gravitation in which general covariance is strictly obeyed. Writing:

${\displaystyle \delta \int ds=0\,}$

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

${\displaystyle g_{\mu \nu }=\psi ^{2}\eta _{\mu \nu }\,}$

they relate Einstein-Grossmann (1913) to Nordström (1913). They also state:

${\displaystyle T\,\propto \,R\,.}$

That is, the trace of the stress energy tensor is proportional to the curvature of space.

Einstein (1916, 1917)

This theory 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. Hilbert was the first to correctly state the Einstein–Hilbert action for GR, 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.

GR 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. Later in this article you will see scalar-tensor theories that contain a scalar field in addition to the tensors of GR, and other variants containing vector fields as well have been developed recently.

Theories from 1917 to the 1980s

This section includes alternatives to GR published after GR but before the observations of galaxy rotation that led to the hypothesis of "dark matter". Those considered here include (see Will (1981), Lang (2002)):

Theories from 1917 to the 1980s.

Publication year(s)	Author(s)	Theory type
1922	Whitehead	Quasilinear
1922, 1923	Cartan	Non-metric
1939	Fierz and Pauli
1943	Birkhov
1948	Milne
1948	Thiry
1954	Papapetrou	Scalar field
1953	Littlewood	Scalar field
1955	Jordan
1956	Bergman	Scalar field
1957	Belinfante and Swihart
1958, 1973	Yilmaz
1961	Brans & Dicke	Scalar-tensor
1960, 1965	Whitrow & Morduch	Scalar field
1966	Kustaanheimo
1967	Kustaanheimo and Nuotio
1968	Deser and Laurent	Quasilinear
1968	Page and Tupper	Scalar field
1968	Bergmann	Scalar-tensor
1970	Bollini-Giambiagi-Tiomno	Quasilinear
1970	Nordtveldt
1970	Wagoner	Scalar-tensor
1971	Rosen	Scalar field
1975	Rosen	Bimetric
1972, 1973	Wei-Tou Ni	Scalar field
1972	Will and Nordtveldt	Vector-tensor
1973	Hellings and Nordtveldt	Vector-tensor
1973	Lightman and Lee	Scalar field
1974	Lee, Lightman and Ni
1977	Bekenstein	Scalar-tensor
1978	Barker	Scalar-tensor
1979	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 recognised 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.

Scalar field theories

The scalar field theories of Nordström (1912, 1913) have already been discussed. Those of Littlewood (1953), Bergman (1956), Yilmaz (1958), Whitrow and Morduch (1960, 1965) and Page and Tupper (1968) follow the general formula give by Page and Tupper.

According to Page and Tupper (1968), who discuss all these except Nordström (1913), the general scalar field theory comes from the principle of least action:

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

where the scalar field is,

${\displaystyle \phi =GM/r}$

and c may or may not depend on ${\displaystyle \phi }$.
In Nordström (1912),

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

In Littlewood (1953) and Bergmann (1956),

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

In Whitrow and Morduch (1960),

${\displaystyle f(\phi /c^{2})=1,\qquad c^{2}=c_{\infty }^{2}-2\phi \,}$

In Whitrow and Morduch (1965),

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

In Page and Tupper (1968),

${\displaystyle f(\phi /c^{2})=\phi /c^{2}+\alpha (\phi /c^{2})^{2},\qquad c_{\infty }^{2}/c^{2}=1+4(\phi /c_{\infty }^{2})+(15+2\alpha )(\phi /c_{\infty }^{2})^{2}}$

Page and Tupper (1968) matches Yilmaz (1958) (see also Yilmaz theory of gravitation) 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 (1972) summarised 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:

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

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

Misner et al. (1973) gives this without the ${\displaystyle \phi R}$ term. ${\displaystyle S_{m}}$ is the matter action.

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

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 (1972) there are two arbitrary functions ${\displaystyle f(\phi )}$ and ${\displaystyle k(\phi )}$ that are related to the metric by:

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

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

Ni (1972) quotes Rosen (1971) as having two scalar fields ${\displaystyle \phi }$ and ${\displaystyle \psi }$ that are related to the metric by:

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

In Papapetrou (1954a) the gravitational part of the Lagrangian is:

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

In Papapetrou (1954b) there is a second scalar field ${\displaystyle \chi }$. The gravitational part of the Lagrangian is now:

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

Bimetric theories

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 (1973, 1975) Bimetric Theory The action is:

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

where the vertical line "|" denotes covariant derivative with respect to ${\displaystyle \eta \,}$. The field equations may be written in the form:

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

Lightman-Lee (1973) developed a metric theory based on the non-metric theory of Belinfante and Swihart (1957a, 1957b). 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:

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

In Rastall (1979), the metric is an algebraic function of the Minkowski metric and a Vector field. 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

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

(see Will (1981) for the field equation for ${\displaystyle T^{\mu \nu }\;}$ and ${\displaystyle K_{\mu }\;}$).

Quasilinear theories

In Whitehead (1922), 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:

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

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

${\displaystyle (y^{\mu })^{-}=x^{\mu }-(x^{\mu })^{-}\;}$, ${\displaystyle (y^{\mu })^{-}(y_{\mu })^{-}=0,\;}$

${\displaystyle w^{-}=(y^{\mu })^{-}(u_{\mu })^{-}\;}$, ${\displaystyle (u_{\mu })=dx^{\mu }/d\sigma ,\;}$

${\displaystyle d\sigma ^{2}=\eta _{\mu \nu }dx^{\mu }dx^{\nu }\;}$

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

Deser and Laurent (1968) and Bollini-Giambiagi-Tiomno (1970) are Linear Fixed Gauge (LFG) 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:

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

The Bianchi identity associated with this partial gauge invariance is wrong. LFG 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.

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

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.

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 GR 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 generalisation 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,

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

and

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

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 GR 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 generalisation of GR.

Scalar-tensor theories

These all contain at least one free parameter, as opposed to GR 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 recognised by Thiry (1948).

Scalar-Tensor theories include Thiry (1948), Jordan (1955), Brans and Dicke (1961), Bergman (1968), Nordtveldt (1970), Wagoner (1970), Bekenstein (1977) and Barker (1978).
The action ${\displaystyle S\;}$ is based on the integral of the Lagrangian ${\displaystyle L_{\phi }\;}$.

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

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

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

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

where ${\displaystyle \omega (\phi )\;}$ is a different dimensionless function for each different scalar-tensor theory. The function ${\displaystyle \lambda (\phi )\;}$ plays the same role as the cosmological constant in GR. ${\displaystyle G_{N}\;}$ 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 (1968) and Wagoner (1970). Special cases are:
Nordtvedt (1970), ${\displaystyle \lambda =0\;}$.
 
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 (1961), ${\displaystyle \omega \;}$ is constant.

Bekenstein (1977) Variable Mass Theory Starting with parameters ${\displaystyle r\;}$ and ${\displaystyle q\;}$, found from a cosmological solution, ${\displaystyle \phi =[1-qf(\phi )]f(\phi )^{-r}\;}$ determines function ${\displaystyle f\;}$ then

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

Barker (1978) Constant G Theory

${\displaystyle \omega (\phi )=(4-3\phi )/(2\phi -2)\;}$

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

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

The above examples are particular cases of Horndeski's theory, 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.

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 (1973) and Will and Nordtvedt (1972) 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

${\displaystyle F_{\mu \nu }=K_{\nu ;\mu }-K_{\mu ;\nu }.}$

Will and Nordtvedt (1972) is a special case where

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

Hellings and Nordtvedt (1973) 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 GR so, so long as GR 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 (2004) is discussed under Modern Theories.

Non-metric theories

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 (1981) claims that all non-metric theories are eliminated by Einstein's Equivalence Principle (EEP). Will (2001) tempers that by explaining experimental criteria for testing non-metric theories against EEP. Misner et al. (1973) claims that Cartan's theory is the only non-metric theory to survive all experimental tests up to that date and Turyshev (2006) 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 (1972).

Cartan (1922, 1923) 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 GR, the Lagrangian is made up of a massless and a mass part. The Lagrangian for the massless part is:

${\displaystyle L={1 \over 32\pi G}\Omega _{\nu }^{\mu }g^{\nu \xi }x^{\eta }x^{\zeta }\varepsilon _{\xi \mu \eta \zeta }\;}$

${\displaystyle \Omega _{\nu }^{\mu }=d\omega _{\nu }^{\mu }+\omega _{\xi }^{\eta }\;}$

${\displaystyle \nabla x^{\mu }=-\omega _{\nu }^{\mu }x^{\nu }\;}$

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 GR.

Some equations of the non-metric theory of Belinfante and Swihart (1957a, 1957b) 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 GR 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 (2004), Moffat (1995), Moffat (2002), Moffat (2005a, b).
These theories are presented with a cosmological constant or added scalar or vector potential.

Motivations

Motivations for the more recent alternatives to GR 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 GR at the present epoch but may have been quite different in the early universe.

There was a slow dawning realisation in the physics world that there were several problems inherent in the then big bang scenario, two of these were 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. Another alternative was constructing an alternative to GR in which the speed of light was larger 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 might hold the answer.

In the 1990s, supernova surveys discovered the accelerated expansion of the universe, 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 GR 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 explanation for the accelerated expansion.

Another observation that sparked recent interest in alternatives to General Relativity is the Pioneer anomaly. It was quickly discovered that alternatives to GR could explain this anomaly. This is now believed to be accounted for by non-uniform thermal radiation.

Cosmological constant and quintessence

The cosmological constant ${\displaystyle \Lambda \;}$ is a very old idea, going back to Einstein in 1917. 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:

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

to

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

In GR, 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 GR. We've already seen how the scalar potential ${\displaystyle \lambda (\phi )\;}$ can be added to scalar tensor theories. This can also be done in every alternative the GR that contains a scalar field ${\displaystyle \phi \;}$ by adding the term ${\displaystyle \lambda (\phi )\;}$ inside the Lagrangian for the gravitational part of the action, the ${\displaystyle L_{\phi }\;}$ part of

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

Because ${\displaystyle \lambda (\phi )\;}$ 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 GR that use vector fields, including Rastall (1979) 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.

Relativistic MOND

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:

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

The relativistic version of this has:

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

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 (2004) introduced a tensor-vector-scalar model (TeVeS). This has two scalar fields ${\displaystyle \phi }$ 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 GR.

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

where

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

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

${\displaystyle k,K}$ 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 PPN parameters of this theory are calculated in, which shows that all its parameters are equal to GR's, except for

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

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 (1995) developed a non-symmetric gravitation theory (NGT). This is not a metric theory. It was first claimed that it does not contain a black hole horizon, but Burko and Ori (1995) have found that NGT 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 and MaCarthy (1993) have criticised NGT, 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 GR.

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

where ${\displaystyle R(W)\;}$ is a curvature term analogous to but not equal to the Ricci curvature in GR, ${\displaystyle \lambda \;}$ 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 (1996) used polarization measurements of the light emitted by galaxies to impose sharp constraints on the magnitude of some of NGT'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 (2005a) 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 GR 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.

Moffat (2005b) Scalar-tensor-vector gravity (STVG) theory

The theory 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:

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

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

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. This avoids a black hole singularity near the origin, while recovering the 1/r fall of the GR potential at large distances. Lousto and Mazzitelli (1997) found an exact solution to this theories representing a gravitational shock-wave.

Testing of alternatives to 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. (1973) Ch.39, Will (1981) Table 2.1, and Ni (1972). 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 (1939); 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, 1973) 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 (1955) are incomplete unless Jordan's parameter ${\displaystyle \eta \;}$ is set to -1, in which case they match the theory of Brans–Dicke (1961) and so are worthy of further consideration.

Milne (1948) is incomplete because it makes no gravitational red-shift prediction.

The theories of Whitrow and Morduch (1960, 1965), Kustaanheimo (1966) and Kustaanheimo and Nuotio (1967) 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 GR) but light as waves is not.

Classical tests

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)
• anomalous perihelion advance of the planets (see Tests of general relativity)

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 (1943) 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.

The Einstein equivalence principle (EEP)

The EEP has three components.
The first is the uniqueness of free fall, also known as the Weak Equivalence Principle (WEP). This is satisfied if inertial mass is equal to gravitational mass. η is a parameter used to test the maximum allowable violation of the WEP. The first tests of the WEP 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.

Schiff's conjecture states that any complete, self-consistent theory of gravity that embodies the WEP necessarily embodies EEP. 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 (1957) is eliminated by the THεμ formalism for testing EEP. 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 (PPN) formalism

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 PPN numbers in Nordtvedt and Will (1972) and Will and Nordtvedt (1972). 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 GR.
• ${\displaystyle \beta }$ is a measure of nonlinearity in the addition of gravitational fields, one for GR.
• ${\displaystyle \eta }$ 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

PPN 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 GR.

GR predicts that gravitational waves travel at the speed of light. Many alternatives to GR say that gravitational waves travel faster than light, possibly breaking of causality. After the multi-messanging 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 theory of gravity were excluded.

Cosmological tests

Many of these have been developed recently. For those theories that aim to replace dark matter, 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 GR, the combination of baryonic matter, dark matter and dark energy add up to make the universe exactly flat. As the accuracy of experimental tests improve, alternatives to GR that aim to replace dark matter or dark energy will have to explain why.

Results of testing theories

PPN parameters for a range of theories

(See Will (1981) and Ni (1972) for more details. Misner et al. (1973) 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 (PPN).

The following table lists PPN 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 (1916) GR	1	1	0	0	0	0	0	0	0	0
Scalar-tensor theories
Bergmann (1968), Wagoner (1970)	${\displaystyle \textstyle {\frac {1+\omega }{2+\omega }}}$	${\displaystyle \beta }$	0	0	0	0	0	0	0	0
Nordtvedt (1970), Bekenstein (1977)	${\displaystyle \textstyle {\frac {1+\omega }{2+\omega }}}$	${\displaystyle \beta }$	0	0	0	0	0	0	0	0
Brans–Dicke (1961)	${\displaystyle \textstyle {\frac {1+\omega }{2+\omega }}}$	1	0	0	0	0	0	0	0	0
Vector-tensor theories
Hellings-Nordtvedt (1973)	${\displaystyle \gamma }$	${\displaystyle \beta }$	0	${\displaystyle \alpha _{1}}$	${\displaystyle \alpha _{2}}$	0	0	0	0	0
Will-Nordtvedt (1972)	1	1	0	0	${\displaystyle \alpha _{2}}$	0	0	0	0	0
Bimetric theories
Rosen (1975)	1	1	0	0	${\displaystyle c_{0}/c_{1}-1}$	0	0	0	0	0
Rastall (1979)	1	1	0	0	${\displaystyle \alpha _{2}}$	0	0	0	0	0
Lightman-Lee (1973)	${\displaystyle \gamma }$	${\displaystyle \beta }$	0	${\displaystyle \alpha _{1}}$	${\displaystyle \alpha _{2}}$	0	0	0	0	0
Stratified theories
Lee-Lightman-Ni (1974)	${\displaystyle ac_{0}/c_{1}}$	${\displaystyle \beta }$	${\displaystyle \xi }$	${\displaystyle \alpha _{1}}$	${\displaystyle \alpha _{2}}$	0	0	0	0	0
Ni (1973)	${\displaystyle ac_{0}/c_{1}}$	${\displaystyle bc_{0}}$	0	${\displaystyle \alpha _{1}}$	${\displaystyle \alpha _{2}}$	0	0	0	0	0
Scalar field theories
Einstein (1912) {Not GR}	0	0		-4	0	-2	0	-1	0	0†
Whitrow-Morduch (1965)	0	-1		-4	0	0	0	-3	0	0†
Rosen (1971)	${\displaystyle \lambda }$	${\displaystyle \textstyle {\frac {3}{4}}+\textstyle {\frac {\lambda }{4}}}$		${\displaystyle -4-4\lambda }$	0	-4	0	-1	0	0
Papetrou (1954a, 1954b)	1	1		-8	-4	0	0	2	0	0
Ni (1972) (stratified)	1	1		-8	0	0	0	2	0	0
Yilmaz (1958, 1962)	1	1		-8	0	-4	0	-2	0	-1†
Page-Tupper (1968)	${\displaystyle \gamma }$	${\displaystyle \beta }$		${\displaystyle -4-4\gamma }$	0	${\displaystyle -2-2\gamma }$	0	${\displaystyle \zeta _{2}}$	0	${\displaystyle \zeta _{4}}$
Nordström (1912)	${\displaystyle -1}$	${\displaystyle \textstyle {\frac {1}{2}}}$		0	0	0	0	0	0	0†
Nordström (1913), Einstein-Fokker (1914)	${\displaystyle -1}$	${\displaystyle \textstyle {\frac {1}{2}}}$		0	0	0	0	0	0	0
Ni (1972) (flat)	${\displaystyle -1}$	${\displaystyle 1-q}$		0	0	0	0	${\displaystyle \zeta _{2}}$	0	0†
Whitrow-Morduch (1960)	${\displaystyle -1}$	${\displaystyle 1-q}$		0	0	0	0	q	0	0†
Littlewood (1953), Bergman(1956)	${\displaystyle -1}$	${\displaystyle \textstyle {\frac {1}{2}}}$		0	0	0	0	-1	0	0†

† The theory is incomplete, and ${\displaystyle \zeta _{4}}$ can take one of two values. The value closest to zero is listed.

All experimental tests agree with GR so far, and so PPN analysis immediately eliminates all the scalar field theories in the table.

A full list of PPN parameters is not available for Whitehead (1922), Deser-Laurent (1968), Bollini-Giambiagi-Tiomino (1970), but in these three cases ${\displaystyle \beta =\xi }$, which is in strong conflict with GR and experimental results. In particular, these theories predict incorrect amplitudes for the Earth's tides. (A minor modification of Whitehead's theory avoids this problem. However, the modification predicts the Nordtvedt effect, which has been experimentally constrained.)

Theories that fail other tests

The stratified theories of Ni (1973), Lee Lightman and Ni (1974) are non-starters because they all fail to explain the perihelion advance of Mercury.

The bimetric theories of Lightman and Lee (1973), Rosen (1975), Rastall (1979) all fail some of the tests associated with strong gravitational fields.

The scalar-tensor theories include GR as a special case, but only agree with the PPN values of GR when they are equal to GR to within experimental error. As experimental tests get more accurate, the deviation of the scalar-tensor theories from GR is being squashed to zero.

The same is true of vector-tensor theories, the deviation of the vector-tensor theories from GR is being squashed to zero. Further, vector-tensor theories are semi-conservative; they have a nonzero value for ${\displaystyle \alpha _{2}}$ which can have a measurable effect on the Earth's tides.

Non-metric theories, such as Belinfante and Swihart (1957a, 1957b), usually fail to agree with experimental tests of Einstein's equivalence principle.

And that leaves, as a likely valid alternative to GR, nothing except possibly Cartan (1922).

That was the situation until cosmological discoveries pushed the development of modern alternatives.