Alternatives to general relativity arephysical theories that attempt to describe the phenomenon ofgravitation in competition withEinstein's theory ofgeneral relativity. There have been many different attempts at constructing an ideal theory ofgravity.^[1] These attempts can be split into four broad categories based on their scope:

1. Classical theories of gravity, which do not involve quantum mechanics or force unification.
2. Theories using the principles ofquantum mechanics resulting inquantized gravity.
3. Theories which attempt to explain gravity and other forces at the same time; these are known asclassical unified field theories.
4. Theories which attempt to both put gravity in quantum mechanical terms and unify forces; these are calledtheories of everything.

None of these alternatives to general relativity have gained wide acceptance.

General relativity has withstood manytests over a large range of mass and size scales.^[2][3] When applied to interpret astronomical observations, cosmological models based on general relativity introduce two components to the universe,^[4]dark matter^[5] anddark energy,^[6] the nature of which is currently anunsolved problem in physics. The many successful, high precision predictions of thestandard model of cosmology has led astrophysicists to conclude it and thus general relativity will be the basis for future progress.^[7][8] However, dark matter is not supported by thestandard model of particle physics, physical models for dark energy do not match cosmological data, and some cosmological observations are inconsistent.^[8] These issues have led to the study of alternative theories of gravity.^[9][10]

${\displaystyle c}$ is thespeed of light,${\displaystyle G}$ is thegravitational constant. "Geometric variables" are not used.

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

${\displaystyle {\eta }_{\mu \nu }}$ is theMinkowski metric.${\displaystyle g_{\mu \nu }}$ is a tensor, usually themetric tensor. These havesignature (−,+,+,+).

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

For comparison with alternatives, the formulas of General Relativity^[11][12] are:

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

${\displaystyle {ds}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }}$

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

which can also be written

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

TheEinstein–Hilbert action for general relativity is:

${\displaystyle S=\frac{c^4}{16\pi G}\int R\sqrt{-g}{d}^{4}x+S_m}$

where${\displaystyle G}$ is Newton's gravitational constant,${\displaystyle R=R_{\mu }^{\mu }}$ is theRicci curvature of space,${\displaystyle g=det(g_{\mu \nu })}$ and${\displaystyle S_m}$ is theaction 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.

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

• an 'action' (see theprinciple of least action, avariational principle based on the concept of action)
• aLagrangian density
• ametric

A further word here aboutMach's principle is appropriate because a few of these theories rely on Mach's principle (e.g. Whitehead^[13]), and many mention it in passing (e.g. Einstein–Grossmann,^[14] Brans–Dicke^[15]). Mach's principle can be thought of as a half-way-house between Newton and Einstein. An explanation follows:

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

This isn't exactly the way Mach originally stated it, see other variants inMach principle.

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, theEinstein–Hilbert action uses${\displaystyle L\propto R}$ whereR is thescalar 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.^[16] The original 1983 version ofMOND did not have an action.

A few theories have an action but not a Lagrangian density. A good example is Whitehead,^[13] 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 symmetricmetric tensor${\displaystyle g_{\mu \nu }}$ ofsignature (−, +, +, +), which governs proper-length and proper-time measurements in the usual manner of special and general relativity:

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

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:

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

where${\displaystyle T^{\mu \nu }}$ is thestress–energy tensor for all matter and non-gravitational fields, and where${\displaystyle {\mathrm{\nabla }}_{\nu }}$ is thecovariant derivative with respect to the metric and${\displaystyle {\mathrm{\Gamma }}_{\sigma \nu }^{\alpha }}$ is theChristoffel symbol. The stress–energy tensor should also satisfy anenergy 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 eliminateevent horizons from the theory.)
• Quasilinear theories (includes Linear fixed gauge)
  • Bollini–Giambiagi–Tiomno
  • Deser–Laurent
  • Whitehead's theory of gravity (intended to use onlyretarded 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
  • TeVeS byBekenstein
  • 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 sectionModern 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

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 ofgeneral relativity in 1915 are discussed inhistory of gravitational theory.

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^[17][18] Lang^[19]):

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 theSupernova Cosmology Project andHigh-Z Supernova Search Team. How to add acosmological constant or quintessence to a theory is discussed under Modern Theories (see also Einstein–Hilbert action).

The scalar field theories of Nordström^[55][56] have already been discussed. Those of Littlewood,^[28] Bergman,^[30] Yilmaz,^[33] Whitrow and Morduch^[35][36] and Page and Tupper^[40] follow the general formula give by Page and Tupper.

According to Page and Tupper,^[40] who discuss all these except Nordström,^[56] the general scalar field theory comes from the principle of least action:

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

where the scalar field is,

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

andc may or may not depend on${\displaystyle \phi }$.

In Nordström,^[55]

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

In Littlewood^[28] and Bergmann,^[30]

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

In Whitrow and Morduch,^[35]

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

In Whitrow and Morduch,^[36]

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

In Page and Tupper,^[40]

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

Page and Tupper^[40] matches Yilmaz's theory^[33] to second order when${\displaystyle \alpha =-7/2}$.

The gravitational deflection of light has to be zero whenc 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^[18] 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:

${\displaystyle S=\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}{L}_{\phi }+S_m}$

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

Misner et al.^[57] gives this without the${\displaystyle \phi R}$ term.${\displaystyle S_m}$ is the matter action.

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

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

In the second theory of Ni^[18] there are two arbitrary functions${\displaystyle f(\phi )}$ and${\displaystyle k(\phi )}$ that are related to the metric by:

${\displaystyle d{s}^2=e^{-2f(\phi )}d{t}^2-e^{2f(\phi )}\left[d{x}^2+d{y}^2+d{z}^2\right]}$

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

Ni^[18] quotes Rosen^[45] as having two scalar fields${\displaystyle \phi }$ and${\displaystyle \psi }$ that are related to the metric by:

${\displaystyle d{s}^2={\phi }^{2}d{t}^2-{\psi }^2\left[d{x}^2+d{y}^2+d{z}^2\right]}$

In Papapetrou^[26] the gravitational part of the Lagrangian is:

${\displaystyle L_{\phi }=e^{\phi }\left(\frac{1}{2}{e}^{-\phi }{\mathrm{\partial }}_{\alpha }\phi {\mathrm{\partial }}_{\alpha }\phi +\frac{3}{2}{e}^{\phi }{\mathrm{\partial }}_0\phi {\mathrm{\partial }}_0\phi \right)}$

In Papapetrou^[27] there is a second scalar field${\displaystyle \chi }$. The gravitational part of the Lagrangian is now:

${\displaystyle L_{\phi }=e^{\frac{1}{2}(3\phi +\chi )}\left(-\frac{1}{2}{e}^{-\phi }{\mathrm{\partial }}_{\alpha }\phi {\mathrm{\partial }}_{\alpha }\phi -e^{-\phi }{\mathrm{\partial }}_{\alpha }\phi {\mathrm{\partial }}_{\chi }\phi +\frac{3}{2}{e}^{-\chi }{\mathrm{\partial }}_0\phi {\mathrm{\partial }}_0\phi \right)}$

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^[58] (1975) developed a bimetric theory. The action is:

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

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

Lightman–Lee^[50] developed a metric theory based on the non-metric theory of Belinfante and Swihart.^[31][32] 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:

${\displaystyle S=\frac{1}{16\pi G}\int d^{4}x\sqrt{-\eta }(a{B}^{\mu \nu |\alpha }{B}_{\mu \nu |\alpha }+f{B}_{,\alpha }{B}^{,\alpha })+S_m}$

and the stress–energy tensor comes from:

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

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

${\displaystyle S=\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}F(N)K^{\mu ;\nu }{K}_{\mu ;\nu }+S_m}$

where

${\displaystyle F(N)=-\frac{N}{2+N}}$ and${\displaystyle N=g^{\mu \nu }{K}_{\mu }{K}_{\nu }}$.

InWhitehead,^[13] the physical metric${\displaystyle g}$ is constructed (bySynge) algebraically from the Minkowski metric${\displaystyle \eta }$ and matter variables, so it doesn't even have a scalar field. The construction is:

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

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

Nevertheless, the metric construction (from a non-metric theory) using the "length contraction" ansatz is criticised.^[59]

Deser and Laurent^[39] and Bollini–Giambiagi–Tiomno^[42] 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:

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

TheBianchi 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${\displaystyle h_{\mu \nu }}$.

Acosmological constant can be introduced into a quasilinear theory by changing the Minkowski background to ade Sitter oranti-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.^[60]

Einstein's general relativity is the simplest plausible theory of gravity that can be based on just one symmetric tensor field (themetric tensor).^{[citation needed]} Others include: Starobinsky (R+R^2) gravity,Gauss–Bonnet gravity,f(R) gravity, andLovelock theory of gravity.

Starobinsky gravity, proposed byAlexei 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 ofStarobinsky inflation. Here${\displaystyle M}$ is a constant.

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, 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) 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 is a covariant theory of gravity, quadratic in curvature, torsion free and parity invariant,^[61]

${\displaystyle \mathcal{L}=\sqrt{-g}\left[M_p^{2}R+R{f}_1\left(\frac{◻}{M_s^2}\right)R+R^{\mu \nu }{f}_2\left(\frac{◻}{M_s^2}\right)R_{\mu \nu }+R^{\mu \nu \rho \sigma }{f}_3\left(\frac{◻}{M_s^2}\right)R_{\mu \nu \rho \sigma }\right].}$

and

${\displaystyle 2f_1\left(\frac{◻}{M_s^2}\right)+f_2\left(\frac{◻}{M_s^2}\right)+2f_3\left(\frac{◻}{M_s^2}\right)=0,}$

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 ininfinite derivative theories of gravity.

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.

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 ofKaluza–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 thenKaluza–Klein can be considered the progenitor of Scalar–Tensor theories of gravity. This was recognized by Thiry.^[25]

Scalar–Tensor theories include Thiry,^[25] Jordan,^[29] Brans and Dicke,^[15] Bergman,^[41] Nordtveldt (1970), Wagoner,^[44] Bekenstein^[52] and Barker.^[53]

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

${\displaystyle S=\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}{L}_{\phi }+S_m}$

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

${\displaystyle S_m=\int d^{4}x\sqrt{g}{G}_N{L}_m}$

${\displaystyle T^{\mu \nu }\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{2}{\sqrt{g}}\frac{\delta S_m}{\delta g_{\mu \nu }}}$

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 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^[41] and Wagoner.^[44] Special cases are:

Nordtvedt,^[43]${\displaystyle \lambda =0}$

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 underCosmological constant.

Brans–Dicke,^[15]${\displaystyle \omega }$ is constant

Bekenstein^[52] variable mass theoryStarting 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

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

Barker^[53] constant G theory

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

Adjustment of${\displaystyle \omega (\phi )}$ allows Scalar Tensor Theories to tend to general relativity in the limit of${\displaystyle \omega \to \mathrm{\infty }}$ 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^[15]) 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 ofHorndeski's theory,^[62][63] 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.^[64][65][66]

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."^[17]

Hellings and Nordtvedt^[49] and Will and Nordtvedt^[48] are both vector–tensor theories. In addition to the metric tensor there is a timelike vector field${\displaystyle K_{\mu }.}$ The gravitational action is:

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

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

${\displaystyle F_{\mu \nu }=K_{\nu ;\mu }-K_{\mu ;\nu }.}$ (See Will^[17] for the field equations for${\displaystyle T^{\mu \nu }}$ and${\displaystyle K_{\mu }.}$)

Will and Nordtvedt^[48] is a special case where

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

Hellings and Nordtvedt^[49] is a special case where

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

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${\displaystyle \omega =\eta =ϵ=\tau =0}$ they reduce to general relativity so, so long as general relativity is confirmed by experiment, general vector–tensor theories can never be ruled out.

Others metric theories have been proposed; that ofBekenstein^[67] is discussed under Modern 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^[17] claims that all non-metric theories are eliminated by Einstein's Equivalence Principle. Will tempers that by explaining experimental criteria for testing non-metric theories against Einstein's Equivalence Principle in his 2001 edition.^[17] Misner et al.^[57] claims that Cartan's theory is the only non-metric theory to survive all experimental tests up to that date and Turyshev^[68] 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.^[69]

Cartan^[20][21] 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:

The${\displaystyle {\omega }_{\nu }^{\mu }}$ is the linear connection.${\displaystyle {\epsilon }_{\xi \mu \eta \zeta }}$ is the completely antisymmetric pseudo-tensor (Levi-Civita symbol) with${\displaystyle {\epsilon }_{0123}=\sqrt{-g}}$, 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:

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

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^[31][32] have already been discussed in the section onbimetric theories.

A distinctively non-metric theory is given bygauge 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 withgeometric algebra.

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,^[67] Moffat,^[70] Moffat,^[71] Moffat.^[72][73] These theories are presented with a cosmological constant or added scalar or vector potential.

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.^{[citation needed]}

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 thehorizon problem and the observation that at early times when quarks were first forming there was not enough space in 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.^{[citation needed]}

In the 1990s, supernova surveys discovered the accelerated expansion of the universe, now usually attributed todark energy. This led to the rapid reinstatement of Einstein's cosmological constant, and quintessence arrived as an alternative to the cosmological constant.^{[citation needed]} 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 eventGW170817 ruled out many alternative theories of gravity as explanations for the accelerated expansion.^[74][75][76]

Another observation that sparked recent interest in alternatives to General Relativity is thePioneer 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.^{[citation needed]}

The cosmological constant${\displaystyle \mathrm{\Lambda }}$ is a very old idea, going back to Einstein in 1917.^[12] The success of the Friedmann model of the universe in which${\displaystyle \mathrm{\Lambda }=0}$ led to the general acceptance that it is zero, but the use of a non-zero value came back when data from supernovae indicated that the expansion of the universe is accelerating.^{[citation needed]}

In Newtonian gravity, the addition of the cosmological constant changes the Newton–Poisson equation from:

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

to

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

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

${\displaystyle S=\frac{1}{16\pi G}\int R\sqrt{-g}{d}^{4}x+S_m}$

to

${\displaystyle S=\frac{1}{16\pi G}\int (R-2\mathrm{\Lambda })\sqrt{-g}{d}^{4}x+S_m}$

which changes the field equation from:

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

to:

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

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

More generally a scalar potential${\displaystyle \lambda (\phi )}$ can be added to scalar tensor theories. This can be done in every alternative the general relativity 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

${\displaystyle S=\frac{1}{16\pi G}\int d^{4}x\sqrt{-g}{L}_{\phi }+S_m}$

Because${\displaystyle \lambda (\phi )}$ is an arbitrary function of the scalar field rather than a constant, 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^[54] 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.

In December 2018, the astrophysicistJamie Farnes from theUniversity of Oxford proposed adark fluid theory, related to notions of gravitationally repulsive negative masses that were presented earlier byAlbert Einstein. The theory may help to better understand the considerable amounts of unknown dark matter and dark energy in theuniverse.^[77]

The theory relies on the concept ofnegative mass and reintroducesFred Hoyle's creation tensor in order to allowmatter 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 acosmological constant. Einstein's field equations are modified to:

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

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 should be directly testable using theSquare Kilometre Array radio telescope now under construction.^[78]

The original theory of MOND by Milgrom was developed in 1983 as an alternative to "dark matter".^[79] Departures from Newton's law of gravitation are governed by an acceleration scale, not a distance scale. MOND successfully predicted the Tully–Fisher observation that the baryonic mass of a galaxy scale as the fourth power of the flat rotation speed. Many attempts at a relativistic version of MOND exist, as reviewed by Famaey and McGaugh.^[80] In so far as these theories actually reduce to non-relativistic MOND in the weak field limit they inherit its apparent failure to reproduce the correct gravitational potentials of galaxy clusters.^[81]

RAQUAL, the relativistic version of MOND's field equationAQUAL has a three part action:^[82]: 13

${\displaystyle S=S_g+S_s+S_m}$

${\displaystyle S_g=\frac{c^4}{16\pi G}\int e^{-2{\varphi }^2}\left[R({\tilde{g}}_{\mu \nu })+{\displaystyle \frac{6}{c^4}}{\varphi }_{,\alpha }{\varphi }_{,}^{\alpha }\right]\sqrt{-g}{d}^{4}x}$

${\displaystyle S_{\varphi }={\displaystyle \frac{-a_0^2\beta (1+\beta {)}^2}{8\pi G}}\int e^{-4{\varphi }^2}f\left[{\displaystyle \frac{e^{-2{\varphi }^2}{\varphi }_{,\mu }{\varphi }_{,}^{\mu }}{a_0^2(1+{\beta }^2)}}\right]\sqrt{-g}{d}^{4}x}$

with a standard mass action. Here${\displaystyle f}$ is an arbitrary function selected to give Newtonian and MOND behaviour in the correct limits. In the strong field limit this becomes a Brans-Dicke scalar-tensor theory with${\displaystyle \beta =2\omega +3}$. This theory was soon rejected because it allowed waves in the scalar field to propagate faster than light.^[83]: 123 By 1988, a second scalar field (PCC) fixed problems with this 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 apreferred frame theory it has problems of its own. Despite these problems core concepts of RAQUAL such as a weak field limit that follows${\displaystyle f(\chi )\approx {\chi }^{\frac{3}{2}}}$ have been adopted under the name "extended gravity".Jacob Bekenstein developed a relativistic generalization of MOND in 2004,TeVeS, which however had its own set of problems (see below). An attempt by Skordis and Złośnik in 2021 has been claimed to be compatible with cosmic microwave background observations, but appears to be highly contrived.^[9][84]

Bekenstein^[67] introduced atensor–vector–scalar model (TeVeS) that attempted to reproduce MOND in 2004. 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 general relativity.

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), andL is a Lagrangian translated from flat spacetime onto the metric${\displaystyle {\tilde{g}}^{\alpha \beta }}$. Note thatG need not equal the observed gravitational constant${\displaystyle G_{Newton}}$.F is an arbitrary function, and

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

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,^[85] which shows that all its parameters are equal to general relativity's, except for

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

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

TeVeS faces problems when confronted with data on theanisotropy of thecosmic microwave background,^[86] the lifetime of compact objects,^[87] and the relationship between the lensing and matter overdensity potentials.^[88] TeVeS also appears inconsistent with the speed of gravitational waves according to LIGO.^[89]

J. W. Moffat^[70] developed anon-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^[90] 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 & McCarthy^[91] 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^[92] 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^[72]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:

${\displaystyle S_F=\int d^{4}x\sqrt{-g}\left(\frac{1}{12}{F}_{\mu \nu \rho }{F}^{\mu \nu \rho }-\frac{1}{4}{\mu }^2{A}_{\mu \nu }{A}^{\mu \nu }\right)}$

${\displaystyle S_{FM}=\int d^{4}x{ϵ}^{\alpha \beta \mu \nu }{A}_{\alpha \beta }{\mathrm{\partial }}_{\mu }{J}_{\nu }}$

where

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

and${\displaystyle {ϵ}^{\alpha \beta \mu \nu }}$ is theLevi-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'sScalar–tensor–vector gravity^[73] 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.

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

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

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.

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

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

where${\displaystyle F(◻)}$ is the exponential of an entire function of theD'Alembertian operator.^[93][94] This avoids a black hole singularity near the origin, while recovering the 1/r fall of the general relativity potential at large distances.^[95]Lousto and Mazzitelli (1997) found an exact solution to this theories representing a gravitational shock-wave.^[96]

The General Relativity Self-interaction orGRSI model^[97] is an attempt to explain astrophysical and cosmological observations withoutdark matter,dark energy by addingself-interaction terms when calculating the gravitational effects ingeneral relativity, analogous to the self-interaction terms inquantum chromodynamics.^[98]Additionally, the model explains theTully-Fisher relation,^[99]theradial acceleration relation,^[100] observations that are currently challenging to understand withinLambda-CDM.

The model was proposed in a series of articles, the first dating from 2003.^[101] The basic point is that since within General Relativity, gravitational fields couple to each other, this can effectively increase the gravitational interaction between massive objects. The additional gravitational strength then avoid the need for dark matter. This field coupling is the origin of General Relativity'snon-linear behavior. It can be understood, in particle language, asgravitons interacting with each other (despite beingmassless) because they carryenergy-momentum.

A natural implication of this model is its explanation of theaccelerating expansion of the universe without resorting todark energy.^[98] The increasedbinding energy within a galaxy requires, byenergy conservation, a weakening of gravitational attraction outside said galaxy. This mimics the repulsion of dark energy.

The GRSI model is inspired from theStrong Nuclear Force, where a comparable phenomenon occurs. The interaction betweengluons emitted by static or nearly staticquarks dramatically strengthens quark-quark interaction, ultimately leading toquark confinement on the one hand (analogous to the need of stronger gravity to explain away dark matter) and thesuppression of the Strong Nuclear Force outside hadrons (analogous to the repulsion of dark energy that balances gravitational attraction at large scales.) Two other parallel phenomena are theTully-Fisher relation in galaxy dynamics that is analogous to theRegge trajectories emerging from the strong force. In both cases, the phenomenological formulas describing these observations are similar, albeit with different numerical factors.

These parallels are expected from a theoretical point of view: General Relativity and the Strong InteractionLagrangians have the same form.^[102][103] The validity of the GRSI model then simply hinges on whether the coupling of the gravitational fields is large enough so that the same effects that occur inhadrons also occur in very massive systems. This coupling is effectively given by${\displaystyle \sqrt{GM/L}}$, where${\displaystyle G}$ is thegravitational constant,${\displaystyle M}$ is the mass of the system, and${\displaystyle L}$ is a characteristic length of the system. The claim of the GRSI proponents, based either onlattice calculations,^[103] a background-field model.^[104] or the coincidental phenomenologies in galactic or hadronic dynamics mentioned in the previous paragraph, is that${\displaystyle \sqrt{GM/L}}$ is indeed sufficiently large for large systems such as galaxies.

The main observations that appear to require dark matter and/or dark energy can be explained within this model. Namely,

• Theflat rotation curves of galaxies.^{[103][104][105]} These results, however, have been challenged.^[106][107]
• TheCosmic Microwave Backgroundanisotropies.^[108]
• Thefainter luminosities of distant supernovae and their consequence on theaccelerating expansion of the universe.^[98]
• Theformation of the Universe's large structures.^[109]
• Thematter power spectrum.^[108]
• The internal dynamics ofgalaxy clusters, including that of theBullet Cluster.^[103]

Additionally, the model explains observations that are currently challenging to understand withinLambda-CDM:

• TheTully-Fisher relation.^[103][104]
• Theradial acceleration relation.^[110]
• TheHubble tension.^[111]
• Thecosmic coincidence, that is the fact that at present time, the purported repulsion of dark energy nearly exactly cancels the action of gravity in theoverall dynamics of the universe.^[105]

Finally, the model made a prediction that the amount of missing mass (i.e., the dark mass in dark matter approaches) in elliptical galaxies correlates with the ellipticity of the galaxies.^[103] This was tested and verified.^[112][113]

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.^[57] Ch.39, Will^[17] Table 2.1, and Ni.^[18] Most such tests can be categorized as in the following subsections.

Self-consistency among non-metric theories includes eliminating theories allowingtachyons, 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;^[22] 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)^[34] 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.

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 thefour-velocity, and${\displaystyle \delta }$ is theKronecker delta. The theories of Thirry (1948) and Jordan^[29] are incomplete unless Jordan's parameter${\displaystyle \eta }$ is set to -1, in which case they match the theory of Brans–Dicke^[15] and so are worthy of further consideration. Milne^[24] is incomplete because it makes no gravitational red-shift prediction. The theories of Whitrow and Morduch,^[35][36] Kustaanheimo^[37] and Kustaanheimo and Nuotio^[38] 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.