Alternatives to general relativity

c\; is the speed of light, 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. \eta_{\mu\nu}\; is the Minkowski metric. g_{\mu\nu}\; is a tensor, usually the metric tensor. These have signature (−,+,+,+). Partial differentiation is written \partial_\mu \varphi\; or \varphi_{,\mu}\;. Covariant differentiation is written \nabla_\mu \varphi\; or \varphi_{;\mu}\;. == General relativity ==

For comparison with alternatives, the formulas of General Relativity are: :\delta \int ds = 0 \, :{ds}^2 = g_{\mu \nu} \, dx^\mu \, dx^\nu \, :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 :T^{\mu\nu} = {c^4 \over 8 \pi G} \left( R^{\mu \nu}-\frac {1}{2} g^{\mu \nu} R \right) \,. The Einstein–Hilbert action for general relativity is: :S = {c^4 \over 16 \pi G} \int R \sqrt{-g} \ d^4 x + S_m \, where G \, is Newton's gravitational constant, R = R_{\mu}^{~\mu} \, is the Ricci curvature of space, g = \det ( g_{\mu \nu} ) \, and S_m \, is the action due to mass. General relativity is a tensor theory; the equations all contain tensors. Nordström's theories, on the other hand, are scalar theories because the gravitational field is a scalar. Other proposed alternatives include scalar–tensor theories that contain a scalar field in addition to the tensors of general relativity. Other variants containing vector fields have been developed recently as well. == 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 A further word here about Mach's principle is appropriate because a few of these theories rely on Mach's principle (e.g. Whitehead Brans–Dicke). 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 in Mach principle. Classification based on the action If a theory has a Lagrangian density for gravity, say L\,, then the gravitational part of the action S\, is the integral of that: :S = \int L \sqrt{-g} \, \mathrm{d}^4x . In this equation it is usual, though not essential, to have g = -1\, at spatial infinity when using Cartesian coordinates. For example, the Einstein–Hilbert action uses L\,\propto\, R where R is the scalar curvature, a measure of the curvature of space. Almost every theory described in this article has an action. It is the most efficient known way to guarantee that the necessary conservation laws of energy, momentum and angular momentum are incorporated automatically; although it is easy to construct an action where those conservation laws are violated. Canonical methods provide another way to construct systems that have the required conservation laws, but this approach is more cumbersome to implement. The original 1983 version of MOND did not have an action. Classification based on the Lagrange density A few theories have an action but not a Lagrangian density. A good example is Whitehead, the action there is termed non-local. Classification based on metricity A theory of gravity is a "metric theory" if and only if it can be given a mathematical representation in which two conditions hold: Condition 1: There exists a symmetric metric tensor g_{\mu\nu}\, of signature (−, +, +, +), which governs proper-length and proper-time measurements in the usual manner of special and general relativity: :{d\tau}^2 = - g_{\mu \nu} \, dx^\mu \, dx^\nu \, where there is a summation over indices \mu and \nu. Condition 2: Stressed matter and fields being acted upon by gravity respond in accordance with the equation: :0 = \nabla_\nu T^{\mu \nu} = {T^{\mu \nu}}_{,\nu} + \Gamma^{\mu}_{\sigma \nu} T^{\sigma \nu} + \Gamma^{\nu}_{\sigma \nu} T^{\mu \sigma} \, where T^{\mu \nu} \, is the stress–energy tensor for all matter and non-gravitational fields, and where \nabla_{\nu} is the covariant derivative with respect to the metric and \Gamma^{\alpha}_{\sigma \nu} \, is the Christoffel symbol. The stress–energy tensor should also satisfy an energy condition. Metric theories include (from simplest to most complex): • Scalar field theories (includes conformally flat theories & Stratified theories with conformally flat space slices) • Bergman • Coleman • Einstein (1912) • Einstein–Fokker theory • Lee–Lightman–Ni • Littlewood • Ni • Nordström's theory of gravitation (first metric theory of gravity to be developed) • Page–Tupper • Papapetrou • Rosen (1971) • Whitrow–Morduch • Yilmaz theory of gravitation (attempted to eliminate event horizons from the theory.) • Quasilinear theories (includes Linear fixed gauge) • Bollini–Giambiagi–Tiomno • Deser–Laurent • Whitehead's theory of gravity (intended to use only retarded potentials) • Tensor theories • Einstein's general relativity • Fourth-order gravity (allows the Lagrangian to depend on second-order contractions of the Riemann curvature tensor) • f(R) gravity (allows the Lagrangian to depend on higher powers of the Ricci scalar) • Gauss–Bonnet gravity • Lovelock theory of gravity (allows the Lagrangian to depend on higher-order contractions of the Riemann curvature tensor) • Infinite derivative gravity • Scalar–tensor theories • TeVeS by Bekenstein • Bergmann–Wagoner • Brans–Dicke theory (the most well-known alternative to general relativity, intended to be better at applying Mach's principle) • Jordan • Nordtvedt • Thiry • Chameleon • Pressuron • Vector–tensor theories • Hellings–Nordtvedt • Will–Nordtvedt • Bimetric theories • Lightman–Lee • Rastall • Rosen (1975) • Other metric theories (see section Modern theories below) Non-metric theories include • Belinfante–Swihart • Einstein–Cartan theory (intended to handle spin-orbital angular momentum interchange) • Kustaanheimo (1967) • Teleparallelism • Gauge theory gravity == Theories from 1917 to the 1980s ==

At the time it was published in the 17th century, Isaac Newton's theory of gravity was the most accurate theory of gravity. Since then, a number of alternatives were proposed. The theories which predate the formulation of general relativity in 1915 are discussed in history of gravitational theory. 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 Lang): These theories are presented here without a cosmological constant or added scalar or vector potential unless specifically noted, for the simple reason that the need for one or both of these was not recognized before the supernova observations by the Supernova Cosmology Project and High-Z Supernova Search Team. How to add a cosmological constant or quintessence to a theory is discussed under Modern Theories (see also Einstein–Hilbert action). Scalar field theories The scalar field theories of Nordström gives this without the \varphi R term. S_m is the matter action. : \Box\varphi=4\pi T^{\mu\nu} \left [\eta_{\mu\nu}e^{-2\varphi}+ \left (e^{2\varphi}+e^{-2\varphi} \right ) \, \partial_\mu t \, \partial_\nu t \right ] 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 (1975) developed a bimetric theory. The action is: : 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 : \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 Deser and Laurent 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 :\mathcal{L}= \sqrt{-g}\left[R+\frac{R^2}{6M^2}\right] and has been used to explain inflation, in the form of Starobinsky inflation. Here M is a constant. Gauss–Bonnet Gauss–Bonnet gravity has the action : \mathcal{L} =\sqrt{-g}\left[R+ R^2 - 4R^{\mu\nu}R_{\mu\nu} + R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma} \right]. where the coefficients of the extra terms are chosen so that the action reduces to general relativity in 4 spacetime dimensions and the extra terms are only non-trivial when more dimensions are introduced. Stelle's 4th derivative gravity Stelle's 4th derivative gravity, which is a generalization of Gauss–Bonnet gravity, has the action : \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 : \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 f(R) theory. Infinite derivative gravity Infinite derivative gravity is a covariant theory of gravity, quadratic in curvature, torsion free and parity invariant, : \mathcal{L} =\sqrt{-g} \left[ M_p^2 R + Rf_1\left( \frac \Box {M_s^2}\right)R + R^{\mu\nu}f_2 \left( \frac \Box {M_s^2} \right) R_{\mu\nu} + R^{\mu\nu\rho\sigma} f_3\left( \frac \Box {M_s^2}\right) R_{\mu\nu\rho\sigma} \right]. and : 2f_1 \left( \frac \Box {M_s^2} \right) + f_2 \left( \frac \Box {M_s^2} \right) + 2f_3 \left( \frac \Box {M_s^2} \right) = 0, in order to make sure that only massless spin −2 and spin −0 components propagate in the graviton propagator around Minkowski background. The action becomes non-local beyond the scale M_s, and recovers to general relativity in the infrared, for energies below the non-local scale M_s. In the ultraviolet regime, at distances and time scales below non-local scale, 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 : \mathcal{L}=\sqrt{-g}\ (\alpha _{0}+\alpha _{1}R+\alpha _{2}\left( R^{2}+R_{\alpha \beta \mu \nu }R^{\alpha \beta \mu \nu }-4R_{\mu \nu }R^{\mu \nu }\right) +\alpha _{3}\mathcal{O}(R^{3})), and can be thought of as a generalization of general relativity. Scalar–tensor theories These all contain at least one free parameter, as opposed to general relativity which has no free parameters. Although not normally considered a Scalar–Tensor theory of gravity, the 5 by 5 metric of Kaluza–Klein reduces to a 4 by 4 metric and a single scalar. So if the 5th element is treated as a scalar gravitational field instead of an electromagnetic field then Kaluza–Klein can be considered the progenitor of Scalar–Tensor theories of gravity. This was recognized by Thiry. \lambda=0\; Since \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, 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." are both vector–tensor theories. In addition to the metric tensor there is a timelike vector field K_\mu. The gravitational action is: :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 \omega, \eta, \epsilon, \tau are constants and :F_{\mu\nu}=K_{\nu;\mu}-K_{\mu;\nu}. (See Will 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 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. Cartan 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: : \begin{align} L & ={1\over 32\pi G}\Omega_\nu^\mu g^{\nu\xi}x^\eta x^\zeta \varepsilon_{\xi\mu\eta\zeta} \\[5pt] \Omega_\nu^\mu & =d \omega^\mu_\nu + \omega^\eta_\xi \\[5pt] \nabla x^\mu & =-\omega^\mu_\nu x^\nu \end{align} The \omega^\mu_\nu\; is the linear connection. \varepsilon_{\xi\mu\eta\zeta}\; is the completely antisymmetric pseudo-tensor (Levi-Civita symbol) with \varepsilon_{0123}=\sqrt{-g}\;, and 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: :T^{\mu\nu}={1\over 16\pi G} (g^{\mu\nu}\eta^\xi_\eta-g^{\xi\mu}\eta^\nu_\eta-g^{\xi\nu} \eta^\mu_\eta) \Omega^\eta_\xi\; 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 have already been discussed in the section on bimetric theories. A distinctively non-metric theory is given by gauge theory gravity, which replaces the metric in its field equations with a pair of gauge fields in flat spacetime. On the one hand, the theory is quite conservative because it is substantially equivalent to Einstein–Cartan theory (or general relativity in the limit of vanishing spin), differing mostly in the nature of its global solutions. On the other hand, it is radical because it replaces differential geometry with geometric algebra. == Modern theories 1980s to present ==

This section includes alternatives to general relativity published after the observations of galaxy rotation that led to the hypothesis of "dark matter". There is no known reliable list of comparison of these theories. Those considered here include: Bekenstein, Moffat, Moffat. These theories are presented with a cosmological constant or added scalar or vector potential. Motivations Motivations for the more recent alternatives to general relativity are almost all cosmological, associated with or replacing such constructs as "inflation", "dark matter" and "dark energy". The basic idea is that gravity agrees with general relativity at the present epoch but may have been quite different in the early universe. In the 1980s, there was a slowly dawning realisation in the physics world that there were several problems inherent in the then-current big-bang scenario, including the horizon problem and the observation that at early times when quarks were first forming there was not enough space 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. In the 1990s, supernova surveys discovered the accelerated expansion of the universe, now usually attributed to dark energy. This led to the rapid reinstatement of Einstein's cosmological constant, and quintessence arrived as an alternative to the cosmological constant. At least one new alternative to general relativity attempted to explain the supernova surveys' results in a completely different way. The measurement of the speed of gravity with the gravitational wave event GW170817 ruled out many alternative theories of gravity as explanations for the accelerated expansion. Another observation that sparked recent interest in alternatives to General Relativity is the Pioneer anomaly. It was quickly discovered that alternatives to general relativity could explain this anomaly. This is now believed to be accounted for by non-uniform thermal radiation. Cosmological constant and quintessence The cosmological constant \Lambda\; is a very old idea, going back to Einstein in 1917. In Newtonian gravity, the addition of the cosmological constant changes the Newton–Poisson equation from: : \nabla^2\varphi=4\pi\rho\ G; to : \nabla^2\varphi + \frac{1}{2}\Lambda c^2 = 4\pi\rho\ G; In general relativity, it changes the Einstein–Hilbert action from : S={1\over 16\pi G}\int R\sqrt{-g} \, d^4x \, +S_m\; to : S={1\over 16\pi G}\int (R-2\Lambda)\sqrt{-g}\,d^4x \, +S_m\; which changes the field equation from: : T^{\mu\nu}={1\over 8\pi G} \left(R^{\mu\nu}-\frac {1}{2} g^{\mu\nu} R \right)\; to: : 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 the same way. More generally a scalar potential \lambda(\varphi)\; can be added to scalar tensor theories. This can be done in every alternative the general relativity that contains a scalar field \varphi\; by adding the term \lambda(\varphi)\; inside the Lagrangian for the gravitational part of the action, the L_\varphi\; part of : S={1\over 16\pi G}\int d^4x \, \sqrt{-g} \, L_\varphi+S_m\; Because \lambda(\varphi)\; 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 The theory relies on the concept of negative mass and reintroduces Fred Hoyle's creation tensor in order to allow matter creation for only negative mass particles. In this way, the negative mass particles surround galaxies and apply a pressure onto them, thereby resembling dark matter. As these hypothesised particles mutually repel one another, they push apart the Universe, thereby resembling dark energy. The creation of matter allows the density of the exotic negative mass particles to remain constant as a function of time, and so appears like a cosmological constant. Einstein's field equations are modified to: : R_{\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 the Square Kilometre Array radio telescope now under construction. 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 has made a number of successful, "prior" predictions, that is, predictions of unknown facts that were subsequently confirmed by observation.. For example, Milgrom predicted in 1993 that the baryonic mass of a galaxy scales as the fourth power of the flat rotation speed and this was subsequently confirmed by observation. Many attempts at a relativistic version of MOND exist, as reviewed by Famaey and McGaugh. 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. RAQUAL, the relativistic version of MOND's field equation AQUAL has a three part action: : S=S_g+S_s+S_m : S_g={c^4 \over 16 \pi G}\int e^{-2\phi^2} \left[ R(\tilde g_{\mu\nu}) + \dfrac{6}{c^4}\phi_{,\alpha}\phi_{,}^{\alpha} \right] \sqrt{-g}\,d^4x : S_{\phi}=\dfrac{-a_0^2\beta(1+\beta)^2}{8 \pi G}\int e^{-4\phi^2} f \left[ \dfrac{e^{-2\phi^2}\phi_{,\mu}\phi_{,}^{\mu}}{a_0^2(1+\beta^2)} \right] \sqrt{-g}\,d^4x with a standard mass action. Here 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 \beta=2\omega +3. This theory was soon rejected because it allowed waves in the scalar field to propagate faster than light. TeVeS Bekenstein which shows that all its parameters are equal to general relativity's, except for :\begin{align} \alpha_1 &= \frac{4G}{K} \left ((2K-1) e^{-4\varphi_0} - e^{4\varphi_0} + 8 \right ) - 8 \\[5pt] \alpha_2 &= \frac{6 G}{2 - K} - \frac{2 G (K + 4) e^{4 \varphi_0}}{(2 - K)^2} - 1 \end{align} both of which expressed in geometric units where c = G_{Newtonian} = 1; so : G^{-1} = \frac{2}{2-K} + \frac{k}{4\pi}. TeVeS faces problems when confronted with data on the anisotropy of the cosmic microwave background, the lifetime of compact objects, and the relationship between the lensing and matter overdensity potentials. TeVeS also appears inconsistent with the speed of gravitational waves according to LIGO. Moffat's theories J. W. Moffat 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 have criticised nonsymmetric gravitational theory, saying that it has unacceptable asymptotic behaviour. The mathematics is not difficult but is intertwined so the following is only a brief sketch. Starting with a non-symmetric tensor g_{\mu\nu}\;, the Lagrangian density is split into : L=L_R+L_M\; where L_M\; is the same as for matter in general relativity. : L_R = \sqrt{-g} \left[R(W)-2\lambda-\frac14\mu^2g^{\mu\nu}g_{[\mu\nu]}\right] - \frac16g^{\mu\nu}W_\mu W_\nu\; where R(W)\; is a curvature term analogous to but not equal to the Ricci curvature in general relativity, \lambda\; and \mu^2\; are cosmological constants, g_{[\nu\mu]}\; is the antisymmetric part of g_{\nu\mu}\;. W_\mu\; is a connection, and is a bit difficult to explain because it's defined recursively. However, W_\mu\approx-2g^{,\nu}_{[\mu\nu]}\; Haugan and Kauffmann 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 This avoids a black hole singularity near the origin, while recovering the 1/r fall of the general relativity potential at large distances. Lousto and Mazzitelli (1997) found an exact solution to this theories representing a gravitational shock-wave. General relativity self-interaction (GRSI) The General Relativity Self-interaction or GRSI model is an attempt to explain astrophysical and cosmological observations without dark matter, dark energy by adding self-interaction terms when calculating the gravitational effects in general relativity, analogous to the self-interaction terms in quantum chromodynamics. Additionally, the model explains the Tully-Fisher relation, the radial acceleration relation, observations that are currently challenging to understand within Lambda-CDM. The model was proposed in a series of articles, the first dating from 2003. 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's non-linear behavior. It can be understood, in particle language, as gravitons interacting with each other (despite being massless) because they carry energy-momentum. A natural implication of this model is its explanation of the accelerating expansion of the universe without resorting to dark energy. The increased binding energy within a galaxy requires, by energy conservation, a weakening of gravitational attraction outside said galaxy. This mimics the repulsion of dark energy. The GRSI model is inspired from the Strong Nuclear Force, where a comparable phenomenon occurs. The interaction between gluons emitted by static or nearly static quarks dramatically strengthens quark-quark interaction, ultimately leading to quark confinement on the one hand (analogous to the need of stronger gravity to explain away dark matter) and the suppression 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 the Tully-Fisher relation in galaxy dynamics that is analogous to the Regge 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 Interaction Lagrangians have the same form. These results, however, have been challenged. • The Cosmic Microwave Background anisotropies. • The fainter luminosities of distant supernovae and their consequence on the accelerating expansion of the universe. • The matter power spectrum. • The Hubble tension. • The cosmic coincidence, that is the fact that at present time, the purported repulsion of dark energy nearly exactly cancels the action of gravity in the overall dynamics of the universe. == 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. 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 Einstein's Equivalence Principle has three components. The first is the uniqueness of free fall, also known as the Weak Equivalence Principle. This is satisfied if inertial mass is equal to gravitational mass. η is a parameter used to test the maximum allowable violation of the Weak Equivalence Principle. The first tests of the Weak Equivalence Principle were done by Eötvös before 1900 and limited η to less than 5. Modern tests have reduced that to less than 5. 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. Modern tests have reduced this to less than 1. 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. and Will and Nordtvedt. analyzed 70,205 luminous red galaxies with a cross-correlation involving galaxy velocity estimates and gravitational potentials estimated from lensing and yet results are still tentative. For those theories that aim to replace dark matter, observations like the galaxy rotation curve, the Tully–Fisher relation, the faster velocity dispersions of dwarf galaxies, and the gravitational lensing due to galactic clusters act as constraints. For those theories that aim to replace inflation, the size of ripples in the spectrum of the cosmic microwave background radiation is the strictest test. For those theories that incorporate or aim to replace dark energy, the supernova brightness results and the age of the universe can be used as tests. Another test is the flatness of the universe. With general relativity, the combination of baryonic matter, dark matter and dark energy add up to make the universe exactly flat. == Results of testing theories ==

Parametric post-Newtonian parameters for a range of theories (See Will and Ni for more details. Misner et al. gives a table for translating parameters from the notation of Ni to that of Will) General Relativity is now more than 100 years old, during which one alternative theory of gravity after another has failed to agree with ever more accurate observations. One illustrative example is Parameterized post-Newtonian formalism. The following table lists Parametric post-Newtonian values for a large number of theories. If the value in a cell matches that in the column heading then the full formula is too complicated to include here. † The theory is incomplete, and \zeta_{ 4} can take one of two values. The value closest to zero is listed. All experimental tests agree with general relativity so far, and so Parametric post-Newtonian analysis immediately eliminates all the scalar field theories in the table. A full list of Parametric post-Newtonian parameters is not available for Whitehead, Deser-Laurent, Bollini–Giambiagi–Tiomino, but in these three cases \beta=\xi, which is in strong conflict with general relativity 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, Lee Lightman and Ni are non-starters because they all fail to explain the perihelion advance of Mercury. The bimetric theories of Lightman and Lee, Rosen, Rastall all fail some of the tests associated with strong gravitational fields. The scalar–tensor theories include general relativity as a special case, but only agree with the Parametric post-Newtonian values of general relativity when they are equal to general relativity to within experimental error. As experimental tests get more accurate, the deviation of the scalar–tensor theories from general relativity is being squashed to zero. The same is true of vector–tensor theories, the deviation of the vector–tensor theories from general relativity is being squashed to zero. Further, vector–tensor theories are semi-conservative; they have a nonzero value for \alpha_2 which can have a measurable effect on the Earth's tides. Non-metric theories, such as Belinfante and Swihart, usually fail to agree with experimental tests of Einstein's equivalence principle. And that leaves, as a likely valid alternative to general relativity, nothing except possibly Cartan. That was the situation until cosmological discoveries pushed the development of modern alternatives. == References ==