Alternatives to general relativity are physical theories that attempt to describe the phenomena of gravitation in competition to Einstein's theory of general relativity. Gravitation is a phenomenon through which all objects attract each other. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...

There have been hundreds of different attempts at constructing an ideal theory of gravity. Gravity is a force of attraction that acts between bodies that have mass. ...

These attempts fall into four 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 theory, see the article quantum gravity. For the unification of gravity and other forces, see the article classical unified field theories. For those theories that attempt to do several at once, see the article theory of everything. This article is in need of attention from an expert on the subject. ... This article or section is in need of attention from an expert on the subject. ... In general relativity, it is assumed that the distance between two points in spacetime is given by the metric tensor. ... Quantum gravity is the field of theoretical physics attempting to unify the theory of quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ... Loop quantum gravity (LQG), also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the seemingly incompatible theories of quantum mechanics and general relativity. ... This article is in need of attention. ... In physics, Kaluza-Klein theory (or KK theory, for short) is a model that seeks to unify the two fundamental forces of gravitation and electromagnetism. ... This article or section is in need of attention from an expert on the subject. ... M-theory is a solution proposed for the unknown theory of everything which would combine all five superstring theories and 11-dimensional supergravity together. ... Quantum gravity is the field of theoretical physics attempting to unify the theory of quantum mechanics, which describes three of the fundamental forces of nature, with general relativity, the theory of the fourth fundamental force: gravity. ... This article is in need of attention. ... This article or section is in need of attention from an expert on the subject. ...

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. The law of universal gravitation states that gravitational force between masses decreases with the distance between them, according to an inverse-square law. ... Joseph Louis Lagrange (January 25, 1736 – April 10, 1813) was an Italian mathematician and astronomer who later lived in France and Prussia. ... In 1784 Georges-Louis Le Sage (1724-1803) of Geneva proposed a simple kinetic theory for gravity, which offered a physical explanation for Newtons force equation. ... The Lorentz transformation (LT), named after its discoverer, the Dutch physicist and mathematician Hendrik Antoon Lorentz (1853-1928), forms the basis for the special theory of relativity, which has been introduced to remove contradictions between the theories of electromagnetism and classical mechanics. ... ‹The template below has been proposed for deletion. ... Tests of Einsteins general theory of relativity did not provide an experimental foundation for the theory until well after it was introduced in 1915. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...

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. Quantum field theory (QFT) is the application of quantum mechanics to fields. ... The strong interaction or strong force is today understood to represent the interactions between quarks and gluons as detailed by the theory of quantum chromodynamics (QCD). ... The weak nuclear force or weak interaction is one of the four fundamental forces of nature. ...

After general relativity (GR), attempts were made to either 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, and they can be rejected on the grounds of Occam's Razor until an experimental discrepancy shows up. Further, shortly after that, theorists switched to string theory which was starting to look promising. 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 metres. Subsequent experiments eliminated these. William of Ockham. ...

Motivations for the more recent alternative theories 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. Investigation of the Pioneer anomaly has caused renewed public interest in alternatives to General Relativity, but the Pioneer anomaly is too strong to be explained by any such theory of gravity. In astrophysics, dark matter is matter that does not emit or reflect enough electromagnetic radiation to be detected directly, but whose presence may be inferred from its gravitational effects on visible matter. ... In physical cosmology, dark energy is a hypothetical form of energy that permeates all of space and has strong negative pressure. ... The Pioneer anomaly or Pioneer effect refers to the observed deviation from expectations of the trajectories of various unmanned spacecraft visiting the outer solar system, notably Pioneer 10 and 11. ...

Notation in this article

is the speed of light, is the gravitational constant. "Geometric variables" are not used.

Latin indexes go from 1 to 3, Greek indexes go from 1 to 4. The Einstein summation convention is used. For other topics related to Einstein see Einstein (disambig) In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate equations or formulas. ...

is the Minkowski metric. is a tensor, usually the metric tensor. These have signature ( − , + , + , + ). In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. ...

Covariant differentiation is written or . In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ...

You may also wish to visit Mathematics of general relativity Notational point: General relativity articles using tensors will use the abstract index notation . ...

Early theories, 1686 to 1916

Main source: Pais (1982). See also article History of general relativity. // Creation of General Relativity Early investigations The development of general relativity began in 1907 with the publication of an article by Albert Einstein on acceleration under special relativity. ...

Early theories of gravity, those before General Relativity, include those of Newton (1686), Einstein (1912a & b), Einstein and Grossmann (1913), Nordström (1912, 1913) and Einstein and Fokker (1914). The law of universal gravitation states that gravitational force between masses decreases with the distance between them, according to an inverse-square law. ... In theoretical physics, Nordströms theory of gravitation was an early competitor of general relativity. ...

In Newton's (1686) theory (rewritten using more modern mathematics) the density of mass generates a scalar field by:
Using the Nabla operator , this can be conveniently written as: In vector calculus, del is a vector differential operator represented by the symbol ∇. This symbol is sometimes called the nabla operator, after the Greek word for a kind of harp with a similar shape (with related words in Aramaic and Hebrew). ...

and the scalar field governs the motion of a freely falling particle by:

where the scalar field is

Some theories of gravity between those of Newton and Einstein include those of Lagrange (applying the variational principle), Le Sage (a corpuscular model), Tomasina (a wave model) and Lorentz (a wave model). Poincaré (1908) compares these and concludes that Lagrange's version is correct^[1]. The other models predict very large super-luminal velocities for the speed of gravity and this in turn would lead to extremely rapid heating of the Earth, which doesn't happen. In 1784 Georges-Louis Le Sage (1724-1803) of Geneva proposed a simple kinetic theory for gravity, which offered a physical explanation for Newtons force equation. ...

The theory of Newton (1686), and Lagrange's improvement on the calculation, 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. The law of universal gravitation states that gravitational force between masses decreases with the distance between them, according to an inverse-square law. ...

Einstein's two part publication in 1912 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: The Lorentz transformation (LT), named after its discoverer, the Dutch physicist and mathematician Hendrik Antoon Lorentz (1853-1928), forms the basis for the special theory of relativity, which has been introduced to remove contradictions between the theories of electromagnetism and classical mechanics. ... The principle of least action was first formulated by Pierre-Louis Moreau de Maupertuis, who said that Nature is thrifty in all its actions. See action (physics). ...

where is the Minkowski metric, and there is a summation from 1 to 4 over indices and . In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ...

Einstein and Grossmann (1913) includes Riemannian geometry and tensor calculus. In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ... It has been suggested that this article or section be merged with Covariant. ...

The equations of electrodynamics exactly match those of GR. The equation Classical electrodynamics (or classical electromagnetism) is a theory of electromagnetism that was developed over the course of the 19th century, most prominently by James Clerk Maxwell. ...

is not in GR. It expresses the stress-energy tensor as a function of the matter density. This article is in need of attention from an expert on the subject. ...

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. Max Abraham (March 26, 1875 - November 16, 1922) was a German physicist. ...

The first approach of Nordström (1912) was to retain the Minkowski metric and a constant value of but to let mass depend on the gravitational field strength . Allowing this field strength to satisfy In theoretical physics, Nordströms theory of gravitation was an early competitor of general relativity. ...

where is rest mass energy and is the d'Alembertian, In special relativity, electromagnetism and wave theory, the dAlembert operator, also called dAlembertian, is the Laplace operator of Minkowski space. ...

and

where 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): In theoretical physics, Nordströms theory of gravitation was an early competitor of general relativity. ...

where is a scalar field,

This theory is Lorentz invariant, satisfies the conservation laws, correctly reduces to the Newtonian limit and satisfies the weak equivalence principle. In relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ...

Einstein and Fokker (1914) is Einstein's first treatment of gravitation in which general covariance is strictly obeyed. Writing:

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

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

Einstein (1916, 1917) is what we now know of as General Relativity. Discarding the Minkowski metric entirely, Einstein gets:

which can also be written

Five days before Einstein presented the last equation above, Hilbert had submitted a paper containing an almost identical equation. See relativity priority dispute. Hilbert was the first to correctly state the Einstein-Hilbert action for GR, which is: Albert Einstein presented the theories of Special Relativity and General Relativity in groundbreaking publications that did not include references to the work of others. ... In general relativity, Einsteins field equations can be derived from an action principle starting from the Einstein-Hilbert action: where g is the (pseudo)Riemannian metric, R is the Ricci scalar, n is the number of spacetime dimensions and k is a constant which depends on the units chosen...

where is Newton's gravitational constant, is the Ricci curvature of space, and is the action due to mass. In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ... In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ...

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)^[2], Lang (2002)^[3]):

Listed by date (the hyperlinks take you further down this article)

Whitehead (1922), Cartan (1922, 1923), Fierz & Pauli (1939), Birkhov (1943), Milne (1948), Thiry (1948), Papapetrou (1954a, 1954b), Littlewood (1953), Jordan (1955), Bergman (1956), Belinfante & Swihart (1957), Yilmaz (1958, 1973), Brans & Dicke (1961), Whitrow & Morduch (1960, 1965), Kustaanheimo (1966) , Kustaanheimo & Nuotio (1967), Deser & Laurent (1968), Page & Tupper (1968), Bergmann (1968), Bollini-Giambini-Tiomno (1970), Nordtveldt (1970), Wagoner (1970), Rosen (1971, 1975, 1975), Ni (1972, 1973), Will & Nordtveldt (1972), Hellings & Nordtveldt (1973), Lightman & Lee (1973), Lee, Lightman & Ni (1974), Beckenstein (1977), Barker (1978), Rastall (1979)

These theories are presented here without a cosmological constant, how to add a cosmological constant or quintessence is discussed under Modern Theories (see also here) 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 Perlmutter. In general relativity, Einsteins field equations can be derived from an action principle starting from the Einstein-Hilbert action: where g is the (pseudo)Riemannian metric, R is the Ricci scalar, n is the number of spacetime dimensions and k is a constant which depends on the units chosen...

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, say , then the action is the integral of that, for example In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ... The principle of least action was first formulated by Pierre-Louis Moreau de Maupertuis, who said that Nature is thrifty in all its actions. See action (physics). ... A variational principle is a principle in physics which is expressed in terms of the calculus of variations. ... A Lagrangian of a dynamical system, named after Joseph Louis Lagrange, is a functional of the dynamical variables which concisely describes the equations of motion of the system. ... In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ...

where is the curvature of space. In this equation it is usual, though not essential, to have .

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. In physics, the action is an integral quantity that is used to determine the evolution of a physical system between two defined states using the calculus of variations. ... In physics, Modified Newtonian Dynamics (MOND) is a theory that explains the galaxy rotation problem without assuming the existence of dark matter. ...

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 metric tensor of signature 1, which governs proper-length and proper-time measurements in the usual manner of special and general relativity: In mathematics, the metric tensor is a symmetric tensor field of rank 2 that is used to measure distance in a space. ... The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. ...

where there is a summation over indices μ and ν. Condition 2. Stressed matter and fields being acted upon by gravity respond in accordance with the equation:

where is the stress-energy tensor for all matter and non-gravitational fields, and where is the covariant derivative with respect to the metric. This article is in need of attention from an expert on the subject. ... In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. ...

Any theory of gravity in which is always true is not a metric theory, but any metric theory can perfectly well be given a mathematical description that violates conditions i and ii.

Metric theories include (from simplest to most complex):

• Scalar Field Theories (includes Conformally flat theories & Stratified theories with conformally flat space slices)

Nordström, Einstein-Fokker, Whitrow-Morduch, Littlewood, Bergman, Page-Tupper, Einstein (1912), Whitrow-Morduch, Rosen (1971), Papapetrou, Ni, Yilmaz, [Coleman], Lee-Lightman-Ni

• Bimetric Theories

Rosen (1975), Rastall, Lightman-Lee

• Quasilinear Theories (includes Linear fixed gauge)

Whitehead, Deser-Laurent, Bollini-Giambini-Tiomno

• Tensor-Theories

Einstein's GR

• Scalar-Tensor Theories

Thiry, Jordan, Brans-Dicke, Bergmann, Wagoner, Nordtvedt, Beckenstein

• Vector-tensor theories

Will-Nordtvedt, Hellings-Nordvedt

• Other metric theories

(see section Modern Theories below)

Non-metric theories include Cartan, Belinfante-Swihart

A word here about Mach's principle is appropriate because a few of these theories rely on Mach's principle eg. Whitehead (1922), and many mention it in passing eg. 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^[4]: This article is in need of attention from an expert on the subject. ...

• 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.

Scalar Field Theories

(see also Scalar theories of gravitation) Scalar theories of gravitation are field theories of gravitation in which the gravitational field is described using a scalar field, which is required to satisfy some field equation. ...

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:

where the scalar field is,

and may or may not depend on .

In Nordström (1912),

 ;

In Littlewood (1953) and Bergmann (1956),

 ;

In Whitrow and Morduch (1960),

 ;

In Whitrow and Morduch (1965),

 ;

In Page and Tupper (1968),

 ;

Page and Tupper (1968) matches Yilmaz (1958) (see also Yilmaz theory of gravitation) to second order when . The Yilmaz theory of gravitation is an attempt by Hüseyn Yilmaz and a handful of coworkers to formulate a classical field theory of gravitation which closely mimics general relativity in weak-field conditions, but in which event horizons cannot appear. ...

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:

Misner et al. (1973) gives this without the term. is the matter action.

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 and that are related to the metric by:

Ni (1972) quotes Rosen (1971) as having two scalar fields and that are related to the metric by:

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

In Papapetrou (1954b) there is a second scalar field . The gravitational part of the Lagrangian is now:

Bimetric theories

(see also Bimetric theory) In general relativity, it is assumed that the distance between two points in spacetime is given by the metric tensor. ...

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:

where the vertical line "|" denotes covariant derivative with respect to . The field equations may be written in the form: In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. ...

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 , , and two constants and the action is:

and the stress-energy tensor comes from:

In Rastall (1979), the metric is an algebraic function of the Minkowski metric and a Vector field^[2]. The Action is:

where

and

(see Will (1981) for the field equation for and ).

Quasilinear Theories

In Whitehead (1922), the physical metric is constructed algebraically from the Minkowski metric and matter variables, so it doesn't even have a scalar field. The construction is:

where the superscript (-) indicates quantities evaluated along the past light cone of the field point and

, , , ,

Deser and Laurent (1968) and Bollini-Giambini-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) to define

The action is:

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 . In differential geometry, the curvature form describes curvature of principal bundle with connection. ...

Tensor Theories

There is only one theory that possesses a single gravitational field that is the metric tensor. Of course this theory is General Relativity.

Scalar-Tensor Theories

(see also Scalar-tensor theory and Brans-Dicke theory) This article is in need of attention from an expert on the subject. ... This article or section is in need of attention from an expert on the subject. ...

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). In physics, Kaluza–Klein theory (or KK theory, for short) is a model that seeks to unify the two fundamental forces of gravitation and electromagnetism. ... In physics, Kaluza–Klein theory (or KK theory, for short) is a model that seeks to unify the two fundamental forces of gravitation and electromagnetism. ...

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 is based on the integral of the Lagrangian .

where is a different dimensionless function for each different scalar-tensor theory. The function plays the same role as the cosmological constant in GR. is a dimensionless normalization constant that fixes the present-day value of . 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),

I'm not worrying about λ until later in the article, it's discussed under Cosmological Constant.

Brans-Dicke (1961), is constant

Bekenstein (1977) Variable Mass Theory Starting with parameters and , found from a cosmological solution, determines function then

Barker (1978) Constant G Theory

Adjustment of allows Scalar Tensor Theories to tend to GR in the limit of 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.

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 . The gravitational action is:

where , , and are constants and

See Will (1981) for the field equations for and .

Will and Nordtvedt (1972) is a special case where

 ;

Hellings and Nordtvedt (1973) is a special case where

 ;  ;

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 they reduce to GR so, so long as GR is confirmed by evperiment, general vector-tensor theories can never be ruled out.

Other metric theories

Others metric theories have been proposed; that of Beckenstein (2004) is discussed under Modern Theories.

Non-metric theories

(see also Einstein-Cartan theory and Cartan connection) This article is in need of attention from an expert on the subject. ... In mathematics, the Cartan connection construction of differential geometry is a flexible generalisation of the connection concept, developed by Élie Cartan. ...

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 and by Heyl in the years 1958 to 1966.

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:

The is the linear connection. is not related in any way to Minkowski but is derived from a completely antisymmetric pseudo-tensor as follows:

and 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:

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.

Testing of alternatives to General Relativity

Main article: Tests of general relativity

The development of theories and tests has gone hand in hand. Most tests can be classified as (see Will 2001): Tests of Einsteins general theory of relativity did not provide an experimental foundation for the theory until well after it was introduced in 1915. ...

Basic Viability. Einstein's Equivalence Principle (EEP). Parametric Post-Newtonian (PPN). Strong Gravity. Gravitational Waves.

Theories that fail Basic Viability Tests

For more details see Misner et al. (1973) Ch.39 and Will (1981) Table 2.1.

Not all theories of gravity are created equal. Very few, among the multitude in the literature, are sufficiently viable to be worth comparison with General Relativity.

In the early 1970s a group of people at Caltech including Thorne, Will and Ni (see Ni (1972)) complied a list of twentieth-century theories of gravity. Of each theory they asked the following questions: (i) is it self-consistent? (ii) is it complete? (iii) does it agree, to within several standard deviations, with all experiments performed to date? If a theory failed these criteria they did not always reject it out of hand. If the theory was incomplete in its initial statement, they sometimes tried to complete it by making minor modifications, usually by insisting that in the absence of gravity the laws of physics reverted to those of Special Relativity. For seven of the theories here, the density could be calculated from either or is the trace of ; both were considered. Even if that failed, eg. consider the case of Thirry (1948) and Jordan (1955). These are incomplete unless Jordan's parameter is set to -1, in which case they match the theory of Brans-Dicke (1961) and so are worthy of further consideration. In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i. ...

For this section of the present article, the criterion of "all experiments performed to date" is replaced by "with the gross features of Newtonian Mechanics and Special Relativity". Subtler problems are discussed later.

Self-consistency among non-metric theories includes eliminating theories allowing tachyons, ghost poles, 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.

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. 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 twice that of GR) but light as waves is not.

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, which disagrees violently with experiment.

A modern example of the lack of a relativistic component is MOND by Milgrom, more on that later.

The Einstein Equivalence Principle (EEP)

Main article: Equivalence principle

The EEP has three components. In relativity, the equivalence principle is applied to several related concepts dealing with gravitation and the uniformity of physical measurements in different frames of reference. ...

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. 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 to less than 5e-9. Modern tests have reduced that to less than 5e-13.

The second is Lorentz invariance. In the absence of gravitational effects the speed of light is constant. The test parameter for this is . The first tests of Lorentz invariance were done by Michelson and Morley before 1890 and limited to less than 5e-3. Modern tests have reduced this to less than 1e-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. Spacial local position invariance is tested using gravitational redshift measurements. The test parameter for this is . Upper limits on this found by Pound and Rebka in 1960 limited to less than 0.1. Modern tests have reduced this to less than 1e-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.

One method for testing EEP is called formalism. This is beyond the scope of this article but in a static spherically symmetric gravitational field EEP is satisfied if .

Theories of gravity may be metric or non-metric. In metric theories, space-time is endowed with a symmetric metric (usually written ) and trajectories of freely falling bodies are geodesics of that metric. Metric theories satisfy the Einstein Equivalence Principle. Extremely few non-metric theories, if any, satisfy this.

One theory eliminated by formalism is the non-metric theory of Belinfante & Swihart (1957).

Parametric Post-Newtonian (PPN) Formalism

Main article: Parameterized post-Newtonian formalism

See also Tests of general relativity, Misner et al. (1973) and Will (1981) for more information. The parameterized post-Newtonian formalism or PPN formalism is a tool used to compare classical theories of gravitation in the limit most important for everyday gravitational experiments: the limit in which the gravitational field is weak and generated by objects moving slowly compared to the speed of light. ... Tests of Einsteins general theory of relativity did not provide an experimental foundation for the theory until well after it was introduced in 1915. ...

Work on developing a standard rather than ah-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 is a measure of how much a theory departs from Newtonian gravity in a different way. Because we're talking about deviation form Newtonian theory here, this only a measure of weak-field effects. The effects of strong gravitational fields are examined later.

These ten are called : , , , , , , , , ,

is a measure of space curvature, being zero for Newtonian gravity and one for GR.

is a measure of nonlinearity in the addition of gravitational fields, one for GR.

is a check for preferred location effects.

, , measure the extent and nature of "preferred-frame effects". Any theory of gravity with at least one α nonzero is called a preferred-frame theory.

, , , , measure the extent and nature of breakdowns in global conservation laws. A theory of gravity possesses 4 conservation laws for energy-momentum and 6 for angular momentum only if all five are zero.

Strong Gravity and Gravitational Waves

Main article: Tests of general relativity

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. Tests of Einsteins general theory of relativity did not provide an experimental foundation for the theory until well after it was introduced in 1915. ...

GR predicts that gravitational waves travel at the speed of light. Many alternatives to GR say that gravitational waves travel faster than light. If true, this could result in failure of causality.

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. In astrophysics, dark matter is matter that does not emit or reflect enough electromagnetic radiation to be detected directly, but whose presence may be inferred from its gravitational effects on visible matter. ... Rotation curve of a typical spiral galaxy: predicted (A) and observed (B). ... In astronomy, the Tully-Fisher relation, published by astronomers R. Brent Tully and J. Richard Fisher in 1977, is a standard candle that measures the distance to rotating spiral galaxies by the width of the galaxys spectral lines. ... This article is in need of attention from an expert on the subject. ...

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. In cosmology, the cosmic microwave background radiation (most often abbreviated CMB but occasionally CMBR, CBR or MBR, also referred as relic radiation) is a form of electromagnetic radiation discovered in 1965 that fills the entire universe. ...

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. In physical cosmology, dark energy is a hypothetical form of energy that permeates all of space and has strong negative pressure. ...

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 90 years old, but while one alternative theory of gravity after another has failed to agree with ever more accurate observations. Nothing illustrates this more clearly than Parameterized post-Newtonian formalism (PPN). The parameterized post-Newtonian formalism or PPN formalism is a tool used to compare classical theories of gravitation in the limit most important for everyday gravitational experiments: the limit in which the gravitational field is weak and generated by objects moving slowly compared to the speed of light. ...

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.

† The theory is incomplete, and ζ₄ 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-Giamiago-Tiomino (1970), but in these three cases β = γ, which is in strong conflict with GR and experimental results. In particular, these theories predict incorrect amplitudes for the Earth's tides..

Theories that Fail other tests

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

The stratified theories of Ni (1973), Lee Lightman and Ni (1974) 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. 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 α₂ which can have a measurable effect on the Earth's tides.

And that leaves, as a likely valid alternative to GR, nothing [except possibly Cartan (1922), which may violate EEP].

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

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: Beckenstein (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 of opinion 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.

The discovery of the accelerated expansion of the universe by Perlmutter 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 Rerlmutter's results in a completely different way.

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 the qualitative features of the anomaly but not the magnitude. Any alternative model of gravity that could explain the Pioneer anomaly would have to depart so strongly from GR that it would fail to satisfy other experimental observations. The Pioneer anomaly or Pioneer effect refers to the observed deviation from expectations of the trajectories of various unmanned spacecraft visiting the outer solar system, notably Pioneer 10 and 11. ...

Cosmological Constant and Quintessence

(also see Cosmological constant, Einstein-Hilbert action, Quintessence (physics)) The cosmological constant (usually denoted by the Greek capital letter lambda: Λ) was proposed by Albert Einstein as part of his theory of general relativity to achieve a stationary universe. ... In general relativity, Einsteins field equations can be derived from an action principle starting from the Einstein-Hilbert action: where g is the (pseudo)Riemannian metric, R is the Ricci scalar, n is the number of spacetime dimensions and k is a constant which depends on the units chosen... In physics, quintessence is a hypothetical form of dark energy postulated as an explanation of observations of an accelerating universe. ...

The cosmological constant is a very old idea, going back to Einstein in 1917. The success of the Friedmann model of the universe in which led to the general acceptance that it is zero, but the use of a non-zero value came back with a vengeance when Perlmutter discovered 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:

to

In GR, it changes the Einstein-Hilbert action from

to

which changes the field equation

to

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 can be added to scalar tensor theories. This can also be done in every alternative the GR that contains a scalar field by adding the term inside the Lagrangian for the gravitational part of the action, the part of

Because 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

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

Relativistic MOND

(see Modified Newtonian dynamics, Tensor-vector-scalar gravity, and Beckenstein (2004) for more details). In physics, Modified Newtonian Dynamics (MOND) is a theory that explains the galaxy rotation problem without assuming the existence of dark matter. ... This article or section is in need of attention from an expert on the subject. ...

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

By 1984, problems ii. and iii. 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:

The relativistic version of this has:

with a nonstandard mass action. Here and are arbitrary functions selected to give Newtonian and MOND behaviour in the correct limits.

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.

Beckenstein (2004) introduced a tensor-vector-scalar model (TeVeS). This has two scalar fields and and vector field . The action is split into parts for gravity, scalars, vector and mass. This article or section is in need of attention from an expert on the subject. ...

The gravity part is the same as in GR.

where , is a length scale, and are constants, square brackets in indices represent anti-symmetrization is a Lagrange multiplier (calculated elsewhere), , and is a Lagrangian translated from flat spacetime onto the metric .

is an arbitrary function, and is given as an example with the right asymptotic behaviour; note how it becomes undefined when

Moffat's Theories

Moffat (1995) developed a non-symmetric gravitation theory (NGT). This is not a metric theory. It is claimed that it does not contain a black hole horizon. Burko and Ori 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 & MaCarthy (1993) have criticised NGT, saying that it has unacceptable asymptotic behaviour. Nonsymmetric Gravitational Theory is a modification of Einsteins theory of General Relativity that tries to explain the mystery of Dark Matter. ...

The mathematics is not difficult but is intertwined so the following is only a brief sketch. Starting with a non-symmetric tensor , the Lagrangian density is split into

where is the same as for matter in GR.

where is a curvature term analogous to but not equal to the Ricci curvature in GR, and are cosmological constants, is the antisymmetric part of . is a connection, and is a bit difficult to explain because it's defined recursively. However,

Moffat (2002) is a claimed to be scalar-tensor bimetric gravity theory (BGT) and is one of the many theories of gravity in which the speed of light is faster in the early universe. These theories were motivated partly be the desire to avoid the "horizon problem" without invoking inflation. It has a variable . The theory also attempts to explain the dimming of supernovae from a perspective other than the acceleration of the universe and so runs the risk of predicting an age for the universe that is too small.

Overall, this theory looks poor. The action is split into gravity, scalar field, and matter parts. The gravity and scalar field equations are standard for a Brans-Dicke theory with cosmological constant and scalar potential, but applied with a Minkowski metric! Only the matter term uses a non-flat metric, and that is where has dimensions of length squared. This theory is going to fail Lorentz invariance and deflection of light tests at the very least.

Moffat (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 , increasing to a final constant value about a million years after the big bang.

The theory seems to contain an asymmetric tensor field and a source current vector. The action is split into:

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

where

and 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 (SVTG) theory. To meet Wikipedias quality standards, this article or section may require cleanup. ...

The theory contains a tensor, vector and three scalar fields. But the equations are quite straightforward. The action is split into: with terms for gravity, vector field K_μ, scalar fields , & , and mass. is the standard gravity term with the exception that is moved inside the integral.

where

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

where is a coupling constant. The functions assumed for the scalar potentials are not stated.

Other Modern Theories

Footnotes

1. ^ Le Sage's theory of gravitation
2. ^ ^{a b} A later edition is Will (1993). See also Ni (1972)
3. ^ Although an important source for this article, the presentations of Turyshev (2006) and Lang (2002) contain many errors of fact
4. ^ this isn't exactly the way Mach originally stated it, see other variants in Mach principle

In 1784 Georges-Louis Le Sage (1724-1803) of Geneva proposed a simple kinetic theory for gravity, which offered a physical explanation for Newtons force equation. ... Machs principle is a hypothesis first stated by the physicist and philosopher Ernst Mach in 1893, which states that: The inertia of any system is the result of the interaction of that system and the rest of the universe. ...

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