Alternatives to general relativity
Encyclopedia
Alternatives to general relativity are physical theories that attempt to describe the phenomena of gravitation
in competition to Einstein's theory of general relativity
.
There have been many different attempts at constructing an ideal theory of gravity. These attempts can be split into four broad categories:
This article deals only with straightforward alternatives to GR. For quantized gravity theories, see the article quantum gravity
. For the unification of gravity and other forces, see the article classical unified field theories
. For those theories that attempt to do several at once, see the article theory of everything
.
). Then came unsuccessful attempts to combine gravity and either wave or corpuscular theories
of gravity. The whole landscape of physics was changed with the discovery of Lorentz transformations, and this led to attempts to reconcile it with gravity. At the same time, experimental physicists started testing the foundations of gravity and relativity – Lorentz invariance
, the gravitational deflection of light
, the Eötvös experiment
. These considerations led to and past the development of general relativity
.
After that, motivations differ. Two major concerns were the development of quantum theory
and the discovery of the strong
and weak
nuclear forces. Attempts to quantize and unify gravity are outside the scope of this article, and so far none has been completely successful.
After general relativity (GR), attempts were made either to improve on theories developed before GR, or to improve GR itself. Many different strategies were attempted, for example the addition of spin to GR, combining a GR-like metric with a space-time that is static with respect to the expansion of the universe, getting extra freedom by adding another parameter. At least one theory was motivated by the desire to develop an alternative to GR that is completely free from singularities.
Experimental tests improved along with the theories. Many of the different strategies that were developed soon after GR were abandoned, and there was a push to develop more general forms of the theories that survived, so that a theory would be ready the moment any test showed a disagreement with GR.
By the 1980s, the increasing accuracy of experimental tests had all led to confirmation of GR, no competitors were left except for those that included GR as a special case. Further, shortly after that, theorists switched to string theory which was starting to look promising, but has since lost popularity. In the mid 1980s a few experiments were suggesting that gravity was being modified by the addition of a fifth force (or, in one case, of a fifth, sixth and seventh force) acting on the scale of meters. Subsequent experiments eliminated these.
Motivations for the more recent alternative theories are almost all cosmological, associated with or replacing such constructs as "inflation
", "dark matter
" and "dark energy
". Investigation of the Pioneer anomaly
has caused renewed public interest in alternatives to General Relativity.
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.
is the Minkowski metric
.
is a tensor, usually the metric tensor
. These have signature
(−,+,+,+).
Partial differentiation
is written
or 
. Covariant differentiation is written 
or 
.
If a theory has a Lagrangian density for gravity, say
, then the gravitational part of the action 
is the integral of that.
In this equation it is usual, though not essential, to have
at spatial infinity when using Cartesian coordinates. For example the Einstein–Hilbert action uses
where R is the scalar curvature
, a measure of the curvature of space.
Almost every theory described in this article has an action
. It is the only known way to guarantee that the necessary conservation laws of energy, momentum and angular momentum are incorporated automatically; although it is easy to construct an action where those conservation laws are violated. The original 1983 version of MOND
did not have an action.
A few theories have an action but not a Lagrangian density. A good example is Whitehead (1922), the action there is termed non-local.
A theory of gravity is a "metric theory" if and only if it can be given a mathematical representation in which two conditions hold:
Condition 1: There exists a symmetric metric tensor
of signature
(−, +, +, +), which governs proper-length and proper-time measurements in the usual manner of special and general relativity:
where there is a summation over indices
and 
.
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 and
is the Christoffel symbol. The stress-energy tensor should also satisfy an energy condition
.
Metric theories include (from simplest to most complex):
(see section Modern theories below)
Non-metric theories include
A word here about Mach's principle
is appropriate because a few of these theories rely on Mach's principle (e.g. Whitehead (1922)), and many mention it in passing (e.g. Einstein–Grossmann (1913), Brans–Dicke (1961)). Mach's principle can be thought of a half-way-house between Newton and Einstein. It goes this way:
So far, all the experimental evidence points to Mach's principle being wrong, but it has not entirely been ruled out.
In Newton's (1686) theory (rewritten using more modern mathematics) the density of mass
generates a scalar field, the gravitational potential 
in joules per kilogram, by
Using the Nabla operator
for the gradient
and divergence
(partial derivatives), this can be conveniently written as:
This scalar field governs the motion of a free-fall
ing particle by:
At distance, r, from an isolated mass, M, the scalar field is
The theory of Newton, and Lagrange's improvement on the calculation (applying the variational principle), completely fails to take into account relativistic effects of course, and so can be rejected as a viable theory of gravity. Even so, Newton's theory is thought to be exactly correct in the limit of weak gravitational fields and low speeds and all other theories of gravity need to reproduce Newton's theory in the appropriate limits.
Mechanical explanations (1650–1900)
To explain Newton's theory, some mechanical explanations of gravitation
(incl. Le Sage's theory
) were created between 1650 and 1900, but they were overthrown because most of them lead to an unacceptable amount of drag
, which is not observed. Other models are violating the energy conservation law and are incompatible with modern thermodynamics
.
Electrostatic models (1870–1900)
At the end of the 19th century, many tried to combine Newton's force law with the established laws of electrodynamics, like those of Weber
, Carl Friedrich Gauß, Bernhard Riemann
and James Clerk Maxwell
. Those models were used to explain the perihelion advance of Mercury. In 1890, Lévy succeeded in doing so by combining the laws of Weber and Riemann, whereby the speed of gravity
is equal to the speed of light in his theory. And in another attempt, Paul Gerber
(1898) even succeeded in deriving the correct formula for the Perihelion shift (which was identical to that formula later used by Einstein). However, because the basic laws of Weber and others were wrong (for example, Weber's law was superseded by Maxwell's theory), those hypothesis were rejected. In 1900, Hendrik Lorentz
tried to explain gravity on the basis of his Lorentz ether theory
and the Maxwell equations. He assumed, like Ottaviano Fabrizio Mossotti and Johann Karl Friedrich Zöllner
, that the attraction of opposite charged particles is stronger than the repulsion of equal charged particles. The resulting net force is exactly what is known as universal gravitation, in which the speed of gravity is that of light. But Lorentz calculated that the value for the perihelion advance of Mercury was much too low.
Lorentz-invariant models (1905–1910)
Based on the principle of relativity
, Henri Poincaré
(1905, 1906), Hermann Minkowski
(1908), and Arnold Sommerfeld
(1910) tried to modify Newton's theory and to establish a Lorentz invariant
gravitational law, in which the speed of gravity is that of light. However, like in Lorentz's model the value for the perihelion advance of Mercury was much too low.
Einstein (1908, 1912)
Einstein's two part publication in 1912 (and before in 1908) is really only important for historical reasons. By then he knew of the gravitational redshift and the deflection of light. He had realized that Lorentz transformations are not generally applicable, but retained them. The theory states that the speed of light is constant in free space but varies in the presence of matter. The theory was only expected to hold when the source of the gravitational field is stationary. It includes the principle of least action
:
where
is the Minkowski metric
, and there is a summation from 1 to 4 over indices
and 
.
Einstein and Grossmann (1913) includes Riemannian geometry
and tensor calculus
.
The equations of electrodynamics
exactly match those of GR. The equation
is not in GR. It expresses the stress-energy tensor
as a function of the matter density.
Abraham (1912)
While this was going on, Abraham
was developing an alternative model of gravity in which the speed of light depends on the gravitational field strength and so is variable almost everywhere. Abraham's 1914 review of gravitation models is said to be excellent, but his own model was poor.
Nordström (1912)
The first approach of Nordström (1912)
was to retain the Minkowski metric and a constant value of
but to let mass depend on the gravitational field strength 
. Allowing this field strength to satisfy
where
is rest mass energy and 
is the d'Alembertian,
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):
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
.
Einstein and Fokker (1914)
This theory 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)
This theory 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:
where
is Newton's gravitational constant, 
is the Ricci curvature
of space,
and 
is the action
due to mass.
GR is a tensor theory, the equations all contain tensors. Nordström's theories, on the other hand, are scalar theories because the gravitational field is a scalar. Later in this article you will see scalar-tensor theories that contain a scalar field in addition to the tensors of GR, and other variants containing vector fields as well have been developed recently.
Those considered here include (see Will (1981), Lang (2002)):
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), Wei-Tou 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 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. How to add a cosmological constant or quintessence to a theory is discussed under Modern Theories (see also here
).
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 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:
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:
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. The Action is:
where
(see Will (1981) for the field equation for
and 
).
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
.
A cosmological constant
can be introduced into a quasilinear theory by the simple expedient of changing the Minkowski background to a de Sitter
or anti-de Sitter spacetime, as suggested by G. Temple in 1923. Temple's suggestions on how to do this were criticized by C. B. Rayner in 1955.
is the simplest plausible theory of gravity that can be based on just one symmetric tensor field (the metric tensor
). Others include: Gauss–Bonnet gravity, f(R) gravity
, and Lovelock theory of gravity
.
Although not normally considered a Scalar-Tensor theory of gravity, the 5 by 5 metric of Kaluza-Klein
reduces to a 4 by 4 metric and a single scalar. So if the 5th element is treated as a scalar gravitational field instead of an electromagnetic field then Kaluza-Klein
can be considered the progenitor of Scalar-Tensor theories of gravity. This was recognised by Thiry (1948).
Scalar-Tensor theories include Thiry (1948), Jordan (1955), Brans and Dicke (1961), Bergman (1968), Nordtveldt (1970), Wagoner (1970), Bekenstein (1977) and Barker (1978).
The action
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),
Since
was thought to be zero at the time anyway, this would not have been considered a significant difference. The role of the cosmological constant in more modern work is discussed under Cosmological constant.
Brans-Dicke (1961),
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.
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 experiment, general vector-tensor theories can never be ruled out.
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 the completely antisymmetric pseudo-tensor (Levi-Civita symbol
) with
, 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.
s, ghost poles and higher order poles, and those that have problems with behaviour at infinity.
Among metric theories, self-consistency is best illustrated by describing several theories that fail this test. The classic example is the spin-two field theory of Fierz and Pauli (1939); the field equations imply that gravitating bodies move in straight lines, whereas the equations of motion insist that gravity deflects bodies away from straight line motion. Yilmaz (1971, 1973) contains a tensor gravitational field used to construct a metric; it is mathematically inconsistent because the functional dependence of the metric on the tensor field is not well defined.
Many early theories are incomplete in that it is unclear whether the density
used by the theory should be calculated from the stress-energy tensor 
as 
or as 
, where 
is the four-velocity
, and
is the Kronecker delta.
The theories of Thirry (1948) and Jordan (1955) 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.
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.
Each theory should reproduce the observed results in these areas, which have to date always aligned with the predictions of general relativity.
In 1964, Irwin I. Shapiro
found a fourth test, called the Shapiro delay
. It is usually regarded as a "classical" test as well.
A modern example of the lack of a relativistic component is MOND by Milgrom, as will be discussed below.
The first is the uniqueness of free fall, also known as the Weak Equivalence Principle (WEP). This is satisfied if inertial mass is equal to gravitational mass. η is a parameter used to test the maximum allowable violation of the WEP. The first tests of the WEP were done by Eötvös before 1900 and limited η to less than 5. 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.
Schiff
's conjecture states that any complete, self-consistent theory of gravity that embodies the WEP necessarily embodies EEP. This is likely to be true if the theory has full energy conservation.
Metric theories satisfy the Einstein Equivalence Principle. Extremely few non-metric theories, if any, satisfy this. For example, the non-metric theory of Belinfante & Swihart (1957) is eliminated by the THεμ formalism for testing EEP.
, Misner et al. (1973) and Will (1981) for more information.
Work on developing a standardized rather than ad-hoc set of tests for evaluating alternative gravitation models began with Eddington in 1922 and resulted in a standard set of PPN numbers in Nordtvedt and Will (1972) and Will and Nordtvedt (1972). Each parameter measures a different aspect of how much a theory departs from Newtonian gravity. Because we are talking about deviation from Newtonian theory here, these only measure weak-field effects. The effects of strong gravitational fields are examined later.
These ten are 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 
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.
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.
, the galaxy rotation curve
, the Tully-Fisher relation
, the faster rotation rate of dwarf galaxies, and the gravitational lens
ing due to galactic clusters act as constraints.
For those theories that aim to replace inflation
, the size of ripples in the spectrum of the cosmic microwave background radiation
is the strictest test.
For those theories that incorporate or aim to replace dark energy
, the supernova brightness results and the age of the universe can be used as tests.
Another test is the flatness of the universe. With GR, the combination of baryonic matter, dark matter and dark energy add up to make the universe exactly flat. As the accuracy of experimental tests improve, alternatives to GR that aim to replace dark matter or dark energy will have to explain why.
General Relativity is now more than 90 years old, during which 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 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
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
, which is in strong conflict with GR and experimental results. In particular, these theories predict incorrect amplitudes for the Earth's tides.
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
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.
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.
There was a slow dawning realisation in the physics world that there were several problems inherent in the then big bang scenario, two of these were the horizon problem
and the observation that at early times when quarks were first forming there was not enough space on the universe to contain even one quark. Inflation theory was developed to overcome these. Another alternative was constructing an alternative to GR in which the speed of light was larger in the early universe.
The discovery of unexpected rotation curves for galaxies took everyone by surprise. Could there be more mass in the universe than we are aware of, or is the theory of gravity itself wrong? The consensus now is that the missing mass is "cold dark matter", but that consensus was only reached after trying alternatives to general relativity and some physicists still believe that alternative models of gravity might hold the answer.
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 Perlmutter'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 this anomaly.
, Einstein–Hilbert action, Quintessence (physics)
)
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.
, Tensor-vector-scalar gravity
, and Beckenstein (2004) for more details).
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, and 
is the MOND length scale.
By 1988, a second scalar field (PCC) fixed problems with the earlier scalar-tensor version but is in conflict with the perihelion precession of Mercury and gravitational lensing by galaxies and clusters.
By 1997, MOND had been successfully incorporated in a stratified relativistic theory [Sanders], but as this is a preferred frame theory it has problems of its own.
Bekenstein (2004) introduced a tensor-vector-scalar
model (TeVeS). This has two scalar fields
and 
and vector field 
. The action is split into parts for gravity, scalars, vector and mass.
The gravity part is the same as in GR.
where
, 
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 
Milgrom proposed a "bimetric MOND" or "BIMOND" theory, with action
with
and 
the (noninteracting) matter actions attached to the two metrics, 
a tensor derived from the difference in the metrics' connections, 
the ratio between the two metric traces, and 
are free parameters. 
is a function which depends on some contractions of the 
tensors.
Assuming that
depends only on the scalar contraction of 
, Milgrom obtained as a nonrelativistic limit his bi-potential version of MOND with action
Here
should scale as 
in the deep-MOND limit and as 
in the Newtonian limit.
(NGT). This is not a metric theory. It was first claimed that it does not contain a black hole horizon, but Burko and Ori (1995) have found that NGT can contain black holes. Later, Moffat claimed that it has also been applied to explain rotation curves of galaxies without invoking "dark matter". Damour, Deser & MaCarthy (1993) have criticised NGT, saying that it has unacceptable asymptotic behaviour.
The mathematics is not difficult but is intertwined so the following is only a brief sketch. Starting with a non-symmetric tensor
, 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's (2002) theory is a 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.
Moffat's (2005a) metric-skew-tensor-gravity (MSTG) theory is able to predict rotation curves for galaxies without either dark matter or MOND, and claims that it can also explain gravitational lensing of galaxy clusters without dark matter. It has variable
, 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.
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 
, scalar fields 
, 
& 
, and mass. 
is the standard gravity term with the exception that 
is moved inside the integral.
where
Gravitation
Gravitation, or gravity, is a natural phenomenon by which physical bodies attract with a force proportional to their mass. Gravitation is most familiar as the agent that gives weight to objects with mass and causes them to fall to the ground when dropped...
in competition to Einstein's theory of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
.
There have been many different attempts at constructing an ideal theory of gravity. These attempts can be split into four broad categories:
- Straightforward alternatives to general relativity (GR), such as the CartanEinstein–Cartan theoryIn theoretical physics, the Einstein–Cartan theory, also known as the Einstein–Cartan–Sciama–Kibble theory or the Cartan–Sciama–Kibble theory is a classical theory of gravitation similar to general relativity but relaxing the assumption that the metric be torsion-free. Introducing torsion allows...
, Brans–Dicke and Rosen bimetricBimetric theoryBimetric theory refers to a class of modified theories of gravity in which two metric tensors are used instead of one. Often the second metric is introduced at high energies, with the implication that the speed of light may be energy dependent....
theories. - Those that attempt to construct a quantized gravityQuantum gravityQuantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
theory such as loop quantum gravityLoop quantum gravityLoop quantum gravity , also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics and general relativity...
. - Those that attempt to unify gravity and other forcesClassical unified field theoriesSince the 19th century, some physicists have attempted to develop a single theoretical framework that can account for the fundamental forces of nature – a unified field theory. Classical unified field theories are attempts to create a unified field theory based on classical physics...
such as Kaluza–KleinKaluza–Klein theoryIn physics, Kaluza–Klein theory is a model that seeks to unify the two fundamental forces of gravitation and electromagnetism. The theory was first published in 1921. It was proposed by the mathematician Theodor Kaluza who extended general relativity to a five-dimensional spacetime...
. - Those that attempt to do several at onceTheory of everythingA theory of everything is a putative theory of theoretical physics that fully explains and links together all known physical phenomena, and predicts the outcome of any experiment that could be carried out in principle....
, such as M-theoryM-theoryIn theoretical physics, M-theory is an extension of string theory in which 11 dimensions are identified. Because the dimensionality exceeds that of superstring theories in 10 dimensions, proponents believe that the 11-dimensional theory unites all five string theories...
.
This article deals only with straightforward alternatives to GR. For quantized gravity theories, see the article quantum gravity
Quantum gravity
Quantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
. For the unification of gravity and other forces, see the article classical unified field theories
Classical unified field theories
Since the 19th century, some physicists have attempted to develop a single theoretical framework that can account for the fundamental forces of nature – a unified field theory. Classical unified field theories are attempts to create a unified field theory based on classical physics...
. For those theories that attempt to do several at once, see the article theory of everything
Theory of everything
A theory of everything is a putative theory of theoretical physics that fully explains and links together all known physical phenomena, and predicts the outcome of any experiment that could be carried out in principle....
.
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. LagrangeJoseph Lagrange
Count Joseph Lagrange was a French soldier who rose through the ranks and gained promotion to the rank of general officer during the French Revolutionary Wars, subsequently pursuing a successful career during the Napoleonic Wars and winning promotion to the top military rank of General of Division....
). Then came unsuccessful attempts to combine gravity and either wave or corpuscular theories
Le Sage's theory of gravitation
Le Sage's theory of gravitation is a kinetic theory of gravity originally proposed by Nicolas Fatio de Duillier in 1690 and later by Georges-Louis Le Sage in 1748. The theory proposed a mechanical explanation for Newton's gravitational force in terms of streams of tiny unseen particles impacting...
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
Lorentz covariance
In standard physics, Lorentz symmetry is "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space"...
, the gravitational deflection of light
Tests of general relativity
At its introduction in 1915, the general theory of relativity did not have a solid empirical foundation. It was known that it correctly accounted for the "anomalous" precession of the perihelion of Mercury and on philosophical grounds it was considered satisfying that it was able to unify Newton's...
, the Eötvös experiment
Eötvös experiment
The Eötvös experiment was a famous physics experiment that measured the correlation between inertial mass and gravitational mass, demonstrating that the two were one and the same, something that had long been suspected but never demonstrated with the same accuracy. The earliest experiments were...
. These considerations led to and past the development of general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
.
After that, motivations differ. Two major concerns were the development of quantum theory
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
and the discovery of the strong
Strong interaction
In particle physics, the strong interaction is one of the four fundamental interactions of nature, the others being electromagnetism, the weak interaction and gravitation. As with the other fundamental interactions, it is a non-contact force...
and weak
Weak interaction
Weak interaction , is one of the four fundamental forces of nature, alongside the strong nuclear force, electromagnetism, and gravity. It is responsible for the radioactive decay of subatomic particles and initiates the process known as hydrogen fusion in stars...
nuclear forces. Attempts to quantize and unify gravity are outside the scope of this article, and so far none has been completely successful.
After general relativity (GR), attempts were made either to improve on theories developed before GR, or to improve GR itself. Many different strategies were attempted, for example the addition of spin to GR, combining a GR-like metric with a space-time that is static with respect to the expansion of the universe, getting extra freedom by adding another parameter. At least one theory was motivated by the desire to develop an alternative to GR that is completely free from singularities.
Experimental tests improved along with the theories. Many of the different strategies that were developed soon after GR were abandoned, and there was a push to develop more general forms of the theories that survived, so that a theory would be ready the moment any test showed a disagreement with GR.
By the 1980s, the increasing accuracy of experimental tests had all led to confirmation of GR, no competitors were left except for those that included GR as a special case. Further, shortly after that, theorists switched to string theory which was starting to look promising, but has since lost popularity. In the mid 1980s a few experiments were suggesting that gravity was being modified by the addition of a fifth force (or, in one case, of a fifth, sixth and seventh force) acting on the scale of meters. Subsequent experiments eliminated these.
Motivations for the more recent alternative theories are almost all cosmological, associated with or replacing such constructs as "inflation
Cosmic inflation
In physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized extremely rapid exponential expansion of the early universe by a factor of at least 1078 in volume, driven by a negative-pressure vacuum energy density. The inflationary epoch comprises the first part...
", "dark matter
Dark matter
In astronomy and cosmology, dark matter is matter that neither emits nor scatters light or other electromagnetic radiation, and so cannot be directly detected via optical or radio astronomy...
" and "dark energy
Dark energy
In physical cosmology, astronomy and celestial mechanics, dark energy is a hypothetical form of energy that permeates all of space and tends to accelerate the expansion of the universe. Dark energy is the most accepted theory to explain recent observations that the universe appears to be expanding...
". Investigation of the Pioneer anomaly
Pioneer anomaly
The Pioneer anomaly or Pioneer effect is the observed deviation from predicted accelerations of the Pioneer 10 and Pioneer 11 spacecraft after they passed about on their trajectories out of the Solar System....
has caused renewed public interest in alternatives to General Relativity.
Notation in this article
is the speed of lightSpeed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
,
is the gravitational constantGravitational constant
The gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
. "Geometric variables
Natural units
In physics, natural units are physical units of measurement based only on universal physical constants. For example the elementary charge e is a natural unit of electric charge, or the speed of light c is a natural unit of speed...
" are not used.
Latin indexes go from 1 to 3, Greek indexes go from 1 to 4. The Einstein summation convention
Einstein notation
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 formulae...
is used.
is the Minkowski metricMinkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
.
is a tensor, usually the metric tensorMetric tensor (general relativity)
In general relativity, the metric tensor is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational field familiar from Newtonian gravitation...
. These have signature
Metric signature
The signature of a metric tensor is the number of positive and negative eigenvalues of the metric. That is, the corresponding real symmetric matrix is diagonalised, and the diagonal entries of each sign counted...
(−,+,+,+).
Partial differentiation
Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant...
is written
or . Covariant differentiation is written or .Classification of theories
Theories of gravity can be classified, loosely, into several categories. Most of the theories described here have:- an 'actionAction (physics)In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
' (see the principle of least actionPrinciple of least actionIn physics, the principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system...
, a variational principleVariational principleA variational principle is a scientific principle used within the calculus of variations, which develops general methods for finding functions which minimize or maximize the value of quantities that depend upon those functions...
based on the concept of action) - a Lagrangian density
- a metricMetric tensorIn the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
If a theory has a Lagrangian density for gravity, say
, then the gravitational part of the action is the integral of that.In this equation it is usual, though not essential, to have
at spatial infinity when using Cartesian coordinates. For example the Einstein–Hilbert action useswhere R is the scalar curvature
Scalar curvature
In Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point...
, a measure of the curvature of space.
Almost every theory described in this article has an action
Action (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
. 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
Modified Newtonian dynamics
In physics, Modified Newtonian dynamics is a hypothesis that proposes a modification of Newton's law of gravity to explain the galaxy rotation problem. When the uniform velocity of rotation of galaxies was first observed, it was unexpected because Newtonian theory of gravity predicts that objects...
did not have an action.
A few theories have an action but not a Lagrangian density. A good example is Whitehead (1922), the action there is termed non-local.
A theory of gravity is a "metric theory" if and only if it can be given a mathematical representation in which two conditions hold:
Condition 1: There exists a symmetric metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
of signatureMetric signature
The signature of a metric tensor is the number of positive and negative eigenvalues of the metric. That is, the corresponding real symmetric matrix is diagonalised, and the diagonal entries of each sign counted...
(−, +, +, +), which governs proper-length and proper-time measurements in the usual manner of special and general relativity:
where there is a summation over indices
and .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 derivativeCovariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...
with respect to the metric and
is the Christoffel symbol. The stress-energy tensor should also satisfy an energy conditionEnergy condition
In relativistic classical field theories of gravitation, particularly general relativity, an energy condition is one of various alternative conditions which can be applied to the matter content of the theory, when it is either not possible or desirable to specify this content explicitly...
.
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
- LeeDavid L. LeeDavid Li Lee is a Taiwanese-American business executive and venture capitalist, best known as a co-founder of Global Crossing Ltd.Lee is a graduate of McGill University. He received a Ph.D. in theoretical physics from Caltech in 1974, with a minor in economics...
–LightmanAlan LightmanAlan Lightman is an American physicist, writer, and social entrepreneur. He is a professor at the Massachusetts Institute of Technology and the author of the international bestseller Einstein's Dreams. He was the first professor at MIT to receive a joint appointment in the sciences and the...
–NiWei-Tou NiNi Wei-Tou is a Taiwanese physicist, who graduated from the Department of Physics of National Taiwan University , and got his PhD of Physics & Mathematics from California Institute of Technology... - Littlewood
- Ni
- Nordström's theory of gravitationNordström's theory of gravitationIn theoretical physics, Nordström's theory of gravitation was a predecessor of general relativity. Strictly speaking, there were actually two distinct theories proposed by the Finnish theoretical physicist Gunnar Nordström, in 1912 and 1913 respectively...
(first metric theory of gravity to be developed) - Page–Tupper
- Papapetrou
- Rosen (1971)
- Whitrow–Morduch
- Yilmaz theory of gravitationYilmaz theory of gravitationThe Yilmaz theory of gravitation is an attempt by Huseyin Yilmaz and his coworkers to formulate a classical field theory of gravitation which is similar to general relativity in weak-field conditions, but in which event horizons cannot appear.Yilmaz's work has been criticized on various grounds,...
(attempted to eliminate event horizons from the theory.)
- Bimetric theories
- LightmanAlan LightmanAlan Lightman is an American physicist, writer, and social entrepreneur. He is a professor at the Massachusetts Institute of Technology and the author of the international bestseller Einstein's Dreams. He was the first professor at MIT to receive a joint appointment in the sciences and the...
–LeeDavid L. LeeDavid Li Lee is a Taiwanese-American business executive and venture capitalist, best known as a co-founder of Global Crossing Ltd.Lee is a graduate of McGill University. He received a Ph.D. in theoretical physics from Caltech in 1974, with a minor in economics... - Rastall
- Rosen (1975)
- Lightman
- Quasilinear theories (includes Linear fixed gauge)
- Bollini–Giambini–Tiomno
- Deser–Laurent
- Whitehead's theory of gravity (intended to use only retarded potentialRetarded potentialThe retarded potential formulae describe the scalar or vector potential for electromagnetic fields of a time-varying current or charge distribution. The retardation of the influence connecting cause and effect is thereby essential; e.g...
s)
- Tensor theories
- Einstein's GR
- Fourth order gravity (allows the Lagrangian to depend on second-order contractions of the Riemann curvature tensor)
- f(R) gravityF(R) gravityf gravity is a type of modified gravity theory first proposed in 1970by Buchdahl as a generalisation of Einstein's General Relativity. Although it is an active field of research, there are known problems with the theory...
(allows the Lagrangian to depend on higher powers of the Ricci scalar) - Gauss–Bonnet gravity
- Lovelock theory of gravityLovelock theory of gravityIn physics, Lovelock's theory of gravity is a generalization of Einstein's theory of general relativity introduced by David Lovelock in 1971. It is the most general metric theory of gravity yielding conserved second order equations of motion in arbitrary number of spacetime dimensions D...
(allows the Lagrangian to depend on higher-order contractions of the Riemann curvature tensor)
- Scalar-tensor theories
- Beckenstein
- Bergmann-Wagoner
- Brans–Dicke theory (the most well-known alternative to GR, intended to be better at applying Mach's principle)
- Jordan
- Nordtvedt
- Thiry
- Vector-tensor theories
- Hellings–Nordtvedt
- WillClifford Martin WillClifford Martin Will is a Canadian born mathematical physicist who is well known for his contributions to the theory of general relativity....
–Nordtvedt
- 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)
- TeleparallelismTeleparallelismTeleparallelism , was an attempt by Einstein to unify electromagnetism and gravity...
- Weyl's gauge theory
A word here about Mach's principle
Mach's principle
In theoretical physics, particularly in discussions of gravitation theories, Mach's principle is the name given by Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach....
is appropriate because a few of these theories rely on Mach's principle (e.g. Whitehead (1922)), and many mention it in passing (e.g. Einstein–Grossmann (1913), Brans–Dicke (1961)). Mach's principle can be thought of a half-way-house between Newton and Einstein. It goes this way:
- Newton: Absolute space and time.
- Mach: The reference frame comes from the distribution of matter in the universe.
- Einstein: There is no reference frame.
So far, all the experimental evidence points to Mach's principle being wrong, but it has not entirely been ruled out.
Early theories, 1686 to 1916
Newton (1686)In Newton's (1686) theory (rewritten using more modern mathematics) the density of mass
generates a scalar field, the gravitational potential in joules per kilogram, byUsing the Nabla operator
for the gradientGradient
In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change....
and divergence
Divergence
In vector calculus, divergence is a vector operator that measures the magnitude of a vector field's source or sink at a given point, in terms of a signed scalar. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around...
(partial derivatives), this can be conveniently written as:
This scalar field governs the motion of a free-fall
Free-fall
Free fall is any motion of a body where gravity is the only force acting upon it, at least initially. These conditions produce an inertial trajectory so long as gravity remains the only force. Since this definition does not specify velocity, it also applies to objects initially moving upward...
ing particle by:
At distance, r, from an isolated mass, M, the scalar field is
The theory of Newton, and Lagrange's improvement on the calculation (applying the variational principle), completely fails to take into account relativistic effects of course, and so can be rejected as a viable theory of gravity. Even so, Newton's theory is thought to be exactly correct in the limit of weak gravitational fields and low speeds and all other theories of gravity need to reproduce Newton's theory in the appropriate limits.
Mechanical explanations (1650–1900)
To explain Newton's theory, some mechanical explanations of gravitation
Mechanical explanations of gravitation
Mechanical explanations of gravitation are attempts to explain the action of gravity by aid of basic mechanical processes, such as pressure forces caused by pushes, and without the use of any action at a distance. These theories were developed from the 16th until the 19th century in connection...
(incl. Le Sage's theory
Le Sage's theory of gravitation
Le Sage's theory of gravitation is a kinetic theory of gravity originally proposed by Nicolas Fatio de Duillier in 1690 and later by Georges-Louis Le Sage in 1748. The theory proposed a mechanical explanation for Newton's gravitational force in terms of streams of tiny unseen particles impacting...
) were created between 1650 and 1900, but they were overthrown because most of them lead to an unacceptable amount of drag
Drag (physics)
In fluid dynamics, drag refers to forces which act on a solid object in the direction of the relative fluid flow velocity...
, which is not observed. Other models are violating the energy conservation law and are incompatible with modern thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
.
Electrostatic models (1870–1900)
At the end of the 19th century, many tried to combine Newton's force law with the established laws of electrodynamics, like those of Weber
Wilhelm Eduard Weber
Wilhelm Eduard Weber was a German physicist and, together with Carl Friedrich Gauss, inventor of the first electromagnetic telegraph.-Early years:...
, Carl Friedrich Gauß, Bernhard Riemann
Bernhard Riemann
Georg Friedrich Bernhard Riemann was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity....
and James Clerk Maxwell
James Clerk Maxwell
James Clerk Maxwell of Glenlair was a Scottish physicist and mathematician. His most prominent achievement was formulating classical electromagnetic theory. This united all previously unrelated observations, experiments and equations of electricity, magnetism and optics into a consistent theory...
. Those models were used to explain the perihelion advance of Mercury. In 1890, Lévy succeeded in doing so by combining the laws of Weber and Riemann, whereby the speed of gravity
Speed of gravity
In the context of classical theories of gravitation, the speed of gravity is the speed at which changes in a gravitational field propagate. This is the speed at which a change in the distribution of energy and momentum of matter results in subsequent alteration, at a distance, of the gravitational...
is equal to the speed of light in his theory. And in another attempt, Paul Gerber
Paul Gerber
Paul Gerber was a German physicist. He studied in Berlin from 1872-1875. In 1877 he became a teacher at the Realgymnasium in Stargard in Pommern...
(1898) even succeeded in deriving the correct formula for the Perihelion shift (which was identical to that formula later used by Einstein). However, because the basic laws of Weber and others were wrong (for example, Weber's law was superseded by Maxwell's theory), those hypothesis were rejected. In 1900, Hendrik Lorentz
Hendrik Lorentz
Hendrik Antoon Lorentz was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect...
tried to explain gravity on the basis of his Lorentz ether theory
Lorentz ether theory
What is now often called Lorentz Ether theory has its roots in Hendrik Lorentz's "Theory of electrons", which was the final point in the development of the classical aether theories at the end of the 19th and at the beginning of the 20th century....
and the Maxwell equations. He assumed, like Ottaviano Fabrizio Mossotti and Johann Karl Friedrich Zöllner
Johann Karl Friedrich Zöllner
Johann Karl Friedrich Zöllner was a German astrophysicist who studied optical illusions. He invented the Zöllner illusion where lines that are parallel appear diagonal. The lunar Zöllner crater is named in his honor...
, that the attraction of opposite charged particles is stronger than the repulsion of equal charged particles. The resulting net force is exactly what is known as universal gravitation, in which the speed of gravity is that of light. But Lorentz calculated that the value for the perihelion advance of Mercury was much too low.
Lorentz-invariant models (1905–1910)
Based on the principle of relativity
Principle of relativity
In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference....
, Henri Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
(1905, 1906), Hermann Minkowski
Hermann Minkowski
Hermann Minkowski was a German mathematician of Ashkenazi Jewish descent, who created and developed the geometry of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.- Life and work :Hermann Minkowski was born...
(1908), and Arnold Sommerfeld
Arnold Sommerfeld
Arnold Johannes Wilhelm Sommerfeld was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and groomed a large number of students for the new era of theoretical physics...
(1910) tried to modify Newton's theory and to establish a Lorentz invariant
Lorentz covariance
In standard physics, Lorentz symmetry is "the feature of nature that says experimental results are independent of the orientation or the boost velocity of the laboratory through space"...
gravitational law, in which the speed of gravity is that of light. However, like in Lorentz's model the value for the perihelion advance of Mercury was much too low.
Einstein (1908, 1912)
Einstein's two part publication in 1912 (and before in 1908) is really only important for historical reasons. By then he knew of the gravitational redshift and the deflection of light. He had realized that Lorentz transformations are not generally applicable, but retained them. The theory states that the speed of light is constant in free space but varies in the presence of matter. The theory was only expected to hold when the source of the gravitational field is stationary. It includes the principle of least action
Principle of least action
In physics, the principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system...
:
where
is the Minkowski metricMinkowski space
In physics and mathematics, Minkowski space or Minkowski spacetime is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated...
, and there is a summation from 1 to 4 over indices
and .Einstein and Grossmann (1913) includes Riemannian geometry
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
and tensor calculus
Covariant transformation
In physics, a covariant transformation is a rule , that describes how certain physical entities change under a change of coordinate system....
.
The equations of electrodynamics
Classical electromagnetism
Classical electromagnetism is a branch of theoretical physics that studies consequences of the electromagnetic forces between electric charges and currents...
exactly match those of GR. The equation
is not in GR. It expresses the stress-energy tensor
Stress-energy tensor
The stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...
as a function of the matter density.
Abraham (1912)
While this was going on, Abraham
Max Abraham
Max Abraham was a German physicist.Abraham was born in Danzig, Imperial Germany to a family of Jewish merchants. His father was Moritz Abraham and his mother was Selma Moritzsohn. Attending the University of Berlin, he studied under Max Planck. He graduated in 1897...
was developing an alternative model of gravity in which the speed of light depends on the gravitational field strength and so is variable almost everywhere. Abraham's 1914 review of gravitation models is said to be excellent, but his own model was poor.
Nordström (1912)
The first approach of Nordström (1912)
Nordström's theory of gravitation
In theoretical physics, Nordström's theory of gravitation was a predecessor of general relativity. Strictly speaking, there were actually two distinct theories proposed by the Finnish theoretical physicist Gunnar Nordström, in 1912 and 1913 respectively...
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 satisfywhere
is rest mass energy and is the d'Alembertian,and
where
is the four-velocity and the dot is a differential with respect to time.The second approach of Nordström (1913)
Nordström's theory of gravitation
In theoretical physics, Nordström's theory of gravitation was a predecessor of general relativity. Strictly speaking, there were actually two distinct theories proposed by the Finnish theoretical physicist Gunnar Nordström, in 1912 and 1913 respectively...
is remembered as the first logically consistent relativistic field theory of gravitation ever formulated. From (note, notation of Pais (1982) not Nordström):
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
Equivalence principle
In the physics of general relativity, the equivalence principle is any of several related concepts dealing with the equivalence of gravitational and inertial mass, and to Albert Einstein's assertion that the gravitational "force" as experienced locally while standing on a massive body is actually...
.
Einstein and Fokker (1914)
This theory 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)
This theory 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
Relativity priority dispute
Albert Einstein presented the theories of Special Relativity and General Relativity in groundbreaking publications that either contained no formal references to previous literature, or referred only to a small number of his predecessors for fundamental results on which he based his theories, most...
. Hilbert was the first to correctly state the Einstein-Hilbert action
Einstein-Hilbert action
The Einstein–Hilbert action in general relativity is the action that yields the Einstein's field equations through the principle of least action...
for GR, which is:
where
is Newton's gravitational constant, is the Ricci curvatureRicci curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space...
of space,
and is the actionAction (physics)
In physics, action is an attribute of the dynamics of a physical system. It is a mathematical functional which takes the trajectory, also called path or history, of the system as its argument and has a real number as its result. Action has the dimension of energy × time, and its unit is...
due to mass.
GR is a tensor theory, the equations all contain tensors. Nordström's theories, on the other hand, are scalar theories because the gravitational field is a scalar. Later in this article you will see scalar-tensor theories that contain a scalar field in addition to the tensors of GR, and other variants containing vector fields as well have been developed recently.
Theories from 1917 to the 1980s
This section includes alternatives to GR published after GR but before the observations of galaxy rotation that led to the hypothesis of "dark matter".Those considered here include (see Will (1981), Lang (2002)):
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), Wei-Tou Ni
Wei-Tou Ni
Ni Wei-Tou is a Taiwanese physicist, who graduated from the Department of Physics of National Taiwan University , and got his PhD of Physics & Mathematics from California Institute of Technology...
(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 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. How to add a cosmological constant or quintessence to a theory is discussed under Modern Theories (see also here
Einstein-Hilbert action
The Einstein–Hilbert action in general relativity is the action that yields the Einstein's field equations through the principle of least action...
).
Scalar field theories
The scalar field theories of Nordström (1912, 1913) have already been discussed. Those of Littlewood (1953), Bergman (1956), Yilmaz (1958), Whitrow and Morduch (1960, 1965) and Page and Tupper (1968) follow the general formula give by Page and Tupper.According to Page and Tupper (1968), who discuss all these except Nordström (1913), the general scalar field theory comes from the principle of least action:
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
Yilmaz theory of gravitation
The Yilmaz theory of gravitation is an attempt by Huseyin Yilmaz and his coworkers to formulate a classical field theory of gravitation which is similar to general relativity in weak-field conditions, but in which event horizons cannot appear.Yilmaz's work has been criticized on various grounds,...
) to second order when
.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
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
Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given...
with respect to
. The field equations may be written in the form:
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. 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
.A cosmological constant
Cosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
can be introduced into a quasilinear theory by the simple expedient of changing the Minkowski background to a de Sitter
De Sitter space
In mathematics and physics, a de Sitter space is the analog in Minkowski space, or spacetime, of a sphere in ordinary, Euclidean space. The n-dimensional de Sitter space , denoted dS_n, is the Lorentzian manifold analog of an n-sphere ; it is maximally symmetric, has constant positive curvature,...
or anti-de Sitter spacetime, as suggested by G. Temple in 1923. Temple's suggestions on how to do this were criticized by C. B. Rayner in 1955.
Tensor theories
Einstein's general relativityGeneral relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
is the simplest plausible theory of gravity that can be based on just one symmetric tensor field (the metric tensor
Metric tensor (general relativity)
In general relativity, the metric tensor is the fundamental object of study. It may loosely be thought of as a generalization of the gravitational field familiar from Newtonian gravitation...
). Others include: Gauss–Bonnet gravity, f(R) gravity
F(R) gravity
f gravity is a type of modified gravity theory first proposed in 1970by Buchdahl as a generalisation of Einstein's General Relativity. Although it is an active field of research, there are known problems with the theory...
, and Lovelock theory of gravity
Lovelock theory of gravity
In physics, Lovelock's theory of gravity is a generalization of Einstein's theory of general relativity introduced by David Lovelock in 1971. It is the most general metric theory of gravity yielding conserved second order equations of motion in arbitrary number of spacetime dimensions D...
.
Scalar-tensor theories
These all contain at least one free parameter, as opposed to GR which has no free parameters.Although not normally considered a Scalar-Tensor theory of gravity, the 5 by 5 metric of Kaluza-Klein
Kaluza–Klein theory
In physics, Kaluza–Klein theory is a model that seeks to unify the two fundamental forces of gravitation and electromagnetism. The theory was first published in 1921. It was proposed by the mathematician Theodor Kaluza who extended general relativity to a five-dimensional spacetime...
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
Kaluza–Klein theory
In physics, Kaluza–Klein theory is a model that seeks to unify the two fundamental forces of gravitation and electromagnetism. The theory was first published in 1921. It was proposed by the mathematician Theodor Kaluza who extended general relativity to a five-dimensional spacetime...
can be considered the progenitor of Scalar-Tensor theories of gravity. This was recognised by Thiry (1948).
Scalar-Tensor theories include Thiry (1948), Jordan (1955), Brans and Dicke (1961), Bergman (1968), Nordtveldt (1970), Wagoner (1970), Bekenstein (1977) and Barker (1978).
The action
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),
Since
was thought to be zero at the time anyway, this would not have been considered a significant difference. The role of the cosmological constant in more modern work is discussed under Cosmological constant.Brans-Dicke (1961),
is constantBekenstein (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 experiment, 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
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 the completely antisymmetric pseudo-tensor (Levi-Civita symbolLevi-Civita symbol
The Levi-Civita symbol, also called the permutation symbol, antisymmetric symbol, or alternating symbol, is a mathematical symbol used in particular in tensor calculus...
) with
, 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
Any putative alternative to general relativity would need to meet a variety of tests for it to become accepted. For in-depth coverage of these tests, see Misner et al. (1973) Ch.39, Will (1981) Table 2.1, and Ni (1972). Most such tests can be categorized as in the following subsections.Self-consistency
Self-consistency among non-metric theories includes eliminating theories allowing tachyonTachyon
A tachyon is a hypothetical subatomic particle that always moves faster than light. In the language of special relativity, a tachyon would be a particle with space-like four-momentum and imaginary proper time. A tachyon would be constrained to the space-like portion of the energy-momentum graph...
s, ghost poles and higher order poles, and those that have problems with behaviour at infinity.
Among metric theories, self-consistency is best illustrated by describing several theories that fail this test. The classic example is the spin-two field theory of Fierz and Pauli (1939); the field equations imply that gravitating bodies move in straight lines, whereas the equations of motion insist that gravity deflects bodies away from straight line motion. Yilmaz (1971, 1973) contains a tensor gravitational field used to construct a metric; it is mathematically inconsistent because the functional dependence of the metric on the tensor field is not well defined.
Completeness
To be complete, a theory of gravity must be capable of analysing the outcome of every experiment of interest. It must therefore mesh with electromagnetism and all other physics. For instance, any theory that cannot predict from first principles the movement of planets or the behaviour of atomic clocks is incomplete.Many early theories are incomplete in that it is unclear whether the density
used by the theory should be calculated from the stress-energy tensor as or as , where is the four-velocityFour-velocity
In physics, in particular in special relativity and general relativity, the four-velocity of an object is a four-vector that replaces classicalvelocity...
, and
is the Kronecker delta.The theories of Thirry (1948) and Jordan (1955) 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.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.
Classical tests
There are three "classical" tests (dating back to the 1910s or earlier) of the ability of gravity theories to handle relativistic effects; they are:- gravitational redshiftGravitational redshiftIn astrophysics, gravitational redshift or Einstein shift describes light or other forms of electromagnetic radiation of certain wavelengths that originate from a source that is in a region of a stronger gravitational field that appear to be of longer wavelength, or redshifted, when seen or...
- gravitational lensing (generally tested around the Sun)
- anomalous perihelion advance of the planets
Each theory should reproduce the observed results in these areas, which have to date always aligned with the predictions of general relativity.
In 1964, Irwin I. Shapiro
Irwin I. Shapiro
Irwin I. Shapiro is an American astrophysicist. Since 1982, he has been a professor at Harvard University. Shapiro was director of the Harvard-Smithsonian Center for Astrophysics from 1982 to 2004.- Biography :Irwin Shapiro was born in New York City in 1929...
found a fourth test, called the Shapiro delay
Shapiro delay
The Shapiro time delay effect, or gravitational time delay effect, is one of the four classic solar system tests of general relativity. Radar signals passing near a massive object take slightly longer to travel to a target and longer to return than it would if the mass of the object were not...
. It is usually regarded as a "classical" test as well.
Agreement with Newtonian mechanics and special relativity
As an example of disagreement with Newtonian experiments, Birkhoff (1943) theory predicts relativistic effects fairly reliably but demands that sound waves travel at the speed of light, which disagrees violently with experiment.A modern example of the lack of a relativistic component is MOND by Milgrom, as will be discussed below.
The Einstein equivalence principle (EEP)
The EEP has three components.The first is the uniqueness of free fall, also known as the Weak Equivalence Principle (WEP). This is satisfied if inertial mass is equal to gravitational mass. η is a parameter used to test the maximum allowable violation of the WEP. The first tests of the WEP were done by Eötvös before 1900 and limited η to less than 5. 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.
Schiff
Leonard I. Schiff
Leonard Isaac Schiff was born in Fall River, Massachusetts on March 29, 1915 and died on Jan 21, 1971.He was a physicist best known for his book Quantum Mechanics.-Education:...
's conjecture states that any complete, self-consistent theory of gravity that embodies the WEP necessarily embodies EEP. This is likely to be true if the theory has full energy conservation.
Metric theories satisfy the Einstein Equivalence Principle. Extremely few non-metric theories, if any, satisfy this. For example, the non-metric theory of Belinfante & Swihart (1957) is eliminated by the THεμ formalism for testing EEP.
Parametric post-Newtonian (PPN) formalism
See also Tests of general relativityTests of general relativity
At its introduction in 1915, the general theory of relativity did not have a solid empirical foundation. It was known that it correctly accounted for the "anomalous" precession of the perihelion of Mercury and on philosophical grounds it was considered satisfying that it was able to unify Newton's...
, Misner et al. (1973) and Will (1981) for more information.
Work on developing a standardized rather than ad-hoc set of tests for evaluating alternative gravitation models began with Eddington in 1922 and resulted in a standard set of PPN numbers in Nordtvedt and Will (1972) and Will and Nordtvedt (1972). Each parameter measures a different aspect of how much a theory departs from Newtonian gravity. Because we are talking about deviation from Newtonian theory here, these only measure weak-field effects. The effects of strong gravitational fields are examined later.
These ten are 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 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
PPN is only a measure of weak field effects. Strong gravity effects can be seen in compact objects such as white dwarfs, neutron stars, and black holes. Experimental tests such as the stability of white dwarfs, spin-down rate of pulsars, orbits of binary pulsars and the existence of a black hole horizon can be used as tests of alternative to GR.GR predicts that gravitational waves travel at the speed of light. Many alternatives to GR say that gravitational waves travel faster than light. 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 matterDark matter
In astronomy and cosmology, dark matter is matter that neither emits nor scatters light or other electromagnetic radiation, and so cannot be directly detected via optical or radio astronomy...
, the galaxy rotation curve
Galaxy rotation curve
The rotation curve of a galaxy can be represented by a graph that plots the orbital velocity of the stars or gas in the galaxy on the y-axis against the distance from the center of the galaxy on the x-axis....
, the Tully-Fisher relation
Tully-Fisher relation
In astronomy, the Tully–Fisher relation, published by astronomers R. Brent Tully and J. Richard Fisher in 1977, is an empirical relationship between the intrinsic luminosity of a spiral galaxy and its velocity width...
, the faster rotation rate of dwarf galaxies, and the gravitational lens
Gravitational lens
A gravitational lens refers to a distribution of matter between a distant source and an observer, that is capable of bending the light from the source, as it travels towards the observer...
ing due to galactic clusters act as constraints.
For those theories that aim to replace inflation
Cosmic inflation
In physical cosmology, cosmic inflation, cosmological inflation or just inflation is the theorized extremely rapid exponential expansion of the early universe by a factor of at least 1078 in volume, driven by a negative-pressure vacuum energy density. The inflationary epoch comprises the first part...
, the size of ripples in the spectrum of the cosmic microwave background radiation
Cosmic microwave background radiation
In cosmology, cosmic microwave background radiation is thermal radiation filling the observable universe almost uniformly....
is the strictest test.
For those theories that incorporate or aim to replace dark energy
Dark energy
In physical cosmology, astronomy and celestial mechanics, dark energy is a hypothetical form of energy that permeates all of space and tends to accelerate the expansion of the universe. Dark energy is the most accepted theory to explain recent observations that the universe appears to be expanding...
, the supernova brightness results and the age of the universe can be used as tests.
Another test is the flatness of the universe. With GR, the combination of baryonic matter, dark matter and dark energy add up to make the universe exactly flat. As the accuracy of experimental tests improve, alternatives to GR that aim to replace dark matter or dark energy will have to explain why.
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, during which 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
Parameterized post-Newtonian formalism
Post-Newtonian formalism is a calculational tool that expresses Einstein's equations of gravity in terms of the lowest-order deviations from Newton's theory. This allows approximations to Einstein's equations to be made in the case of weak fields...
(PPN).
The following table lists PPN values for a large number of theories. If the value in a cell matches that in the column heading then the full formula is too complicated to include here.
 |  |  |  |  |
||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Einstein (1916) GR | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Scalar-Tensor theories | ||||||||||
| Bergmann (1968), Wagoner (1970) |  |
 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Nordtvedt (1970), Bekenstein (1977) |  |
 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Brans-Dicke (1961) |  |
1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| Vector-Tensor theories |  |
 |
0 |  |
 |
0 | 0 | 0 | 0 | 0 |
| Hellings-Nordtvedt (1973) |  |
 |
0 |  |
 |
0 | 0 | 0 | 0 | 0 |
| Will-Nordtvedt (1972) | 1 | 1 | 0 | 0 |  |
0 | 0 | 0 | 0 | 0 |
| Bimetric theories | ||||||||||
| Rosen (1975) | 1 | 1 | 0 | 0 |  |
0 | 0 | 0 | 0 | 0 |
| Rastall (1979) | 1 | 1 | 0 | 0 |  |
0 | 0 | 0 | 0 | 0 |
| Lightman-Lee (1973) |  |
 |
0 |  |
 |
0 | 0 | 0 | 0 | 0 |
| Stratified theories | ||||||||||
| Lee-Lightman-Ni (1974) |  |
 |
 |
 |
 |
0 | 0 | 0 | 0 | 0 |
| Ni (1973) |  |
 |
0 |  |
 |
0 | 0 | 0 | 0 | 0 |
| Scalar Field theories | ||||||||||
| Einstein (1912) {Not GR} | 0 | 0 | ||||||||
| 0 | ||||||||||
| 0 | ||||||||||
| 0 | 0† | |||||||||
| Whitrow-Morduch (1965) | 0 | |||||||||
| 0 | 0 | 0 | ||||||||
| 0 | 0† | |||||||||
| Rosen (1971) |  |
 |
 |
0 | ||||||
| 0 | ||||||||||
| 0 | 0 | |||||||||
| Papetrou (1954a, 1954b) | 1 | 1 | ||||||||
| 0 | 0 | 2 | 0 | 0 | ||||||
| Ni (1972) (stratified) | 1 | 1 | ||||||||
| 0 | 0 | 0 | 2 | 0 | 0 | |||||
| Yilmaz (1958, 1962) | 1 | 1 | ||||||||
| 0 | ||||||||||
| 0 | ||||||||||
| 0 | ||||||||||
| Page-Tupper (1968) |  |
 |
 |
0 |  |
0 |  |
0 |  |
|
| Nordström (1912) |  |
 |
0 | 0 | 0 | 0 | 0 | 0 | 0† | |
| Nordström (1913), Einstein-Fokker (1914) |  |
 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | |
| Ni (1972) (flat) |  |
 |
0 | 0 | 0 | 0 |  |
0 | 0† | |
| Whitrow-Morduch (1960) |  |
 |
0 | 0 | 0 | 0 | q | 0 | 0† | |
| Littlewood (1953), Bergman(1956) |  |
 |
0 | 0 | 0 | 0 | ||||
| 0 | 0† |
† The theory is incomplete, and
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
, 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
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
Horizon problem
The horizon problem is a problem with the standard cosmological model of the Big Bang which was identified in the 1970s. It points out that different regions of the universe have not "contacted" each other because of the great distances between them, but nevertheless they have the same temperature...
and the observation that at early times when quarks were first forming there was not enough space on the universe to contain even one quark. Inflation theory was developed to overcome these. Another alternative was constructing an alternative to GR in which the speed of light was larger in the early universe.
The discovery of unexpected rotation curves for galaxies took everyone by surprise. Could there be more mass in the universe than we are aware of, or is the theory of gravity itself wrong? The consensus now is that the missing mass is "cold dark matter", but that consensus was only reached after trying alternatives to general relativity and some physicists still believe that alternative models of gravity might hold the answer.
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 Perlmutter's results in a completely different way.
Another observation that sparked recent interest in alternatives to General Relativity is the Pioneer anomaly
Pioneer anomaly
The Pioneer anomaly or Pioneer effect is the observed deviation from predicted accelerations of the Pioneer 10 and Pioneer 11 spacecraft after they passed about on their trajectories out of the Solar System....
. It was quickly discovered that alternatives to GR could explain this anomaly.
Cosmological constant and quintessence
(also see Cosmological constantCosmological constant
In physical cosmology, the cosmological constant was proposed by Albert Einstein as a modification of his original theory of general relativity to achieve a stationary universe...
, Einstein–Hilbert action, Quintessence (physics)
Quintessence (physics)
In physics, quintessence is a hypothetical form of dark energy postulated as an explanation of observations of an accelerating universe. It has been proposed by some physicists to be a fifth fundamental force...
)
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 acceleratingFirst, let's see how it influences the equations of Newtonian gravity and General Relativity.
In Newtonian gravity, the addition of the cosmological constant changes the Newton-Poisson equation from:
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 dynamicsModified Newtonian dynamics
In physics, Modified Newtonian dynamics is a hypothesis that proposes a modification of Newton's law of gravity to explain the galaxy rotation problem. When the uniform velocity of rotation of galaxies was first observed, it was unexpected because Newtonian theory of gravity predicts that objects...
, Tensor-vector-scalar gravity
Tensor-vector-scalar gravity
Tensor–vector–scalar gravity , developed by Jacob Bekenstein, is a relativistic generalization of Mordehai Milgrom's MOdified Newtonian Dynamics paradigm.The main features of TeVeS can be summarized as follows:...
, and Beckenstein (2004) for more details).
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
AQUAL
AQUAL is a theory of gravity based on Modified Newtonian Dynamics , but using a Lagrangian. It was developed by Jacob Bekenstein and Mordehai Milgrom in their 1984 paper, "Does the missing mass problem signal the breakdown of Newtonian gravity?"...
). 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, and is the MOND length scale.By 1988, a second scalar field (PCC) fixed problems with the earlier scalar-tensor version but is in conflict with the perihelion precession of Mercury and gravitational lensing by galaxies and clusters.
By 1997, MOND had been successfully incorporated in a stratified relativistic theory [Sanders], but as this is a preferred frame theory it has problems of its own.
Bekenstein (2004) introduced a tensor-vector-scalar
Tensor-vector-scalar gravity
Tensor–vector–scalar gravity , developed by Jacob Bekenstein, is a relativistic generalization of Mordehai Milgrom's MOdified Newtonian Dynamics paradigm.The main features of TeVeS can be summarized as follows:...
model (TeVeS). This has two scalar fields
and and vector field . The action is split into parts for gravity, scalars, vector and mass.
The gravity part is the same as in GR.
where
, 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 Milgrom proposed a "bimetric MOND" or "BIMOND" theory, with action
with
and the (noninteracting) matter actions attached to the two metrics, a tensor derived from the difference in the metrics' connections, the ratio between the two metric traces, and are free parameters. is a function which depends on some contractions of the tensors.Assuming that
depends only on the scalar contraction of , Milgrom obtained as a nonrelativistic limit his bi-potential version of MOND with action
Here
should scale as in the deep-MOND limit and as in the Newtonian limit.Moffat's theories
J. W. Moffat (1995) developed a non-symmetric gravitation theoryNonsymmetric gravitational theory
In theoretical physics, the nonsymmetric gravitational theory of John Moffat is a classical theory of gravitation which tries to explain the observation of the flat rotation curves of galaxies....
(NGT). This is not a metric theory. It was first claimed that it does not contain a black hole horizon, but Burko and Ori (1995) have found that NGT can contain black holes. Later, Moffat claimed that it has also been applied to explain rotation curves of galaxies without invoking "dark matter". Damour, Deser & MaCarthy (1993) have criticised NGT, saying that it has unacceptable asymptotic behaviour.
The mathematics is not difficult but is intertwined so the following is only a brief sketch. Starting with a non-symmetric tensor
, 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's (2002) theory is a 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.Moffat's (2005a) metric-skew-tensor-gravity (MSTG) theory is able to predict rotation curves for galaxies without either dark matter or MOND, and claims that it can also explain gravitational lensing of galaxy clusters without dark matter. It has variable
, 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
Scalar-tensor-vector gravity
Scalar–tensor–vector gravity is a modified theory of gravity developed by John Moffat, a researcher at the Perimeter Institute for Theoretical Physics in Waterloo, Ontario...
(SVTG) theory.
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 , 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.
-
