Alternatives to general relativity

Alternatives to general relativity are physical theories that try to explain how gravitation works, instead of using Einstein's idea of general relativity. Many scientists have tried to create a better theory of gravity, and these attempts can be grouped into four main types.

The first type includes classical theories of gravity, which do not use quantum mechanics or try to combine forces. The second type uses quantum mechanics to create quantized gravity. The third type tries to explain gravity and other forces together, called classical unified field theories. The fourth type aims to put gravity into quantum terms and unify all forces, known as theories of everything.

Even though many ideas have been suggested, none of these alternatives to general relativity have become widely accepted. General relativity has been tested many times and works very well for explaining observations in space. However, because of some problems with explaining dark matter and dark energy, scientists keep looking for better theories.

Notation in this article

Main articles: Mathematics of general relativity and Ricci calculus

This section explains some special symbols and rules used in the study of gravity. The symbol c stands for the speed of light, and G represents the gravitational constant. Scientists use letters with special shapes, like Greek letters, to keep track of different directions in space and time. They also use certain rules for adding and changing these symbols, which help them describe how gravity works.

General relativity

Main article: General relativity

General relativity is Einstein's theory that explains gravity as the bending of space and time. It uses special mathematical tools called tensors to describe how objects move and how space behaves. One key idea is that the shape of space changes based on the amount of matter and energy present.

Some other theories try to explain gravity in different ways. For example, some use a single number (a scalar) instead of tensors, while others mix scalar numbers with tensors or add new kinds of fields called vectors. These ideas are still being studied to see if they can better explain what we observe in space.

Classification of theories

Theories of gravity can be grouped into different types. Many of these theories share some basic ideas, like having an 'action' — a way to measure change — and using something called a Lagrangian density. They also often use a 'metric,' which helps describe the shape of space.

One important idea is Mach's principle. Some theories use this principle, which suggests that the way space behaves depends on the matter in the universe. This idea sits between Newton's view of absolute space and Einstein's view that there is no fixed reference frame.

Theories can also be classified by how they use math to describe gravity. Some use a 'metric tensor,' which helps measure distances and times, just like in Einstein's theory. Others might not follow this exactly, leading to different ways of understanding gravity. There are many types of metric theories, including scalar field theories, quasilinear theories, tensor theories, and more. Each type has its own special features and ideas about how gravity works.

Theories from 1917 to the 1980s

Main article: History of gravitational theory

At the time it was published in the 17th century, Isaac Newton's theory of gravity was the most accurate theory of gravity. Since then, a number of alternatives were proposed. The theories which predate the formulation of general relativity in 1915 are discussed in history of gravitational theory.

This section includes alternatives to general relativity published after general relativity but before the observations of galaxy rotation that led to the hypothesis of "dark matter". These theories are presented here without a cosmological constant or added scalar or vector potential unless specifically noted, for the simple reason that the need for one or both of these was not recognized before the supernova observations by the Supernova Cosmology Project and High-Z Supernova Search Team. How to add a cosmological constant or quintessence to a theory is discussed under Modern Theories (see also Einstein–Hilbert action).

Scalar field theories

The scalar field theories of Nordström have already been discussed. Those of Littlewood, Bergman, Yilmaz, Whitrow and Morduch and Page and Tupper follow the general formula give by Page and Tupper.

According to Page and Tupper, who discuss all these except Nordström, the general scalar field theory comes from the principle of least action.

In Nordström,

In Littlewood and Bergmann,

In Whitrow and Morduch,

In Page and Tupper,

Page and Tupper matches Yilmaz's theory 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 summarized some theories and also created two more. In the first, a pre-existing special relativity space-time and universal time coordinate acts with matter and non-gravitational fields to generate a scalar field. This scalar field acts together with all the rest to generate the metric.

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

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 there are two arbitrary functions and that are related to the metric by:

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

In Papapetrou the gravitational part of the Lagrangian is:

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

Bimetric theories

Quasilinear theories

In Whitehead, the physical metric is constructed (by Synge) 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

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

Deser and Laurent and Bollini–Giambiagi–Tiomno are Linear Fixed Gauge theories. Taking an approach from quantum field theory, combine a Minkowski spacetime with the gauge invariant action of a spin-two tensor field (i.e. graviton) to define

The action is:

The Bianchi identity associated with this partial gauge invariance is wrong. Linear Fixed Gauge theories seek to remedy this by breaking the gauge invariance of the gravitational action through the introduction of auxiliary gravitational fields that couple to .

A cosmological constant can be introduced into a quasilinear theory by changing the Minkowski background to a de Sitter or anti-de Sitter spacetime, as suggested by G. Temple in 1923. Temple's suggestions on how to do this were criticized by C. B. Rayner in 1955.

Tensor theories

Einstein's general relativity is the simplest plausible theory of gravity that can be based on just one symmetric tensor field (the metric tensor). Others include: Starobinsky (R+R^2) gravity, Gauss–Bonnet gravity, f(R) gravity, and Lovelock theory of gravity.

Starobinsky

Gauss–Bonnet

Gauss–Bonnet gravity has the action

where the coefficients of the extra terms are chosen so that the action reduces to general relativity in 4 spacetime dimensions and the extra terms are only non-trivial when more dimensions are introduced.

Stelle's 4th derivative gravity

Stelle's 4th derivative gravity, which is a generalization of Gauss–Bonnet gravity, has the action

f(R)

f(R) gravity has the action

and is a family of theories, each defined by a different function of the Ricci scalar. Starobinsky gravity is actually an f ( R ) theory.

Infinite derivative gravity

Infinite derivative gravity is a covariant theory of gravity, quadratic in curvature, torsion free and parity invariant,

and

in order to make sure that only massless spin −2 and spin −0 components propagate in the graviton propagator around Minkowski background. The action becomes non-local beyond the scale , and recovers to general relativity in the infrared, for energies below the non-local scale . In the ultraviolet regime, at distances and time scales below non-local scale, , the gravitational interaction weakens enough to resolve point-like singularity, which means Schwarzschild's singularity can be potentially resolved in infinite derivative theories of gravity.

Lovelock

Lovelock gravity has the action

and can be thought of as a generalization of general relativity.

Scalar–tensor theories

These all contain at least one free parameter, as opposed to general relativity which has no free parameters.

Although not normally considered a Scalar–Tensor theory of gravity, the 5 by 5 metric of Kaluza–Klein reduces to a 4 by 4 metric and a single scalar. So if the 5th element is treated as a scalar gravitational field instead of an electromagnetic field then Kaluza–Klein can be considered the progenitor of Scalar–Tensor theories of gravity. This was recognized by Thiry.

Scalar–Tensor theories include Thiry, Jordan, Brans and Dicke, Bergman, Nordtveldt (1970), Wagoner, Bekenstein and Barker.

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 general relativity. 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 and Wagoner. Special cases are:

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, is constant

Bekenstein variable mass theory Starting with parameters and , found from a cosmological solution, determines function then

Barker constant G theory

Adjustment of allows Scalar Tensor Theories to tend to general relativity in the limit of in the current epoch. However, there could be significant differences from general relativity in the early universe.

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

The above examples are particular cases of Horndeski's theory, the most general Lagrangian constructed out of the metric tensor and a scalar field leading to second order equations of motion in 4-dimensional space. Viable theories beyond Horndeski (with higher order equations of motion) have been shown to exist.

Vector–tensor theories

Before we start, Will (2001) has said: "Many alternative metric theories developed during the 1970s and 1980s could be viewed as "straw-man" theories, invented to prove that such theories exist or to illustrate particular properties. Few of these could be regarded as well-motivated theories from the point of view, say, of field theory or particle physics. Examples are the vector–tensor theories studied by Will, Nordtvedt and Hellings."

Hellings and Nordtvedt and Will and Nordtvedt are both vector–tensor theories. In addition to the metric tensor there is a timelike vector field . The gravitational action is:

where , , , are constants and

(See Will for the field equations for and .)

Will and Nordtvedt is a special case where

Hellings and Nordtvedt 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 general relativity so, so long as general relativity is confirmed by experiment, general vector–tensor theories can never be ruled out.

Other metric theories

Others metric theories have been proposed; that of Bekenstein 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 claims that all non-metric theories are eliminated by Einstein's Equivalence Principle. Will tempers that by explaining experimental criteria for testing non-metric theories against Einstein's Equivalence Principle in his 2001 edition. Misner et al. claims that Cartan's theory is the only non-metric theory to survive all experimental tests up to that date and Turyshev lists Cartan's theory among the few that have survived all experimental tests up to that date. The following is a quick sketch of Cartan's theory as restated by Trautman.

Cartan suggested a simple generalization of Einstein's theory of gravitation. He proposed a model of space time with a metric tensor and a linear "connection" compatible with the metric but not necessarily symmetric. The torsion tensor of the connection is related to the density of intrinsic angular momentum. Independently of Cartan, similar ideas were put forward by Sciama, by Kibble in the years 1958 to 1966, culminating in a 1976 review by Hehl et al.

The original description is in terms of differential forms, but for the present article that is replaced by the more familiar language of tensors (risking loss of accuracy). As in general relativity, the Lagrangian is made up of a massless and a mass part. The Lagrangian for the massless part is:

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 general relativity.

Some equations of the non-metric theory of Belinfante and Swihart have already been discussed in the section on bimetric theories.

A distinctively non-metric theory is given by gauge theory gravity, which replaces the metric in its field equations with a pair of gauge fields in flat spacetime. On the one hand, the theory is quite conservative because it is substantially equivalent to Einstein–Cartan theory (or general relativity in the limit of vanishing spin), differing mostly in the nature of its global solutions. On the other hand, it is radical because it replaces differential geometry with geometric algebra.

Theories from 1917 to the 1980s.
Publication year(s)	Author(s)	Theory name	Theory type
1922	Alfred North Whitehead	Whitehead's theory of gravitation	Quasilinear
1922, 1923	Élie Cartan	Einstein–Cartan theory	Non-metric
1939	Markus Fierz, Wolfgang Pauli
1943	George David Birkhoff
1948	Edward Arthur Milne	Kinematic Relativity
1948	Yves Thiry
1954	Achilles Papapetrou		Scalar field
1953	Dudley E. Littlewood		Scalar field
1955	Pascual Jordan
1956	Otto Bergmann		Scalar field
1957	Frederik Belinfante, James C. Swihart
1958, 1973	Huseyin Yilmaz	Yilmaz theory of gravitation
1961	Carl H. Brans, Robert H. Dicke	Brans–Dicke theory	Scalar–tensor
1960, 1965	Gerald James Whitrow, G. E. Morduch		Scalar field
1966	Paul Kustaanheimo
1967	Paul Kustaanheimo, V. S. Nuotio
1968	Stanley Deser, B. E. Laurent		Quasilinear
1968	C. Page, B. O. J. Tupper		Scalar field
1968	Peter Bergmann		Scalar–tensor
1970	C. G. Bollini, J. J. Giambiagi, J. Tiomno		Quasilinear
1970	Kenneth Nordtvedt
1970	Robert V. Wagoner		Scalar–tensor
1971	Nathan Rosen		Scalar field
1975	Nathan Rosen		Bimetric
1972, 1973	Ni Wei-tou		Scalar field
1972	Clifford Martin Will, Kenneth Nordtvedt		Vector–tensor
1973	Ronald Hellings, Kenneth Nordtvedt		Vector–tensor
1973	Alan Lightman, David L. Lee		Scalar field
1974	David L. Lee, Alan Lightman, Ni Wei-tou
1977	Jacob Bekenstein		Scalar–tensor
1978	B. M. Barker		Scalar–tensor
1979	P. Rastall		Bimetric

Modern theories 1980s to present

This section explores ideas about gravity that came up after scientists observed how galaxies spin, which led to the idea of "dark matter." There isn't a complete list of all these theories, but some well-known ones include Bekenstein, Moffat, and others. These theories often include a cosmological constant or add special forces.

The main reason for these new ideas about gravity is to explain things like "inflation," "dark matter," and "dark energy." These are big mysteries in how the universe works. Scientists think gravity works like Einstein's theory now, but maybe it worked differently in the early universe.

In the 1980s, scientists realized there were problems with the common Big Bang theory, like the horizon problem. Some suggested that the speed of light was faster in the early universe. Then, they saw that galaxies spin in surprising ways. This made people wonder if there is more mass in the universe than we can see, or if our theory of gravity is wrong. Most scientists now think the missing mass is "cold dark matter," but some still think new gravity theories might have the answer.

In the 1990s, scientists found that the universe is expanding faster and faster, which many now think is because of dark energy. This brought back Einstein's idea of a cosmological constant, and another idea called quintessence appeared. Some new gravity theories tried to explain this expansion in different ways. An event called GW170817 showed that many of these theories don't work. Another puzzle is the Pioneer anomaly, which some thought new gravity theories could explain, but it's now thought to be because of heat radiation.

Cosmological constant and quintessence

The idea of a cosmological constant goes back to Einstein in 1917. It was thought to be zero, but data from exploding stars showed the universe's expansion is speeding up, bringing this idea back. In simple gravity, adding a cosmological constant changes some equations. In Einstein's theory, it changes the equations that describe gravity. In newer theories, a similar idea can be added.

Quintessence is like a special force that can change over time, making the universe's expansion speed up more in the early universe and slow down now.

Farnes' theories

In 2018, an astronomer named Jamie Farnes suggested a theory using "negative mass" to explain dark matter and dark energy with one idea. This theory changes Einstein's equations in a specific way.

Relativistic MOND

Modified Newtonian Dynamics (MOND) was created in 1983 to explain galaxy motions without dark matter. Many attempts have been made to make MOND work with Einstein's theory. One version, called TeVeS, tries to include extra forces but has some problems.

Moffat's theories

J. W. Moffat created a theory called non-symmetric gravitation. It uses a special kind of space and can explain galaxy motions without dark matter. Another theory, called MSTG, also tries to explain galaxy motions and lensing without dark matter.

Infinite derivative gravity

Some theories add many extra terms to Einstein's equations to avoid problems and explain how gravity works at very small distances.

General relativity self-interaction (GRSI)

The GRSI model tries to explain observations about the universe without dark matter or dark energy by adding special interactions to Einstein's theory. This model can explain things like the way galaxies spin and the expansion of the universe, similar to how strong forces work between tiny particles.

The GRSI model can explain several observations that are hard to understand with the usual theory, such as the way galaxies spin, the cosmic microwave background, and the expansion of the universe. It also addresses puzzles like the Tully-Fisher relation and the Hubble tension.

Testing of alternatives to general relativity

Main article: Tests of general relativity

For any new theory to replace Einstein's general relativity, it must pass many tests. These tests check if the theory can explain things we already know about gravity.

One important test is to see if the theory is self-consistent, meaning it doesn't contain contradictions. For example, some older theories had problems because they allowed impossible situations or gave unclear results. Another test is to see if the theory can explain all known experiments, like how planets move or how time passes in strong gravity. Theories that cannot explain these things are considered incomplete.

Scientists also check if theories agree with well-known tests, like how light bends around the Sun or how the orbits of planets change over time. These tests have always matched Einstein's theory so far. Finally, theories must also work well with other areas of physics, like the study of electricity and magnetism, to be considered complete.

Results of testing theories

General Relativity has been tested for over 100 years, and many alternative theories of gravity have not matched observations. One way scientists compare these theories is using something called Parametric post-Newtonian formalism. This helps show how different theories predict the behavior of gravity.

So far, all experiments support general relativity. Some theories, like Whitehead's, have issues with predicting correct tides unless changed. Other theories, such as those by Ni and Lee Lightman, cannot explain certain movements of planets. As tests become more precise, differences between these theories and general relativity become smaller and smaller. Currently, no other theory matches general relativity as well, except possibly one called Cartan.

	γ {\displaystyle \gamma }	β {\displaystyle \beta }	ξ {\displaystyle \xi }	α 1 {\displaystyle \alpha _{1}}	α 2 {\displaystyle \alpha _{2}}	α 3 {\displaystyle \alpha _{3}}	ζ 1 {\displaystyle \zeta _{1}}	ζ 2 {\displaystyle \zeta _{2}}	ζ 3 {\displaystyle \zeta _{3}}	ζ 4 {\displaystyle \zeta _{4}}
Newton	0	0	0	0	0	0	0	0	0	0
Einstein general relativity	1	1	0	0	0	0	0	0	0	0
Scalar–tensor theories
Bergmann, Wagoner	1 + ω 2 + ω {\displaystyle \textstyle {\frac {1+\omega }{2+\omega }}}	β {\displaystyle \beta }	0	0	0	0	0	0	0	0
Nordtvedt, Bekenstein	1 + ω 2 + ω {\displaystyle \textstyle {\frac {1+\omega }{2+\omega }}}	β {\displaystyle \beta }	0	0	0	0	0	0	0	0
Brans–Dicke	1 + ω 2 + ω {\displaystyle \textstyle {\frac {1+\omega }{2+\omega }}}	1	0	0	0	0	0	0	0	0
Vector–tensor theories
Hellings–Nordtvedt	γ {\displaystyle \gamma }	β {\displaystyle \beta }	0	α 1 {\displaystyle \alpha _{1}}	α 2 {\displaystyle \alpha _{2}}	0	0	0	0	0
Will–Nordtvedt	1	1	0	0	α 2 {\displaystyle \alpha _{2}}	0	0	0	0	0
Bimetric theories
Rosen	1	1	0	0	c 0 / c 1 − 1 {\displaystyle c_{0}/c_{1}-1}	0	0	0	0	0
Rastall	1	1	0	0	α 2 {\displaystyle \alpha _{2}}	0	0	0	0	0
Lightman–Lee	γ {\displaystyle \gamma }	β {\displaystyle \beta }	0	α 1 {\displaystyle \alpha _{1}}	α 2 {\displaystyle \alpha _{2}}	0	0	0	0	0
Stratified theories
Lee–Lightman–Ni	a c 0 / c 1 {\displaystyle ac_{0}/c_{1}}	β {\displaystyle \beta }	ξ {\displaystyle \xi }	α 1 {\displaystyle \alpha _{1}}	α 2 {\displaystyle \alpha _{2}}	0	0	0	0	0
Ni	a c 0 / c 1 {\displaystyle ac_{0}/c_{1}}	b c 0 {\displaystyle bc_{0}}	0	α 1 {\displaystyle \alpha _{1}}	α 2 {\displaystyle \alpha _{2}}	0	0	0	0	0
Scalar field theories
Einstein (1912) {Not general relativity}	0	0		-4	0	-2	0	-1	0	0†
Whitrow–Morduch	0	-1		-4	0	0	0	−3	0	0†
Rosen	λ {\displaystyle \lambda }	3 4 + λ 4 {\displaystyle \textstyle {\frac {3}{4}}+\textstyle {\frac {\lambda }{4}}}		− 4 − 4 λ {\displaystyle -4-4\lambda }	0	-4	0	-1	0	0
Papapetrou	1	1		-8	-4	0	0	2	0	0
Ni (stratified)	1	1		-8	0	0	0	2	0	0
Yilmaz (1962)	1	1		-8	0	-4	0	-2	0	-1†
Page–Tupper	γ {\displaystyle \gamma }	β {\displaystyle \beta }		− 4 − 4 γ {\displaystyle -4-4\gamma }	0	− 2 − 2 γ {\displaystyle -2-2\gamma }	0	ζ 2 {\displaystyle \zeta _{2}}	0	ζ 4 {\displaystyle \zeta _{4}}
Nordström	− 1 {\displaystyle -1}	1 2 {\displaystyle \textstyle {\frac {1}{2}}}		0	0	0	0	0	0	0†
Nordström, Einstein–Fokker	− 1 {\displaystyle -1}	1 2 {\displaystyle \textstyle {\frac {1}{2}}}		0	0	0	0	0	0	0
Ni (flat)	− 1 {\displaystyle -1}	1 − q {\displaystyle 1-q}		0	0	0	0	ζ 2 {\displaystyle \zeta _{2}}	0	0†
Whitrow–Morduch	− 1 {\displaystyle -1}	1 − q {\displaystyle 1-q}		0	0	0	0	q	0	0†
Littlewood, Bergman	− 1 {\displaystyle -1}	1 2 {\displaystyle \textstyle {\frac {1}{2}}}		0	0	0	0	-1	0	0†