Alternatives to general relativity — Safekipedia Adventurer

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 part talks about special symbols and rules used to study gravity. The symbol c means the speed of light, and G stands for the gravitational constant. Scientists use letters with special shapes, like Greek letters, to show different directions in space and time. They also have rules for adding and changing these symbols to explain how gravity works.

General relativity

Main article: General relativity

General relativity is a theory made by Einstein. It explains gravity as the bending of space and time. It uses special math tools called tensors to show how objects move and how space acts. One big idea is that the shape of space changes depending on how much matter and energy is there.

Some other theories try to explain gravity in different ways. For example, some use a single number instead of tensors. Others mix that number 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 see in space.

Classification of theories

Theories of gravity can be sorted into different groups. Many of these theories share 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 key idea is Mach's principle. Some theories use this idea, suggesting that how space behaves depends on the matter in the universe. This idea fits between Newton's view of fixed space and Einstein's view that space and time change.

Theories can also be grouped by the math they use to describe gravity. Some use a 'metric tensor,' which helps measure distances and times, just like in Einstein's theory. Others might work differently, leading to new ways to understand gravity. There are many types of metric theories, each with its own features and ideas about gravity.

Theories from 1917 to the 1980s

Main article: History of gravitational theory

At the time it was published, Isaac Newton's theory of gravity was the best way to understand gravity. Since then, many other ideas were suggested. The theories that came before general relativity in 1915 are talked about in the history of gravitational theory.

This part looks at ideas about gravity that came after general relativity but before we learned about "dark matter" from how stars move in groups. These ideas do not include extra parts like a cosmological constant or special fields, because those were not thought about before later space observations by the Supernova Cosmology Project and High-Z Supernova Search Team. How to add a cosmological constant or quintessence to a theory is talked about in Modern Theories (see also Einstein–Hilbert action).

Scalar field theories

Scalar field theories were suggested by scientists like Nordström, Littlewood, Bergman, Yilmaz, Whitrow, Morduch, and Page and Tupper. Page and Tupper say that all these theories come from a basic idea called the principle of least action.

For Nordström,

For Littlewood and Bergmann,

For Whitrow and Morduch,

For Page and Tupper,

Page and Tupper’s theory matches Yilmaz’s theory in some cases when

The bending of light by gravity must be zero when a certain value c is constant. But both changing c and zero bending of light do not match what we see, so scalar theories of gravity seem unlikely to work. Also, if we change the numbers in a scalar theory to make the bending of light right, then another effect called gravitational redshift may not be right.

Ni suggested some theories and made two new ones. In the first, space-time and time work with matter and other fields to make a scalar field. This scalar field works with everything else to make the shape of space-time.

Misner and others give this without a certain term. is the action for matter.

This theory fits together well. But the way the Solar System moves through the universe does not match what we see.

In Ni’s second theory, there are two special functions and that relate to the shape of space-time by:

Ni says Rosen had two scalar fields and that relate to the shape of space-time by:

In Papapetrou, the part of the theory that relates to gravity is:

In Papapetrou there is a second scalar field . The part of the theory that relates to gravity is now:

Bimetric theories

Quasilinear theories

In Whitehead, the shape of space-time is made (by Synge) from the Minkowski shape and matter, so it does not need a scalar field. The way it is made is:

where the superscript (−) means values along the past light cone of the point.

But the way the shape of space-time is made has been criticized.

Deser and Laurent and Bollini–Giambiagi–Tiomno are Linear Fixed Gauge theories. They mix a Minkowski space with a special math idea from particles to make

The main idea is:

The Bianchi identity for this special idea is not right. Linear Fixed Gauge theories try to fix this by adding extra parts that connect to .

We can add a cosmological constant to a quasilinear theory by changing the Minkowski background to a de Sitter or anti-de Sitter spacetime, as G. Temple suggested in 1923. Temple’s ideas on how to do this were criticized by C. B. Rayner in 1955.

Tensor theories

Einstein's general relativity is the simplest idea about gravity based on 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 main idea

where the numbers for extra parts are chosen so that the idea becomes general relativity in 4 parts of space-time and the extra parts only matter when more parts are added.

Stelle's 4th derivative gravity

Stelle's 4th derivative gravity, which is a version of Gauss–Bonnet gravity, has the main idea

f(R)

f(R) gravity has the main idea

and is a group of theories, each with a different use for the Ricci scalar. Starobinsky gravity is actually an f ( R ) theory.

Infinite derivative gravity

Infinite derivative gravity is a theory of gravity that is quadratic in curves, does not twist, and keeps left-right balance,

and

to make sure only certain parts move in the graviton around Minkowski space. The main idea becomes spread out at big distances, and goes back to general relativity for low energies. At very small distances and times, the force of gravity becomes weaker, which might mean that a problem called Schwarzschild's singularity could be solved in infinite derivative theories of gravity.

Lovelock

Lovelock gravity has the main idea

and can be seen as a version of general relativity.

Scalar–tensor theories

These theories all have at least one free number, unlike general relativity which has none.

Although not usually thought of as a Scalar–Tensor theory, the 5 by 5 shape of Kaluza–Klein becomes a 4 by 4 shape and one scalar. So if the 5th part is seen as a scalar gravity field instead of an electric field then Kaluza–Klein can be seen as the first Scalar–Tensor theory. This was noticed by Thiry.

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

The main idea comes from the math rule of the Lagrangian .

where is a different math rule for each Scalar–Tensor theory. The rule does the same job as the cosmological constant in general relativity. is a number that fixes the value of today. We can add a special rule for the scalar.

The full idea is used in Bergman and Wagoner. Special cases are:

Since was thought to be zero at the time, this would not have been seen as important. The job of the cosmological constant in newer work is talked about under Cosmological constant.

Brans–Dicke, is constant

Bekenstein variable mass theory Starting with numbers and , found from space, decides the rule then

Barker constant G theory

Changing lets Scalar Tensor Theories become like general relativity today. But there could be big differences from general relativity in the very early universe.

As long as general relativity matches what we see, general Scalar–Tensor theories (including Brans–Dicke) can never be fully ruled out, but tests keep showing general relativity is right and the numbers have to be set very close to what general relativity needs.

The examples above are special cases of Horndeski's theory, the most general math rule made from the shape of space-time and a scalar field that gives second order rules in 4 parts of space. Theories beyond Horndeski (with higher order rules) have been shown to exist.

Vector–tensor theories

Will (2001) said: "Many new ideas about gravity from the 1970s and 1980s were made to show that such ideas exist or to show special points. Few of these were seen as strong ideas from the view of particles or fields. Examples are the vector–tensor theories studied by Will, Nordtvedt and Hellings."

Hellings and Nordtvedt and Will and Nordtvedt are both vector–tensor theories. Besides the shape of space-time there is a time-like vector field . The main rule for gravity is:

where , , , are numbers and

(See Will for the rules 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, meaning they follow rules for keeping balance but can show special effects from a preferred frame. When they become general relativity then, as long as general relativity matches what we see, general vector–tensor theories can never be fully ruled out.

Other metric theories

Other ideas about gravity have been suggested; that of Bekenstein is talked about under Modern Theories.

Non-metric theories

Cartan's theory is interesting because it is a non-metric theory and it is old. The state of Cartan's theory is not clear. Will says all non-metric theories are ruled out by Einstein's Equivalence Principle. But Will adds that tests can be done to check non-metric theories against Einstein's Equivalence Principle. Misner and others say that Cartan's theory is the only non-metric theory that has passed all tests up to that time and Turyshev lists Cartan's theory among the few that have passed all tests up to that time. This is a short view of Cartan's theory as restated by Trautman.

Cartan suggested a simple change to Einstein's theory of gravity. He suggested a model of space-time with a shape and a line "connection" that fits the shape but may not be steady. The twist of the connection is linked to the amount of turning in matter. Other scientists like Sciama, and Kibble from 1958 to 1966 also suggested similar ideas, ending in a 1976 review by Hehl and others.

The first way to talk about it uses math shapes, but this part uses tensors (risking losing some detail). Like in general relativity, the main idea is made of a part with no mass and a part with mass. The main idea for the part with no mass is:

The is the line connection. is the completely turning pseudo-tensor (Levi-Civita symbol) with , and is the shape as usual. By saying that the line connection fits the shape, we can remove extra parts in the non-metric theory. The force from matter is found from:

The curves of space are not common curves, but on a common space-time the main idea would become the main idea of general relativity.

Some rules of the non-metric theory of Belinfante and Swihart have been talked about in the part on bimetric theories.

A special non-metric theory is gauge theory gravity, which changes the shape in its rules with two gauge fields in straight space. One way, the theory is careful because it is almost the same as Einstein–Cartan theory (or general relativity when spin is zero), mostly different in how answers come out. The other way, it is new because it changes math shapes 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 talks about new ideas about gravity that started after scientists saw how galaxies spin. This led to the idea of "dark matter." There isn't a full list of all these theories, but some well-known ones include Bekenstein, Moffat, and others. These theories often use 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 a long time ago.

In the 1980s, scientists found problems with the common Big Bang theory, like the horizon problem. Some thought 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 a new idea to replace Einstein's general relativity, it must pass many tests. These tests check if the idea can explain what we know about gravity.

One test is to see if the idea makes sense and does not have mistakes. Some older ideas had problems because they did not work well. Another test is to see if the idea can explain things we have observed, like how planets move or how time changes in strong gravity. Ideas that cannot explain these things are not finished.

Scientists also check if the ideas match well-known tests, like how light bends around the Sun or how planets move over time. These tests have matched Einstein's theory so far. Finally, new ideas must also work well with other parts of physics, like the study of electricity and magnetism, to be considered good enough.

Results of testing theories

General Relativity has been tested for more than 100 years. Many other ideas about gravity have not worked as well. Scientists use something called Parametric post-Newtonian formalism to compare these ideas. It helps show how each one predicts gravity.

So far, all tests support general relativity. Some theories, like Whitehead's, have trouble matching what we see with tides unless they are changed. Other theories, such as those by Ni and Lee Lightman, cannot explain some planet movements. As tests get better, the differences between these theories and general relativity get smaller. Right now, no other theory matches general relativity as well, except maybe 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†