Alternatives to general relativity

Alternatives to general relativity are physical theories that try to explain how gravitation works, instead of using Einstein's idea called general relativity. Scientists have created many different theories to better understand gravity. These theories can be grouped into four main types.

Some theories are based on older ideas about physics and do not include the modern idea of quantum mechanics. Others try to use quantum mechanics to break gravity into tiny pieces called quantized gravity. Some theories try to explain gravity and other forces in the universe together; these are known as classical unified field theories. And finally, some theories try to do both: use quantum mechanics for gravity and also explain all forces together. These are called 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 things we see in space. It even helps explain mysterious parts of the universe called dark matter and dark energy, though we still do not fully understand what they are. Because general relativity works so well, many scientists think it will continue to be the best way to understand gravity, even though there are still some puzzles to solve.

Notation in this article

Main articles: Mathematics of general relativity and Ricci calculus

This section explains the special symbols and rules used in the study of gravity. The letter c represents the speed of light, and G stands for the gravitational constant. Scientists use letters with special shapes, called indices, to describe how space and time work together. Latin letters (like A, B, C) count from 1 to 3, while Greek letters (like α, β, γ) count from 0 to 3.

Two important symbols are η_μν (the Minkowski metric) and g_μν (usually the metric tensor). These help describe the shape of space and time. The article also explains how to show changes in values using symbols like ∂_μφ (which means partial differentiation) and ∇_μφ (which means covariant differentiation).

General relativity

Main article: General relativity

General relativity is a way scientists explain how gravity works. It uses special kinds of math to describe how space and time change because of mass and energy. These math ideas help us understand big questions about the universe, like how stars move and how black holes form.

Some other ideas about gravity are different. For example, Nordström’s theories use a simpler math idea called a scalar. Scientists have also tried mixing this math with other ideas or adding something called vector fields to better explain gravity. These different theories help scientists explore new ways to understand the forces that pull objects together.

Classification of theories

Theories of gravity can be sorted into different groups. Many of these theories share some basic ideas. They often talk about an 'action,' which is a way to explain how objects move and change. They also use something called a Lagrangian density.

These theories often use a tool called a metric. The metric helps describe the shape and distance in space.

One key idea is Mach's principle. Some theories follow this idea. It is a middle ground between Newton's view and Einstein's view of space and time. Newton believed space and time were fixed. Einstein said there was no fixed reference frame. Mach suggested that how things move depends on the matter around them in the universe.

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

See also: Scalar theories of gravitation

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.

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 φ R term. S_m is the matter action.

t 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 there are two arbitrary functions f ( φ ) and k ( φ ) that are related to the metric by:

η μ ν ∂ μ ∂ ν φ = 4 π ρ ∗ k ( φ )

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

ds^2 = φ^2 dt^2 − ψ^2 [ dx^2 + dy^2 + dz^2 ]

In Papapetrou the gravitational part of the Lagrangian is:

L_φ = e^φ ( 1/2 e^−φ ∂_α φ ∂_α φ + 3/2 e^φ ∂_0 φ ∂_0 φ )

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

L_φ = e^{1/2 (3 φ + χ )} ( −1/2 e^−φ ∂_α φ ∂_α φ − e^−φ ∂_α φ ∂_χ φ + 3/2 e^−χ ∂_0 φ ∂_0 φ )

Bimetric theories

See also: Bimetric theory

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 (1975) developed a bimetric theory. The action is:

Box_η g_μν − g^αβ η^γδ g_μα|γ g_νβ|δ = −16 π G √(g/η) (T_μν − 1/2 g_μν T)

Lightman–Lee developed a metric theory based on the non-metric theory of Belinfante and Swihart. The result is known as BSLL theory. Given a tensor field B_μν , B = B_μν η^μν , and two constants a and f the action is:

S = 1/(16πG) ∫ d^4x √(−η) (a B^μν|α B_μν|α + f B_,α B_,α) + S_m

and the stress–energy tensor comes from:

a Box_η B^μν + f η^μν Box_η B = −4πG √(g/η) T^αβ ( ∂g_αβ/∂B_μν )

In Rastall, the metric is an algebraic function of the Minkowski metric and a Vector field. The action is:

S = 1/(16πG) ∫ d^4x √(−g) F(N) K^μ;ν K_μ;ν + S_m

where

F(N) = −N/(2+N) and N = g^μν K_μ K_ν

Quasilinear theories

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

g_μν(x^α) = η_μν − 2 ∫_Σ^− y_μ^− y_ν^− / (w^−)^3 [ √(−g) ρ u^α dΣ_α ]^−

where the superscript (−) indicates quantities evaluated along the past η light cone of the field point x^α and

( y^μ )^− = x^μ − ( x^μ )^− , ( y^μ )^− ( y_μ )^− = 0 , w^− = ( y^μ )^− ( u_μ )^− , ( u_μ ) = dx^μ/dσ , dσ^2 = η_μν dx^μ dx^ν

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) h_μν to define

g_μν = η_μν + h_μν

The action is:

S = 1/(16πG) ∫ d^4x √(−η) [ 2 h|^μν_ν h_μλ|^λ − 2 h|^μν_ν h_λ|^μ_λ + h_ν|^μ_ν h_λλ|^μ − h^μν|^λ h_μν|_λ ] + S_m

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 h_μν .

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

See also: Starobinsky inflation

Starobinsky gravity, proposed by Alexei Starobinsky has the Lagrangian

L = √(−g) [ R + R^2/(6M^2) ]

and has been used to explain inflation, in the form of Starobinsky inflation. Here M is a constant.

Gauss–Bonnet

Gauss–Bonnet gravity has the action

L = √(−g) [ R + R^2 − 4 R^μν R_μν + R^μνρσ R_μνρσ ] .

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

L = √(−g) [ R + f_1 R^2 + f_2 R^μν R_μν + f_3 R^μνρσ R_μνρσ ] .

f(R)

f(R) gravity has the action

L = √(−g) f(R)

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

Infinite derivative gravity

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

L = √(−g) [ M_p^2 R + R f_1 ( □/M_s^2 ) R + R^μν f_2 ( □/M_s^2 ) R_μν + R^μνρσ f_3 ( □/M_s^2 ) R_μνρσ ] .

and

2 f_1 ( □/M_s^2 ) + f_2 ( □/M_s^2 ) + 2 f_3 ( □/M_s^2 ) = 0 ,

in order to make sure that only massless spin −2 and spin −0 components propagate in the graviton propagator around Minkowski background. The action becomes non-local beyond the scale M_s , and recovers to general relativity in the infrared, for energies below the non-local scale M_s . In the ultraviolet regime, at distances and time scales below non-local scale, M_s^−1 , the gravitational interaction weakens enough to resolve point-like singularity, which means Schwarzschild's singularity can be potentially resolved in infinite derivative theories of gravity.

Lovelock

Lovelock gravity has the action

L = √(−g) ( α_0 + α_1 R + α_2 ( R^2 + R_αβμν R^αβμν − 4 R_μν R^μν ) + α_3 O(R^3) ) ,

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

Scalar–tensor theories

See also: Scalar–tensor theory, Brans–Dicke theory, Dilaton, Chameleon particle, Pressuron, and Horndeski's theory

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 S is based on the integral of the Lagrangian L_φ .

S = 1/(16πG) ∫ d^4x √(−g) L_φ + S_m

L_φ = φ R − ω(φ)/φ g^μν , ∂_μ φ ∂_ν φ + 2 φ λ(φ)

S_m = ∫ d^4x √g G_N L_m

T^μν   = d/d(2 √g) δS_m/δg_μν

where ω(φ) is a different dimensionless function for each different scalar–tensor theory. The function λ(φ) plays the same role as the cosmological constant in general relativity. G_N is a dimensionless normalization constant that fixes the present-day value of G . An arbitrary potential can be added for the scalar.

The full version is retained in Bergman and Wagoner. Special cases are:

Nordtvedt, λ = 0

Since λ 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, ω is constant

Bekenstein variable mass theory Starting with parameters r and q , found from a cosmological solution, φ = [1−qf(φ)] f(φ)^−r determines function f then

ω(φ) = −3/2 − 1/4 f(φ) [(1−6q)qf(φ)−1] [r+(1−r)qf(φ)]^−2

Barker constant G theory

ω(φ) = (4−3φ)/(2φ−2)

Adjustment of ω(φ) allows Scalar Tensor Theories to tend to general relativity in the limit of ω → ∞ 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 K_μ .

The gravitational action is:

S = 1/(16πG) ∫ d^4x √(−g) [ R + ω K_μ K^μ R + η K^μ K^ν R_μν − ε F_μν F^μν + τ K_μ;ν K^μ;ν ] + S_m

where ω, η, ε, τ are constants and

F_μν = K_ν;μ − K_μ;ν . (See Will for the field equations for T^μν and K_μ .)

Will and Nordtvedt is a special case where

ω = η = ε = 0 ; τ = 1

Hellings and Nordtvedt is a special case where

τ = 0 ; ε = 1 ; η = −2ω

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 ω = η = ε = τ = 0 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

See also: Einstein–Cartan theory and Cartan connection

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:

L = 1/(32πG) Ω^μ_ν g^νξ x^η x^ζ ε_ξμηζ

Ω^μ_ν = dω^μ_ν + ω^ξ_η ∇_x^μ = −ω^ν_μ x_ν

The ω^ν_μ is the linear connection. ε_ξμηζ is the completely antisymmetric pseudo-tensor (Levi-Civita symbol) with ε_0123 = √(−g) , and g^νξ is the metric tensor as usual. By assuming that the linear connection is metric, it is possible to remove the unwanted freedom inherent in the non-metric theory. The stress–energy tensor is calculated from:

T^μν = 1/(16πG) ( g^μν η^ξ_η − g^ξμ η^ν_η − g^ξν η^μ_η ) Ω^η_ξ

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 one hand, the theory is quite conservative because it is substantially equivalent to Einstein–Cartan theory (or general relativity in the limit of vanishing spin), differing mostly in the nature of its global solutions. On the other hand, it is radical because it replaces differential geometry with geometric algebra.

Modern theories 1980s to present

This section talks about new ideas about gravity that scientists thought of after they saw how galaxies spin. This made them think about something called "dark matter." There isn’t a full list of all these ideas, but some important ones were created by scientists named Bekenstein and Moffat. These theories often include extra ideas like a cosmological constant or special math.

These new ideas about gravity mainly try to understand the universe on a very large scale. They try to explain things like "inflation" (a quick expansion of the very early universe), "dark matter," and "dark energy." The basic idea is that gravity works like Einstein’s theory now, but might have worked very differently when the universe was very young.

In the 1980s, scientists found puzzles in the old ideas about the universe starting with a "big bang." One puzzle was that the universe didn’t seem big enough to hold even tiny particles in the very early moments. Some scientists suggested that the universe expanded very fast to solve this, called "inflation." Another idea was that maybe gravity works differently, with light moving faster in the early universe. When scientists saw how galaxies spin, they wondered if there is more stuff in the universe than we can see, or if our ideas about gravity are wrong. Most now think there is unseen matter called "cold dark matter," but some scientists still think new gravity ideas might have the answer.

In the 1990s, studies of very bright stars called supernovae showed that the universe is expanding faster and faster. This is often explained by something called "dark energy," which is like a constant in Einstein’s equations. Some new gravity ideas tried to explain these results in different ways. Measurements of how fast gravity moves also helped rule out many older gravity theories.

Another thing that made scientists interested in new gravity ideas was something called the "Pioneer anomaly." This was an unexpected push on old space probes called the Pioneer spacecraft. Some gravity theories could explain this, but it is now thought to be caused by heat from the sun affecting the spacecraft.

Cosmological constant and quintessence

The idea of a cosmological constant goes back to Einstein in 1917. It was mostly ignored because it made the universe unstable, but it came back when supernovae showed the universe’s expansion is speeding up. In simple gravity, adding a cosmological constant changes some math equations. In Einstein’s theory, it changes the equations that describe how space and time work.

In newer gravity theories, a cosmological constant can be added in similar ways. Sometimes, a special math idea called "quintessence" is used instead, which can change over time and might explain why the universe’s expansion is speeding up.

Farnes' theories

In 2018, a scientist named Jamie Farnes suggested a new idea about gravity using something called "negative mass." This idea might help explain both dark matter and dark energy with one simple concept. Negative mass particles could push on galaxies from outside, acting like dark matter, and also push the universe apart, acting like dark energy.

Relativistic MOND

An idea from the 1980s called MOND tried to explain galaxy spins without dark matter by changing Newton’s ideas about gravity. Many newer versions of this idea exist, trying to make it work with Einstein’s theory. One version, called TeVeS, adds extra math to try to fit with what we see, but it has some problems when tested against observations.

Moffat's theories

A scientist named J. W. Moffat created theories that don’t use the usual way of describing space and time. These theories can explain how galaxies spin without dark matter and also how light bends around big objects. These ideas are still being studied and tested.

Infinite derivative gravity

Some scientists added extra math terms to Einstein’s theory to avoid problems with old ideas. These new terms help avoid singularities (points where math breaks down) and make the theory work better at very small distances.

General relativity self-interaction (GRSI)

The GRSI model tries to explain things like galaxy spins and the speeding up of the universe’s expansion by adding extra interactions to Einstein’s theory. This is similar to how particles called gluons interact in the theory of the strong force. These extra interactions can make gravity stronger in big objects like galaxies and weaker over very large distances, which might explain why the universe’s expansion is speeding up without needing dark energy.

Testing of alternatives to general relativity

Main article: Tests of general relativity

To see if a new idea about gravity could replace Einstein's theory, scientists use many different tests. These tests help make sure the new idea works well with what we already know about space, time, and how things move.

One important test is to see if the new idea is clear and does not have mistakes. For example, some older ideas did not work well when tested with simple experiments, like how planets move around the Sun. Another test is to see if the idea can explain things we see in space, like how light bends when it passes close to big objects. Scientists also check if the new idea matches what we know from everyday physics and from experiments that show how things move without gravity. All these tests help scientists find the best ideas about how gravity works.

Results of testing theories

Many ideas about how gravity might work have been tested. So far, all of these ideas agree with Einstein's theory called general relativity. Scientists compare these ideas using something called Parametric post-Newtonian values. These values help show how well different theories match what we see in space.

Some theories do not work well with certain observations. For example, some theories cannot explain the way Mercury moves around the Sun. Others do not match what we see in very strong gravity, like near big stars. As tests become more exact, the differences between these theories and general relativity get smaller. This means that, so far, general relativity still fits all the observations best.