Alternatives to general relativity are physical theories that attempt to describe the phenomenon of gravitation in competition with 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 based on their scope:
Classical theories of gravity, which do not involve quantum mechanics or force unification. Theories using the principles of quantum mechanics resulting in quantized gravity. Theories which attempt to explain gravity and other forces at the same time; these are known as classical unified field theories. Theories which attempt to both put gravity in quantum mechanical terms and unify forces; these are called theories of everything. None of these alternatives to general relativity have gained wide acceptance. General relativity has withstood many tests over a large range of mass and size scales. When applied to interpret astronomical observations, cosmological models based on general relativity introduce two components to the universe, dark matter and dark energy, the nature of which is currently an unsolved problem in physics. The many successful, high precision predictions of the standard model of cosmology has led astrophysicists to conclude it and thus general relativity will be the basis for future progress. However, dark matter is not supported by the Standard Model of particle physics, physical models for dark energy do not match cosmological data, and some cosmological observations are inconsistent. These issues have led to the study of alternative theories of gravity.
Notation in this article
c {\displaystyle c\;} is the speed of light, G {\displaystyle G\;} is the gravitational constant. "Geometric variables" are not used. Latin indices go from 1 to 3, Greek indices go from 0 to 3. The Einstein summation convention is used.
η μ ν {\displaystyle \eta _{\mu \nu }\;} is the Minkowski metric. g μ ν {\displaystyle g_{\mu \nu }\;} is a tensor, usually the metric tensor. These have signature (−,+,+,+). Partial differentiation is written ∂ μ φ {\displaystyle \partial _{\mu }\varphi \;} or φ , μ {\displaystyle \varphi _{,\mu }\;} . Covariant differentiation is written ∇ μ φ {\displaystyle \nabla _{\mu }\varphi \;} or φ ; μ {\displaystyle \varphi _{;\mu }\;} .
General relativity
For comparison with alternatives, the formulas of General Relativity are:
δ ∫ d s = 0 {\displaystyle \delta \int ds=0\,}
d s 2 = g μ ν d x μ d x ν {\displaystyle {ds}^{2}=g_{\mu \nu }\,dx^{\mu }\,dx^{\nu }\,}
R μ ν = 8 π G c 4 ( T μ ν − 1 2 g μ ν T ) {\displaystyle R_{\mu \nu }={\frac {8\pi G}{c^{4}}}\left(T_{\mu \nu }-{\frac {1}{2}}g_{\mu \nu }T\right)\,}
which can also be written
T μ ν = c 4 8 π G ( R μ ν − 1 2 g μ ν R ) . {\displaystyle T^{\mu \nu }={c^{4} \over 8\pi G}\left(R^{\mu \nu }-{\frac {1}{2}}g^{\mu \nu }R\right)\,.}
The Einstein–Hilbert action for general relativity is:
S = c 4 16 π G ∫ R − g d 4 x + S m {\displaystyle S={c^{4} \over 16\pi G}\int R{\sqrt {-g}}\ d^{4}x+S_{m}\,}
where G {\displaystyle G\,} is Newton's gravitational constant, R = R μ μ {\displaystyle R=R_{\mu }^{~\mu }\,} is the Ricci curvature of space, g = det ( g μ ν ) {\displaystyle g=\det(g_{\mu \nu })\,} and S m {\displaystyle S_{m}\,} is the action due to mass. General relativity 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. Other proposed alternatives include scalar–tensor theories that contain a scalar field in addition to the tensors of general relativity. Other variants containing vector fields have been developed recently as well.
Classification of theories Theories of gravity can be classified, loosely, into several categories. Most of the theories described here have:
an 'action' (see the principle of least action, a variational principle based on the concept of action) a Lagrangian density a metric A further word here about Mach's principle is appropriate because a few of these theories rely on Mach's principle (e.g. Whitehead), and many mention it in passing (e.g. Einstein–Grossmann, Brans–Dicke). Mach's principle can be thought of as a half-way-house between Newton and Einstein. An explanation follows:
Newton: Absolute space and time. Mach: The reference frame comes from the distribution of matter in the universe. Einstein: There is no reference frame. This isn't exactly the way Mach originally stated it, see other variants in Mach principle.
Classification based on the action If a theory has a Lagrangian density for gravity, say L {\displaystyle L\,} , then the gravitational part of the action S {\displaystyle S\,} is the integral of that:
S = ∫ L − g d 4 x {\displaystyle S=\int L{\sqrt {-g}}\,\mathrm {d} ^{4}x} . In this equation it is usual, though not essential, to have g = − 1 {\displaystyle g=-1\,} at spatial infinity when using Cartesian coordinates. For example, the Einstein–Hilbert action uses L ∝ R {\displaystyle L\,\propto \,R} 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 most efficient 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. Canonical methods provide another way to construct systems that have the required conservation laws, but this approach is more cumbersome to implement. The original 1983 version of MOND did not have an action.
Classification based on the Lagrange density A few theories have an action but not a Lagrangian density. A good example is Whitehead, the action there is termed non-local.
Classification based on metricity 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 g μ ν {\displaystyle g_{\mu \nu }\,} of signature (−, +, +, +), which governs proper-length and proper-time measurements in the usual manner of special and general relativity:
d τ 2 = − g μ ν d x μ d x ν {\displaystyle {d\tau }^{2}=-g_{\mu \nu }\,dx^{\mu }\,dx^{\nu }\,}
where there is a summation over indices μ {\displaystyle \mu } and ν {\displaystyle \nu } . Condition 2: Stressed matter and fields being acted upon by gravity respond in accordance with the equation:
0 = ∇ ν T μ ν = T μ ν , ν + Γ σ ν μ T σ ν + Γ σ ν ν T μ σ {\displaystyle 0=\nabla _{\nu }T^{\mu \nu }={T^{\mu \nu }}_{,\nu }+\Gamma _{\sigma \nu }^{\mu }T^{\sigma \nu }+\Gamma _{\sigma \nu }^{\nu }T^{\mu \sigma }\,}
where T μ ν {\displaystyle T^{\mu \nu }\,} is the stress–energy tensor for all matter and non-gravitational fields, and where ∇ ν {\displaystyle \nabla _{\nu }} is the covariant derivative with respect to the metric and Γ σ ν α {\displaystyle \Gamma _{\sigma \nu }^{\alpha }\,} is the Christoffel symbol. The stress–energy tensor should also satisfy an energy condition. 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 Lee–Lightman–Ni Littlewood Ni Nordström's theory of gravitation (first metric theory of gravity to be developed) Page–Tupper Papapetrou Rosen (1971) Whitrow–Morduch Yilmaz theory of gravitation (attempted to eliminate event horizons from the theory.) Quasilinear theories (includes Linear fixed gauge) Bollini–Giambiagi–Tiomno Deser–Laurent Whitehead's theory of gravity (intended to use only retarded potentials) Tensor theories Einstein's general relativity Fourth-order gravity (allows the Lagrangian to depend on second-order contractions of the Riemann curvature tensor) f(R) gravity (allows the Lagrangian to depend on higher powers of the Ricci scalar) Gauss–Bonnet gravity Lovelock theory of gravity (allows the Lagrangian to depend on higher-order contractions of the Riemann curvature tensor) Infinite derivative gravity Scalar–tensor theories TeVeS by Bekenstein Bergmann–Wagoner Brans–Dicke theory (the most well-known alternative to general relativity, intended to be better at applying Mach's principle) Jordan Nordtvedt Thiry Chameleon Pressuron Vector–tensor theories Hellings–Nordtvedt Will–Nordtvedt Bimetric theories Lightman–Lee Rastall Rosen (1975) 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) Teleparallelism Gauge theory gravity
Theories from 1917 to the 1980s
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". Those considered here include (see Will Lang):
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:
δ ∫ f ( φ c 2 ) d s = 0 {\displaystyle \delta \int f\left({\tfrac {\varphi }{c^{2}}}\right)\,ds=0}
where the scalar field is,
φ = G M r {\displaystyle \varphi ={\frac {GM}{r}}}
and c may or may not depend on φ {\displaystyle \varphi } . In Nordström,
f ( φ / c 2 ) = exp ( − φ / c 2 ) , c = c ∞ {\displaystyle f(\varphi /c^{2})=\exp(-\varphi /c^{2}),\qquad c=c_{\infty }}
In Littlewood and Bergmann,
f ( φ c 2 ) = exp ( − φ c 2 − ( c / φ 2 ) 2 2 ) c = c ∞ {\displaystyle f\left({\frac {\varphi }{c^{2}}}\right)=\exp \left(-{\frac {\varphi }{c^{2}}}-{\frac {(c/\varphi ^{2})^{2}}{2}}\right)\qquad c=c_{\infty }\,}
In Whitrow and Morduch,
f ( φ c 2 ) = 1 , c 2 = c ∞ 2 − 2 φ {\displaystyle f\left({\frac {\varphi }{c^{2}}}\right)=1,\qquad c^{2}=c_{\infty }^{2}-2\varphi \,}
In Whitrow and Morduch,
f ( φ c 2 ) = exp ( − φ c 2 ) , c 2 = c ∞ 2 − 2 φ {\displaystyle f\left({\frac {\varphi }{c^{2}}}\right)=\exp \left(-{\frac {\varphi }{c^{2}}}\right),\qquad c^{2}=c_{\infty }^{2}-2\varphi \,}
In Page and Tupper,
f ( φ c 2 ) = φ c 2 + α ( φ c 2 ) 2 , c ∞ 2 c 2 = 1 + 4 ( φ c ∞ 2 ) + ( 15 + 2 α ) ( φ c ∞ 2 ) 2 {\displaystyle f\left({\frac {\varphi }{c^{2}}}\right)={\frac {\varphi }{c^{2}}}+\alpha \left({\frac {\varphi }{c^{2}}}\right)^{2},\qquad {\frac {c_{\infty }^{2}}{c^{2}}}=1+4\left({\frac {\varphi }{c_{\infty }^{2}}}\right)+(15+2\alpha )\left({\frac {\varphi }{c_{\infty }^{2}}}\right)^{2}}
Page and Tupper matches Yilmaz's theory to second order when α = − 7 / 2 {\displaystyle \alpha =-7/2} . 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. The action is:
S = 1 16 π G ∫ d 4 x − g L φ + S m {\displaystyle S={1 \over 16\pi G}\int d^{4}x{\sqrt {-g}}L_{\varphi }+S_{m}}
L φ = φ R − 2 g μ ν ∂ μ φ ∂ ν φ {\displaystyle L_{\varphi }=\varphi R-2g^{\mu \nu }\,\partial _{\mu }\varphi \,\partial _{\nu }\varphi }
Misner et al. gives this without the φ R {\displaystyle \varphi R} term. S m {\displaystyle S_{m}} is the matter action.
◻ φ = 4 π T μ ν [ η μ ν e − 2 φ + ( e 2 φ + e − 2 φ ) ∂ μ t ∂ ν t ] {\displaystyle \Box \varphi =4\pi T^{\mu \nu }\left[\eta _{\mu \nu }e^{-2\varphi }+\left(e^{2\varphi }+e^{-2\varphi }\right)\,\partial _{\mu }t\,\partial _{\nu }t\right]}
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 ) {\displaystyle f(\varphi )} and k ( φ ) {\displaystyle k(\varphi )} that are related to the metric by:
d s 2 = e − 2 f ( φ ) d t 2 − e 2 f ( φ ) [ d x 2 + d y 2 + d z 2 ] {\displaystyle ds^{2}=e^{-2f(\varphi )}dt^{2}-e^{2f(\varphi )}\left[dx^{2}+dy^{2}+dz^{2}\right]}
η μ ν ∂ μ ∂ ν φ = 4 π ρ ∗ k ( φ ) {\displaystyle \eta ^{\mu \nu }\partial _{\mu }\partial _{\nu }\varphi =4\pi \rho ^{*}k(\varphi )}
Ni quotes Rosen as having two scalar fields φ {\displaystyle \varphi } and ψ {\displaystyle \psi } that are related to the metric by:
d s 2 = φ 2 d t 2 − ψ 2 [ d x 2 + d y 2 + d z 2 ] {\displaystyle ds^{2}=\varphi ^{2}\,dt^{2}-\psi ^{2}\left[dx^{2}+dy^{2}+dz^{2}\right]}
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 ) {\displaystyle L_{\varphi }=e^{\varphi }\left({\tfrac {1}{2}}e^{-\varphi }\,\partial _{\alpha }\varphi \,\partial _{\alpha }\varphi +{\tfrac {3}{2}}e^{\varphi }\,\partial _{0}\varphi \,\partial _{0}\varphi \right)}
In Papapetrou there is a second scalar field χ {\displaystyle \chi } . 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 ) {\displaystyle L_{\varphi }=e^{{\frac {1}{2}}(3\varphi +\chi )}\left(-{\tfrac {1}{2}}e^{-\varphi }\,\partial _{\alpha }\varphi \,\partial _{\alpha }\varphi -e^{-\varphi }\,\partial _{\alpha }\varphi \,\partial _{\chi }\varphi +{\tfrac {3}{2}}e^{-\chi }\,\partial _{0}\varphi \,\partial _{0}\varphi \right)\,}
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 (1975) developed a bimetric theory. The action is:
S = 1 64 π G ∫ d 4 x − η η μ ν g α β g γ δ ( g α γ | μ g α δ | ν − 1 2 g α β | μ g γ δ | ν ) + S m {\displaystyle S={1 \over 64\pi G}\int d^{4}x\,{\sqrt {-\eta }}\eta ^{\mu \nu }g^{\alpha \beta }g^{\gamma \delta }(g_{\alpha \gamma |\mu }g_{\alpha \delta |\nu }-\textstyle {\frac {1}{2}}g_{\alpha \beta |\mu }g_{\gamma \delta |\nu })+S_{m}}
◻ η g μ ν − g α β η γ δ g μ α | γ g ν β | δ = − 16 π G g / η ( T μ ν − 1 2 g μ ν T ) {\displaystyle \Box _{\eta }g_{\mu \nu }-g^{\alpha \beta }\eta ^{\gamma \delta }g_{\mu \alpha |\gamma }g_{\nu \beta |\delta }=-16\pi G{\sqrt {g/\eta }}(T_{\mu \nu }-\textstyle {\frac {1}{2}}g_{\mu \nu }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 μ ν {\displaystyle B_{\mu \nu }\,} , B = B μ ν η μ ν {\displaystyle B=B_{\mu \nu }\eta ^{\mu \nu }\,} , and two constants a {\displaystyle a\,} and f {\displaystyle f\,} the action is:
S = 1 16 π G ∫ d 4 x − η ( a B μ ν | α B μ ν | α + f B , α B , α ) + S m {\displaystyle S={1 \over 16\pi G}\int d^{4}x{\sqrt {-\eta }}(aB^{\mu \nu |\alpha }B_{\mu \nu |\alpha }+fB_{,\alpha }B^{,\alpha })+S_{m}}
and the stress–energy tensor comes from:
a ◻ η B μ ν + f η μ ν ◻ η B = − 4 π G g / η T α β ( ∂ g α β ∂ B μ ν ) {\displaystyle a\Box _{\eta }B^{\mu \nu }+f\eta ^{\mu \nu }\Box _{\eta }B=-4\pi G{\sqrt {g/\eta }}\,T^{\alpha \beta }\left({\frac {\partial g_{\alpha \beta }}{\partial B_{\mu }\nu }}\right)}
In Rastall, the metric is an algebraic function of the Minkowski metric and a Vector field. The action is:
S = 1 16 π G ∫ d 4 x − g F ( N ) K μ ; ν K μ ; ν + S m {\displaystyle S={1 \over 16\pi G}\int d^{4}x\,{\sqrt {-g}}F(N)K^{\mu ;\nu }K_{\mu ;\nu }+S_{m}}
where
F ( N ) = − N 2 + N {\displaystyle F(N)=-{\frac {N}{2+N}}} and N = g