Alternatives to general relativity

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Alternatives to general relativity are physical theories that attempt to describe the phenomenon 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:

• Straightforward alternatives to general relativity (GR), such as the Cartan, Brans–Dicke and Rosen bimetric theories.
• Those that attempt to construct a quantized gravity theory such as loop quantum gravity.
• Those that attempt to unify gravity and other forces such as Kaluza–Klein.
• Those that attempt to do several at once, such as M-theory.

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.

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. Lagrange). 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.

Notation in this article

is the speed of light, ${\displaystyle G\;}$ 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.


${\displaystyle \eta _{\mu \nu }\;}$ is the Minkowski metric. ${\displaystyle g_{\mu \nu }\;}$ is a tensor, usually the metric tensor. These have signature (−,+,+,+).


Partial differentiation is written ${\displaystyle \partial _{\mu }\phi \;}$ or ${\displaystyle \phi _{,\mu }\;}$. Covariant differentiation is written ${\displaystyle \nabla _{\mu }\phi \;}$ or ${\displaystyle \phi _{;\mu }\;}$.

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

If a theory has a Lagrangian density for gravity, say ${\displaystyle L\,}$, then the gravitational part of the action ${\displaystyle S\,}$ is the integral of that.

${\displaystyle S=\int L{\sqrt {-g}}\,\mathrm {d} ^{4}x}$

In this equation it is usual, though not essential, to have ${\displaystyle g=-1\,}$ at spatial infinity when using Cartesian coordinates. For example, the Einstein–Hilbert action uses

${\displaystyle L\,\propto \,R}$

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 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 ${\displaystyle g_{\mu \nu }\,}$ of signature (−, +, +, +), which governs proper-length and proper-time measurements in the usual manner of special and general relativity:

${\displaystyle {d\tau }^{2}=-g_{\mu \nu }\,dx^{\mu }\,dx^{\nu }\,}$

where there is a summation over indices ${\displaystyle \mu }$ and ${\displaystyle \nu }$.

Condition 2: Stressed matter and fields being acted upon by gravity respond in accordance with the equation:

${\displaystyle 0=\nabla _{\nu }T^{\mu \nu }={T^{\mu \nu }}_{,\nu }+\Gamma _{\sigma \nu }^{\mu }T^{\sigma \nu }+\Gamma _{\sigma \nu }^{\nu }T^{\mu \sigma }\,}$

where ${\displaystyle T^{\mu \nu }\,}$ is the stress–energy tensor for all matter and non-gravitational fields, and where ${\displaystyle \nabla _{\nu }}$ is the covariant derivative with respect to the metric and ${\displaystyle \Gamma _{\sigma \nu }^{\alpha }\,}$ 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 GR
  • 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 theorem gravity
• Scalar-tensor theories
  • Bekenstein
  • Bergmann-Wagoner
  • Brans–Dicke theory (the most well-known alternative to GR, 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

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:

• 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 ${\displaystyle \rho \,}$ generates a scalar field, the gravitational potential ${\displaystyle \phi \,}$ in joules per kilogram, by

${\displaystyle {\partial ^{2}\phi \over \partial x^{j}\partial x^{j}}=4\pi G\rho \,.}$

Using the Nabla operator ${\displaystyle \nabla }$ for the gradient and divergence (partial derivatives), this can be conveniently written as:

${\displaystyle \nabla ^{2}\phi =4\pi G\rho \,.}$

This scalar field governs the motion of a free-falling particle by:

${\displaystyle {d^{2}x^{j} \over dt^{2}}=-{\partial \phi \over \partial x^{j}\,}.}$

At distance, r, from an isolated mass, M, the scalar field is

${\displaystyle \phi =-GM/r\,.}$

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, as 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:

${\displaystyle \delta \int d\tau =0\,}$

${\displaystyle {d\tau }^{2}=-\eta _{\mu \nu }dx^{\mu }dx^{\nu }\,}$

where ${\displaystyle \eta _{\mu \nu }\,}$ is the Minkowski metric, and there is a summation from 1 to 4 over indices ${\displaystyle \mu \,}$ and ${\displaystyle \nu \,}$.
Einstein and Grossmann (1913) includes Riemannian geometry and tensor calculus.

${\displaystyle \delta \int d\tau =0\,}$

${\displaystyle {d\tau }^{2}=-g_{\mu \nu }dx^{\mu }dx^{\nu }\,}$

The equations of electrodynamics exactly match those of GR. The equation

${\displaystyle T^{\mu \nu }=\rho {dx^{\mu } \over d\tau }{dx^{\nu } \over d\tau }\,}$

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 ${\displaystyle c\,}$ but to let mass depend on the gravitational field strength ${\displaystyle \phi \,}$. Allowing this field strength to satisfy

${\displaystyle \Box \phi =\rho \,}$

where ${\displaystyle \rho \,}$ is rest mass energy and ${\displaystyle \Box \,}$ is the d'Alembertian,

${\displaystyle m=m_{0}\exp(\phi /c^{2})\,}$

and

${\displaystyle -{\partial \phi \over \partial x^{\mu }}={\dot {u}}_{\mu }+{u_{\mu } \over c^{2}{\dot {\phi }}}\,}$

where ${\displaystyle u\,}$ 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):

${\displaystyle \delta \int \psi d\tau =0\,}$

${\displaystyle {d\tau }^{2}=-\eta _{\mu \nu }dx^{\mu }dx^{\nu }\,}$

where ${\displaystyle \psi \,}$ is a scalar field,

${\displaystyle -{\partial T^{\mu \nu } \over \partial x^{\nu }}=T{1 \over \psi }{\partial \psi \over \partial x_{\mu }}\,}$

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:

${\displaystyle \delta \int ds=0\,}$

${\displaystyle {ds}^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }\,}$

${\displaystyle g_{\mu \nu }=\psi ^{2}\eta _{\mu \nu }\,}$

they relate Einstein-Grossmann (1913) to Nordström (1913). They also state:

${\displaystyle T\,\propto \,R\,.}$

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 call "general relativity" (included here for comparison). Discarding the Minkowski metric entirely, Einstein gets:

${\displaystyle \delta \int ds=0\,}$

${\displaystyle {ds}^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }\,}$

${\displaystyle R_{\mu \nu }={\frac {8\pi G}{c^{4}}}\left(T_{\mu \nu }-{\frac {1}{2}}g_{\mu \nu }T\right)\,}$

which can also be written

${\displaystyle T^{\mu \nu }={c^{4} \over 8\pi G}\left(R^{\mu \nu }-{\frac {1}{2}}g^{\mu \nu }R\right)\,.}$

Five days before Einstein presented the last equation above, Hilbert had submitted a paper containing an almost identical equation. Hilbert was the first to correctly state the Einstein–Hilbert action for GR, which is:

${\displaystyle S={c^{4} \over 16\pi G}\int R{\sqrt {-g}}\ d^{4}x+S_{m}\,}$

where ${\displaystyle G\,}$ is Newton's gravitational constant, ${\displaystyle R=R_{\mu }^{~\mu }\,}$ is the Ricci curvature of space, ${\displaystyle g=\det(g_{\mu \nu })\,}$ and ${\displaystyle S_{m}\,}$ 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.

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)):

Theories from 1917 to the 1980s.

Publication year(s)	Author(s)	Theory type
1922	Whitehead	Quasilinear
1922, 1923	Cartan	Non-metric
1939	Fierz and Pauli
1943	Birkhov
1948	Milne
1948	Thiry
1954	Papapetrou	Scalar field
1953	Littlewood	Scalar field
1955	Jordan
1956	Bergman	Scalar field
1957	Belinfante and Swihart
1958, 1973	Yilmaz
1961	Brans & Dicke	Scalar-tensor
1960, 1965	Whitrow & Morduch	Scalar field
1966	Kustaanheimo
1967	Kustaanheimo and Nuotio
1968	Deser and Laurent	Quasilinear
1968	Page and Tupper	Scalar field
1968	Bergmann	Scalar-tensor
1970	Bollini-Giambiagi-Tiomno	Quasilinear
1970	Nordtveldt
1970	Wagoner	Scalar-tensor
1971	Rosen	Scalar field
1975	Rosen	Bimetric
1972, 1973	Wei-Tou Ni	Scalar field
1972	Will and Nordtveldt	Vector-tensor
1973	Hellings and Nordtveldt	Vector-tensor
1973	Lightman and Lee	Scalar field
1974	Lee, Lightman and Ni
1977	Bekenstein	Scalar-tensor
1978	Barker	Scalar-tensor
1979	Rastall	Bimetric

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 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.

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:

${\displaystyle \delta \int f\left({\tfrac {\phi }{c^{2}}}\right)ds=0}$

where the scalar field is,

${\displaystyle \phi =GM/r}$

and c may or may not depend on ${\displaystyle \phi }$.
In Nordström (1912),

${\displaystyle f(\phi /c^{2})=\exp(-\phi /c^{2}),\qquad c=c_{\infty }}$

In Littlewood (1953) and Bergmann (1956),

${\displaystyle f(\phi /c^{2})=\exp(-\phi /c^{2}-(\phi /c^{2})^{2}/2),\qquad c=c_{\infty }\,}$

In Whitrow and Morduch (1960),

${\displaystyle f(\phi /c^{2})=1,\qquad c^{2}=c_{\infty }^{2}-2\phi \,}$

In Whitrow and Morduch (1965),

${\displaystyle f(\phi /c^{2})=\exp(-\phi /c^{2}),\qquad c^{2}=c_{\infty }^{2}-2\phi \,}$

In Page and Tupper (1968),

${\displaystyle f(\phi /c^{2})=\phi /c^{2}+\alpha (\phi /c^{2})^{2},\qquad c_{\infty }^{2}/c^{2}=1+4(\phi /c_{\infty }^{2})+(15+2\alpha )(\phi /c_{\infty }^{2})^{2}}$

Page and Tupper (1968) matches Yilmaz (1958) (see also Yilmaz theory of gravitation) to second order when ${\displaystyle \alpha =-7/2}$.

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 (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:

${\displaystyle S={1 \over 16\pi G}\int d^{4}x{\sqrt {-g}}L_{\phi }+S_{m}}$

${\displaystyle L_{\phi }=\phi R-2g^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi }$

Misner et al. (1973) gives this without the ${\displaystyle \phi R}$ term. ${\displaystyle S_{m}}$ is the matter action.

${\displaystyle \Box \phi =4\pi T^{\mu \nu }\left[\eta _{\mu \nu }e^{-2\phi }+\left(e^{2\phi }+e^{-2\phi }\right)\partial _{\mu }t\partial _{\nu }t\right]}$

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 (1972) there are two arbitrary functions ${\displaystyle f(\phi )}$ and ${\displaystyle k(\phi )}$ that are related to the metric by:

${\displaystyle ds^{2}=e^{-2f(\phi )}dt^{2}-e^{2f(\phi )}\left[dx^{2}+dy^{2}+dz^{2}\right]}$

${\displaystyle \eta ^{\mu \nu }\partial _{\mu }\partial _{\nu }\phi =4\pi \rho ^{*}k(\phi )}$

Ni (1972) quotes Rosen (1971) as having two scalar fields ${\displaystyle \phi }$ and ${\displaystyle \psi }$ that are related to the metric by:

${\displaystyle ds^{2}=\phi ^{2}dt^{2}-\psi ^{2}\left[dx^{2}+dy^{2}+dz^{2}\right]}$

In Papapetrou (1954a) the gravitational part of the Lagrangian is:

${\displaystyle L_{\phi }=e^{\phi }\left({\tfrac {1}{2}}e^{-\phi }\partial _{\alpha }\phi \partial _{\alpha }\phi +{\tfrac {3}{2}}e^{\phi }\partial _{0}\phi \partial _{0}\phi \right)}$

In Papapetrou (1954b) there is a second scalar field ${\displaystyle \chi }$. The gravitational part of the Lagrangian is now:

${\displaystyle L_{\phi }=e^{{\frac {1}{2}}(3\phi +\chi )}\left(-{\tfrac {1}{2}}e^{-\phi }\partial _{\alpha }\phi \partial _{\alpha }\phi -e^{-\phi }\partial _{\alpha }\phi \partial _{\chi }\phi +{\tfrac {3}{2}}e^{-\chi }\partial _{0}\phi \partial _{0}\phi \right)\,}$

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 (1973, 1975) Bimetric Theory The action is:

${\displaystyle S={1 \over 64\pi G}\int d^{4}x{\sqrt {-\eta }}\eta ^{\mu \nu }g^{\alpha \beta }g^{\gamma \delta }(g_{\alpha \gamma |\mu }g_{\alpha \delta |\nu }-\textstyle {\frac {1}{2}}g_{\alpha \beta |\mu }g_{\gamma \delta |\nu })+S_{m}}$

where the vertical line "|" denotes covariant derivative with respect to ${\displaystyle \eta \,}$. The field equations may be written in the form:

${\displaystyle \Box _{\eta }g_{\mu \nu }-g^{\alpha \beta }\eta ^{\gamma \delta }g_{\mu \alpha |\gamma }g_{\nu \beta |\delta }=-16\pi G{\sqrt {g/\eta }}(T_{\mu \nu }-\textstyle {\frac {1}{2}}g_{\mu \nu }T)\,}$

Lightman-Lee (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 ${\displaystyle B_{\mu \nu }\,}$, ${\displaystyle B=B_{\mu \nu }\eta ^{\mu \nu }\,}$, and two constants ${\displaystyle a\,}$ and ${\displaystyle f\,}$ the action is:

${\displaystyle S={1 \over 16\pi G}\int d^{4}x{\sqrt {-\eta }}(aB^{\mu \nu |\alpha }B_{\mu \nu |\alpha }+fB_{,\alpha }B^{,\alpha })+S_{m}}$

and the stress–energy tensor comes from:

${\displaystyle a\Box _{\eta }B^{\mu \nu }+f\eta ^{\mu \nu }\Box _{\eta }B=-4\pi G{\sqrt {g/\eta }}T^{\alpha \beta }(\partial g_{\alpha \beta }/\partial B_{\mu }\nu )}$

In Rastall (1979), the metric is an algebraic function of the Minkowski metric and a Vector field. The Action is:

${\displaystyle S={1 \over 16\pi G}\int d^{4}x{\sqrt {-g}}F(N)K^{\mu ;\nu }K_{\mu ;\nu }+S_{m}}$

where

${\displaystyle F(N)=-N/(2+N)\;}$ and ${\displaystyle N=g^{\mu \nu }K_{\mu }K_{\nu }\;}$

(see Will (1981) for the field equation for ${\displaystyle T^{\mu \nu }\;}$ and ${\displaystyle K_{\mu }\;}$).

Quasilinear theories

In Whitehead (1922), the physical metric ${\displaystyle g\;}$ is constructed (by Synge) algebraically from the Minkowski metric ${\displaystyle \eta \;}$ and matter variables, so it doesn't even have a scalar field. The construction is:

${\displaystyle g_{\mu \nu }(x^{\alpha })=\eta _{\mu \nu }-2\int _{\Sigma ^{-}}{y_{\mu }^{-}y_{\nu }^{-} \over (w^{-})^{3}}[{\sqrt {-g}}\rho u^{\alpha }d\Sigma _{\alpha }]^{-}}$

where the superscript (-) indicates quantities evaluated along the past ${\displaystyle \eta \;}$ light cone of the field point ${\displaystyle x^{\alpha }\;}$ and

${\displaystyle (y^{\mu })^{-}=x^{\mu }-(x^{\mu })^{-}\;}$, ${\displaystyle (y^{\mu })^{-}(y_{\mu })^{-}=0,\;}$

${\displaystyle w^{-}=(y^{\mu })^{-}(u_{\mu })^{-}\;}$, ${\displaystyle (u_{\mu })=dx^{\mu }/d\sigma ,\;}$

${\displaystyle d\sigma ^{2}=\eta _{\mu \nu }dx^{\mu }dx^{\nu }\;}$

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

Deser and Laurent (1968) and Bollini-Giambiagi-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) ${\displaystyle h_{\mu \nu }\;}$ to define

${\displaystyle g_{\mu \nu }=\eta _{\mu \nu }+h_{\mu \nu }\;}$

The action is:

${\displaystyle S={1 \over 16\pi G}\int d^{4}x{\sqrt {-\eta }}[2h_{|\nu }^{\mu \nu }h_{\mu \lambda }^{|\lambda }-2h_{|\nu }^{\mu \nu }h_{\lambda |\mu }^{\lambda }+h_{\nu |\mu }^{\nu }h_{\lambda }^{\lambda |\mu }-h^{\mu \nu |\lambda }h_{\mu \nu |\lambda }]+S_{m}\;}$

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 ${\displaystyle h_{\mu \nu }\;}$.

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.

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

Starobinsky gravity, proposed by Alexei Starobinsky has the Lagrangian

${\displaystyle {\mathcal {L}}={\sqrt {-g}}\left[R+{\frac {R^{2}}{6M^{2}}}\right]}$

and has been used to explain inflation, in the form of Starobinsky inflation.

Gauss-Bonnet

Gauss–Bonnet gravity has the action

${\displaystyle {\mathcal {L}}={\sqrt {-g}}\left[R+R^{2}-4R^{\mu \nu }R_{\mu \nu }+R^{\mu \nu \rho \sigma }R_{\mu \nu \rho \sigma }\right].}$

where the coefficients of the extra terms are chosen so that the action reduces to GR 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 generalisation of Gauss-Bonnet gravity, has the action

${\displaystyle {\mathcal {L}}={\sqrt {-g}}\left[R+f_{1}R^{2}+f_{2}R^{\mu \nu }R_{\mu \nu }+f_{3}R^{\mu \nu \rho \sigma }R_{\mu \nu \rho \sigma }\right].}$

f(r)

f(R) gravity has the action

${\displaystyle {\mathcal {L}}={\sqrt {-g}}f(R)}$

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

Infinite derivative gravity

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

${\displaystyle {\mathcal {L}}={\sqrt {-g}}\left[M_{p}^{2}R+Rf_{1}(\Box /M_{s}^{2})R+R^{\mu \nu }f_{2}(\Box /M_{s}^{2})R_{\mu \nu }+R^{\mu \nu \rho \sigma }f_{3}(\Box /M_{s}^{2})R_{\mu \nu \rho \sigma }\right].}$

and

${\displaystyle 2f_{1}(\Box /M_{s}^{2})+f_{2}(\Box /M_{s}^{2})+2f_{3}(\Box /M_{s}^{2})=0,}$

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 ${\displaystyle M_{s}}$, and recovers to GR in the infrared, for energies below the non-local scale ${\displaystyle M_{s}}$. In the ultraviolet regime, at distances and time scales below non-local scale, ${\displaystyle M_{s}^{-1}}$, 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

${\displaystyle {\mathcal {L}}={\sqrt {-g}}\ (\alpha _{0}+\alpha _{1}R+\alpha _{2}\left(R^{2}+R_{\alpha \beta \mu \nu }R^{\alpha \beta \mu \nu }-4R_{\mu \nu }R^{\mu \nu }\right)+\alpha _{3}{\mathcal {O}}(R^{3})),}$

and can be thought of as a generalisation of GR.

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 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 ${\displaystyle S\;}$ is based on the integral of the Lagrangian ${\displaystyle L_{\phi }\;}$.

${\displaystyle S={1 \over 16\pi G}\int d^{4}x{\sqrt {-g}}L_{\phi }+S_{m}\;}$

${\displaystyle L_{\phi }=\phi R-{\omega (\phi ) \over \phi }g^{\mu \nu }\partial _{\mu }\phi \partial _{\nu }\phi +2\phi \lambda (\phi )\;}$

${\displaystyle S_{m}=\int d^{4}x{\sqrt {g}}G_{N}L_{m}\;}$

${\displaystyle T^{\mu \nu }\ {\stackrel {\mathrm {def} }{=}}\ {2 \over {\sqrt {g}}}{\delta S_{m} \over \delta g_{\mu \nu }}}$

where ${\displaystyle \omega (\phi )\;}$ is a different dimensionless function for each different scalar-tensor theory. The function ${\displaystyle \lambda (\phi )\;}$ plays the same role as the cosmological constant in GR. ${\displaystyle G_{N}\;}$ is a dimensionless normalization constant that fixes the present-day value of ${\displaystyle G\;}$. 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), ${\displaystyle \lambda =0\;}$.
 
Since ${\displaystyle \lambda }$ 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), ${\displaystyle \omega \;}$ is constant.

Bekenstein (1977) Variable Mass Theory Starting with parameters ${\displaystyle r\;}$ and ${\displaystyle q\;}$, found from a cosmological solution, ${\displaystyle \phi =[1-qf(\phi )]f(\phi )^{-r}\;}$ determines function ${\displaystyle f\;}$ then

${\displaystyle \omega (\phi )=-\textstyle {\frac {3}{2}}-\textstyle {\frac {1}{4}}f(\phi )[(1-6q)qf(\phi )-1][r+(1-r)qf(\phi )]^{-2}\;}$

Barker (1978) Constant G Theory

${\displaystyle \omega (\phi )=(4-3\phi )/(2\phi -2)\;}$

Adjustment of ${\displaystyle \omega (\phi )\;}$ allows Scalar Tensor Theories to tend to GR in the limit of ${\displaystyle \omega \rightarrow \infty \;}$ 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.

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 (1973) and Will and Nordtvedt (1972) are both vector-tensor theories. In addition to the metric tensor there is a timelike vector field ${\displaystyle K_{\mu }.}$ The gravitational action is:

${\displaystyle S={\frac {1}{16\pi G}}\int d^{4}x{\sqrt {-g}}\left[R+\omega K_{\mu }K^{\mu }R+\eta K^{\mu }K^{\nu }R_{\mu \nu }-\epsilon F_{\mu \nu }F^{\mu \nu }+\tau K_{\mu ;\nu }K^{\mu ;\nu }\right]+S_{m}}$

where ${\displaystyle \omega ,\eta ,\epsilon ,\tau }$ are constants and

${\displaystyle F_{\mu \nu }=K_{\nu ;\mu }-K_{\mu ;\nu }.}$

Will and Nordtvedt (1972) is a special case where

${\displaystyle \omega =\eta =\epsilon =0;\quad \tau =1}$

Hellings and Nordtvedt (1973) is a special case where

${\displaystyle \tau =0;\quad \epsilon =1;\quad \eta =-2\omega }$

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 ${\displaystyle \omega =\eta =\epsilon =\tau =0}$ 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 Bekenstein (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 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 GR, the Lagrangian is made up of a massless and a mass part. The Lagrangian for the massless part is:

${\displaystyle L={1 \over 32\pi G}\Omega _{\nu }^{\mu }g^{\nu \xi }x^{\eta }x^{\zeta }\varepsilon _{\xi \mu \eta \zeta }\;}$

${\displaystyle \Omega _{\nu }^{\mu }=d\omega _{\nu }^{\mu }+\omega _{\xi }^{\eta }\;}$

${\displaystyle \nabla x^{\mu }=-\omega _{\nu }^{\mu }x^{\nu }\;}$

The ${\displaystyle \omega _{\nu }^{\mu }\;}$ is the linear connection. ${\displaystyle \varepsilon _{\xi \mu \eta \zeta }\;}$ is the completely antisymmetric pseudo-tensor (Levi-Civita symbol) with ${\displaystyle \varepsilon _{0123}={\sqrt {-g}}\;}$, and ${\displaystyle g^{\nu \xi }\,}$ 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:

${\displaystyle T^{\mu \nu }={1 \over 16\pi G}(g^{\mu \nu }\eta _{\eta }^{\xi }-g^{\xi \mu }\eta _{\eta }^{\nu }-g^{\xi \nu }\eta _{\eta }^{\mu })\Omega _{\xi }^{\eta }\;}$

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.

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.

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: Bekenstein (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 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.

In the 1990s, supernova surveys discovered the accelerated expansion of the universe, usually attributed to dark energy. This 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 the supernova surveys' results in a completely different way. The measurement of the speed of gravity with the gravitational wave event GW170817 ruled out many alternative theories of gravity as explanation for the accelerated expansion.

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. This is now believed to be accounted for by non-uniform thermal radiation.

Cosmological constant and quintessence

The cosmological constant ${\displaystyle \Lambda \;}$ is a very old idea, going back to Einstein in 1917. The success of the Friedmann model of the universe in which ${\displaystyle \Lambda =0\;}$ led to the general acceptance that it is zero, but the use of a non-zero value came back with a vengeance when data from supernovae indicated 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:

${\displaystyle \nabla ^{2}\phi =4\pi \rho \ G;}$

to

${\displaystyle \nabla ^{2}\phi +{\frac {1}{2}}\Lambda c^{2}=4\pi \rho \ G;}$

In GR, it changes the Einstein–Hilbert action from

${\displaystyle S={1 \over 16\pi G}\int R{\sqrt {-g}}\,d^{4}x\,+S_{m}\;}$

to

${\displaystyle S={1 \over 16\pi G}\int (R-2\Lambda ){\sqrt {-g}}\,d^{4}x\,+S_{m}\;}$

which changes the field equation

${\displaystyle T^{\mu \nu }={1 \over 8\pi G}\left(R^{\mu \nu }-{\frac {1}{2}}g^{\mu \nu }R\right)\;}$

to

${\displaystyle T^{\mu \nu }={1 \over 8\pi G}\left(R^{\mu \nu }-{\frac {1}{2}}g^{\mu \nu }R+g^{\mu \nu }\Lambda \right)\;}$

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 ${\displaystyle \lambda (\phi )\;}$ can be added to scalar tensor theories. This can also be done in every alternative the GR that contains a scalar field ${\displaystyle \phi \;}$ by adding the term ${\displaystyle \lambda (\phi )\;}$ inside the Lagrangian for the gravitational part of the action, the ${\displaystyle L_{\phi }\;}$ part of

${\displaystyle S={1 \over 16\pi G}\int d^{4}x\,{\sqrt {-g}}L_{\phi }+S_{m}\;}$

Because ${\displaystyle \lambda (\phi )\;}$ 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

${\displaystyle K^{\mu }K^{\nu }g_{\mu \nu }\;}$

is added to the Lagrangian for the gravitational part of the action.

Relativistic MOND

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.

1. It did not include relativistic effects
2. It violated the conservation of energy, momentum and angular momentum
3. It was inconsistent in that it gives different galactic orbits for gas and for stars
4. It did not state how to calculate gravitational lensing from galaxy clusters.

By 1984, problems 2 and 3 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:

${\displaystyle L=-{a_{0}^{2} \over 8\pi G}f\left\lbrack {\frac {|\nabla \phi |^{2}}{a_{0}^{2}}}\right\rbrack -\rho \phi }$

The relativistic version of this has:

${\displaystyle L=-{a_{0}^{2} \over 8\pi G}{\tilde {f}}\left(l_{0}^{2}g^{\mu \nu }\,\partial _{\mu }\phi \,\partial _{\nu }\phi \right)}$

with a nonstandard mass action. Here ${\displaystyle f}$ and ${\displaystyle {\tilde {f}}}$ are arbitrary functions selected to give Newtonian and MOND behaviour in the correct limits, and ${\displaystyle l_{0}=c^{2}/a_{0}\;}$ 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 ${\displaystyle \phi }$ and ${\displaystyle \sigma \;}$ and vector field ${\displaystyle U_{\alpha }}$. The action is split into parts for gravity, scalars, vector and mass.

${\displaystyle S=S_{g}+S_{s}+S_{v}+S_{m}}$

The gravity part is the same as in GR.

${\displaystyle {\begin{aligned}S_{s}&=-{\frac {1}{2}}\int \left[\sigma ^{2}h^{\alpha \beta }\phi _{,\alpha }\phi _{,\beta }+{\frac {1}{2}}Gl_{0}^{-2}\sigma ^{4}F(kG\sigma ^{2})\right]{\sqrt {-g}}\,d^{4}x\\S_{v}&=-{\frac {K}{32\pi G}}\int \left[g^{\alpha \beta }g^{\mu \nu }U_{[\alpha ,\mu ]}U_{[\beta ,\nu ]}-{\frac {2\lambda }{K}}\left(g^{\mu \nu }U_{\mu }U_{\nu }+1\right)\right]{\sqrt {-g}}\,d^{4}x\\S_{m}&=\int L\left({\tilde {g}}_{\mu \nu },f^{\alpha },f_{|\mu }^{\alpha },\cdots \right){\sqrt {-g}}\,d^{4}x\end{aligned}}}$

where

${\displaystyle h^{\alpha \beta }=g^{\alpha \beta }-U^{\alpha }U^{\beta }}$

${\displaystyle {\tilde {g}}^{\alpha \beta }=e^{2\phi }g^{\alpha \beta }+2U^{\alpha }U^{\beta }\sinh(2\phi )}$

${\displaystyle k,K}$ are constants, square brackets in indices ${\displaystyle U_{[\alpha ,\mu ]}}$ represent anti-symmetrization, ${\displaystyle \lambda }$ is a Lagrange multiplier (calculated elsewhere), and L is a Lagrangian translated from flat spacetime onto the metric ${\displaystyle {\tilde {g}}^{\alpha \beta }}$. Note that G need not equal the observed gravitational constant ${\displaystyle G_{Newton}}$. F is an arbitrary function, and

${\displaystyle F(\mu )={\frac {3}{4}}{\mu ^{2}(\mu -2)^{2} \over 1-\mu }}$

is given as an example with the right asymptotic behaviour; note how it becomes undefined when ${\displaystyle \mu =1}$.
 
The PPN parameters of this theory are calculated in, which shows that all its parameters are equal to GR's, except for

${\displaystyle {\begin{aligned}\alpha _{1}&={\frac {4G}{K}}\left((2K-1)e^{-4\phi _{0}}-e^{4\phi _{0}}+8\right)-8\\\alpha _{2}&={\frac {6G}{2-K}}-{\frac {2G(K+4)e^{4\phi _{0}}}{(2-K)^{2}}}-1\end{aligned}}}$

both of which expressed in geometric units where ${\displaystyle c=G_{Newtonian}=1}$; so

${\displaystyle G^{-1}={\frac {2}{2-K}}+{\frac {k}{4\pi }}.}$

Moffat's theories

J. W. Moffat (1995) developed a non-symmetric gravitation theory (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 and 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 ${\displaystyle g_{\mu \nu }\;}$, the Lagrangian density is split into

${\displaystyle L=L_{R}+L_{M}\;}$

where ${\displaystyle L_{M}\;}$ is the same as for matter in GR.

${\displaystyle L_{R}={\sqrt {-g}}\left[R(W)-2\lambda -{\frac {1}{4}}\mu ^{2}g^{\mu \nu }g_{[\mu \nu ]}\right]-{\frac {1}{6}}g^{\mu \nu }W_{\mu }W_{\nu }\;}$

where ${\displaystyle R(W)\;}$ is a curvature term analogous to but not equal to the Ricci curvature in GR, ${\displaystyle \lambda \;}$ and ${\displaystyle \mu ^{2}\;}$ are cosmological constants, ${\displaystyle g_{[\nu \mu ]}\;}$ is the antisymmetric part of ${\displaystyle g_{\nu \mu }\;}$. ${\displaystyle W_{\mu }\;}$ is a connection, and is a bit difficult to explain because it's defined recursively. However, ${\displaystyle W_{\mu }\approx -2g_{[\mu \nu ]}^{,\nu }\;}$.
 
Haugan and Kauffmann (1996) used polarization measurements of the light emitted by galaxies to impose sharp constraints on the magnitude of some of NGT's parameters. They also used Hughes-Drever experiments to constrain the remaining degrees of freedom. Their constraint is eight orders of magnitude sharper than previous estimates.

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 ${\displaystyle G\;}$, increasing to a final constant value about a million years after the big bang.

The theory seems to contain an asymmetric tensor ${\displaystyle A_{\mu \nu }\;}$ field and a source current ${\displaystyle J_{\mu }\;}$ vector. The action is split into:

${\displaystyle S=S_{G}+S_{F}+S_{FM}+S_{M}\;}$

Both the gravity and mass terms match those of GR with cosmological constant. The skew field action and the skew field matter coupling are:

${\displaystyle S_{F}=\int d^{4}x\,{\sqrt {-g}}\left({\frac {1}{12}}F_{\mu \nu \rho }F^{\mu \nu \rho }-{\frac {1}{4}}\mu ^{2}A_{\mu \nu }A^{\mu \nu }\right)\;}$

${\displaystyle S_{FM}=\int d^{4}x\,\epsilon ^{\alpha \beta \mu \nu }A_{\alpha \beta }\partial _{\mu }J_{\nu }\;}$

where

${\displaystyle F_{\mu \nu \rho }=\partial _{\mu }A_{\nu \rho }+\partial _{\rho }A_{\mu \nu }}$

and ${\displaystyle \epsilon ^{\alpha \beta \mu \nu }\;}$ 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 (STVG) theory

The theory contains a tensor, vector and three scalar fields. But the equations are quite straightforward. The action is split into: ${\displaystyle S=S_{G}+S_{K}+S_{S}+S_{M}}$ with terms for gravity, vector field ${\displaystyle K_{\mu },}$ scalar fields ${\displaystyle G,\omega ,\mu }$ and mass. ${\displaystyle S_{G}}$ is the standard gravity term with the exception that ${\displaystyle G}$ is moved inside the integral.

${\displaystyle S_{K}=-\int d^{4}x\,{\sqrt {-g}}\omega \left({\frac {1}{4}}B_{\mu \nu }B^{\mu \nu }+V(K)\right),\qquad {\text{where }}\quad B_{\mu \nu }=\partial _{\mu }K_{\nu }-\partial _{\nu }K_{\mu }.}$

${\displaystyle S_{S}=-\int d^{4}x\,{\sqrt {-g}}{\frac {1}{G^{3}}}\left({\frac {1}{2}}g^{\mu \nu }\,\nabla _{\mu }G\,\nabla _{\nu }G-V(G)\right)+{\frac {1}{G}}\left({\frac {1}{2}}g^{\mu \nu }\,\nabla _{\mu }\omega \,\nabla _{\nu }\omega -V(\omega )\right)+{1 \over \mu ^{2}G}\left({\frac {1}{2}}g^{\mu \nu }\,\nabla _{\mu }\mu \,\nabla _{\nu }\mu -V(\mu )\right).}$

The potential function for the vector field is chosen to be:

${\displaystyle V(K)=-{\frac {1}{2}}\mu ^{2}\phi ^{\mu }\phi _{\mu }-{\frac {1}{4}}g\left(\phi ^{\mu }\phi _{\mu }\right)^{2}}$

where ${\displaystyle g}$ is a coupling constant. The functions assumed for the scalar potentials are not stated.

Infinite derivative gravity

In order to remove ghosts in the modified propagator, as well as to obtain asymptotic freedom, Biswas, Mazumdar and Siegel (2005) considered a string-inspired infinite set of higher derivative terms

${\displaystyle S=\int \mathrm {d} ^{4}x{\sqrt {-g}}\left({\frac {R}{2}}+RF(\Box )R\right)}$

where ${\displaystyle F(\Box )}$ is the exponential of an entire function of the D'Alembertian operator. This avoids a black hole singularity near the origin, while recovering the 1/r fall of the GR potential at large distances. Lousto and Mazzitelli (1997) found an exact solution to this theories representing a gravitational shock-wave.

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 tachyons, 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 ${\displaystyle \rho }$ used by the theory should be calculated from the stress–energy tensor ${\displaystyle T}$ as ${\displaystyle \rho =T_{\mu \nu }u^{\mu }u^{\nu }}$ or as ${\displaystyle \rho =T_{\mu \nu }\delta ^{\mu \nu }}$, where ${\displaystyle u}$ is the four-velocity, and ${\displaystyle \delta }$ is the Kronecker delta.

The theories of Thirry (1948) and Jordan (1955) are incomplete unless Jordan's parameter ${\displaystyle \eta \;}$ 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 half 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 redshift
• gravitational lensing (generally tested around the Sun)
• anomalous perihelion advance of the planets (see Tests of general relativity)

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.

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. This was the consequence of an assumption made to simplify handling the collision of masses.

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 5×10⁻⁹. Modern tests have reduced that to less than 5×10⁻¹³.

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 5×10⁻³. Modern tests have reduced this to less than 1×10⁻²¹.

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 1×10⁻⁴.

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 satisfy this. For example, the non-metric theory of Belinfante & Swihart (1957) is eliminated by the THεμ formalism for testing EEP. Gauge theory gravity is a notable exception, where the strong equivalence principle is essentially the minimal coupling of the gauge covariant derivative.

Parametric post-Newtonian (PPN) formalism

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: ${\displaystyle \gamma ,\beta ,\eta ,\alpha _{1},\alpha _{2},\alpha _{3},\zeta _{1},\zeta _{2},\zeta _{3},\zeta _{4}.}$

• ${\displaystyle \gamma }$ is a measure of space curvature, being zero for Newtonian gravity and one for GR.
• ${\displaystyle \beta }$ is a measure of nonlinearity in the addition of gravitational fields, one for GR.
• ${\displaystyle \eta }$ is a check for preferred location effects.
• ${\displaystyle \alpha _{1},\alpha _{2},\alpha _{3}}$ measure the extent and nature of "preferred-frame effects". Any theory of gravity in which at least one of the three is nonzero is called a preferred-frame theory.
• ${\displaystyle \zeta _{1},\zeta _{2},\zeta _{3},\zeta _{4},\alpha _{3}}$ 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, possibly breaking of causality. After the multi-messanging detection of the GW170817 coalescence of neutron stars, where light and gravitational waves were measured to travel at the same speed with an error of 1/10¹⁵, many of those modified theory of gravity were excluded.

Cosmological tests

Many of these have been developed recently. For those theories that aim to replace dark matter, the galaxy rotation curve, the Tully-Fisher relation, the faster rotation rate of dwarf galaxies, and the gravitational lensing 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.

Results of testing theories

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 100 years old, during which one alternative theory of gravity after another has failed to agree with ever more accurate observations. One illustrative example is 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.

	${\displaystyle \gamma }$	${\displaystyle \beta }$	${\displaystyle \xi }$	${\displaystyle \alpha _{1}}$	${\displaystyle \alpha _{2}}$	${\displaystyle \alpha _{3}}$	${\displaystyle \zeta _{1}}$	${\displaystyle \zeta _{2}}$	${\displaystyle \zeta _{3}}$	${\displaystyle \zeta _{4}}$
Einstein (1916) GR	1	1	0	0	0	0	0	0	0	0
Scalar-tensor theories
Bergmann (1968), Wagoner (1970)	${\displaystyle \textstyle {\frac {1+\omega }{2+\omega }}}$	${\displaystyle \beta }$	0	0	0	0	0	0	0	0
Nordtvedt (1970), Bekenstein (1977)	${\displaystyle \textstyle {\frac {1+\omega }{2+\omega }}}$	${\displaystyle \beta }$	0	0	0	0	0	0	0	0
Brans–Dicke (1961)	${\displaystyle \textstyle {\frac {1+\omega }{2+\omega }}}$	1	0	0	0	0	0	0	0	0
Vector-tensor theories
Hellings-Nordtvedt (1973)	${\displaystyle \gamma }$	${\displaystyle \beta }$	0	${\displaystyle \alpha _{1}}$	${\displaystyle \alpha _{2}}$	0	0	0	0	0
Will-Nordtvedt (1972)	1	1	0	0	${\displaystyle \alpha _{2}}$	0	0	0	0	0
Bimetric theories
Rosen (1975)	1	1	0	0	${\displaystyle c_{0}/c_{1}-1}$	0	0	0	0	0
Rastall (1979)	1	1	0	0	${\displaystyle \alpha _{2}}$	0	0	0	0	0
Lightman-Lee (1973)	${\displaystyle \gamma }$	${\displaystyle \beta }$	0	${\displaystyle \alpha _{1}}$	${\displaystyle \alpha _{2}}$	0	0	0	0	0
Stratified theories
Lee-Lightman-Ni (1974)	${\displaystyle ac_{0}/c_{1}}$	${\displaystyle \beta }$	${\displaystyle \xi }$	${\displaystyle \alpha _{1}}$	${\displaystyle \alpha _{2}}$	0	0	0	0	0
Ni (1973)	${\displaystyle ac_{0}/c_{1}}$	${\displaystyle bc_{0}}$	0	${\displaystyle \alpha _{1}}$	${\displaystyle \alpha _{2}}$	0	0	0	0	0
Scalar field theories
Einstein (1912) {Not GR}	0	0		-4	0	-2	0	-1	0	0†
Whitrow-Morduch (1965)	0	-1		-4	0	0	0	-3	0	0†
Rosen (1971)	${\displaystyle \lambda }$	${\displaystyle \textstyle {\frac {3}{4}}+\textstyle {\frac {\lambda }{4}}}$		${\displaystyle -4-4\lambda }$	0	-4	0	-1	0	0
Papetrou (1954a, 1954b)	1	1		-8	-4	0	0	2	0	0
Ni (1972) (stratified)	1	1		-8	0	0	0	2	0	0
Yilmaz (1958, 1962)	1	1		-8	0	-4	0	-2	0	-1†
Page-Tupper (1968)	${\displaystyle \gamma }$	${\displaystyle \beta }$		${\displaystyle -4-4\gamma }$	0	${\displaystyle -2-2\gamma }$	0	${\displaystyle \zeta _{2}}$	0	${\displaystyle \zeta _{4}}$
Nordström (1912)	${\displaystyle -1}$	${\displaystyle \textstyle {\frac {1}{2}}}$		0	0	0	0	0	0	0†
Nordström (1913), Einstein-Fokker (1914)	${\displaystyle -1}$	${\displaystyle \textstyle {\frac {1}{2}}}$		0	0	0	0	0	0	0
Ni (1972) (flat)	${\displaystyle -1}$	${\displaystyle 1-q}$		0	0	0	0	${\displaystyle \zeta _{2}}$	0	0†
Whitrow-Morduch (1960)	${\displaystyle -1}$	${\displaystyle 1-q}$		0	0	0	0	q	0	0†
Littlewood (1953), Bergman(1956)	${\displaystyle -1}$	${\displaystyle \textstyle {\frac {1}{2}}}$		0	0	0	0	-1	0	0†

† The theory is incomplete, and ${\displaystyle \zeta _{4}}$ 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-Giambiagi-Tiomino (1970), but in these three cases ${\displaystyle \beta =\xi }$, which is in strong conflict with GR and experimental results. In particular, these theories predict incorrect amplitudes for the Earth's tides. (A minor modification of Whitehead's theory avoids this problem. However, the modification predicts the Nordtvedt effect, which has been experimentally constrained.)

Theories that fail other tests

The stratified theories of Ni (1973), Lee Lightman and Ni (1974) are non-starters because they 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 to within experimental error. 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 ${\displaystyle \alpha _{2}}$ which can have a measurable effect on the Earth's tides.

Non-metric theories, such as Belinfante and Swihart (1957a, 1957b), usually fail to agree with experimental tests of Einstein's equivalence principle.

And that leaves, as a likely valid alternative to GR, nothing except possibly Cartan (1922).

That was the situation until cosmological discoveries pushed the development of modern alternatives.

Single-payer healthcare

From Wikipedia, the free encyclopedia

Single-payer healthcare is a healthcare system financed by taxes that covers the costs of essential healthcare for all residents, with costs covered by a single public system (hence 'single-payer'). Alternatively, a multi-payer healthcare system is one in which private, qualified individuals or their employers pay for health insurance with various limits on healthcare coverage via multiple private or public sources.

Single-payer systems may contract for healthcare services from private organizations (as is the case in Canada) or may own and employ healthcare resources and personnel (as is the case in the United Kingdom). "Single-payer" describes the mechanism by which healthcare is paid for by a single public authority, not the type of delivery or for whom physicians work, which may be public, private, or a mix of both.

Description

Single-payer healthcare systems pay for all covered healthcare-related services by a single government or government-related source. It is a strategy employed by governments to achieve several goals, including universal healthcare, decreased economic burden of health care, and improved health outcomes for the population. Universal health care worldwide was established as a goal of the World Health Organization in 2010 and adopted by the United Nations General Assembly in 2015 for the 2030 Agenda for Sustainable Development.

A single-payer health system establishes one health risk pool consisting of the entire population of a geographic or political region. It also establishes one set of rules for services offered, reimbursement rates, drug prices, and minimum standards for required services.

In wealthy nations, that kind of publicly managed insurance is typically extended to all citizens and legal residents. Examples include the United Kingdom's National Health Service, Australia's Medicare, Canada's Medicare, and Taiwan's National Health Insurance.

The standard usage of the term "single-payer healthcare" refers to health insurance, as opposed to healthcare delivery, operating as a public service and offered to citizens and legal residents towards providing nearly universal or universal healthcare. The fund can be managed by the government directly or as a publicly owned and regulated agency. Single-payer contrasts with other funding mechanisms like 'multi-payer' (multiple public and/or private sources), 'two-tiered' (defined either as a public source with the option to use qualifying private coverage as a substitute, or as a public source for catastrophic care backed by private insurance for common medical care), and 'insurance mandate' (citizens are required to buy private insurance which meets a national standard and which is generally subsidized). Some systems combine elements of these four funding mechanisms.

In contrast to the standard usage of the term, some writers describe all publicly administered systems as "single-payer plans," and others have described any system of healthcare which intends to cover the entire population, such as voucher plans, as "single-payer plans," although these usages generally don't meet strict definitions of the term.

States with single-payer systems

Several nations worldwide have single-payer health insurance programs. These programs generally provide some form of universal healthcare, which is implemented in a variety of ways. In some cases doctors are employed, and hospitals are run by the government such as in the UK or Spain. Alternatively, the government may purchase healthcare services from outside organizations, such as the approach taken in Canada.

Canada

Healthcare in Canada is delivered through a publicly funded healthcare system, which is mostly free at the point of use and has most services provided by private entities. The system was established by the provisions of the Canada Health Act of 1984. The government assures the quality of care through federal standards. The government does not participate in day-to-day care or collect any information about an individual's health, which remains confidential between a person and his or her physician.

Canada's provincially based Medicare systems are cost-effective partly because of their administrative simplicity. In each province, each doctor handles the insurance claim against the provincial insurer. There is no need for the person who accesses healthcare to be involved in billing and reclaim. Private insurance represents a minimal part of the overall system.

In general, costs are paid through funding from income taxes, except in British Columbia, the only province to impose a fixed monthly premium which is waived or reduced for those on low incomes. A health card is issued by the Provincial Ministry of Health to each individual who enrolls for the program and everyone receives the same level of care.

There is no need for a variety of plans because virtually all essential basic care is covered, including maternity and infertility problems. Depending on the province, dental and vision care may not be covered but are often insured by employers through private companies. In some provinces, private supplemental plans are available for those who desire private rooms if they are hospitalized.

Cosmetic surgery and some forms of elective surgery are not considered essential care and are generally not covered. These can be paid out-of-pocket or through private insurers. Health coverage is not affected by loss or change of jobs, as long as premiums are up to date, and there are no lifetime limits or exclusions for pre-existing conditions.

Pharmaceutical medications are covered by public funds or through employment-based private insurance. Drug prices are negotiated with suppliers by the federal government to control costs. Family physicians (often known as general practitioners or GPs in Canada) are chosen by individuals. If a patient wishes to see a specialist or is counseled to see a specialist, a referral can be made by a GP.

Canadians do wait for some treatments and diagnostic services. Survey data shows that the median wait time to see a special physician is a little over four weeks with 89.5% waiting less than three months. The median wait time for diagnostic services such as MRI and CAT scans is two weeks, with 86.4% waiting less than three months. The median wait time for surgery is four weeks, with 82.2% waiting less than three months.

While physician income initially boomed after the implementation of a single-payer program, a reduction in physician salaries followed, which many feared would be a long-term result of government-run healthcare. However, by the beginning of the 21st century, medical professionals were again among Canada's top earners.

Taiwan

Healthcare in Taiwan is administrated by the Department of Health of the Executive Yuan. As with other developed economies, Taiwanese people are well-nourished but face such health problems as chronic obesity and heart disease.

In 2002, Taiwan had nearly 1.6 physicians and 5.9 hospital beds per 1,000 population, and there were a total of 36 hospitals and 2,601 clinics in the country. Health expenditures constituted 5.8 percent of the GDP in 2001, 64.9% of which coming from public funds.

Despite the initial shock on Taiwan's economy from increased costs of expanded healthcare coverage, the single-payer system has provided protection from greater financial risks and has made healthcare more financially accessible for the population, resulting in a steady 70% public satisfaction rating.

The current healthcare system in Taiwan, known as National Health Insurance (NHI), was instituted in 1995. NHI is a single-payer compulsory social insurance plan which centralizes the disbursement of health care funds. The system promises equal access to health care for all citizens, and the population coverage had reached 99% by the end of 2004.

NHI is mainly financed through premiums, which are based on the payroll tax, and is supplemented with out-of-pocket payments and direct government funding. In the initial stage, fee-for-service predominated for both public and private providers. Most health providers operate in the private sector and form a competitive market on the health delivery side. However, many healthcare providers took advantage of the system by offering unnecessary services to a larger number of patients and then billing the government.

In the face of increasing loss and the need for cost containment, NHI changed the payment system from fee-for-service to a global budget, a kind of prospective payment system, in 2002. Taiwan's success with a single-payer health insurance program is owed, in part, to the country's human resources and the government's organizational skills, allowing for the effective and efficient management of the government-run health insurance program.

South Korea

South Korea used to have a multipayer Social health insurance universal healthcare system, similar to systems used in countries like Japan and Germany, with healthcare societies providing coverage for whole populace. Prior to 1977, the country had voluntary private health insurance, but reforms initiated in 1977 resulted in universal coverage by 1989. A major healthcare financing reform in 2000 merged all medical societies into the National Health Insurance Service. This new service became a single-payer healthcare system in 2004.

Regions with 'Beveridge Model' systems

Scandinavia

The countries of Scandinavia are sometimes considered to have a single-payer health care services, as opposed to single-payer national health care insurance like Taiwan or Canada. This is a form of the 'Beveridge Model' of health care systems that features public health providers in addition to public health insurance.

The term 'Scandinavian model' of health care systems has a few common features: largely public providers, limited private health coverage, and regionally-run, devolved systems with limited involvement from the central government. Due to this third characteristic, they can also be argued to be single-payer only on a regional level, or to be multi-payer systems, as opposed to the nationally run health coverage found in Canada, Taiwan and South Korea.

United Kingdom

As in Scandinavia, healthcare in the United Kingdom is a devolved matter, meaning England, Northern Ireland, Scotland and Wales each have their own systems of private and publicly funded healthcare, generally referred to as the National Health Service (NHS). With largely public or government owned providers, this also fits into the 'Beveridge Model' of health care systems, sometimes considered to be single-payer, although unlike Scandinavia, there is a more significant role for both private coverage and providers. Each country's having different policies and priorities has resulted in a variety of differences existing between the systems. That said, each country provides public healthcare to all UK permanent residents that is free at the point of use, being paid for from general taxation.

In addition, each also has a private sector which is considerably smaller than its public equivalent, with provision of private healthcare acquired by means of private health insurance, funded as part of an employer funded healthcare scheme or paid directly by the customer, though provision can be restricted for those with conditions such as AIDS/HIV.

The individual systems are:

• England: National Health Service
• Northern Ireland: Health and Social Care in Northern Ireland (HSCNI)
• Scotland: NHS Scotland
• Wales: NHS Wales

In England, funding from general taxation is channeled through NHS England, which is responsible for commissioning mainly specialist services and primary care, and Clinical Commissioning Groups (CCGs), which manage 60% of the budget and are responsible for commissioning health services for their local populations.

These commissioning bodies do not provide services themselves directly, but procure these from NHS Trusts and Foundation Trusts, as well as private, voluntary and social enterprise sector providers.

Regions with hybrid single-payer/private insurance systems

Australia

Healthcare in Australia is provided by both private and government institutions. Medicare is the publicly funded universal health care venture in Australia. It was instituted in 1984 and coexists with a private health system. Medicare is funded partly by a 2% income tax levy (with exceptions for low-income earners), but mostly out of general revenue. An additional levy of 1% is imposed on high-income earners without private health insurance.

As well as Medicare, there is a separate Pharmaceutical Benefits Scheme that considerably subsidises a range of prescription medications. The Minister for Health administers national health policy, elements of which (such as the operation of hospitals) are overseen by individual states.

Spain

Building upon less structured foundations, in 1963 the existence of a single-payer healthcare system in Spain was established by the Spanish government. The system was sustained by contributions from workers, and covered them and their dependants.

The universality of the system was established later in 1986. At the same time, management of public healthcare was delegated to the different autonomous communities in the country. While previously this was not the case, in 1997 it was established that public authorities can delegate management of publicly funded healthcare to private companies.

Additionally, in parallel to the single-payer healthcare system there are private insurers, which provide coverage for some private doctors and hospitals. Employers will sometimes offer private health insurance as a benefit, with 14.8% of the Spanish population being covered under private health insurance in 2013.

In 2000, the Spanish healthcare system was rated by the World Health Organization as the 7th best in the world.

United States

Medicare in the United States is a single-payer healthcare system, but is restricted to persons over the age of 65, people under 65 who have specific disabilities, and anyone with End-Stage Renal Disease.

A number of proposals have been made for a universal single-payer healthcare system in the United States, among them the United States National Health Care Act (popularly known as H.R. 676 or "Medicare for All") originally introduced in the House in February 2003 and repeatedly since.

On July 18, 2018, it was announced that over 60 House Democrats would be forming a Medicare For All Caucus.

Advocates argue that preventive healthcare expenditures can save several hundreds of billions of dollars per year because publicly funded universal healthcare would benefit employers and consumers, that employers would benefit from a bigger pool of potential customers and that employers would likely pay less, would be spared administrative costs, and inequities between employers would be reduced. Prohibitively high cost is the primary reason Americans give for problems accessing health care. At over 27 million, the number of people without health insurance coverage in the United States is one of the primary concerns raised by advocates of health care reform. Lack of health insurance is associated with increased mortality, about sixty thousand preventable deaths per year, depending on the study. A study done at Harvard Medical School with Cambridge Health Alliance showed that nearly 45,000 annual deaths are associated with a lack of patient health insurance. The study also found that uninsured, working Americans have a risk of death about 40% higher compared to privately insured working Americans.

Advocates also argue that single-payer could benefit from a more fluid economy with increasing economic growth, aggregate demand, corporate profit, and quality of life. Others have estimated a long-term savings amounting to 40% of all national health expenditures due to the extended preventive health care, although estimates from the Congressional Budget Office and The New England Journal of Medicine have found that preventive care is more expensive due to increased utilization.

Any national system would be paid for in part through taxes replacing insurance premiums, but advocates also believe savings would be realized through preventive care and the elimination of insurance company overhead and hospital billing costs.

A 2008 analysis of a single-payer bill by Physicians for a National Health Program estimated the immediate savings at $350 billion per year. The Commonwealth Fund believes that, if the United States adopted a universal health care system, the mortality rate would improve and the country would save approximately $570 billion a year.

Opponents argue single-payer does not translate into better health care. Instead, access to health care diminishes under single-payer systems, and the overall quality of care suffers. Opponents also claim that single-payer systems cause shortages of general physicians and specialists and reduce access to medical technology.

National policies and proposals

Government is increasingly involved in U.S. health care spending, paying about 45% of the $2.2 trillion the nation spent on individuals' medical care in 2004. However, studies have shown that the publicly administered share of health spending in the U.S. may be closer to 60% as of 2002.

According to Princeton University health economist Uwe Reinhardt, U.S. Medicare, Medicaid, and State Children's Health Insurance Program (SCHIP) represent "forms of 'social insurance' coupled with a largely private health-care delivery system" rather than forms of "socialized medicine." In contrast, he describes the Veterans Administration healthcare system as a pure form of socialized medicine because it is "owned, operated and financed by government."

In a peer-reviewed paper published in the Annals of Internal Medicine, researchers of the RAND Corporation reported that the quality of care received by Veterans Administration patients scored significantly higher overall than did comparable metrics for patients currently using United States Medicare.

The United States National Health Care Act is a perennial piece of legislation introduced many times in the United States House of Representatives by then Representative John Conyers (D-MI). The act would establish a universal single-payer health care system in the United States, the rough equivalent of Canada's Medicare, the United Kingdom's National Health Service, and Taiwan's Bureau of National Health Insurance, among other examples. The bill was first introduced in 2003 and has been reintroduced in each Congress since. During the 2009 health care debates over the bill that became the Patient Protection and Affordable Care Act, H.R. 676 was expected to be debated and voted upon by the House in September 2009, but was never debated. In the wake of Bernie Sanders' 2016 presidential campaign, in which a push for universal healthcare featured prominently, single-payer proposals gained traction. Conyers reintroduced his bill in the House of Representatives in January 2017. Four months later, the bill was supported by 112 co-sponsors, surpassing for the first time the 25% mark of co-sponsorship. In September of the same year, Sanders himself, together with 16 co-sponsors, introduced a Medicare-for-all bill in the Senate (S. 1804). An analysis of a Mercatus Center study of the 2017 proposal by economist Jeffrey Sachs found that "it rightfully and straightforwardly concludes that M4A would provide more health care coverage at lower cost than the status quo, projecting a net reduction in national health expenditures of roughly $2 trillion over a 10-year period (2022-2031), while also enabling increased health care coverage."

The Congressional Budget Office and related government agencies scored the cost of a single-payer health care system several times since 1991. The General Accounting Office published a report in 1991 noting that "[I]f the US were to shift to a system of universal coverage and a single payer, as in Canada, the savings in administrative costs [10 percent of health spending] would be more than enough to offset the expense of universal coverage."

The CBO scored the cost in 1991, noting that "the population that is currently uninsured could be covered without dramatically increasing national spending on health" and that "all US residents might be covered by health insurance for roughly the current level of spending or even somewhat less, because of savings in administrative costs and lower payment rates for services used by the privately insured."

A CBO report in 1993 stated that "[t]he net cost of achieving universal insurance coverage under this single payer system would be negative" in part because "consumer payments for health would fall by $1,118 per capita, but taxes would have to increase by $1,261 per capita" in order to pay for the plan.  A July 1993 scoring also resulted in positive outcomes, with the CBO stating that, "[a]s the program was phased in, the administrative savings from switching to a single-payer system would offset much of the increased demand for health care services.

Later, the cap on the growth of the national health budget would hold the rate of growth of spending below the baseline." The CBO also scored Sen. Paul Wellstone's American Health and Security Act of 1993 in December 1993, finding that "by year five (and in subsequent years) the new system would cost less than baseline."

A 2014 study published in the journal BMC Medical Services Research by James Kahn, et al., found that the actual administrative burden of health care in the United States was 27% of all national health expenditures. The study examined both direct costs charged by insurers for profit, administration and marketing but also the indirect burden placed on health care providers like hospitals, nursing homes and doctors for costs they incurred in working with private health insurers including contract negotiations, financial and clinical record-keeping (variable and idiosyncratic for each payer).

Kahn, et al. estimate that the added cost for the private insurer health system in the US was about $471 billion in 2012 compared to a single-payer system like Canada's. This represents just over 20% of the total national healthcare expenditure in 2012. Kahn asserts that this excess administrative cost will increase under the Affordable Care Act with its reliance on the provision of health coverage through a multi-payer system.

State proposals

Several single-payer state referendums and bills from state legislatures have been proposed, but with the exception of Vermont, all have failed. In December 2014, Vermont canceled its plan for single-payer health care.

California

California attempted passage of a single-payer bill as early as 1994, and the first successful passages of legislation through the California State Legislature, SB 840 or "The California Universal Healthcare Act" (authored by Sheila Kuehl), occurred in 2006 and again in 2008. Both times, Governor Arnold Schwarzenegger vetoed the bill. State Senator Mark Leno has reintroduced the bill in each legislative session since.

On February 17, 2017, SB 562, which is also known as "The Healthy California Act" was introduced to the California State Senate. This bill is a $400 billion plan that was sponsored by the California Nurses Association to implement single-payer healthcare in California. Under this bill, which was co-authored by State Senators Ricardo Lara (D-Bell Gardens) and Toni Atkins (D-San Diego), Californians would have health coverage without having to pay any premiums, co-pays, or deductibles. Under this proposed bill, all California residents will be covered in the Healthy California Act SB 562 regardless of their immigration status. This bill will also include transient students that attend California institutions whom, purchased their healthcare program through the school. Services that will be covered by this bill will need to determine as medically necessary by the patient’s chosen health care provider. These services will range from preventable services to emergency services, in addition to prescription drugs services. SB 562 passed in the State Senate on June 1, 2017 with a vote of 23-14. When the bill was sent to the State Assembly, it did not get approved and was put on hold since there were flaws that did not address issues like how to fund for this bill and how care would be delivered to patients. Although the bill is currently put on hold, there are hopes it will be revived in 2018 with the necessary changes so it can be reviewed again by both the State Senate and State Assembly.

According to SB-562, a Healthy California Trust Fund would be established to provide funding for the bill. Currently, states receive funding from the federal government for certain healthcare services such as Medicaid and Medicare. In addition to taxes, these funds would be pooled into the new trust fund and provide the sources of funding needed to implement The Healthy California Act. However, California must first obtain a waiver from the federal government which would allow California to pool all the money received from these federal programs into one central fund.

Colorado

The Colorado State Health Care System Initiative, Amendment 69, was a citizen-initiated constitutional amendment proposal in November 2016 to vote on a single-payer healthcare system funded by a 10% payroll tax split 2:1 between employers and employees. This would have replaced the private health insurance premiums currently paid by employees and companies. The ballot was rejected by 79% of the electorate.

Hawaii

In 2009, the Hawaii state legislature passed a single-payer healthcare bill that was vetoed by Republican Governor Linda Lingle. While the veto was overridden by the legislature, the bill was not implemented.

Illinois

In 2007, the Health Care for All Illinois Act was introduced and the Illinois House of Representatives' Health Availability Access Committee passed the single-payer bill favorably out of committee by an 8–4 vote. The legislation was eventually referred back to the House rules committee and not taken up again during that session.

Massachusetts

Massachusetts had passed a universal healthcare program in 1986, but budget constraints and partisan control of the legislature resulted in its repeal before the legislation could be enacted.

Question 4, a nonbinding referendum, was on the ballot in 14 state districts in November 2010, asking voters, "[S]hall the representative from this district be instructed to support legislation that would establish healthcare as a human right regardless of age, state of health or employment status, by creating a single payer health insurance system like Medicare that is comprehensive, cost effective, and publicly provided to all residents of Massachusetts?" The ballot question passed in all 14 districts that offered the question.

Minnesota

The Minnesota Health Act, which would establish a statewide single-payer health plan, has been presented to the Minnesota legislature regularly since 2009. The bill was passed out of both the Senate Health Housing and Family Security Committee and the Senate Commerce and Consumer Protection Committee in 2009, but the House version was ultimately tabled.

In 2010, the bill passed the Senate Judiciary Committee on a voice vote as well as the House Health Care & Human Services Policy and Oversight Committee. In 2011, the bill was introduced as a two-year bill in both the Senate and House, but did not progress. It has been introduced again in the 2013 session in both chambers.

Montana

In September 2011, Governor Brian Schweitzer announced his intention to seek a waiver from the federal government allowing Montana to set up a single-payer healthcare system. Governor Schweitzer was unable to implement single-payer health care in Montana, but did make moves to open government-run clinics, and in his final budget as governor, increased coverage for lower-income Montana residents.

New York

New York State has been attempting passage of the New York Health Act, which would establish a statewide single-payer health plan, since 1992. The New York Health Act passed the Assembly four times: once in 1992 and again in 2015, 2016, and 2017, but has not yet advanced through the Senate after referrals to the Health Committee. On all occasions, the legislation passed the Assembly by an almost two-to-one ratio of support.

Oregon

The state of Oregon attempted to pass single-payer healthcare via Oregon Ballot Measure 23 in 2002, and the measure was rejected by a significant majority.

Pennsylvania

The Family Business and Healthcare Security Act has been introduced in the Pennsylvania legislature numerous times, but has never been able to pass.

Vermont

In December 2014, Vermont canceled its plan for single-payer healthcare. Vermont passed legislation in 2011 creating Green Mountain Care. When Governor Peter Shumlin signed the bill into law, Vermont became the first state to functionally have a single-payer health care system. While the bill is considered a single-payer bill, private insurers can continue to operate in the state indefinitely, meaning it does not fit the strict definition of single-payer.

Representative Mark Larson, the initial sponsor of the bill, has described Green Mountain Care's provisions "as close as we can get [to single-payer] at the state level." Vermont abandoned the plan in 2014, citing costs and tax increases as too high to implement.

Public opinion

Advocates for single-payer point to support in polls, although the polling is mixed depending on how the question is asked. Polls from Harvard University in 1988, the Los Angeles Times in 1990, and the Wall Street Journal in 1991 all showed strong support for a health care system comparable to the system in Canada.

More recently, however, polling support has declined. A 2007 Yahoo/AP poll showed a majority of respondents considered themselves supporters of "single-payer health care," and a plurality of respondents in a 2009 poll for Time Magazine showed support for "a national single-payer plan similar to Medicare for all." Polls by Rasmussen Reports in 2011 and 2012 showed pluralities opposed to single-payer healthcare.

A 2001 article in the public health journal Health Affairs studied fifty years of American public opinion of various health care plans and concluded that, while there appears to be general support of a "national health care plan," poll respondents "remain satisfied with their current medical arrangements, do not trust the federal government to do what is right, and do not favor a single-payer type of national health plan."

Politifact rated a statement by Michael Moore "false" when he stated that "[t]he majority actually want single-payer health care." According to Politifact, responses on these polls largely depend on the wording. For example, people respond more favorably when they are asked if they want a system "like Medicare."

Advocacy groups

Physicians for a National Health Program, the American Medical Student Association, Healthcare-NOW!, and the California Nurses Association are among advocacy groups that have called for the introduction of a single-payer healthcare program in the United States.

A 2007 study published in the Annals of Internal Medicine found that 59% of physicians "supported legislation to establish national health insurance" while 9% were neutral on the topic, and 32% opposed it.