Alternatives to general relativity

Alternatives to general relativity

Alternatives to general relativity are physical theories that attempt to describe the phenomena 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. For those theories that attempt to do several at once, see the article theory of everything.


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 to either 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, and they can be rejected on the grounds of Occam's Razor until an experimental discrepancy shows up. Further, shortly after that, theorists switched to string theory which was starting to look promising. 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 metres. 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". The basic idea is that gravity agrees with GR at the present epoch but may have been quite different in the early universe. Investigation of the Pioneer anomaly has caused renewed public interest in alternatives to General Relativity, but the Pioneer anomaly is too strong to be explained by any such theory of gravity.

Notation in this article

c; is the speed of light, G; is the gravitational constant. "Geometric variables" are not used.

Latin indexes go from 1 to 3, Greek indexes go from 1 to 4. The Einstein summation convention is used.

eta_{mu u}; is the Minkowski metric. g_{mu u}; is a tensor, usually the metric tensor. These have signature (-,+,+,+).

Partial differentiation is written partial_mu phi; or phi_{,mu};. Covariant differentiation is written abla_mu phi; or 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, say L,, then the action S, is the integral of that, for example

:S,propto,int d^4 x R sqrt{-g}L,

where R, is the curvature of space. In this equation it is usual, though not essential, to have g=-1,.

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 metric tensor g_{mu u}, of signature 1, which governs proper-length and proper-time measurements in the usual manner of special and general relativity:

:ds^2=g_{mu u}dx^mu dx^ u,

where there is a summation over indices mu and u.Condition 2. Stressed matter and fields being acted upon by gravity respond in accordance with the equation:

: ablacdot T=0,

where T, is the stress-energy tensor for all matter and non-gravitational fields, and where abla is the covariant derivative with respect to the metric.

Any theory of gravity in which g_{mu u} e g_{ umu} is always true is not a metric theory, but any metric theory can perfectly well be given a mathematical description that violates conditions 1 and 2.

Metric theories include (from simplest to most complex):
* Scalar Field Theories (includes Conformally flat theories & Stratified theories with conformally flat space slices)Nordström, Einstein-Fokker, Whitrow-Morduch, Littlewood, Bergman, Page-Tupper, Einstein (1912), Whitrow-Morduch, Rosen (1971), Papapetrou, Ni, Yilmaz, [Coleman] , Lee-Lightman-Ni
* Bimetric TheoriesRosen (1975), Rastall, Lightman-Lee
* Quasilinear Theories (includes Linear fixed gauge)Whitehead, Deser-Laurent, Bollini-Giambini-Tiomno
* Tensor-TheoriesEinstein's GR
* Scalar-Tensor TheoriesThiry, Jordan, Brans-Dicke, Bergmann, Wagoner, Nordtvedt, Beckenstein
* Vector-tensor theoriesWill-Nordtvedt, Hellings-Nordvedt
* Other metric theories(see section Modern Theories below)

Non-metric theories includeCartan, Belinfante-Swihart

A word here about Mach's principle is appropriate because a few of these theories rely on Mach's principle eg. Whitehead (1922), and many mention it in passing eg. 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:this isn't exactly the way Mach originally stated it, see other variants in Mach principle]

* 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 ho, generates a scalar field, the gravitational potential phi, in Joules per kilogram, by:{partial^2 phi over partial x^j partial x^j} = 4 pi G ho ,.

Using the Nabla operator abla for the gradient and divergence (partial derivatives), this can be conveniently written as:: abla^2 phi = 4 pi G ho ,.

This scalar field governs the motion of a free-falling particle by::{ d^2x^jover dt^2} = -{partialphioverpartial x^j,}.

At distance, "r", from an isolated mass, "M", the scalar field is: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. [cite journal | author=Zenneck, J. | authorlink=Jonathan Zenneck | title=Gravitation | journal=Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen | year=1903 | volume=5 | pages=25–67 | url= | language=German] 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. [cite journal | author=Lorentz, H.A. | authorlink = Hendrik Lorentz | title=Considerations on Gravitation | journal=Proc. Acad. Amsterdam | year=1900 | volume=2 | pages=559–574 | url=]

;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, like in Lorentz's model the value for the perihelion advance of Mercury was much too low. [Citation | author=Walter, S. | year=2007 | editor=Renn, J. | contribution= [ Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905–1910] | journal=The Genesis of General Relativity |pages=193–252 |volume=3 |place=Berlin | publisher=Springer]

;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:

deltaint ds=0,

ds^2=eta_{mu u}dx^mu dx^ u,

where eta_{mu u}, is the Minkowski metric, and there is a summation from 1 to 4 over indices mu, and u,.

Einstein and Grossmann (1913) includes Riemannian geometry and tensor calculus.

deltaint ds=0,

ds^2=g_{mu u}dx^mu dx^ u,

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

T_{mu u}=kappa ho{dx^muover ds}{dx^ uover ds},

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 c, but to let mass depend on the gravitational field strength phi,. Allowing this field strength to satisfy

Boxphi= ho,

where ho, is rest mass energy and Box, is the d'Alembertian,

m=m_0 exp(phi/c^2),


-{partialphioverpartial x^mu}=dot{u}_mu+{u_muover c^2dot{phi,

where 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):

deltaintpsi ds=0,

ds^2=eta_{mu u}dx^mu dx^ u,

where psi, is a scalar field,

-{partial T^{mu u}overpartial x^ u}=T{1overpsi}{partialpsioverpartial 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:

deltaintpsi ds=0,

ds^2=g_{mu u}dx^mu dx^ u,

g_{mu u}=psi^2eta_{mu u},

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


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 know of as General Relativity. Discarding the Minkowski metric entirely, Einstein gets:

deltaintpsi ds=0,

ds^2=g_{mu u}dx^mu dx^ u,

R_{mu u}=8pi G(T_{mu u}-g_{mu u}T/2),

which can also be written T^{mu u}={1over 8pi G}(R^{mu u}-g^{mu u}R/2),

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

S={1over 16pi G}int R sqrt{-g}d^4 x+S_m,

where G, is Newton's gravitational constant, R=R_{mumu}, is the Ricci curvature of space, g=g_{mumu}, and 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),A later edition is Will (1993). See also Ni (1972)] Lang (2002)Although an important source for this article, the presentations of Turyshev (2006) and Lang (2002) contain many errors of fact] ):

Listed by date (the hyperlinks take you further down this article)

Whitehead (1922), Cartan (1922, 1923), Fierz & Pauli (1939), Birkhov (1943), Milne (1948), Thiry (1948), Papapetrou (1954a, 1954b), Littlewood (1953), Jordan (1955), Bergman (1956), Belinfante & Swihart (1957), Yilmaz (1958, 1973), Brans & Dicke (1961), Whitrow & Morduch (1960, 1965), Kustaanheimo (1966) , Kustaanheimo & Nuotio (1967), Deser & Laurent (1968), Page & Tupper (1968), Bergmann (1968), Bollini-Giambini-Tiomno (1970), Nordtveldt (1970), Wagoner (1970), Rosen (1971, 1975, 1975), Ni (1972, 1973), Will & Nordtveldt (1972), Hellings & Nordtveldt (1973), Lightman & Lee (1973), Lee, Lightman & Ni (1974), Beckenstein (1977), Barker (1978), Rastall (1979)

These theories are presented here without a cosmological constant, how to add a cosmological constant or quintessence is discussed under Modern Theories (see also here) 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 Perlmutter.

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:

deltaint f(phi/c^2)ds=0,

where the scalar field is,


and c, may or may not depend on phi,.

In Nordström (1912),

f(phi/c^2)=exp(-phi/c^2), ; c=c_infty,

In Littlewood (1953) and Bergmann (1956),

f(phi/c^2)=exp(-phi/c^2-(phi/c^2)^2/2), ; c=c_infty,

In Whitrow and Morduch (1960),

f(phi/c^2)=1, ; c^2=c_infty^2-2phi,

In Whitrow and Morduch (1965),

f(phi/c^2)=exp(-phi/c^2), ; c^2=c_infty^2-2phi,

In Page and Tupper (1968),

f(phi/c^2)=phi/c^2+alpha(phi/c^2)^2, ; c_infty^2/c^2=1+4(phi/c_infty^2)+(15+2alpha)(phi/c_infty^2)^2,

Page and Tupper (1968) matches Yilmaz (1958) (see also Yilmaz theory of gravitation) to second order when 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:

S={1over 16pi G}int d^4 x sqrt{-g}L_phi+S_m,

L_phi=phi R-2g^{mu u}partial_muphipartial_ uphi,

Misner et al. (1973) gives this without the phi R, term. S_m, is the matter action.

Boxphi=4pi T^{mu u} [eta_{mu u}e^{-2phi}+(e^{2phi}+e^{-2phi})partial_mu tpartial_ u t] , 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 f(phi), and k(phi), that are related to the metric by:

ds^2=e^{-2f(phi)}dt^2-e^{2f(phi)} [dx^2+dy^2+dz^2] ,

eta^{mu u}partial_mupartial_ uphi=4pi ho^*k(phi),

Ni (1972) quotes Rosen (1971) as having two scalar fields phi, and psi, that are related to the metric by:

ds^2=phi^2dt^2-psi^2 [dx^2+dy^2+dz^2] ,

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

L_phi=e^phi( extstylefrac{1}{2}e^{-phi}partial_alphaphipartial_alphaphi + extstylefrac{3}{2}e^{phi}partial_0phipartial_0phi),

In Papapetrou (1954b) there is a second scalar field chi,. The gravitational part of the Lagrangian is now:

L_phi=e^{(3phi+chi)/2}(- extstylefrac{1}{2}e^{-phi}partial_alphaphipartial_alphaphi -e^{-phi}partial_alphaphipartial_chiphi+ extstylefrac{3}{2}e^{-chi}partial_0phipartial_0phi),

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 TheoryThe action is:

S={1over 64pi G}int d^4 xsqrt{-eta}eta^{mu u}g^{alphaeta}g^{gammadelta} (g_{alphagamma |mu}g_{alphadelta | u} - extstylefrac{1}{2}g_{alphaeta |mu}g_{gammadelta | u})+S_m

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

Box_eta g_{mu u}-g^{alphaeta}eta^{gammadelta}g_{mualpha |gamma}g_{ ueta |delta}=-16pi Gsqrt{g/eta}(T_{mu u}- extstylefrac{1}{2}g_{mu u} 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 B_{mu u},, B=B_{mu u}eta^{mu u},, and two constants a, and f, the action is:

S={1over 16pi G}int d^4 xsqrt{-eta}(aB^{mu u|alpha}B_{mu u|alpha} +fB_{,alpha}B^{,alpha})+S_m

and the stress-energy tensor comes from:

aBox_eta B^{mu u}+feta^{mu u}Box_eta B=-4pi Gsqrt{g/eta}T^{alphaeta} (partial g_{alphaeta}/partial B_mu u)

In Rastall (1979), the metric is an algebraic function of the Minkowski metric and a Vector field.Will (1981) lists this as bimetric but I don't see why it isn't just a vector field theory] The Action is:

S={1over 16pi G}int d^4 x sqrt{-g} F(N)K^{mu; u}K_{mu; u}+S_m


F(N)=-N/(2+N); and N=g^{mu u}K_mu K_ u;

(see Will (1981) for the field equation for T^{mu u}; and K_mu;).

Quasilinear Theories

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

g_{mu u}(x^alpha)=eta_{mu u}-2int_{Sigma^-}{y_mu^- y_ u^-over(w^-)^3} [sqrt{-g} ho u^alpha dSigma_alpha] ^-

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

(y^mu)^-=x^mu-(x^mu)^-; , (y^mu)^-(y_mu)^-=0; , w^-=(y^mu)^-(u_mu)^-; , (u_mu)=dx^mu/dsigma;, dsigma^2=eta_{mu u}dx^mu dx^ u;

Deser and Laurent (1968) and Bollini-Giambini-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) h_{mu u}; to define

g_{mu u} = eta_{mu u}+h_{mu u};

The action is:

S={1over 16pi G} int d^4 xsqrt{-eta} [2h_

† The theory is incomplete, and 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-Giamiago-Tiomino (1970), but in these three cases eta=gamma, which is in strong conflict with GR and experimental results. In particular, these theories predict incorrect amplitudes for the Earth's tides..

Theories that fail other tests

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

The stratified theories of Ni (1973), Lee Lightman and Ni (1974) 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. 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 alpha_2 which can have a measurable effect on the Earth's tides.

And that leaves, as a likely valid alternative to GR, nothing [except possibly Cartan (1922), which may violate EEP] .

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

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:Beckenstein (2004), Moffat (1995), Moffat (2002), Moffat (2005a, b).

These theories are presented with a cosmological constant or added scalar or vector potential.


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 of opinion 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.

The discovery of the accelerated expansion of the universe by Perlmutter 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 Perlmutter's results in a completely different way.

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 the qualitative features of the anomaly but not the magnitude. Any alternative model of gravity that could explain the Pioneer anomaly would have to depart so strongly from GR that it would fail to satisfy other experimental observations.

Cosmological Constant and Quintessence

(also see Cosmological constant, Einstein-Hilbert action, Quintessence (physics))

The cosmological constant Lambda; is a very old idea, going back to Einstein in 1917. The success of the Friedmann model of the universe in which 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 Perlmutter discovered 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:

abla^2phi=4pi ho;


abla^2phi-Lambdaphi=4pi ho;

In GR, it changes the Einstein-Hilbert action from

S={1over 16pi G}int Rsqrt{-g}d^4x+S_m;


S={1over 16pi G}int (R-2Lambda)sqrt{-g}d^4x+S_m;

which changes the field equation

T^{mu u}={1over 8pi G}(R^{mu u}-g^{mu u} R/2); to

T^{mu u}={1over 8pi G}(R^{mu u}-g^{mu u} R/2+Lambda g^{mu u});

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

S={1over 16pi G}int d^4xsqrt{-g}L_phi+S_m;

Because 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

K^mu K^ u g_{mu u};

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

Relativistic MOND

(see Modified Newtonian dynamics, Tensor-vector-scalar gravity, and Beckenstein (2004) for more details).

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.i. It did not include relativistic effectsii. It violated the conservation of energy, momentum and angular momentumiii. It was inconsistent in that it gives different galactic orbits for gas and for starsiv. It did not state how to calculate gravitational lensing from galaxy clusters.

By 1984, problems ii. and iii. 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:

L=-{a_0^2over 8pi G}fleftlbrack{| ablaphi|^2over a_0^2} ight brack- hophi;

The relativistic version of this has:

L=-{a_0^2over 8pi G} ilde f(L^2 g^{mu u}partial_muphipartial_ uphi);

with a nonstandard mass action. Here f; and ilde f are arbitrary functions selected to give Newtonian and MOND behaviour in the correct limits.

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.

Beckenstein (2004) introduced a tensor-vector-scalar model (TeVeS). This has two scalar fields phi; and sigma; and vector field U_alpha;. The action is split into parts for gravity, scalars, vector and mass.


The gravity part is the same as in GR.

S_s=- extstylefrac12int [sigma^2 h^{alphaeta}phi_{,alpha}phi{,eta} + extstylefrac12G l^{-2}sigma^4F(kGsigma^2)] sqrt{-g}d^4x;

S_v=-{Kover 32pi G}int [g^{alphaeta}g^{mu u}U_{ [alpha,mu] }U_{ [eta, u] } -2(lambda/K)(g^mu u U_mu U_nu+1)] sqrt{-g}d^4x;

S_m=int L( ilde g_{mu u},f^alpha,f^alpha_{|mu},cdots)sqrt{-g}d^4x;

where h^{alphaeta} stackrel{mathrm{def{=} g^{alphaeta}-U^alpha U^eta;, l; is a length scale, k; and K; are constants, square brackets in indices U_{ [alpha,mu] }; represent anti-symmetrization lambda; is a Lagrange multiplier (calculated elsewhere), ilde g^{alphaeta}=e^{2phi}g^{alphaeta}+2U^alpha U^etasinh(2phi);, and L; is a Lagrangian translated from flat spacetime onto the metric ilde g^{alphaeta};.

F; is an arbitrary function, and F(mu)= extstylefrac34{mu^2(mu-2)^2over 1-mu}; is given as an example with the right asymptotic behaviour; note how it becomes undefined when mu=1;

Moffat's Theories

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 & 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 g_{mu u};, the Lagrangian density is split into


where L_M; is the same as for matter in GR.

L_R=sqrt{-g} [R(W)-2lambda- extstylefrac14mu^2g^{mu u}g_{ [mu u] }] - extstylefrac16g^{mu u}W_mu W_ u;

where R(W); is a curvature term analogous to but not equal to the Ricci curvature in GR, lambda; and mu^2; are cosmological constants, g_{ [ umu] }; is the antisymmetric part of g_{ umu};.W_mu; is a connection, and is a bit difficult to explain because it's defined recursively. However, W_muapprox-2g^{, u}_{ [mu u] };

Moffat (2002) is a claimed to be scalar-tensor bimetric gravity theory (BGT) and is one of the many theories of gravity in which the speed of light is faster in the early universe. These theories were motivated partly be the desire to avoid the "horizon problem" without invoking inflation. It has a variable G;. The theory also attempts to explain the dimming of supernovae from a perspective other than the acceleration of the universe and so runs the risk of predicting an age for the universe that is too small.

Overall, this theory looks poor. The action is split into gravity, scalar field, and matter parts. The gravity and scalar field equations are standard for a Brans-Dicke theory with cosmological constant and scalar potential, but applied with a Minkowski metric! Only the matter term uses a non-flat metric, and that is g_{mu u}=eta_{mu u}+Bpartial_muphipartial uphi; where B; has dimensions of length squared. This theory is going to fail Lorentz invariance and deflection of light tests at the very least.Fact|date=October 2007

Moffat (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 G;, increasing to a final constant value about a million years after the big bang.

The theory seems to contain an asymmetric tensor A_{mu u}; field and a source current J_mu; vector. The action is split into:


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

S_F=int d^4xsqrt{-g}( extstylefrac1{12}F_{mu u ho}F^{mu u ho} - extstylefrac14mu^2 A_{mu u}A^{mu u});

S_{FM}=int d^4xepsilon^{alphaetamu u}A_{alphaeta}partial_mu J_ u;


F_{mu u ho}=partial_mu A_{ u ho}+partial_ ho A_{mu u}

and epsilon^{alphaetamu u}; 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 (SVTG) theory.

The theory contains a tensor, vector and three scalar fields. But the equations are quite straightforward. The action is split into:S=S_G+S_K+S_S+S_M; with terms for gravity, vector field K_mu, scalar fields G;, omega; & mu;, and mass. S_G; is the standard gravity term with the exception that G; is moved inside the integral.

S_K=-int d^4xsqrt{-g}omega( extstylefrac14B_{mu u}B^{mu u}+V(K));

where B_{mu u}=partial_mu K_ u-partial_ u K_mu;

egin{matrix}S_S=-int d^4xsqrt{-g}({1over G^3}( extstylefrac12g^{mu u} abla_mu G abla_ u G -V(G))+{1over G}( extstylefrac12g^{mu u} abla_muomega abla_ uomega -V(omega))\ +{1overmu^2G}( extstylefrac12g^{mu u} abla_mumu abla_ umu-V(mu)))end{matrix}

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

V(K)=- extstylefrac12mu^2phi^muphi_mu- extstylefrac14g(phi^mu phi_mu)^2;

where g; is a coupling constant. The functions assumed for the scalar potentials are not stated.



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