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:
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
is the speed of light, 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.
is the Minkowski metric. is a tensor, usually the metric tensor. These have signature .
Partial differentiation is written or . Covariant differentiation is written or .
Classification of Theories
Theories of gravity can be classified, loosely, into several categories. Most of the theories described here have:
If a theory has a Lagrangian density, say , then the action is the integral of that, for example
where is the curvature of space. In this equation it is usual, though not essential, to have .
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 of signature 1, which governs proper-length and proper-time measurements in the usual manner of special and general relativity:
where there is a summation over indices and .
Condition 2. Stressed matter and fields being acted upon by gravity respond in accordance with the equation:
where is the stress-energy tensor for all matter and non-gravitational fields, and where is the covariant derivative with respect to the metric.
Any theory of gravity in which 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
Rosen (1975), Rastall, Lightman-Lee
Whitehead, Deser-Laurent, Bollini-Giambini-Tiomno
Thiry, Jordan, Brans-Dicke, Bergmann, Wagoner, Nordtvedt, Beckenstein
(see section Modern Theories below)
Non-metric theories include
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:
- 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
In Newton's (1686) theory (rewritten using more modern mathematics) the density of mass
generates a scalar field, the gravitational potential
in Joules per kilogram, by
Using the Nabla operator for the gradient and divergence (partial derivatives), this can be conveniently written as:
This scalar field governs the motion of a free-falling particle by:
At distance, r, from an isolated mass, M, the scalar field is
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, like 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:
where is the Minkowski metric, and there is a summation from 1 to 4 over indices and .
Einstein and Grossmann (1913) includes Riemannian geometry and tensor calculus.
The equations of electrodynamics exactly match those of GR. The equation
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 but to let mass depend on the gravitational field strength . Allowing this field strength to satisfy
where is rest mass energy and is the d'Alembertian,
where 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):
where is a scalar field,
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:
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:
which can also be written
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:
where is Newton's gravitational constant, is the Ricci curvature of space, and 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)):
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:
where the scalar field is,
and may or may not depend on .
In Nordström (1912),
In Littlewood (1953) and Bergmann (1956),
In Whitrow and Morduch (1960),
In Whitrow and Morduch (1965),
In Page and Tupper (1968),
Page and Tupper (1968) matches Yilmaz (1958) (see also Yilmaz theory of gravitation) to second order when .
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:
Misner et al. (1973) gives this without the term. is the matter action.
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 and that are related to the metric by:
Ni (1972) quotes Rosen (1971) as having two scalar fields and that are related to the metric by:
In Papapetrou (1954a) the gravitational part of the Lagrangian is:
In Papapetrou (1954b) there is a second scalar field . The gravitational part of the Lagrangian is now:
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:
where the vertical line "|" denotes covariant derivative with respect to . The field equations may be written in the form:
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 , , and two constants and the action is:
and the stress-energy tensor comes from:
In Rastall (1979), the metric is an algebraic function of the Minkowski metric and a Vector field.
The Action is:
(see Will (1981) for the field equation for and ).
In Whitehead (1922), the physical metric is constructed algebraically from the Minkowski metric and matter variables, so it doesn't even have a scalar field. The construction is:
where the superscript (-) indicates quantities evaluated along the past light cone of the field point and
, , , ,
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) to define
The action is: