Alternatives to general relativity
Proposed theories of gravity / From Wikipedia, the free encyclopedia
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Alternatives to general relativity are physical theories that attempt to describe the phenomenon of gravitation in competition with Einstein's theory of general relativity. There have been many different attempts at constructing an ideal theory of gravity.[1]
These attempts can be split into four broad categories based on their scope. In this article, straightforward alternatives to general relativity are discussed, which do not involve quantum mechanics or force unification. Other theories which do attempt to construct a theory using the principles of quantum mechanics are known as theories of quantized gravity. Thirdly, there are theories which attempt to explain gravity and other forces at the same time; these are known as classical unified field theories. Finally, the most ambitious theories attempt to both put gravity in quantum mechanical terms and unify forces; these are called theories of everything.
None of these alternatives to general relativity have gained wide acceptance. General relativity has withstood many tests,[2][3] remaining consistent with all observations so far. In contrast, many of the early alternatives have been definitively disproven. However, some of the alternative theories of gravity are supported by a minority of physicists, and the topic remains the subject of intense study in theoretical physics.
At the time it was published in the 17th century, Isaac Newton's theory of gravity was the most accurate theory of gravity. Since then, a number of alternatives were proposed. The theories which predate the formulation of general relativity in 1915 are discussed in history of gravitational theory.
General relativity
This theory[4][5] is what we now call "general relativity" (included here for comparison). 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 General relativity priority dispute. Hilbert was the first to correctly state the Einstein–Hilbert action for general relativity, which is:
where is Newton's gravitational constant, is the Ricci curvature of space, and is the action due to mass.
General relativity is a tensor theory, the equations all contain tensors. Nordström's theories, on the other hand, are scalar theories because the gravitational field is a scalar. Other proposed alternatives include scalar–tensor theories that contain a scalar field in addition to the tensors of general relativity, and other variants containing vector fields as well have been developed recently.
After general relativity, attempts were made either to improve on theories developed before general relativity, or to improve general relativity itself. Many different strategies were attempted, for example the addition of spin to general relativity, combining a general relativity-like metric with a spacetime 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 general relativity that is free of singularities.
Experimental tests improved along with the theories. Many of the different strategies that were developed soon after general relativity were abandoned, and there was a push to develop more general forms of the theories that survived, so that a theory would be ready when any test showed a disagreement with general relativity.
By the 1980s, the increasing accuracy of experimental tests had all confirmed general relativity; no competitors were left except for those that included general relativity 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 in the range of a few 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.
is the speed of light, 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.
is the Minkowski metric. is a tensor, usually the metric tensor. These have signature (−,+,+,+).
Partial differentiation is written or . Covariant differentiation is written or .
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 , then the gravitational part of the action is the integral of that:
- .
In this equation it is usual, though not essential, to have at spatial infinity when using Cartesian coordinates. For example, the Einstein–Hilbert action uses
where R is the scalar curvature, a measure of the curvature of space.
Almost every theory described in this article has an action. It is the most efficient known way to guarantee that the necessary conservation laws of energy, momentum and angular momentum are incorporated automatically; although it is easy to construct an action where those conservation laws are violated. Canonical methods provide another way to construct systems that have the required conservation laws, but this approach is more cumbersome to implement.[6] 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,[7] 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 of signature (−, +, +, +), 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 and is the Christoffel symbol. The stress–energy tensor should also satisfy an energy condition.
Metric theories include (from simplest to most complex):
- Scalar field theories (includes conformally flat theories & Stratified theories with conformally flat space slices)
- Bergman
- Coleman
- Einstein (1912)
- Einstein–Fokker theory
- Lee–Lightman–Ni
- Littlewood
- Ni
- Nordström's theory of gravitation (first metric theory of gravity to be developed)
- Page–Tupper
- Papapetrou
- Rosen (1971)
- Whitrow–Morduch
- Yilmaz theory of gravitation (attempted to eliminate event horizons from the theory.)
- Quasilinear theories (includes Linear fixed gauge)
- Bollini–Giambiagi–Tiomno
- Deser–Laurent
- Whitehead's theory of gravity (intended to use only retarded potentials)
- Tensor theories
- Einstein's general relativity
- Fourth-order gravity (allows the Lagrangian to depend on second-order contractions of the Riemann curvature tensor)
- f(R) gravity (allows the Lagrangian to depend on higher powers of the Ricci scalar)
- Gauss–Bonnet gravity
- Lovelock theory of gravity (allows the Lagrangian to depend on higher-order contractions of the Riemann curvature tensor)
- Infinite derivative gravity
- Scalar–tensor theories
- Bekenstein
- Bergmann–Wagoner
- Brans–Dicke theory (the most well-known alternative to general relativity, intended to be better at applying Mach's principle)
- Jordan
- Nordtvedt
- Thiry
- Chameleon
- Pressuron
- Vector–tensor theories
- Bimetric theories
- 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[7]), and many mention it in passing (e.g. Einstein–Grossmann,[8] Brans–Dicke[9]). Mach's principle can be thought of a half-way-house between Newton and Einstein. It goes this way:[10]
- Newton: Absolute space and time.
- Mach: The reference frame comes from the distribution of matter in the universe.
- Einstein: There is no reference frame.