Scalar theories of gravitation are field theories of
gravitation
In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the stron ...
in which the gravitational field is described using a
scalar field
In mathematics and physics, a scalar field is a function (mathematics), function associating a single number to every point (geometry), point in a space (mathematics), space – possibly physical space. The scalar may either be a pure Scalar ( ...
, which is required to satisfy some field equation.
''Note:'' This article focuses on
relativistic classical field theories
A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations, without considering effects of quantization; theories that incorporate quantum mechanics are called quantum ...
of gravitation. The best known relativistic classical field theory of gravitation,
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, is a tensor theory, in which the gravitational interaction is described using a
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
field.
Newtonian gravity
The prototypical scalar theory of gravitation is
Newtonian gravitation
Newton's law of universal gravitation is usually stated as that every particle attracts every other particle in the universe with a force that is proportional to the product of their masses and inversely proportional to the square of the distanc ...
. In this theory, the gravitational interaction is completely described by the
potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple re ...
, which is required to satisfy the
Poisson equation
Poisson's equation is an elliptic partial differential equation of broad utility in theoretical physics. For example, the solution to Poisson's equation is the potential field caused by a given electric charge or mass density distribution; with t ...
(with the mass density acting as the source of the field). To wit:
, where
* ''G'' is the gravitational constant and
*
is the mass density.
This field theory formulation leads directly to the familiar law of universal gravitation,
.
Nordström's theories of gravitation
The first attempts to present a relativistic (classical) field theory of gravitation were also scalar theories.
Gunnar Nordström
Gunnar Nordström (12 March 1881 – 24 December 1923) was a Finnish theoretical physicist best remembered for his theory of gravitation, which was an early competitor of general relativity. Nordström is often designated by modern writers as '' ...
created two such theories.
Nordström's first idea (1912) was to simply replace the divergence operator in the field equation of Newtonian gravity with the
d'Alembertian
In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of ...
operator
. This gives the field equation
:
.
However, several theoretical difficulties with this theory quickly arose, and Nordström dropped it.
A year later, Nordström tried again, presenting the field equation
:
,
where
is the trace of the
stress–energy tensor
The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress ...
.
Solutions of Nordström's second theory are
conformally flat
A (pseudo-)Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation.
In practice, the metric g of the manifold M has to be conformal to the flat metric \eta, i.e., the ...
Lorentzian spacetimes. That is, the metric tensor can be written as
, where
* η
μν is the
Minkowski metric
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
, and
*
is a scalar which is a function of position.
This suggestion signifies that the inertial mass should depend on the scalar field.
Nordström's second theory satisfies the weak
equivalence principle
In the theory of general relativity, the equivalence principle is the equivalence of gravitational and inertial mass, and Albert Einstein's observation that the gravitational "force" as experienced locally while standing on a massive body (suc ...
. However:
* The theory fails to predict any deflection of light passing near a massive body (contrary to observation)
* The theory predicts an anomalous
perihelion
An apsis (; ) is the farthest or nearest point in the orbit of a planetary body about its primary body. For example, the apsides of the Earth are called the aphelion and perihelion.
General description
There are two apsides in any ellip ...
precession
Precession is a change in the orientation of the rotational axis of a rotating body. In an appropriate reference frame it can be defined as a change in the first Euler angle, whereas the third Euler angle defines the rotation itself. In othe ...
of
Mercury
Mercury commonly refers to:
* Mercury (planet), the nearest planet to the Sun
* Mercury (element), a metallic chemical element with the symbol Hg
* Mercury (mythology), a Roman god
Mercury or The Mercury may also refer to:
Companies
* Merc ...
, but this disagrees in both sign and magnitude with the observed anomalous precession (the part which cannot be explained using Newtonian gravitation).
Despite these disappointing results, Einstein's critiques of Nordström's second theory played an important role in his development of general relativity.
Einstein's scalar theory
In 1913, Einstein (erroneously) concluded from his
hole argument In general relativity, the hole argument is an apparent paradox that much troubled Albert Einstein while developing his famous field equations.
Some philosophers of physics take the argument to raise a problem for '' manifold substantialism'', a ...
that general covariance was not viable. Inspired by Nordström's work, he proposed his own scalar theory. This theory employs a massless scalar field coupled to the stress–energy tensor, which is the sum of two terms. The first,
:
represents the stress–momentum–energy of the scalar field itself. The second represents the stress-momentum-energy of any matter which may be present:
:
where
is the
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
vector of an observer, or
tangent vector
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are eleme ...
to the world line of the observer. (Einstein made no attempt, in this theory, to take account of possible gravitational effects of the field energy of the
electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
.)
Unfortunately, this theory is not
diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable.
Definition
Given two m ...
covariant. This is an important consistency condition, so Einstein dropped this theory in late 1914. Associating the scalar field with the metric leads to Einstein's later conclusions that the theory of gravitation he sought could not be a scalar theory. Indeed, the theory he finally arrived at in 1915,
general relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
, is a tensor theory, not a scalar theory, with a 2-tensor, the metric, as the potential. Unlike his 1913 scalar theory, it is
generally covariant
In theoretical physics, general covariance, also known as diffeomorphism covariance or general invariance, consists of the invariance of the ''form'' of physical laws under arbitrary differentiable coordinate transformations. The essential idea is ...
, and it does take into account the field energy–momentum–stress of the electromagnetic field (or any other nongravitational field).
Additional variations
*
Kaluza–Klein theory involves the use of a scalar gravitational field in addition to the
electromagnetic field
An electromagnetic field (also EM field or EMF) is a classical (i.e. non-quantum) field produced by (stationary or moving) electric charges. It is the field described by classical electrodynamics (a classical field theory) and is the classical c ...
potential
in an attempt to create a five-dimensional unification of gravity and electromagnetism. Its generalization with a 5th variable component of the metric that leads to a variable gravitational constant was first given by
Pascual Jordan
Ernst Pascual Jordan (; 18 October 1902 – 31 July 1980) was a German theoretical and mathematical physicist who made significant contributions to quantum mechanics and quantum field theory. He contributed much to the mathematical form of matrix ...
.
[
* Brans–Dicke theory is a scalar-tensor theory, not a scalar theory, meaning that it represents the gravitational interaction using both a scalar field and a tensor field. We mention it here because one of the field equations of this theory involves only the scalar field and the trace of the stress–energy tensor, as in Nordström's theory. Moreover, the Brans–Dicke theory is equal to the independently derived theory of Jordan (hence it is often referred to as the Jordan-Brans–Dicke or JBD theory). The Brans–Dicke theory couples a scalar field with the curvature of space-time and is self-consistent and, assuming appropriate values for a tunable constant, this theory has not been ruled out by observation. The Brans–Dicke theory is generally regarded as a leading competitor of general relativity, which is a pure tensor theory. However, the Brans–Dicke theory seems to need too high a parameter, which favours general relativity).]
* Zee combined the idea of the BD theory with the Higgs-Mechanism of Symmetry Breakdown for mass generation, which led to a scalar-tensor theory with Higgs field as scalar field, in which the scalar field is massive (short-ranged). An example of this theory was proposed by H. Dehnen and H. Frommert 1991, parting from the nature of Higgs field interacting gravitational- and Yukawa (long-ranged)-like with the particles that get mass through it.
* The Watt–Misner theory (1999) is a recent example of a scalar theory of gravitation. It is not intended as a viable theory of gravitation (since, as Watt and Misner point out, it is not consistent with observation), but as a toy theory which can be useful in testing numerical relativity schemes. It also has pedagogical value.
See also
*Nordström's theory of gravitation
In theoretical physics, Nordström's theory of gravitation was a predecessor of general relativity. Strictly speaking, there were actually ''two'' distinct theories proposed by the Finnish theoretical physicist Gunnar Nordström, in 1912 and 1913 ...
References
External links
*Goenner, Hubert F. M., "On the History of Unified Field Theories"; ''Living Rev. Relativ. 7(2)'', 2004
lrr-2004-2
Retrieved August 10, 2005.
*
*P. Jordan, ''Schwerkraft und Weltall'', Vieweg (Braunschweig) 1955.
{{DEFAULTSORT:Scalar Theories Of Gravitation
Theories of gravity