In
theoretical physicsTheoretical physics is a branch of physics which employs mathematical models and abstractions of physics in an attempt to explain natural phenomena. Its central core is mathematical physics,[Sometimes mathematical physics and theoretical physics are used synonymously to refer to the...]
, the
Brans–Dicke theory of gravitation (sometimes called the
Jordan–Brans–Dicke theory) is a theoretical framework to explain
gravitationGravitation is a natural phenomenon by which objects with mass attract one another. In everyday life, gravitation is most commonly thought of as the agency which lends weight to objects with mass. Gravitation causes dispersed matter to coalesce, thus accounting for the existence of the Earth, the...
. It is a well-known competitor of
EinsteinAlbert Einstein was a theoretical physicist. His many contributions to physics include the special and general theories of relativity, the founding of relativistic cosmology, the first post-Newtonian expansion, explaining the perihelion advance of Mercury, prediction of the deflection of...
's more popular theory of
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
. It is an example of a
scalar-tensor theoryScalar-tensor theories are theories that include a scalar field as well as a tensor field to represent an interaction, especially the gravitational one.- Tensor fields and field theory :...
, a gravitational theory in which the gravitational interaction is mediated by a
scalar fieldIn mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity...
as well as the
tensor fieldIn mathematics, physics and engineering, a tensor field assigns a tensor to each point of a mathematical space . Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical...
of general relativity. The
gravitational constantThe gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
G is not presumed to be constant but instead 1/G is replaced by a scalar field which can vary from place to place and with time.
The theory was developed in 1961 by
Robert H. DickeRobert Henry Dicke was an American physicist, who made important contributions to the fields of astrophysics, atomic physics, cosmology and gravity.-Biography:...
and
Carl H. BransCarl Henry Brans is an American mathematical physicist best known for his research into the theoretical underpinnings of gravitation elucidated in his most widely publicized work, the Brans–Dicke theory....
building upon, among others, the earlier 1959 work of
Pascual JordanPascual Jordan was a theoretical and mathematical physicist who made significant contributions to quantum mechanics and quantum field theory. He contributed much to the mathematical form of matrix mechanics, and developed canonical anticommutation relations for fermions...
.
At present, both Brans–Dicke theory and general relativity are generally held to be in agreement with observation, although the experiments of the
golden age of general relativityThe Golden Age of General Relativity is the period roughly from 1960 to 1975 during which the study of general relativity, which had previously been regarded as something of a curiosity, entered the mainstream of theoretical physics...
have considerably constrained the allowed parameters of Brans-Dicke theory.
In
theoretical physicsTheoretical physics is a branch of physics which employs mathematical models and abstractions of physics in an attempt to explain natural phenomena. Its central core is mathematical physics,[Sometimes mathematical physics and theoretical physics are used synonymously to refer to the...]
, the
Brans–Dicke theory of gravitation (sometimes called the
Jordan–Brans–Dicke theory) is a theoretical framework to explain
gravitationGravitation is a natural phenomenon by which objects with mass attract one another. In everyday life, gravitation is most commonly thought of as the agency which lends weight to objects with mass. Gravitation causes dispersed matter to coalesce, thus accounting for the existence of the Earth, the...
. It is a well-known competitor of
EinsteinAlbert Einstein was a theoretical physicist. His many contributions to physics include the special and general theories of relativity, the founding of relativistic cosmology, the first post-Newtonian expansion, explaining the perihelion advance of Mercury, prediction of the deflection of...
's more popular theory of
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
. It is an example of a
scalar-tensor theoryScalar-tensor theories are theories that include a scalar field as well as a tensor field to represent an interaction, especially the gravitational one.- Tensor fields and field theory :...
, a gravitational theory in which the gravitational interaction is mediated by a
scalar fieldIn mathematics and physics, a scalar field associates a scalar value to every point in a space. The scalar may either be a mathematical number, or a physical quantity...
as well as the
tensor fieldIn mathematics, physics and engineering, a tensor field assigns a tensor to each point of a mathematical space . Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical...
of general relativity. The
gravitational constantThe gravitational constant, denoted G, is an empirical physical constant involved in the calculation of the gravitational attraction between objects with mass. It appears in Newton's law of universal gravitation and in Einstein's theory of general relativity. It is also known as the universal...
G is not presumed to be constant but instead 1/G is replaced by a scalar field which can vary from place to place and with time.
The theory was developed in 1961 by
Robert H. DickeRobert Henry Dicke was an American physicist, who made important contributions to the fields of astrophysics, atomic physics, cosmology and gravity.-Biography:...
and
Carl H. BransCarl Henry Brans is an American mathematical physicist best known for his research into the theoretical underpinnings of gravitation elucidated in his most widely publicized work, the Brans–Dicke theory....
building upon, among others, the earlier 1959 work of
Pascual JordanPascual Jordan was a theoretical and mathematical physicist who made significant contributions to quantum mechanics and quantum field theory. He contributed much to the mathematical form of matrix mechanics, and developed canonical anticommutation relations for fermions...
.
At present, both Brans–Dicke theory and general relativity are generally held to be in agreement with observation, although the experiments of the
golden age of general relativityThe Golden Age of General Relativity is the period roughly from 1960 to 1975 during which the study of general relativity, which had previously been regarded as something of a curiosity, entered the mainstream of theoretical physics...
have considerably constrained the allowed parameters of Brans-Dicke theory. Brans-Dicke theory represents a minority viewpoint in physics.
Comparison with general relativity
Both Brans–Dicke theory and general relativity are examples of a class of
relativisticThe theory of relativity, or simply relativity, generally refers specifically to two theories of Albert Einstein: special relativity and general relativity...
classical field theories of
gravitationGravitation is a natural phenomenon by which objects with mass attract one another. In everyday life, gravitation is most commonly thought of as the agency which lends weight to objects with mass. Gravitation causes dispersed matter to coalesce, thus accounting for the existence of the Earth, the...
, called
metric theories. In these theories,
spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions...
is equipped with a
metric tensorIn the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
, , and the gravitational field is represented (in whole or in part) by the
Riemann curvature tensorIn the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds...
, which is determined by the metric tensor.
All metric theories satisfy the
Einstein equivalence principleIn the physics of general relativity, the equivalence principle refers to several related concepts dealing with the equivalence of gravitational and inertial mass, and to Albert Einstein's assertion that the gravitational "force" as experienced locally while standing on a massive body is actually...
, which in modern geometric language states that in a very small region (too small to exhibit measurable
curvatureIn mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line, but this is defined in different ways depending on the context...
effects), all the laws of physics known in
special relativitySpecial relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies"...
are valid in
local Lorentz frames. This implies in turn that metric theories all exhibit the
gravitational redshiftIn physics, light or other forms of electromagnetic radiation of a certain wavelength originating from a source placed in a region of stronger gravitational field will be found to be of longer wavelength when received by an observer in a region of weaker gravitational field...
effect.
As in general relativity, the source of the gravitational field is considered to be the
stress-energy tensorThe stress-energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...
or
matter tensor. However, the way in which the immediate presence of mass-energy in some region affects the gravitational field in that region differs from general relativity. So does the way in which spacetime curvature affects the motion of matter. In the Brans–Dicke theory, in addition to the metric, which is a
rank two tensor field, there is a
scalar field, , which has the physical effect of changing the
effective gravitational constant from place to place. (This feature was actually a key desideratum of Dicke and Brans; see the paper by Brans cited below, which sketches the origins of the theory.)
The field equations of Brans–Dicke theory contain a
parameterIn mathematics, statistics, and the mathematical sciences, a parameter is a quantity that serves to relate functions and variables using a common variable when such a relationship would be difficult to explicate with an equation...
, , called the
Brans–Dicke coupling constant. This is a true dimensionless
constantIn mathematics, a constant is a non-varying value, i.e. completely fixed or fixed in the context of use. The term usually occurs in opposition to variable, which is a symbol that stands for a value that may vary....
which must be chosen once and for all. However, it can be chosen to fit observations. Such parameters are often called
tuneable parameters. In addition, the present ambient value of the effective gravitational constant must be chosen as a boundary condition. General relativity contains no dimensionless parameters whatsoever, and therefore is easier to falsify than Brans–Dicke theory. Theories with tuneable parameters are sometimes deprecated on the principle that, of two theories which both agree with observation, the more parsimonious is preferable. On the other hand, it seems as though they are a necessary feature of some theories, such as the weak mixing angle of the
Standard ModelThe Standard Model of particle physics is a theory of three of the four known fundamental interactions and the elementary particles that take part in these interactions. These particles make up all visible matter in the universe...
.
Brans–Dicke theory is "less stringent" than general relativity in another sense: it admits more solutions. In particular, exact vacuum solutions to the Einstein field equation of general relativity, augmented by the trivial scalar field , become exact vacuum solutions in Brans–Dicke theory, but some spacetimes which are
not vacuum solutions to the Einstein field equation become, with the appropriate choice of scalar field, vacuum solutions of Brans–Dicke theory. Similarly, an important class of spacetimes, the pp-wave metrics, are also exact
null dust solutionA null dust solution is a Lorentzian manifold in which the Einstein tensor is null. Such a spacetime can be interpreted as an exact solution of Einstein's field equation, in which the only mass-energy present in the spacetime is due to some kind of massless radiation.-Mathematical definition:The...
s of both general relativity and Brans–Dicke theory, but here too, Brans–Dicke theory allows additional
wave solutions having geometries which are incompatible with general relativity.
Like general relativity, Brans–Dicke theory predicts
light deflectionA gravitational lens is formed when the light from a very distant, bright source is "bent" around a massive object between the source object and the observer...
and the
precessionPrecession refers to a change in the orientation of the rotation axis of a rotating body. It can be defined as a change in direction of the rotation axis in which the second Euler angle is constant...
of
periheliaIn celestial mechanics, an apsis, plural apsides is the point of greatest or least distance of the elliptical orbit of an object from its center of attraction, which is generally the center of mass of the system....
of planets orbiting the Sun. However, the precise formulas which govern these effects, according to Brans–Dicke theory, depend upon the value of the coupling constant . This means that it is possible to set an observational lower bound on the possible value of from observations of the solar system and other gravitational systems. The value of consistent with experiment has risen with time. In 1973 was consistent with known data. By 1981 was consistent with known data. In 2003 evidence – derived from the Cassini–Huygens experiment – shows that the value of must exceed 40,000.
It is often said that Brans–Dicke theory, but not general relativity, satisfies
Mach's principleIn theoretical physics, particularly in discussions of gravitation theories, Mach's principle is the name given by Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach....
. However, some authors have argued that these claims are wrong (especially when one recognizes that there is no complete agreement upon precisely what "Mach's principle" really means). It is also often taught that general relativity is obtained from the Brans–Dicke theory in the limit . But Faraoni (see references) claims that this is naïve. Some have argued that only general relativity satisfies the strong
equivalence principleIn the physics of general relativity, the equivalence principle refers to several related concepts dealing with the equivalence of gravitational and inertial mass, and to Albert Einstein's assertion that the gravitational "force" as experienced locally while standing on a massive body is actually...
.
The field equations
The field equations of the Brans/Dicke theory are
where
- is the dimensionless Dicke coupling constant,
- is the metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
,
- is the Einstein tensor
In differential geometry, the Einstein tensor , named after Albert Einstein, is used to express the curvature of a Riemannian manifold...
, a kind of average curvature,
- is the Ricci tensor
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space...
, a kind of traceIn linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal of A, i.e.,...
of the curvature tensor,
- is the Ricci scalar
In Riemannian geometry, the scalar curvature is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point...
, the trace of the Ricci tensor,
- is the stress-energy tensor
The stress-energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...
,
- is the trace of ,
- is the scalar field,
- is the Laplace–Beltrami operator or covariant wave operator,
The first equation says that the
trace of the stress-energy tensor acts as the source for the scalar field . Since electromagnetic fields contribute only a traceless term to the stress-energy tensor,
this implies that in a region of spacetime containing only an electromagnetic field (plus the gravitational field), the right hand side vanishes, and obeys the (curved spacetime)
wave equationThe wave equation is an important second-order linear partial differential equation of waves, such as sound waves, light waves and water waves. It arises in fields such as acoustics, electromagnetics, and fluid dynamics...
. Therefore changes in propagate through
electrovacuum regions; in this sense, we say that is a
long-range field.
The second equation describes how the stress-energy tensor and scalar field together affect spacetime curvature. The left hand side, the
Einstein tensorIn differential geometry, the Einstein tensor , named after Albert Einstein, is used to express the curvature of a Riemannian manifold...
, can be thought of as a kind of average curvature. It is a matter of pure mathematics that, in any metric theory, the Riemann tensor can always be written as the sum of the Weyl curvature (or
conformal curvature tensor) plus a piece constructed from the Einstein tensor.
For comparison, the field equation of general relativity is simply
This means that in general relativity, the Einstein curvature at some event is entirely determined by the stress-energy tensor at that event; the other piece, the Weyl curvature, is the part of the gravitational field which can propagate as a gravitational wave across a vacuum region. But in the Brans–Dicke theory, the Einstein tensor is determined partly by the immediate presence of mass-energy and momentum, and partly by the long-range scalar field .
The
vacuum field equations of both theories are obtained when the stress-energy tensor vanishes. This models situations in which no non-gravitational fields are present.
The action principle
The following
LagrangianThe Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics known as Lagrangian mechanics. In classical mechanics, the...
contains the complete description of the Brans/Dicke theory:
where
- is the determinant of the metric,
- is the four-dimensional volume form
In mathematics, a volume form on a differentiable manifold is a nowhere vanishing differential form of top degree. Thus on a manifold M of dimension n, a volume form is an n-form, a section of the line bundle Ωn = Λn, that is nowhere equal to zero...
,
- is the matter term or matter Lagrangian.
The matter term includes the contribution of ordinary matter (e.g. gaseous matter) and also electromagnetic fields. In a vacuum region, the matter term vanishes identically; the remaining term is the
gravitational term. To obtain the vacuum field equations, we must vary the gravitational term in the Lagrangian with respect to the metric ; this gives the second field equation above. When we vary with respect to the scalar field , we obtain the first field equation.
Note that, unlike for the General Relativity field equations, the term does not vanish, as the result is not a total derivative. It can be shown that
To prove this result, use
By evaluating the s in Riemann normal coordinates, 6 individual terms vanish. 6 further terms combine when manipulated using
Stokes' theoremIn differential geometry, Stokes' theorem is a statement about the integration of differential forms on manifolds, which generalizes several theorems from vector calculus. William Thomson first discovered the result and communicated it to George Stokes in July 1850...
to provide the desired .
For comparison, the Lagrangian defining general relativity is
Varying the gravitational term with respect to gives the vacuum Einstein field equation.
In both theories, the full field equations can be obtained by variations of the full Lagrangian.
See also
- Classical theories of gravitation
In theoretical physics, the current Gold Standard Theory of Gravitation is the general theory of relativity. This is a classical, relativistic field theory of gravitation...
- Self-creation cosmology
Self-creation cosmology theories are gravitational theories in which the mass of the universe is created out of its self-contained gravitational and scalar fields, as opposed to the theory of continuous creation cosmology or the steady state theory which depend on an extra 'creation'...
, a modification of the Brans–Dicke theory that allows a non-minimally connected scalar field to interact with matter.
- Mach's principle
In theoretical physics, particularly in discussions of gravitation theories, Mach's principle is the name given by Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach....
- General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...
External links
See
Box 39.1.
Chapter 8: "The Rise and Fall of the Brans–Dicke Theory".
See also the
eprint version on the
ArXivThe arXiv is an archive for electronic preprints of scientific papers in the fields of mathematics, physics, computer science, quantitative biology and statistics which can be accessed via the world wide web. In many fields of mathematics and physics, almost all scientific papers are placed on...
.