Brans–Dicke Theory
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In physics, the Brans–Dicke theory of gravitation (sometimes called the Jordan–Brans–Dicke theory) is a competitor to Einstein's
general theory of relativity General relativity, also known as the general theory of relativity, and as 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 physi ...
. It is an example of a scalar–tensor theory, a gravitational theory in which the gravitational interaction is mediated by a
scalar field In mathematics and physics, a scalar field is a function associating a single number to each point in a region of space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical ...
as well as the
tensor field In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry, ...
of general relativity. The
gravitational constant The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It ...
G is not presumed to be constant but instead 1 / G is replaced by a
scalar field In mathematics and physics, a scalar field is a function associating a single number to each point in a region of space – possibly physical space. The scalar may either be a pure mathematical number ( dimensionless) or a scalar physical ...
\phi which can vary from place to place and with time. The theory was developed in 1961 by Robert H. Dicke and Carl H. Brans building upon, among others, the earlier 1959 work of
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 matri ...
. At present, both Brans–Dicke theory and general relativity are generally held to be in agreement with observation. 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 relativistic classical field theories of
gravitation In physics, gravity (), also known as gravitation or a gravitational interaction, is a fundamental interaction, a mutual attraction between all massive particles. On Earth, gravity takes a slightly different meaning: the observed force b ...
, called '' metric theories''. In these theories,
spacetime In physics, spacetime, also called the space-time continuum, is a mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum. Spacetime diagrams are useful in visualiz ...
is equipped with a
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
, g_, and the gravitational field is represented (in whole or in part) by the
Riemann curvature tensor Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to mathematical analysis, analysis, number theory, and differential geometry. In the field of real analysis, he is mos ...
R_, which is determined by the metric tensor. All metric theories satisfy the Einstein equivalence principle, which in modern geometric language states that in a very small region (too small to exhibit measurable
curvature In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being a straight line or by which a surface deviates from being a plane. If a curve or su ...
effects), all the laws of physics known in
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
are valid in ''local Lorentz frames''. This implies in turn that metric theories all exhibit the
gravitational redshift In physics and general relativity, gravitational redshift (known as Einstein shift in older literature) is the phenomenon that electromagnetic waves or photons travelling out of a gravitational well lose energy. This loss of energy correspo ...
effect. As in general relativity, the source of the gravitational field is considered to be 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 ...
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'', \phi, 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
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
, \omega, called the ''Brans–Dicke coupling constant''. This is a true
dimensionless Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that align with another sy ...
constant which must be chosen once and for all. However, it can be chosen to fit observations. Such parameters are often called ''tunable parameters''. In addition, the present ambient value of the effective gravitational constant must be chosen as a
boundary condition In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...
. General relativity contains no dimensionless parameters whatsoever, and therefore is easier to falsify (show whether false) than Brans–Dicke theory. Theories with tunable parameters are sometimes deprecated on the principle that, of two theories which both agree with observation, the more
parsimonious In philosophy, Occam's razor (also spelled Ockham's razor or Ocham's razor; ) is the problem-solving principle that recommends searching for explanations constructed with the smallest possible set of elements. It is also known as the principle o ...
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 Model The Standard Model of particle physics is the Scientific theory, theory describing three of the four known fundamental forces (electromagnetism, electromagnetic, weak interaction, weak and strong interactions – excluding gravity) in the unive ...
. 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 In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Albert Einstein in 1915 in the ...
of general relativity, augmented by the trivial scalar field \phi=1, 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 solutions 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 deflection and the
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 o ...
of perihelia 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 \omega. This means that it is possible to set an observational lower bound on the possible value of \omega from observations of the
Solar System The Solar SystemCapitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Sola ...
and other gravitational systems. The value of \omega consistent with experiment has risen with time. In 1973 \omega > 5 was consistent with known data. By 1981 \omega > 30 was consistent with known data. In 2003 evidence – derived from the ''
Cassini–Huygens ''Cassini–Huygens'' ( ), commonly called ''Cassini'', was a space research, space-research mission by NASA, the European Space Agency (ESA), and the Italian Space Agency (ASI) to send a space probe to study the planet Saturn and its system, i ...
'' experiment – shows that the value of \omega must exceed 40,000. It is also often taught that general relativity is obtained from the Brans–Dicke theory in the limit \omega \rightarrow \infty. But Faraoni claims that this breaks down when the trace of the stress-energy momentum vanishes, i.e. T^_ = 0, an example of which is the Campanelli- Lousto wormhole solution. Some have argued that only general relativity satisfies the strong
equivalence principle The equivalence principle is the hypothesis that the observed equivalence of gravitational and inertial mass is a consequence of nature. The weak form, known for centuries, relates to masses of any composition in free fall taking the same t ...
.


The field equations

The field equations of the Brans–Dicke theory are :G_ = \frac T_ + \frac \left(\partial_a\phi \partial_b\phi - \frac g_ \partial_c\phi\partial^c\phi\right) + \frac(\nabla_a\nabla_b\phi - g_ \Box\phi) - g_\frac, :\Box\phi = \fracT + \frac where :\omega is the dimensionless Dicke coupling constant; :g_ is the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
; :G_ = R_ - \tfrac R g_ is the
Einstein tensor In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field e ...
, a kind of average curvature; :R_ = R^m_ is the
Ricci tensor In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure ...
, a kind of trace of the curvature tensor; :R = R^m_ is the Ricci scalar, the trace of the Ricci tensor; :T_ is 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 ...
; :T = T_a^a is the trace of the stress–energy tensor; :\phi is the scalar field; :V(\phi) is the scalar potential; :V'(\phi) is the derivative of the scalar potential with respect to \phi; :\Box is the
Laplace–Beltrami operator In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named aft ...
or covariant wave operator, \Box \phi = \phi^_. The first equation describes how the stress–energy tensor and scalar field \phi together affect spacetime curvature. The left-hand side, the
Einstein tensor In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field e ...
, 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'') and a piece constructed from the Einstein tensor. The second equation says that the trace of the stress–energy tensor acts as the source for the scalar field \phi. 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 \phi obeys the (curved spacetime)
wave equation The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light ...
. Therefore, changes in \phi propagate through ''electrovacuum'' regions; in this sense, we say that \phi is a ''long-range field''. For comparison, the field equation of general relativity is simply :G_ = 8 \pi T_. 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 \phi. 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 Lagrangian contains the complete description of the Brans–Dicke theory:Georgios Kofinas, Minas Tsoukalas
On the action of the complete Brans-Dicke theories
on arXiv:1512.04786 r-qc 28. Nov. 2016, DOI:10.1140/epjc/s10052-016-4505-y, equation (2.9) on page 2. Some authors use :S_M = \int d^4x\sqrt \;\mathcal_\mathrm for the matter term, see Brans-Dicke-Theorie: Definition (German).
:S = \frac\int d^4x\sqrt \left(\phi R - \frac\partial_a\phi\partial^a\phi\right) + \int d^4 x \sqrt \,\mathcal_\mathrm, where g is the determinant of the metric, \sqrt \, d^4 x is the four-dimensional
volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of t ...
, and \mathcal_\mathrm is the ''matter term'', or ''matter Lagrangian density''. 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 g_; this gives the first field equation above. When we vary with respect to the scalar field \phi, we obtain the second field equation. Note that, unlike for the General Relativity field equations, the \delta R_/\delta g_ term does not vanish, as the result is not a total derivative. It can be shown that :\frac = \phi R_ + g_g^\phi_ - \phi_. To prove this result, use :\delta (\phi R) = R \delta \phi + \phi R_ \delta g^ + \phi \nabla_s (g^ \delta\Gamma^s_ - g^\delta\Gamma^r_ ). By evaluating the \delta\Gammas in Riemann normal coordinates, 6 individual terms vanish. 6 further terms combine when manipulated using
Stokes' theorem Stokes' theorem, also known as the Kelvin–Stokes theorem after Lord Kelvin and George Stokes, the fundamental theorem for curls, or simply the curl theorem, is a theorem in vector calculus on \R^3. Given a vector field, the theorem relates th ...
to provide the desired (g_g^\phi_ - \phi_)\delta g^. For comparison, the Lagrangian defining general relativity is :S = \int d^4x \sqrt \, \left(\frac + \mathcal_\mathrm\right). Varying the gravitational term with respect to g_ 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 *
Dilaton In particle physics, the hypothetical dilaton is a particle of a scalar field \varphi that appears in theories with extra dimensions when the volume of the compactified dimensions varies. It appears as a radion in Kaluza–Klein theory's compa ...
*
General relativity General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
*
Mach's principle In theoretical physics, particularly in discussions of gravitation theories, Mach's principle (or Mach's conjecture) is the name given by Albert Einstein to an imprecise hypothesis often credited to the physicist and philosopher Ernst Mach. The ...
* Scientific importance of GW170817


Notes


References

* * * See ''Box 39.1''. * *


External links


Scholarpedia article on the subject
by Carl H. Brans * {{DEFAULTSORT:Brans-Dicke Theory Theories of gravity