View source for Brans–Dicke theory

{{Use American English|date = March 2019}}
{{Short description|Proposed theory of gravitation}}
In physics, the '''Brans–Dicke theory of gravitation''' (sometimes called the '''Jordan–Brans–Dicke theory''') is a competitor to Einstein's [[General relativity|general theory of relativity]]. It is an example of a [[scalar–tensor theory]], a gravitational theory in which the gravitational interaction is mediated by a [[scalar field]] as well as the [[tensor field]] of general relativity. The [[gravitational constant]] <math>G</math> is not presumed to be constant but instead <math>1 / G</math> is replaced by a [[scalar field]] <math>\phi</math> which can vary from place to place and with time.

The theory was developed in 1961 by [[Robert H. Dicke]] and [[Carl H. Brans]]<ref>{{cite journal |last1=Brans |first1=C. H. |last2=Dicke |first2=R. H. |date= November 1, 1961 |title=Mach's Principle and a Relativistic Theory of Gravitation |journal=[[Physical Review]] |doi=10.1103/PhysRev.124.925 |bibcode=1961PhRv..124..925B |volume=124 |issue=3 |pages=925–935}}</ref> building upon, among others, the earlier 1959 work of [[Pascual Jordan]]. 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 [[theory of relativity|relativistic]] [[classical field theories]] of [[gravitation]], called ''[[metric theories]]''. In these theories, [[spacetime]] is equipped with a [[metric tensor]], <math>g_{ab}</math>, and the gravitational field is represented (in whole or in part) by the [[Riemann curvature tensor]] <math>R_{abcd}</math>, which is determined by the metric tensor.

All metric theories satisfy the [[equivalence principle|Einstein equivalence principle]], which in modern geometric language states that in a very small region (too small to exhibit measurable [[curvature]] effects), all the laws of physics known in [[special relativity]] are valid in ''local Lorentz frames''. This implies in turn that metric theories all exhibit the [[gravitational redshift]] effect.

As in general relativity, the source of the gravitational field is considered to be the [[stress–energy tensor]] 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'', <math>\phi</math>, which has the physical effect of changing the ''effective gravitational constant'' from place to place.  (This feature was actually a key [[Wiktionary:desideratum|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]], <math>\omega</math>, called the ''Brans–Dicke coupling constant''. This is a true [[dimensionless]] [[Constant (mathematics)|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]]. General relativity contains no dimensionless parameters whatsoever, and therefore is easier to [[falsifiability|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 [[principle of parsimony|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 Model]].

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 <math>\phi=1</math>, 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-waves|pp-wave metrics]], are also exact [[null dust solution]]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 [[gravitational lens|light deflection]] and the [[precession]] of [[apsis|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 <math>\omega</math>. This means that it is possible to set an observational lower bound on the possible value of <math>\omega</math> from observations of the solar system and other gravitational systems. The value of <math>\omega </math> consistent with experiment has risen with time. In 1973 <math>\omega > 5</math> was consistent with known data. By 1981 <math>\omega >  30</math> was consistent with known data. In 2003 evidence – derived from the [[Cassini–Huygens]] experiment – shows that the value of <math>\omega</math> must exceed 40,000.

It is also often taught<ref>{{cite book|last= Weinberg|first= Steven|date= 1971|title= Gravitation and cosmology: principles and applications of the general theory of relativity|url= https://archive.org/details/gravitationcosmo00stev_0/page/160|publisher= Wiley|page= [https://archive.org/details/gravitationcosmo00stev_0/page/160 160]|isbn= 0471925675|url-access= registration}}</ref> that general relativity is obtained from the Brans–Dicke theory in the limit <math>\omega \rightarrow \infty</math>. But Faraoni<ref>{{cite journal |last=Faroni |first=Valerio |year=1999 |title=Illusions of general relativity in Brans-Dicke gravity |journal=Phys. Rev. |arxiv=gr-qc/9902083 |bibcode=1999PhRvD..59h4021F |doi=10.1103/PhysRevD.59.084021 |volume=D59 |issue=8 |pages=084021|s2cid=7558104 }}</ref> claims that this breaks down when the trace of the stress-energy momentum vanishes, i.e. <math>T^{\mu}_{\mu} = 0</math>, an example of which is the [[Manuela Campanelli (scientist)|Campanelli]]-[[Carlos Lousto|Lousto]] wormhole solution.<ref>M. Campanelli, C.O. Lousto, Int. J. Mod. Phys. D 02, 451 (1993) https://doi.org/10.1142/S0218271893000325</ref> Some have argued{{who|date=November 2014}} that only general relativity satisfies the strong [[equivalence principle]].

== The field equations ==
The field equations of the Brans–Dicke theory are

:<math>G_{ab} = \frac{8\pi}{\phi} T_{ab} + \frac{\omega}{\phi^2}
\left(\partial_a\phi \partial_b\phi - \frac{1}{2} g_{ab} \partial_c\phi\partial^c\phi\right)
+ \frac{1}{\phi}(\nabla_a\nabla_b\phi - g_{ab} \Box\phi) - g_{ab}\frac{V(\phi)}{2\phi},</math>

:<math>\Box\phi = \frac{8\pi}{3+2\omega}T + \frac{2V(\phi) - \phi V'(\phi)}{3+2\omega}</math>

where
:<math>\omega</math> is the dimensionless Dicke coupling constant;
:<math>g_{ab}</math> is the [[metric tensor]];
:<math>G_{ab} = R_{ab} - \tfrac{1}{2} R g_{ab}</math> is the [[Einstein tensor]], a kind of average curvature;
:<math>R_{ab} = R^m{}_{a m b}</math> is the [[Ricci curvature|Ricci tensor]], a kind of [[trace (linear algebra)|trace]] of the curvature tensor;
:<math>R = R^m{}_{m}</math> is the [[Scalar curvature|Ricci scalar]], the trace of the Ricci tensor;
:<math>T_{ab}</math> is the [[stress–energy tensor]];
:<math>T = T_a^a</math> is the trace of the stress–energy tensor;
:<math>\phi</math> is the scalar field;
:<math>V(\phi)</math> is the scalar potential;
:<math>V'(\phi)</math> is the derivative of the scalar potential with respect to <math>\phi</math>;
:<math>\Box</math> is the [[Laplace–Beltrami operator]] or covariant wave operator, <math>\Box \phi = \phi^{;a}{}_{;a}</math>.

The first equation describes how the stress–energy tensor and scalar field <math>\phi</math> together affect spacetime curvature. The left-hand side, the [[Einstein tensor]], 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 <math>\phi</math>. 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 <math>\phi</math> obeys the (curved spacetime) [[wave equation]]. Therefore, changes in <math>\phi</math> propagate through ''electrovacuum'' regions; in this sense, we say that <math>\phi</math> is a ''long-range field''.

For comparison, the field equation of general relativity is simply

:<math>G_{ab} = 8 \pi T_{ab}.</math>

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 <math>\phi</math>.

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 (field theory)|Lagrangian]] contains the complete description of the Brans–Dicke theory:<ref>Georgios Kofinas, Minas Tsoukalas: [https://arxiv.org/abs/1512.04786 On the action of the complete Brans-Dicke theories], on arXiv:1512.04786 [gr-qc], 28. Nov. 2016, [[DOI:10.1140/epjc/s10052-016-4505-y]], equation (2.9) on page 2. Some authors use
:<math>S_M = \int d^4x\sqrt{-g} \;\mathcal{L}_\mathrm{M}</math>
for the matter term, see [[:de:Brans-Dicke-Theorie#Definition|Brans-Dicke-Theorie: Definition]] (German).</ref>

:<math>S = \frac{1}{16 \pi}\int d^4x\sqrt{-g}
\left(\phi R - \frac{\omega}{\phi}\partial_a\phi\partial^a\phi\right) + \int d^4 x \sqrt{-g} \,\mathcal{L}_\mathrm{M},</math>

where <math>g</math> is the determinant of the metric, <math>\sqrt{-g} \, d^4 x</math> is the four-dimensional [[volume form]], and <math>\mathcal{L}_\mathrm{M}</math> 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 <math>g_{ab}</math>; this gives the first field equation above. When we vary with respect to the scalar field <math>\phi</math>, we obtain the second field equation.

Note that, unlike for the General Relativity field equations, the <math>\delta R_{ab}/\delta g_{cd}</math> term does not vanish, as the result is not a total derivative. It can be shown that

:<math>\frac{\delta(\phi R)}{\delta g^{ab}} = \phi R_{ab} + g_{ab}g^{cd}\phi_{;c;d} - \phi_{;a;b}.</math>

To prove this result, use

:<math>\delta (\phi R) = R \delta \phi + \phi R_{mn} \delta g^{mn} + \phi \nabla_s (g^{mn} \delta\Gamma^s_{nm} - g^{ms}\delta\Gamma^r_{rm} ).</math>

By evaluating the <math>\delta\Gamma</math>s in Riemann normal coordinates, 6 individual terms vanish. 6 further terms combine when manipulated using [[Stokes' theorem]] to provide the desired <math>(g_{ab}g^{cd}\phi_{;c;d} - \phi_{;a;b})\delta g^{ab}</math>.

For comparison, the Lagrangian defining general relativity is

:<math>S = \int d^4x \sqrt{-g} \, \left(\frac{R}{16\pi G} + \mathcal{L}_\mathrm{M}\right).</math>

Varying the gravitational term with respect to <math>g_{ab}</math> gives the vacuum Einstein field equation.

In both theories, the full field equations can be obtained by variations of the full Lagrangian.

== See also ==
{{Portal|Physics}}
* [[Classical theories of gravitation]]
* [[Dilaton]]
* [[General relativity]]
* [[Mach's principle]]
* [[GW170817#Scientific importance|Scientific importance of GW170817]]

== Notes ==
{{Reflist}}

== References ==
* {{cite journal |last=Bergmann |first=Peter G. |date=May 1968 |title=Comments on the Scalar-Tensor Theory |journal=[[International Journal of Theoretical Physics|Int. J. Theor. Phys.]] |issn=0020-7748 |doi=10.1007/BF00668828 |bibcode=1968IJTP....1...25B|volume=1 |number=1 |pages=25–36|s2cid=119985328 }}
* {{cite journal |last=Wagoner |first=Robert V. |date=June 1970 |title=Scalar-Tensor Theory and Gravitational Waves |journal=Phys. Rev. D |publisher=[[American Physical Society]] |doi=10.1103/PhysRevD.1.3209 |bibcode=1970PhRvD...1.3209W |volume=1 |issue=12 |pages=3209–3216}}
* {{cite book |last1=Misner |first1=Charles W. |author-link1=Charles W. Misner |last2=Thorne |first2=Kip S. |author-link2=Kip Thorne |last3=Wheeler |first3=John Archibald |author-link3=John Archibald Wheeler |year=1973 |title=[[Gravitation (book)|Gravitation]] |publisher=[[W. H. Freeman]] |location=San Francisco |isbn=0-7167-0344-0}} See ''Box 39.1''.
* {{cite book |last=Will |first=Clifford M. |author-link=Clifford Martin Will |year=1986 |title=Was Einstein Right?: Putting General Relativity to the Test |chapter=Chapter 8: The Rise and Fall of the Brans–Dicke Theory |publisher=[[Basic Books]] |location=NY |isbn=0-19-282203-9}}
* {{cite book |last=Faraoni |first=Valerio |year=2004 |title=Cosmology in Scalar-Tensor Gravity |publisher=[[Kluwer Academic]] |location=Dordrecht, The Netherlands |isbn=1-4020-1988-2}}

== External links ==
{{sisterlinks|b=Modern Physics/Brans-Dicke theory}}
* [http://www.scholarpedia.org/article/Jordan-Brans-Dicke_Theory Scholarpedia article on the subject] by [[Carl H. Brans]]
* {{cite arXiv |last=Brans |first=Carl H. |title=The roots of scalar-tensor theory: an approximate history |year=2005 |eprint=gr-qc/0506063}}

{{theories of gravitation}}
{{Relativity}}

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[[Category:Theories of gravity]]