Sub-Riemannian manifold

In theoretical physics, the Brans–Dicke theory of gravitation (sometimes called the Jordan–Brans–Dicke theory) is a theoretical framework to explain gravitation. It is a well-known competitor of Einstein's more popular theory of general 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 G is not presumed to be constant but instead 1/G is replaced by a scalar field $\phi$ 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.

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, called metric theories. In these theories, spacetime is equipped with a metric tensor, $g_{ab}$, and the gravitational field is represented (in whole or in part) by the Riemann curvature tensor $R_{abcd}$, 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 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, $\phi$, 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, $\omega$, called the Brans–Dicke coupling constant. This is a true dimensionless constant 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 (show whether false) 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 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 $\phi =1$, 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 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 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 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 (see references) claims that this breaks down when the trace of the stress-energy momentum vanishes, i.e. $T_{\mu }^{\mu }=0$ . Some have argued that only general relativity satisfies the strong equivalence principle.

The field equations

The field equations of the Brans/Dicke theory are

$\Box \phi ={\frac {8\pi }{3+2\omega }}T$

$G_{ab}={\frac {8\pi }{\phi }}T_{ab}+{\frac {\omega }{\phi ^{2}}}(\partial _{a}\phi \partial _{b}\phi -{\frac {1}{2}}g_{ab}\partial _{c}\phi \partial ^{c}\phi )+{\frac {1}{\phi }}(\nabla _{a}\nabla _{b}\phi -g_{ab}\Box \phi )$,

where

$\omega$ is the dimensionless Dicke coupling constant;

$g_{ab}$ is the metric tensor;

$G_{ab}=R_{ab}-{\tfrac {1}{2}}Rg_{ab}$ is the Einstein tensor, a kind of average curvature;

$R_{ab}={R^{m}}_{amb}$ is the Ricci tensor, a kind of trace of the curvature tensor;

$R={R^{m}}_{m}$ is the Ricci scalar, the trace of the Ricci tensor;

$T_{ab}$ is the stress-energy tensor;

$T=T_{a}^{a}$ is the trace of the stress-energy tensor;

$\phi$ is the scalar field; and

$\Box$ is the Laplace–Beltrami operator or covariant wave operator, $\Box \phi =\phi _{\;\;;a}^{;a}$.

The first 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. Therefore changes in $\phi$ propagate through electrovacuum regions; in this sense, we say that $\phi$ is a long-range field.

The second equation describes how the stress-energy tensor and scalar field $\phi$ 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) plus a piece constructed from the Einstein tensor.

For comparison, the field equation of general relativity is simply

$G_{ab}=8\pi T_{ab}.$

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:

$S=\int d^{4}x{\sqrt {-g}}\;\left({\frac {\phi R-\omega {\frac {\partial _{a}\phi \partial ^{a}\phi }{\phi }}}{16\pi }}+{\mathcal {L}}_{\mathrm {M} }\right)$

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

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

${\frac {\delta (\phi R)}{\delta g^{ab}}}=\phi R_{ab}+g_{ab}g^{cd}\phi _{;cd}-\phi _{;ab}$

To prove this result, use

$\delta (\phi R)=R\delta \phi +\phi R_{mn}\delta g^{mn}+\phi \nabla _{s}(g^{mn}\delta \Gamma _{nm}^{s}-g^{ms}\delta \Gamma _{rm}^{r})$

By evaluating the $\delta \Gamma$s in Riemann normal coordinates, 6 individual terms vanish. 6 further terms combine when manipulated using Stokes' theorem to provide the desired $(g_{ab}g^{cd}\phi _{;cd}-\phi _{;ab})\delta g^{ab}$.

For comparison, the Lagrangian defining general relativity is

$S=\int d^{4}x{\sqrt {-g}}\;\left({\frac {R}{16\pi G}}+{\mathcal {L}}_{\mathrm {M} }\right)$

Varying the gravitational term with respect to $g_{ab}$ 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
• Self-creation cosmology, a modification of the Brans–Dicke theory that allows a non-minimally connected scalar field to interact with matter.
• Mach's principle
• General relativity

References

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• Template:Cite arxiv
• The Brans-Dicke theory
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  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
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  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang