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 theory was developed in 1961 by Robert H. Dicke and Carl H. Brans[1] building upon, among others, the earlier 1959 work of Pascual Jordan.
Brans–Dicke theory represents a minority viewpoint in physics.
In these theories, spacetime is equipped with a metric tensor,
, and the gravitational field is represented (in whole or in part) by the Riemann curvature 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.
So does the way in which spacetime curvature affects the motion of matter.
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.
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.
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
This means that it is possible to set an observational lower bound on the possible value of
It is also often taught[2] that general relativity is obtained from the Brans–Dicke theory in the limit
But Faraoni[3] claims that this breaks down when the trace of the stress-energy momentum vanishes, i.e.
that only general relativity satisfies the strong equivalence principle.
The left-hand side, the Einstein tensor, can be thought of as a kind of average curvature.
The second equation says that the trace of the stress–energy tensor acts as the source for the scalar field
obeys the (curved spacetime) wave equation.
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.
The vacuum field equations of both theories are obtained when the stress–energy tensor vanishes.
To obtain the vacuum field equations, we must vary the gravitational term in the Lagrangian with respect to the metric
Note that, unlike for the General Relativity field equations, the
term does not vanish, as the result is not a total derivative.
s in Riemann normal coordinates, 6 individual terms vanish.
6 further terms combine when manipulated using Stokes' theorem to provide the desired
For comparison, the Lagrangian defining general relativity is Varying the gravitational term with respect to