The gravitational force is dependent of the distance r of the massive objects to each other (more exactly, their centre of mass).
Einstein's theory of gravity, the General Relativity (GR) is of another nature.
That curvature is defined mathematically by the so-called metric, which is a function of the total energy, including mass, in the area.
The derivative of the metric is a function that approximates the classical Newtonian force in most cases.
The metric is a tensorial quantity of degree 2 (it can be given as a 4x4 matrix, an object carrying 2 indices).
The field theoretical start of General Relativity is given through the Lagrange density.
It is a scalar and gauge invariant (look at gauge theories) quantity dependent on the curvature scalar R. This Lagrangian, following Hamilton's principle, leads to the field equations of Hilbert and Einstein.
there, is not the constant of gravitation that can be measured with, for instance, Cavendish type experiments.
The metric and scalar-field equations respectively write: and Also, the theory satisfies the following conservation equation, implying that test-particles follow space-time geodesics such as in general relativity: where
is the stress-energy tensor defined as Developing perturbatively the theory defined by the previous action around a Minkowskian background, and assuming non-relativistic gravitational sources, the first order gives the Newtonian approximation of the theory.
satisfying the following usual Poisson equation at the lowest order of the approximation: where
indicates that the corresponding value is taken at present cosmological time and location).
Therefore, the empirical gravitational constant is a function of the present value of the scalar-field background
[1] However, no deviation from the constancy of the Newtonian gravitational constant has been measured,[2] implying that the scalar-field background
[3] Developing the theory at the next level leads to the so-called first post-Newtonian order.
For a theory without potential and in a system of coordinates respecting the weak isotropy condition[4] (i.e.,
is a function depending on the coordinate gauge It corresponds to the remaining diffeomorphism degree of freedom that is not fixed by the weak isotropy condition.
The sources are defined as the so-called post-Newtonian parameters are and finally the empirical gravitational constant
Seven years of data from the NASA MESSENGER mission constraints the post-Newtonian parameter
[6] Both constraints show that while the theory is still a potential candidate to replace general relativity, the scalar field must be very weakly coupled in order to explain current observations.
Generalized scalar-tensor theories have also been proposed as explanation for the accelerated expansion of the universe but the measurement of the speed of gravity with the gravitational wave event GW170817 has ruled this out.
This theory was modified in 1955 by P. Jordan in his Projective Relativity theory, in which, following group-theoretical reasonings, Jordan took a functional 5th metric component that led to a variable gravitational constant G. In his original work, he introduced coupling parameters of the scalar field, to change energy conservation as well, according to the ideas of Dirac.
This theory becomes Einsteinian for high values for the parameter of the scalar field.
The coupling of a universal scalar field directly to the gravitational field gives rise to potentially observable effects for the motion of matter configurations to which gravitational energy contributes significantly.
This is known as the "Dicke–Nordtvedt" effect, which leads to possible violations of the Strong as well as the Weak Equivalence Principle for extended masses.
He proposed a Broken-Symmetric Theory of Gravitation, combining the idea of Brans and Dicke with the one of Symmetry Breakdown, which is essential within the Standard Model SM of elementary particles, where the so-called Symmetry Breakdown leads to mass generation (as a consequence of particles interacting with the Higgs field).
There, the massive scalar field couples to the masses, which are at the same time the source of the scalar Higgs field, which generates the mass of the elementary particles through Symmetry Breakdown.
Further, they are an option to explain dynamics usually given through the standard cold dark matter models, as well as MOND, Axions (from Breaking of a Symmetry, too), MACHOS,... A generic prediction of all string theory models is that the spin-2 graviton has a spin-0 partner called the dilaton.
However, the precise form of such a theory is not currently known because one does not have the mathematical tools in order to address the corresponding non-perturbative calculations.
Besides, the precise effective 4-dimensional form of the theory is also confronted to the so-called landscape issue.