indices). Another possibility to explain gravitation in this context is by using both tensor (of degree n>1) and scalar fields, i.e. so that gravitation is given neither solely through a scalar field nor solely through a metric. These are scalar–tensor theories of gravitation. 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. If in the Lagrangian the curvature (or a quantity related to it) is multiplied with a square scalar field, field theories of scalar–tensor theories of gravitation are obtained. In them, the gravitational constant of Newton is no longer a real constant but a quantity dependent of the scalar field. Mathematical formulation An action of such a gravitational scalar–tensor theory can be written as follows: where is the metric determinant, is the Ricci scalar constructed from the metric , is a coupling constant with the dimensions , is the scalar-field potential, is the material Lagrangian and represents the non-gravitational fields. Here, the Brans–Dicke parameter has been generalized to a function. Although is often written as being , one has to keep in mind that the fundamental constant there, is not the constant of gravitation that can be measured with, for instance, Cavendish type experiments. Indeed, the empirical gravitational constant is generally no longer a constant in scalar–tensor theories, but a function of the scalar field . 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 The Newtonian approximation of the theory 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. In this approximation, and for a theory without potential, the metric writes with satisfying the following usual Poisson equation at the lowest order of the approximation: where is the density of the gravitational source and (the subscript 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 and therefore theoretically depends on time and location. However, no deviation from the