constancy of the Newtonian gravitational constant has been measured, implying that the scalar-field background is pretty stable over time. Such a stability is not theoretically generally expected but can be theoretically explained by several mechanisms. The first post-Newtonian approximation of the theory 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 (i.e., ), the metric takes the following form: with where 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 is given by where is the (true) constant that appears in the coupling constant defined previously. Observational constraints on the theory Current observations indicate that , which means that . Although explaining such a value in the context of the original Brans–Dicke theory is impossible, Damour and Nordtvedt found that the field equations of the general theory often lead to an evolution of the function toward infinity during the evolution of the universe. Hence, according to them, the current high value of the function could be a simple consequence of the evolution of the universe. The best current constraint on the post-Newtonian parameter comes from Mercury's perihelion shift and is . 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. Higher-dimensional relativity and scalar–tensor theories After the postulation of the General Relativity of Einstein and Hilbert, Theodor Kaluza and Oskar Klein proposed in 1917 a generalization in a 5-dimensional manifold: Kaluza–Klein theory. This theory possesses a 5-dimensional metric (with a compactified and constant 5th metric component, dependent on the gauge potential) and unifies gravitation and electromagnetism, i.e. there is a geometrization of electrodynamics. 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,