5 Theories With Varying Constants

As explained in the introduction, Dirac postulated that G varies as the inverse of the cosmic time. Such an hypothesis is indeed not a theory since the evolution of G with time is postulated and not derived from an equation of evolution12 consistent with the other field equations, that have to take into account that G is no more a constant (in particular in a Lagrangian formulation one needs to take into account that G is no more constant when varying.

The first implementation of Dirac’s phenomenological idea into a field-theory framework (i.e., modifying Einstein’s gravity and incorporating non-gravitational forces and matter) was proposed by Jordan [268]. He realized that the constants have to become dynamical fields and proposed the action

∫ [ ( )2 ] S = √ −-gd4x ϕη R − ξ ∇-ϕ- − ϕF 2 , (156 ) ϕ 2
η and ξ being two parameters. It follows that both G and the fine-structure constant have been promoted to the status of a dynamical field.

Fierz [195Jump To The Next Citation Point] realized that with such a Lagrangian, atomic spectra will be space-time-dependent, and he proposed to fix η to the value − 1 to prevent such a space-time dependence. This led to the definition of a one-parameter (ξ) class of scalar-tensor theories in which only G is assumed to be a dynamical field. This was then further explored by Brans and Dicke [67] (with the change of notation ξ → ω). In this Jordan–Fierz–Brans–Dicke theory the gravitational constant is replaced by a scalar field, which can vary both in space and time. It follows that, for cosmological solutions, −n G ∝ t with −1 n = 2 + 3ωBD ∕2. Thus, Einstein’s gravity is recovered when ωBD → ∞. This kind of theory was further generalized to obtain various functional dependencies for G in the formalisation of scalar-tensor theories of gravitation (see, e.g., Damour and Esposito-Farèse [124Jump To The Next Citation Point] or Will [540]).

 5.1 Introducing new fields: generalities
  5.1.1 The example of scalar-tensor theories
  5.1.2 Making other constants dynamical
 5.2 High-energy theories and varying constants
  5.2.1 Kaluza–Klein
  5.2.2 String theory
 5.3 Relations between constants
  5.3.1 Implication of gauge coupling unification
  5.3.2 Masses and binding energies
  5.3.3 Gyromagnetic factors
 5.4 Models with varying constants
  5.4.1 String dilaton and Runaway dilaton models
  5.4.2 The Chameleon mechanism
  5.4.3 Bekenstein and related models
  5.4.4 Other ideas

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