A generally covariant classical model for gravity, which in the limit of weak fields is essentially equivalent to Fujii's massive dilaton theory, is obtained when the invariance under a group of space-time-dependent geometry-preserving mass-unit transformations of a scalar-tensor theory is broken by ascribing a mass to the scalar field. The experimental consequences are discussed.In recent articles 1 * 2 Fujii has proposed a theory in which a nonzero mass is ascribed to the Nambu-Goldstone boson of scale invariance, the dilaton. The coupling of this particle with matter then leads to a gravitational potential V which has, in addition to the usual long-range Newtonian part, a component with a finite range of order m -1 , where m% (in units with c = 1), is the dilaton mass. From arguments based on the dilatongraviton mixing problem in strong-interaction physics, Fujii suggests 2 that m' 1 should be of the order of 1 km or less. A study 1 of the existing experimental data limits m" 1 to be either between 10 m and 1 km, or less than 1 cm.It is the purpose of this note to present a purely classical (nonquantum) generally covariant model for gravity which, in the weak-field approximation, reduces to a theory essentially equivalent to Fujii's. It is hoped in this way not only to extend the validity domain of the theory beyond weak fields but also to gain some insight into the meaning of scale invariance and units transformations and their connection with gravity.In a previous work 3 the relationship between Mach's principle and a group of space-time-dependent scale changes in the unit of mass was discussed within the context of the Brans-Dicke theory 4 of gravity. Transformations of the scalar field cp and the stress-energy tensor T u under this (mass-gauge) group were defined so as to preserve both (i) the geometry of space-time and (ii) the form of the field equations and hence the law of conservation of energy. A theory fully covariant under this group is obtained when the Brans-Dicke coupling constant a> = 0. The strange situation then results where a knowledge of the space-time geometry (e.g., as deduced from astronomical observations) does not suffice to determine, even qualitatively, the mass-energy content. 3 We now consider the effect of breaking massgauge invariance by adding a scalar-field mass term -rn 2 f(cp) to the Lagrange function. This results in the action principlewith R the curvature scalar, L the Lagrangian for matter. The field equations arewhere a semicolon denotes covariant differentiation, a prime denotes differentiation by (p f Ucp =g tJ cp.ij withgu the metric (signature +2), and Gij is the Einstein tensor. The scalar property of cp prevents us from specifying a priori the form of the function/. We note also that the weak principle of equivalence 4 (i.e., geodesic motion for small particles) remains valid for the action (1). For the treatment of local and astronomical (noncosmological) problems we demand that a weak-field approximation should be possible. For this purpose we writewhere J] u is...

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