Note on the relationship between the speed of light and gravity in the bi-metric theory of gravity

We analyze the relationship between the speed of light ce and the speed of gravity cg in the bi-metric theory of gravity. We reveal that in this theory the speed of light is a function of the speed of gravity which is a primary fundamental constant. Thus, experimental measurement of ce in the framework of the bi-metric theory is merely a convoluted way to measure cg. PACS numbers: 04.20.Cv, 04.80.-y, 06.20.Fn, 06.20.Jr, 95.10.Km Submitted to: Class. Quantum Grav. Note on the relationship between the speed of light and gravity 2 Any plausible bi-metric theory of gravity operates with two metric tensors. One of them, gαβ, makes a gravity null cone along which weak gravitational waves propagate with speed cg, while the other, g̃αβ makes a light null cone along which electromagnetic waves propagate with speed ce. Had the difference between the two null cones existed the Lorentz-invariance of gravity would be violated. Thus, experimental measurement of the difference between the speed of light and that of gravity is crucially important for testing the Lorentz invariance of gravity [1, 2]. To model this difference we use the bi-metric theory of gravity proposed in [3]. It defines the speed of light ce as a function of the speed of gravity cg ce = cg(1− ǫ) 1/2 , (1) where 0 ≤ ǫ ≤ 1 is a sliding parameter characterizing the difference (Lorenzt-invariance breaking) between the gravity and light null cones. Practical measurement of the parameter ǫ can be rendered in gravitational experiments in which the gravitational field is time-dependent, but proper interpretation of this measurement requires to chose a specific system of units [4]. If one assumes the gravitodynamic system of units with cg = 1 [3, 8], then, the speed of light ce is measured and it is expressed in terms of the parameter ǫ as ce = (1− ǫ) . On the other hand, if one assumes the electrodynamic (metric) system of units [1, 2, 5] with the speed of light ce = 1, then the speed of gravity is measured and it is expressed in terms of the parameter ǫ as cg = (1− ǫ) . We emphasize that neither the speed of light nor the speed of gravity can be measured independently of the assumption of the system of units used for measuring astronomical distances. Irrespective of the choice of the system of units the primary measured parameter of the bi-metric theory of gravity is