Bigravity / Bimetric Theories of Gravity
In General > s.a. actions for general relativity [bimetric reformulation]; Whitehead Theory [in which one metric is a fixed background].
* History: 1934, Born-Infeld theory can be considered as a form of bimetric theory (according to Moffat); 1973–1974, New version proposed by N Rosen; 1979–1995, J Moffat's non-symmetric gravitation theory; 1992, J Bekenstein's treatment of gravitational lensing and MOND; 1998–2003, Clayton-Moffat scalar-vector-tensor theory.
* Idea: A theory of gravity with two distinct metrics, a gravitational gab which describes the geometry of spacetime and controls the speed of gravity, and a matter metric ?ab which may be the one coupled to all other fields and controling their propagation; In a scalar-tensor version with a scalar field ? coupled to gravity, it can be given by
?ab = gab + (B/c2) ?a? ?b? ,
where B is a positive number (which will probably be renormalized by quantum effects).
* Dynamics: The action can be of the form S[gab, ?ab] = ?M d4x |g|1/2 R[gab] + ?M d4x |?|1/2 R[?ab] + ?M d4x V[gab, ?ab].
* Properties: One of the main differences with general relativity is that it (or some versions of it) does not produce black holes.
> Quantum aspects: see Boulware-Deser Ghost; effective action; quantum gravity renormalization.> Online resources: see Wikipedia page.Phenomenology > s.a. anomalous acceleration; test-particle motion; variation of constants.
* Variation of c: The different metrics that apply to gravitation and other fields give rise to an effectively changing speed of light; Depending on whether one uses electromagnetic or other (say, gravitational) waves to study the early universe, one gets that the other speed changes in time (the ratio cgrav/c? is dimensionless and meaningful).
https://www.phy.olemiss.edu/~luca/Topics/grav/bimetric.html 
