All existing treatments of bimetric MOND (BIMOND) — a class of relativistic
versions of MOND — have dealt with a rather restricted sub-class: The
Lagrangian of the interaction between the gravitational degrees of freedom —
the two metrics — is a function of a certain {\it single} scalar argument
built from the difference in connections of the two metrics. I show that the
scope of BIMOND is much richer: The two metrics can couple through several
scalars to give theories that all have a “good” nonrelativistic (NR) limit —
one that accounts correctly, a-la MOND, for the dynamics of galactic systems,
{\it including gravitational lensing}. This extended-BIMOND framework exhibits
a qualitative departure from the way we think of MOND at present, as
encapsulated, in all its aspects, by one “interpolating function” of one
acceleration variable. After deriving the general field equations, I pinpoint
the subclass of theories that satisfy the pivotal requirement of a good NR
limit. These involve three independent, quadratic scalar variables. In the NR
limit these scalars all reduce to the same acceleration scalar, and the NR
theory then does hinge on one function of a {\it a single} acceleration
variable — representing the NR MOND “interpolating function”, whose form is
largely dictated by the observed NR galactic dynamics. However, these scalars
behave differently, in different relativistic contexts. So, the full richness
of the multi-variable Lagrangian, as it enters cosmology, for example, is
hardly informed by what we learn from observations of galactic dynamics. In
this paper, I present the formalism, with some generic examples. I also
consider some cosmological solutions where the two metrics are small departures
from one Friedman-Lemaitre-Robertson-Walker metric. This may offer a framework
for describing cosmology within the extended BIMOND.

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