  A modified theory of general relativity is proposed, where the gravitational
constant is replaced by a dynamical variable in space-time. The dynamics of the
gravitational coupling is described by a family of parametrized null geodesics,
implying that the gravitational coupling at a space-time point is determined by
solving transport equations along all null geodesics through this point.
  General relativity with dynamical gravitational coupling (DGC) is introduced.
We motivate DGC from general considerations and explain how it arises in the
context of causal fermion systems. The underlying physical idea is that the
gravitational coupling is determined by microscopic structures on the Planck
scale which propagate with the speed of light.
  In order to clarify the mathematical structure, we analyze the conformal
behavior and prove local existence and uniqueness of the time evolution. The
differences to Einstein's theory are worked out in the examples of the
Friedmann-Robertson-Walker model and the spherically symmetric collapse of a
shell of matter. Potential implications for the problem of dark matter and for
inflation are discussed. It is shown that the effects in the solar system are
too small for being observable in present-day experiments.
