Nonlocal Gravity: Modified Poisson's Equation
  The recent nonlocal generalization of Einstein's theory of gravitation
reduces in the Newtonian regime to a nonlocal and nonlinear modification of
Poisson's equation of Newtonian gravity. The nonlocally modified Poisson
equation implies that nonlocality can simulate dark matter. Observational data
regarding dark matter provide limited information about the functional form of
the reciprocal kernel, from which the original nonlocal kernel of the theory
must be determined. We study this inverse problem of nonlocal gravity in the
linear domain, where the applicability of the Fourier transform method is
critically examined and the conditions for the existence of the nonlocal kernel
are discussed. This approach is illustrated via simple explicit examples for
which the kernels are numerically evaluated. We then turn to a general
discussion of the modified Poisson equation and present a formal solution of
this equation via a successive approximation scheme. The treatment is
specialized to the gravitational potential of a point mass, where in the linear
regime we recover the Tohline-Kuhn approach to modified gravity.
