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Nonlocal gravity: Modified Poisson's equation

### Abstract

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.

© 2012 American Institute of Physics

Received 20 January 2012
Accepted 23 March 2012
Published online 13 April 2012

Acknowledgments:
B.M. is grateful to F.W. Hehl and J.R. Kuhn for valuable comments and helpful correspondence.

Article outline:

I. INTRODUCTION
II. INVERSE PROBLEMS IN LINEAR NONLOCAL GRAVITY
A. Liouville-Neumann method
B. Fourier transform method
III. EXISTENCE OF THE LINEAR KERNEL: EXAMPLES
A. An example
B. A second example
IV. NONLOCAL AND NONLINEAR POISSONâ€™S EQUATION
A. Formal solution via successive approximations
V. GRAVITATIONAL POTENTIAL OF A POINT MASS
A. Linear kernel
B. Nonlinear kernel
VI. DISCUSSION