We discuss the structure of compact bodies – neutron stars and black holes – in metric theories of gravity. We give a recipe for calculating the structure of such bodies, assumed to be non-rotating for simplicity, in generic metric theories, and then describe the results of calculations in specific theories. Turning to the motion of compact bodies, we develop the modified Einstein-Infeld-Hoffmann (EIH) framework, a generalization of the post-Newtonian N-body Lagrangian to systems containing one or more compact bodies, introducing parameters that may depend upon both the theory of gravity and the internal structure of each body. We analyse a number of observable effects in binary systems containing compact bodies and obtain their dependences on these parameters. The modified EIH parameters are calculated in a number of theories of gravity, including general relativity, scalar-tensor theories and Einstein-Aether/Khronometric theories.
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