Post-Newtonian Dynamics with State-Dependent Delay

The retarded potentials appearing in electrodynamics are an example of systems with state-dependent delay. By analogy, Newton's law of gravitation may be generalized using the concept of a retarded gravitational potential. For systems with state-dependent delay (SDD), delay and state or observations are implicitly related. Such an SDD-approach provides an approximation to general relativity for weak (gravitational) fields and slow speeds. While it has been shown that solutions to Einstein's field equations always exist, an existence theorem says nothing about the computation. So a semi-relativistic approximation is beneficial: One where space-time may still be considered as flat, but where Newton's equations (where gravity acts instantaneously) are replaced by a post-Newtonian theory, involving propagation of gravity at the speed of light. In this paper some direct effects (gravitational entrainment, dynamic friction, and orbital velocity for binaries) of such a post-Newtonian theory are described. We show also that a naive delay correction in Newton's gravitation law cannot be correct. Copyright (C) 2021 The Authors.