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We present a particular low-energy limit of the Hamiltonian of free test particle motion in arbitrary relativistic space-times. As it turns out, this limit gives insight into the general Newtonian limit by providing an intermediate, "pseudo-Newtonian" step, which encompasses some pseudo-Newtonian formulas already present in the literature.
If the metric is expressed so as to be diagonal in the coordinate-time components, we are able to derive a description exactly reproducing the spatial shapes of geodesics. In fully general space-times where dragging ("time non-diagonal") terms appear in the metric, the limit at least yields a previously unknown Hamiltonian reproducing exact shapes of null geodesics. Furthermore, if the space-time is stationary, the exact shapes of null geodesics can be also correctly parametrized by coordinate-time, the limit thus provides an alternative Hamiltonian for computations in gravitational lensing.
Relevant astrophysical superpositions of gravitating sources, the addition of electromagnetic fields, and fluid dynamics in the pseudo-Newtonian limit are discussed. Additionally, the method is demonstrated in the case of the Kerr space-time and massive-particle circular orbits, and analogies with respect to the recent pseudo-Kerr Lagrangian by Ghosh et al. (2014) is commented upon.