Abstract
We show, contrary to expectation, that the trajectory predicted by general-relativistic mechanics for a low-speed weak-gravity system is not always well-approximated by the trajectories predicted by special-relativistic and Newtonian mechanics for the same parameters and initial conditions. If the system is dissipative, the breakdown of agreement occurs for chaotic trajectories only. If the system is non-dissipative, the breakdown of agreement occurs for chaotic trajectories and non-chaotic trajectories. The agreement breaks down slowly for non-chaotic trajectories but rapidly for chaotic trajectories. When the predictions are different, general-relativistic mechanics must therefore be used, instead of special-relativistic mechanics (Newtonian mechanics), to correctly study the dynamics of a weak-gravity system (a low-speed weak-gravity system).
Highlights
IntroductionFor dynamical systems where gravity does not play a dynamical role, it is expected (see, for example, [1,2,3]) that, if the speed of the system is low (i.e., much less than the speed of light c), the dynamics predicted by special-relativistic mechanics is always well-approximated by the prediction of Newtonian mechanics for the same parameters and initial conditions
For dynamical systems where gravity does not play a dynamical role, it is expected that, if the speed of the system is low, the dynamics predicted by special-relativistic mechanics is always well-approximated by the prediction of Newtonian mechanics for the same parameters and initial conditions
The loss of agreement means [6,7,8] that special-relativistic mechanics must be used, instead of the standard practice of using Newtonian mechanics, to correctly study the dynamics of a low-speed system
Summary
For dynamical systems where gravity does not play a dynamical role, it is expected (see, for example, [1,2,3]) that, if the speed of the system is low (i.e., much less than the speed of light c), the dynamics predicted by special-relativistic mechanics is always well-approximated by the prediction of Newtonian mechanics for the same parameters and initial conditions. If gravity is weak and the speed of the system is low, the dynamical prediction of general-relativistic mechanics is expected (see, for example, [3,10,11,12,13]) to be always well-approximated by the Newtonian prediction for the same parameters and initial conditions. This is followed by the results and discussion, and concluding remarks on their significance
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