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The Gravitational Two-Body Problem in the Einstein-Infeld-Hoffmann Approximation
The Gravitational Two-Body Problem in the Einstein-Infeld-Hoffmann Approximation
This Demonstration displays the orbits of the two-body problem in general relativity in various ways. Several simplifications are involved: separating off the trivial center of mass motion, restricting to bounded motion in a plane, and neglecting the emission of gravitational radiation. Actually, we use a Hamiltonian formulation of the work by Einstein [1] on gravitating particles. The Hamilton function is the sum of the nonrelativistic term that defines the Kepler problem and a relativistic term of order that takes into account velocity-dependent masses and retardation. The function is simple enough that Hamilton's canonical equations of motion can be solved by Mathematica's built-in function NDSolve for given initial conditions and a given number of revolutions (say four) within fractions of a second.
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In order to make the relativistic effects obvious, the masses of the two bodies are preset to large values on narrow orbits that are more probable for Schwarzschild black holes than for stars. Hover the pointer over the control labels to see explanations in the form of tooltips.