● Domain 02 · Astrophysics
Precession that is physics.
Orbital work is judged over horizons no training window covers, where a per-step error too small to measure becomes the whole answer.
GRMHD accretion flowMagnetised plasma orbiting a black hole. The inner fluid laps the outer, so the flow shears into filaments and the field winds up.General relativistic magnetohydrodynamics · Gammie, McKinney & Tóth, ApJ 589 (2003); Blandford & Znajek, MNRAS 179 (1977); Event Horizon Telescope Collaboration, ApJL 875 (2019)
Precession you cannot attribute
A non-conservative integrator makes an orbit precess. So does general relativity. On the screen the two are identical, so an artefact the size of the effect cannot be told from the effect.
What changes
A symplectic method preserves the geometry of the flow instead of chasing the error in each step. Energy stays bounded over any length of integration, and the precession that remains is physics.
Demonstration
One orbit, four central objects.
The geometry, and what it does to the orbit
Flamm's paraboloid: the embedding of the equatorial slice of the Schwarzschild geometry, not a rubber sheet.
The orbit, brightest where the body is now. The same semi-major axis and eccentricity in every case.
Where each periastron fell. The rate they walk round is the whole of the effect.
The throat, at the surface of the object. It deepens as the object gets more compact.
Periastron advance is geometry here: what an ellipse does when the surface it is drawn on is curved. The same orbit runs over four objects of rising compactness, at the same semi-major axis and the same eccentricity, so curvature is the one thing that changes and the advance rises with it. Compactness is set high so the effect runs in seconds, and the advance printed beside each object is the one measured from the run.
Sources
Δφ = 6πGM / c2a(1 − e2)
Periastron advance at first post-Newtonian order · Einstein (1916); Will, Theory and Experiment in Gravitational Physics (2018)
ω = ∑i dqi ∧ dpi
The two-form a symplectic integrator preserves exactly · Hairer, Lubich & Wanner, Springer (2006)
Is this your problem?
Tell us what has to hold, and what happens if it does not.
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