Problems 11-
Problem 4.11
By directly integrating the first post-Newtonian (1PN) equation of motion
we can compare with the effective-potential quadrature from Problem 4.9 and the analytical approximation. The precession is measured by detecting the next pericenter passage via a zero-crossing event \(v_r = \mathbf{r}\cdot\mathbf{v} = 0\) with direction from negative to positive. The precession per orbit is then
where \((x_p, y_p)\) is the position at the next pericenter.
Three-way comparison
>>> from galactic_dynamics_bovy.chapter04.gr_precession_s2 import compute_precession_1pn, GM_BH, C, A_S2, E_S2
>>> from galactic_dynamics_bovy.chapter04.gr_precession import compute_delta_psi
>>> import numpy as np
>>> prec_1pn = compute_precession_1pn()
>>> r_p = A_S2 * (1.0 - E_S2)
>>> prec_quad = compute_delta_psi(r_p, E_S2, GM=GM_BH, c=C) - 2.0 * np.pi
>>> prec_analytical = 6.0 * np.pi * GM_BH / (C**2 * A_S2 * (1.0 - E_S2**2))
>>> np.degrees(prec_1pn) * 60
12.104969...
>>> np.degrees(prec_quad) * 60
12.135288...
>>> np.degrees(prec_analytical) * 60
12.124627...
| Method | \(\Delta\varpi\) (arcmin) | vs eff. potential |
|---|---|---|
| Effective potential quadrature (Problem 4.9) | 12.135 | — |
| 1PN integration (Eq. 4.11.1) | 12.105 | \(-0.25\%\) |
| Analytical \(6\pi GM_\bullet / [c^2 a(1-e^2)]\) | 12.125 | \(-0.09\%\) |
All three values are consistent with the \(\sim 12'\) per orbit measured by the GRAVITY interferometer.
Why the two GR approaches differ
The effective-potential quadrature from Problem 4.9 adds a correction \(-GM_\bullet L^2/(c^2 r^3)\) to the Newtonian effective potential. This term comes from the exact Schwarzschild geodesic equation** — the full metric solution for a test particle around a non-rotating point mass. The corresponding radial force
is conservative and velocity-independent: it can be derived from a scalar potential.
The 1PN equation of motion (Eq. 4.11.1) comes from a different starting point: it is the test-particle limit of the Einstein-Infeld-Hoffmann (EIH) \(N\)-body equations (Einstein et al., 1938)1, a weak-field slow-motion expansion (\(GM/(c^2 r) \ll 1\), \(v/c \ll 1\)) of the general-relativistic equations of motion. (Will, 2008)2 applies this framework to stellar orbits around Sgr A*. The EIH formalism is designed for the general \(N\)-body problem where no exact metric solution is available; Eq. (4.11.1) is the one-body reduction. The acceleration depends on both position and velocity; in particular the \(v_r \mathbf{v}\) term cannot be derived from a scalar potential, reflecting the fact that in GR the gravitational field couples to kinetic energy (mass-energy equivalence).
Both approaches agree at leading order, but because the effective potential captures the exact one-body Schwarzschild result while the 1PN EOM truncates at first order in \(GM/(c^2 r)\), the effective-potential quadrature implicitly includes the 2PN and all higher-order corrections that the 1PN integration drops. This is why it gives a slightly larger precession (12.135 vs 12.105 arcmin, a ~0.25% difference). For orbits closer to the black hole the gap would grow.
Orbit comparison
Figure 4.11: Radial distance \(r(t)\) for S2 over five orbital periods (\(T \approx 16.2\) yr). The 1PN orbit (black) progressively leads the Newtonian orbit (gray) due to a shorter radial period. The 1PN correction effectively deepens the potential near pericenter, causing the star to fall in and bounce back faster. Black dots mark 1PN pericenter passages.
>>> from galactic_dynamics_bovy.chapter04.gr_precession_s2 import plot_gr_precession_s2
>>> plot_gr_precession_s2()
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Einstein, A., Infeld, L., & Hoffmann, B. (1938). The gravitational equations and the problem of motion. Annals of Mathematics, 39(1), 65--100. https://doi.org/10.2307/1968714 ↩
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Will, C. M. (2008). Testing the general relativistic "no-hair" theorems using the Galactic Center black hole Sagittarius A*. The Astrophysical Journal, 674(1), L25. https://doi.org/10.1086/528847 ↩