# Eikonal Blog

## 2011.02.16

### Post-Newtonian gravity

• Clifford M. Will’s papers on Post-Newtonian approach:
• 0) “Generation of Post-Newtonian Gravitational Radiation via Direct Integration of the Relaxed Einstein Equations” by Clifford M. Will (arXiv:gr-qc/9910057v1; 1999.10.15) – http://arxiv.org/abs/gr-qc/9910057
Abstract: The completion of a network of advanced laser-interferometric gravitational-wave observatories around 2001 will make possible the study of the inspiral and coalescence of binary systems of compact objects (neutron stars and black holes), using gravitational radiation. To extract useful information from the waves, such as the masses and spins of the bodies, theoretical general relativistic gravitational waveform templates of extremely high accuracy will be needed for filtering the data, probably as accurate as $O[(v/c)^6]$ beyond the predictions of the quadrupole formula. We summarize a method, called DIRE, for Direct Integration of the Relaxed Einstein Equations, which extends and improves an earlier framework due to Epstein and Wagoner, in which Einstein’s equations are recast as a flat spacetime wave equation with source composed of matter confined to compact regions and gravitational non-linearities extending to infinity. The new method is free of divergences or undefined integrals, correctly predicts all gravitational wave “tail” effects caused by backscatter of the outgoing radiation off the background curved spacetime, and yields radiation that propagates asymptotically along true null cones of the curved spacetime. The method also yields equations of motion through $O[(v/c)^4]$, radiation-reaction terms at $O[(v/c)^5]$ and $O[(v/c)^7]$, and gravitational waveforms and energy flux through $O[(v/c)^4]$, in agreement with other approaches. We report on progress in evaluating the $O[(v/c)^6]$ contributions.
• 1) “Post-Newtonian Gravitational Radiation and Equations of Motion via Direct Integration of the Relaxed Einstein Equations. I. Foundations” by Michael E. Pati, Clifford M. Will (arXiv:gr-qc/0007087v1; 2000.07.31) – http://arxiv.org/abs/gr-qc/0007087
Abstract: We present a self-contained framework called Direct Integration of the Relaxed Einstein Equations (DIRE) for calculating equations of motion and gravitational radiation emission for isolated gravitating systems based on the post-Newtonian approximation. We cast the Einstein equations into their “relaxed” form of a flat-spacetime wave equation together with a harmonic gauge condition, and solve the equations formally as a retarded integral over the past null cone of the field point (chosen to be within the near zone when calculating equations of motion, and in the far zone when calculating gravitational radiation). The “inner” part of this integral(within a sphere of radius ${\cal R} \sim$ one gravitational wavelength) is approximated in a slow-motion expansion using standard techniques; the “outer” part, extending over the radiation zone, is evaluated using a null integration variable. We show generally and explicitly that all contributions to the inner integrals that depend on ${\cal R}$ cancel corresponding terms from the outer integrals, and that the outer integrals converge at infinity, subject only to reasonable assumptions about the past behavior of the source. The method cures defects that plagued previous “brute-force” slow-motion approaches to motion and gravitational radiation for isolated systems. We detail the procedure for iterating the solutions in a weak-field, slow-motion approximation, and derive expressions for the near-zone field through 3.5 post-Newtonian order in terms of Poisson-like potentials.
• 2) “Post-Newtonian gravitational radiation and equations of motion via direct integration of the relaxed Einstein equations. II. Two-body equations of motion to second post-Newtonian order, and radiation-reaction to 3.5 post-Newtonian order” by Michael E. Pati, Clifford M. Will (arXiv:gr-qc/0201001v1; 2001.12.31) – http://arxiv.org/abs/gr-qc/0201001
Abstract: We derive the equations of motion for binary systems of compact bodies in the post-Newtonian (PN) approximation to general relativity. Results are given through 2PN order (order (v/c)^4 beyond Newtonian theory), and for gravitational radiation reaction effects at 2.5PN and 3.5PN orders. The method is based on a framework for direct integration of the relaxed Einstein equations (DIRE) developed earlier, in which the equations of motion through 3.5PN order can be expressed in terms of Poisson-like potentials that are generalizations of the instantaneous Newtonian gravitational potential, and in terms of multipole moments of the system and their time derivatives. All potentials are well defined and free of divergences associated with integrating quantities over all space. Using a model of the bodies as spherical, non-rotating fluid balls whose characteristic size s is small compared to the bodies’ separation r, we develop a method for carefully extracting only terms that are independent of the parameter s, thereby ignoring tidal interactions, spin effects, and internal self-gravity effects. Through 2.5PN order, the resulting equations agree completely with those obtained by other methods; the new 3.5PN back-reaction results are shown to be consistent with the loss of energy and angular momentum via radiation to infinity.
• 3) “Post-Newtonian gravitational radiation and equations of motion via direct integration of the relaxed Einstein equations. III. Radiation reaction for binary systems with spinning bodies” by Clifford M. Will (arXiv:gr-qc/0502039v2; 2005.04.29) – http://arxiv.org/abs/gr-qc/0502039
Abstract: Using post-Newtonian equations of motion for fluid bodies that include radiation-reaction terms at 2.5 and 3.5 post-Newtonian (PN) order (O[(v/c)^5] and O[(v/c)^7] beyond Newtonian order), we derive the equations of motion for binary systems with spinning bodies. In particular we determine the effects of radiation-reaction coupled to spin-orbit effects on the two-body equations of motion, and on the evolution of the spins. For a suitable definition of spin, we reproduce the standard equations of motion and spin-precession at the first post-Newtonian order. At 3.5PN order, we determine the spin-orbit induced reaction effects on the orbital motion, but we find that radiation damping has no effect on either the magnitude or the direction of the spins. Using the equations of motion, we find that the loss of total energy and total angular momentum induced by spin-orbit effects precisely balances the radiative flux of those quantities calculated by Kidder et al. The equations of motion may be useful for evolving inspiraling orbits of compact spinning binaries.
• 4) “Post-Newtonian gravitational radiation and equations of motion via direct integration of the relaxed Einstein equations. IV. Radiation reaction for binary systems with spin-spin coupling” by Han Wang, Clifford M. Will (arXiv:gr-qc/0701047v2; 2007.03.16) – http://arxiv.org/abs/gr-qc/0701047
Abstract: Using post-Newtonian equations of motion for fluid bodies that include radiation-reaction terms at 2.5 and 3.5 post-Newtonian (PN) order O[(v/c)^5] and O[(v/c)^7] beyond Newtonian order), we derive the equations of motion for binary systems with spinning bodies, including spin-spin effects. In particular we determine the effects of radiation-reaction coupled to spin-spin effects on the two-body equations of motion, and on the evolution of the spins. We find that radiation damping causes a 3.5PN order, spin-spin induced precession of the individual spins. This contrasts with the case of spin-orbit coupling, where there is no effect on the spins at 3.5PN order. Employing the equations of motion and of spin precession, we verify that the loss of total energy and total angular momentum induced by spin-spin effects precisely balances the radiative flux of those quantities calculated by Kidder et al.
• 5) “Post-Newtonian gravitational radiation and equations of motion via direct integration of the relaxed Einstein equations. V. Evidence for the strong equivalence principle to second post-Newtonian order” by Thomas Mitchell, Clifford M. Will (arXiv:0704.2243v2 [gr-qc]; 2007.07.17)- http://arxiv.org/abs/0704.2243
Abstract: Using post-Newtonian equations of motion for fluid bodies valid to the second post-Newtonian order, we derive the equations of motion for binary systems with finite-sized, non-spinning but arbitrarily shaped bodies. In particular we study the contributions of the internal structure of the bodies (such as self-gravity) that would diverge if the size of the bodies were to shrink to zero. Using a set of virial relations accurate to the first post-Newtonian order that reflect the stationarity of each body, and redefining the masses to include 1PN and 2PN self-gravity terms, we demonstrate the complete cancellation of a class of potentially divergent, structure-dependent terms that scale as s^{-1} and s^{-5/2}, where s is the characteristic size of the bodies. This is further evidence of the Strong Equivalence Principle, and supports the use of post-Newtonian approximations to derive equations of motion for strong-field bodies such as neutron stars and black holes. This extends earlier work done by Kopeikin.