Conservative PN Equations of Motion

Conservative post-Newtonian equations of motion describe compact bodies over times short compared with radiation-driven inspiral, while retaining relativistic corrections to their mutual binding. The calculation is an elimination problem: solve the gravitational field perturbatively with time-symmetric boundary conditions, insert that solution into the particle dynamics, and express the result in a chosen coordinate system. Coordinate accelerations change under near-identity transformations; conserved relations among invariant quantities do not.

Required background. Weak Metric Perturbations and Gauge Freedom supplies linearized fields and gauge transformations; Presymplectic Potentials and Currents supplies action variation and reduced-phase-space reasoning. Helpful background. The Einstein Equation supplies the nonlinear field equations; Gravitational Radiation and Waveforms clarifies what is removed when the conservative, time-symmetric sector is selected.

Relative motion at first post-Newtonian order

Restore $c$ for order counting and keep $G$ explicit. In the center-of-mass frame define

$$\mathbf{x}=\mathbf{y}_1-\mathbf{y}_2,\qquad r=\lvert\mathbf{x}\rvert,\qquad \mathbf{n}=\frac{\mathbf{x}}{r},\qquad \mathbf{v}=\dot{\mathbf{x}},\qquad \dot r=\mathbf{n}\cdot\mathbf{v},$$

with total mass $M=m_1+m_2$, reduced mass $\mu=m_1m_2/M$, and symmetric mass ratio $\nu=\mu/M$. In harmonic coordinates the conservative relative acceleration through 1PN is

$$\begin{aligned} \mathbf{a}={}&-\frac{GM}{r^2}\mathbf{n}\\ &+\frac{GM}{c^2r^2} \bigg\{ \mathbf{n}\bigg[ (4+2\nu)\frac{GM}{r} -(1+3\nu)v^2 +\frac32\nu\dot r^2 \bigg] +(4-2\nu)\dot r\,\mathbf{v} \bigg\} +O(c^{-4}). \end{aligned}$$

This equation already displays the complete scalar basis allowed at 1PN by rotations, parity, time reversal, and exchange of the bodies: the acceleration is a linear combination of $\mathbf n$ and $\mathbf v$, with coefficients built from $v^2$, $\dot r^2$, and $GM/r$. General relativity fixes their numerical coefficients. The unreduced individual-body formula and its higher-order continuation are given in Blanchet 2024, § 3.2.2, eq. (339).

The displayed equation assumes nonspinning bodies whose sizes are small compared with $r$, a weak interbody field $GM/(rc^2)\ll1$, and slow relative motion. It does not assume that each body has weak internal gravity. Through the orders at which finite-size operators are absent, strong internal fields are absorbed into the masses. Spin and eccentric solutions require additional variables, developed on Spinning and Eccentric PN Orbits.

From the field equations to an acceleration

One direct route introduces the harmonic-gauge field variable

$$h^{\mu\nu}=\sqrt{-g}\,g^{\mu\nu}-\eta^{\mu\nu}, \qquad \partial_\nu h^{\mu\nu}=0,$$

for which Einstein’s equations take a relaxed wave-equation form,

$$\Box h^{\mu\nu} =\frac{16\pi G}{c^4}\tau^{\mu\nu}[T,h].$$

The effective source $\tau^{\mu\nu}$ contains matter and nonlinear gravitational terms. Begin with the Newtonian potentials, substitute them into the nonlinear source, solve the next Poisson or near-zone wave equations, and repeat. Evaluating the regularized metric on each worldline and using its equation of motion yields the coordinate acceleration. The iteration is triangular: to determine a term at a specified PN order, only a finite set of lower-order potentials and retardation corrections is needed. The smooth-source construction and the subsequent compact-body reduction are separated explicitly in Blanchet 2024, §§ 2.3.1 and 3.2.2, eqs. (147)–(155) and (339)–(340).

A second route starts from the Einstein–Hilbert plus worldline action. Solving the field equation with time-symmetric boundary conditions and substituting the solution back gives a reduced, or Fokker, action for the worldlines. Its variation produces the conservative equations. At sufficiently high order that reduced action can depend on accelerations or be nonlocal in time; lower-order equations are then used only in a controlled order-reduction step. Simply replacing accelerations inside an unreduced action can change the variational problem.

The two routes are not expected to give identical coordinate formulas. Harmonic, ADM-type, and effective-field-theory variables encode different gauge choices and field redefinitions. Agreement means that a permitted transformation relates the actions or equations and that invariant observables coincide. The relation among these formulations is developed on PN Hamiltonians, Lagrangians, and Contact Transformations.

Conserved quantities and circular invariants

Time translation and rotation imply a conserved energy and angular momentum for the conservative sector. Through 1PN, an energy whose total derivative vanishes after using the displayed acceleration is

$$\begin{aligned} \frac{E}{\mu}={}& \frac{v^2}{2}-\frac{GM}{r}\\ &+\frac{1}{c^2}\bigg[ \frac38(1-3\nu)v^4 +\frac{GM}{2r}\Big((3+\nu)v^2+\nu\dot r^2\Big) +\frac{G^2M^2}{2r^2} \bigg] +O(c^{-4}). \end{aligned}$$

Here $E$ is binding energy: the constant rest energy $Mc^2$ has been removed. The corresponding conserved angular momentum is

$$\mathbf J =\mu\,\mathbf x\times\mathbf v \left[1+\frac1{c^2} \left(\frac{1-3\nu}{2}v^2+(3+\nu)\frac{GM}{r}\right)\right] +O(c^{-4}).$$

There is a short check of this less familiar correction. The acceleration gives $d(\mathbf x\times\mathbf v)/dt=(4-2\nu)GM\dot r(\mathbf x\times\mathbf v)/(c^2r^2)$. Using only Newtonian motion inside the already-1PN bracket, its derivative is $-(4-2\nu)GM\dot r/(c^2r^2)$. The two terms cancel through the retained order. Thus the Euclidean quantity $\mathbf x\times\mathbf v$ need not be constant, while the corrected Noether angular momentum is. Both conserved quantities follow from the 1PN Lagrangian.

The energy expression is coordinate dependent when regarded as a function of $r$, $v$, and $\dot r$. For a circular orbit, however, the acceleration is radial and $\dot r=0$. Its radial component gives the harmonic-coordinate frequency relation

$$\Omega^2=\frac{GM}{r^3} \left[ 1+(-3+\nu)\frac{GM}{rc^2} +O(c^{-4}) \right].$$

Eliminate $r$ in favor of the invariant frequency parameter

$$x=\left(\frac{GM\Omega}{c^3}\right)^{2/3}.$$

The circular binding energy becomes

$$\frac{E(x)}{\mu c^2} =-\frac{x}{2} \left[ 1-\left(\frac34+\frac{\nu}{12}\right)x +O(x^2) \right].$$

The square brackets may look surprising because their 1PN coefficient is negative while $E$ itself is negative: the correction makes the binding energy less negative than the Newtonian extrapolation at fixed frequency. Writing the observable as $E(x)$ removes the harmonic-coordinate radius and any gauge length that may occur at higher orders.

For eccentric orbits, the periastron advance supplies another discriminating invariant. At 1PN,

$$K\equiv\frac{\Omega_\phi}{\Omega_r} =1+\frac{3GM}{c^2p}+O(c^{-4}),$$

where $p=a(1-e^2)$ at Newtonian order. The absence of $\nu$ in the leading correction is a nontrivial result, not an assumption; mass-ratio dependence enters at higher PN order.

Coordinate freedom and consistency checks

A near-identity change of variables,

$$\mathbf{x}'=\mathbf{x}+\frac{1}{c^2}\boldsymbol{\xi}(\mathbf{x},\mathbf{v})+O(c^{-4}),$$

changes the displayed acceleration by $c^{-2}d^2\boldsymbol{\xi}/dt^2$ plus terms produced by rewriting its arguments. It can move terms between the $\mathbf n$ and $\mathbf v$ structures without changing a physical orbit. A coordinate separation or the location of a coordinate-defined last circular orbit is therefore not an invariant prediction.

Several checks constrain a high-order result. The Newtonian limit must recover inverse-square motion. The limit $\nu\to0$ must reproduce the PN expansion of Schwarzschild geodesic invariants. The equations must possess the Poincaré generators to the claimed order, including the center-of-energy integral; at 3PN this requirement was historically powerful enough to constrain an ambiguity. Independent harmonic-coordinate, ADM-Hamiltonian, and surface-integral derivations of 3PN motion ultimately agree, as summarized in Schäfer and Jaranowski 2024, §§ 1.2 and 6.1.

At 4PN the conservative dynamics contains a time-symmetric tail integral. It is still conservative, but it is not representable globally by a purely instantaneous acceleration without additional variables or a nonlocal action. At 2.5PN, by contrast, the leading time-antisymmetric radiation-reaction force appears. Removing it defines a useful conservative problem, not the full inspiral. The approximation also breaks down when pericenter velocities or $GM/(rc^2)$ cease to be small, and a formal remainder $O(c^{-4})$ alone does not quantify the numerical error.

Exercise

Starting from the harmonic-coordinate circular frequency relation, derive the invariant 1PN energy $E(x)$. Use

$$x=\gamma\left[1+\left(-1+\frac{\nu}{3}\right)\gamma+O(\gamma^2)\right], \qquad \gamma=\frac{GM}{rc^2}.$$

Solution

For $\dot r=0$, insert $v^2=r^2\Omega^2$ into the conserved energy. Keeping only terms through $\gamma^2$ gives

$$\frac{E}{\mu c^2} =-\frac{\gamma}{2} +\frac{7-\nu}{8}\gamma^2 +O(\gamma^3).$$

Invert the supplied relation:

$$\gamma=x\left[1+\left(1-\frac{\nu}{3}\right)x+O(x^2)\right].$$

Substitution then yields

$$\frac{E}{\mu c^2} =-\frac{x}{2} +\left(\frac38+\frac{\nu}{24}\right)x^2 =-\frac{x}{2} \left[ 1-\left(\frac34+\frac{\nu}{12}\right)x \right] +O(x^3).$$

The coordinate parameter $\gamma$ has disappeared. This is precisely why comparison between two formulations should be made at fixed $x$, not fixed coordinate radius.

References

• Blanchet, Luc. 2024. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27, 4. DOI.
• Schäfer, Gerhard, and Piotr Jaranowski. 2024. “Hamiltonian Formulation of General Relativity and Post-Newtonian Dynamics of Compact Binaries.” Living Reviews in Relativity 27, 2. DOI.