Post-Newtonian Approximation in GR

• Post-Newtonian approximation is a systematic expansion that adds relativistic corrections to Newtonian gravity using powers of 1/c².
• It employs Poisson-type equations to compute gravitoelectric and gravitomagnetic potentials, ensuring precise treatment of fluid dynamics and boundary conditions.
• Applied to non-axisymmetric models like Dedekind ellipsoids, it underpins accurate predictions in compact binary dynamics and gravitational waveforms.

The post-Newtonian (PN) approximation is a perturbative method for solving Einstein’s field equations in the regime where gravitational fields are weak and velocities are slow compared to the speed of light. It provides a hierarchy of corrections, order by order in powers of $1/c$, to Newtonian gravity, enabling precise modeling of astrophysical systems with small but non-negligible relativistic effects. The approach is foundational in gravitational physics, underpinning the theory of compact binary dynamics, gravitational waveforms, and relativistic stellar structure.

1. Mathematical and Physical Foundation

The central concept of the PN approximation is the systematic expansion of the spacetime metric and matter variables in the small parameter $\epsilon \equiv 1/c^2$, reflecting the hierarchy of relativistic corrections to Newtonian gravity. In a quasi-Gaussian or harmonic gauge with coordinates $(x^0 = ct, x^a)$, the metric components are expanded as

$$\begin{aligned} g_{00} &= -1 + 2U\,\epsilon + [2\Phi - 2U^2]\,\epsilon^2 + O(\epsilon^3), \ g_{0i} &= 4U_i\,\epsilon^{3/2} + O(\epsilon^{5/2}), \ g_{ij} &= \delta_{ij}(1 + 2U\,\epsilon) + O(\epsilon^2), \end{aligned}$$

where $U$ is the Newtonian (gravitoelectric) potential, $U_i$ is the gravitomagnetic vector potential, and $\Phi$ is a 1PN-order scalar potential. The field equations reduce at each PN order to elliptic Poisson-type equations for these potentials, with sources constructed from the fluid variables (density $\rho$, velocity $v_i$, pressure $p$) and their lower-order expansions (Gürlebeck et al., 2010).

Formally, the expansion may be written for any physical quantity $Q$ as

$$Q = Q^{(0)} + \epsilon Q^{(2)} + \epsilon^2 Q^{(4)} + \dots,$$

with each term capturing the structure at a corresponding PN order.

2. PN Potentials, Field Equations, and Gauge

The gravitational potentials in the PN hierarchy, and their governing equations, are:

• Newtonian potential: $\Delta U = -4\pi G \rho$,
• Gravitomagnetic vector potential: $\Delta U_i = -4\pi G \rho v_i^{(0)}$,
• Scalar 1PN potential: $$\Delta\Phi = -4\pi G \left[ \rho (v^{(0)})^2 + 3p^{(2)} + 2\rho U \right]$$,

where $\Delta$ is the three-dimensional Laplacian operator, and $v_i^{(0)}$ is the Newtonian velocity field. These potentials are subject to boundary conditions dictated by asymptotic flatness and the matching to the physical surface of self-gravitating fluids (e.g., stars) (Gürlebeck et al., 2010).

Gauge conditions, such as the use of harmonic or quasi-Gaussian coordinates, preserve the elliptic character of the equations and facilitate the separation of the physical degrees of freedom. All quantities are expanded within domains tailored to the geometry of the system, frequently employing adapted coordinate systems (e.g., surface-fitted ellipsoidal coordinates for triaxial bodies).

3. Explicit Construction: Example of Non-Axisymmetric Dedekind Ellipsoids

A concrete illustration is the PN expansion for Dedekind ellipsoids—triaxial, homogeneous, non-axisymmetric equilibrium figures with steady internal flow. The procedure includes:

• Adapting confocal ellipsoidal coordinates $(\lambda, \mu, \nu)$ to the figure’s geometry.
• Expanding the boundary coordinate as $$\lambda' = \lambda [1 + \sum_{n=1}^\infty S^{(2n)}(\mu, \nu)\, \epsilon^{2n}]^{-1}$$ so that the physical surface remains at constant $\lambda'=a_1$, with $a_i$ the principal semi-axes.
• Solving the Newtonian, vector, and scalar 1PN Poisson equations inside the ellipsoid, using expansions in ellipsoidal harmonics and imposing continuity and regularity.
• Expressing fluid variables as

$$\rho = \rho_0 + 0\cdot\epsilon + \dots, \quad p = p^{(2)}\epsilon + O(\epsilon^2), \quad v_i = v_i^{(0)} + v_i^{(2)}\epsilon + O(\epsilon^2),$$

with Newtonian and PN-corrected pressure distributions such that $p=0$ at the (deformed) surface and fluid flow has no normal component there.

• The corrections at $O(\epsilon)$ shift the figure’s axes, $$a_i = a_i^{(0)}[1 + \Delta_i\epsilon + O(\epsilon^2)]$$, and adjust the internal velocity field (Gürlebeck et al., 2010).

This method exemplifies the general PN philosophy: expand about a known Newtonian solution, solve Poisson-type equations for gravitational potentials at each order, and enforce boundary and fluid-mechanical conditions to uniquely determine all corrections.

4. PN Expansion and Solution Strategies

The iterative solution strategy applies generally to isolated gravitating systems:

1. Begin with a Newtonian equilibrium, specifying density and velocity structure.
2. Expand metric and matter variables, ensuring that coordinate systems are adapted so that boundary surfaces map to their PN-corrected locations (e.g., via deformed $\lambda'$ in ellipsoidal coordinates).
3. At each PN order, solve the corresponding Poisson equations for vector and scalar potentials in the fixed background coordinates, using harmonics or separation of variables as appropriate.
4. Use the Bianchi identities (conservation laws and Euler equation) at the relevant order to determine higher-order corrections to the pressure, velocity, and surface shape, typically by imposing $p=0$ and vanishing normal velocity at the boundary.
5. The endpoint is an $O(1/c^2)$-accurate solution for the metric, matter configuration, and internal flow.

For the Dedekind figure, closed-form expressions are available for the axis corrections $\Delta_i$ as functions of the axis ratios, but explicit coefficients require recourse to standard references (e.g., Chandrasekhar & Elbert).

5. Physical Implications and Limits of Validity

The PN approximation, at first order ($O(1/c^2)$), captures leading-order relativistic corrections to the structure and dynamics of self-gravitating fluids beyond Newtonian theory. Notably:

• Non-axisymmetric figures (like PN Dedekind ellipsoids) develop small shape deformations and velocity corrections at 1PN order, while the global geometry and leading-order equilibrium are still controlled by Newtonian physics.
• The inclusion of velocity (gravitomagnetic) and scalar potentials at 1PN is essential for capturing internal circulation and frame-dragging phenomena.
• The method is effective so long as $\epsilon = 1/c^2$ remains a small parameter; for more compact objects or in the strong-field regime, higher-order PN (or fully relativistic numerical) approaches become necessary.
• The PN method’s philosophy—expanding about the Newtonian solution, solving for gravitational potentials at each order, and imposing consistent boundary/physical conditions—is broadly applicable to astrophysical fluid bodies and underlies many analytic and semi-analytic results in relativistic stellar and dynamical modeling.

6. Key Formulas and Summary Table

The essential formulas for the 1PN expansion in this context can be summarized as follows:

Quantity	Formula	PN Order
Expansion parameter	$\epsilon = 1/c^2$	—
Metric components	$$g_{00} = -1 + 2U\, \epsilon + [2\Phi - 2U^2]\, \epsilon^2 + \dots$$	$O(\epsilon^2)$
	$g_{0i} = 4U_i\, \epsilon^{3/2} + \dots$	$O(\epsilon^{3/2})$
	$g_{ij} = \delta_{ij}(1 + 2U\,\epsilon) + \dots$	$O(\epsilon)$
Field equations	$\Delta U = -4\pi G \rho$	Newtonian
	$\Delta U_i = -4\pi G \rho v_i^{(0)}$	$O(\epsilon^{3/2})$
	$$\Delta \Phi = -4\pi G [\rho (v^{(0)})^2 + 3p^{(2)} + 2\rho U]$$	$O(\epsilon^2)$
Adapted boundary	$$\lambda' = \lambda [1 + \sum_{n=1}^\infty S^{(2n)}(\mu, \nu)\, \epsilon^{2n}]^{-1}$$, surface at $\lambda' = a_1$	—
Axis corrections	$$a_i = a_i^{(0)}(1 + \Delta_i \epsilon) + O(\epsilon^2)$$	$O(\epsilon)$

Precise evaluations of coefficients and further closed-form quantities are system-specific (triaxial ellipsoid, axis ratios) and are referenced in the relevant literature (Gürlebeck et al., 2010).

7. Context and Extensions

This surface-adapted, PN expansion has broad applications:

• Non-axisymmetric equilibrium modeling (e.g., triaxial neutron stars, tidal interactions),
• Dynamical stability and mode analysis of rotating fluid figures,
• Higher-order PN extensions for more compact or faster-moving systems,
• Calibration and validation of numerical relativity codes in the near-Newtonian regime.

The surface-fitted coordinate approach, combined with systematic PN expansion and matching conditions, is representative of state-of-the-art analytic techniques for self-gravitating fluids in general relativity (Gürlebeck et al., 2010).