Math 2.0

Relativity and gravitational theory Definition

Post-Newtonian expansion

Builds on Einstein field equationsNewton's law of universal gravitationMinkowski spacePoisson's equationSchwarzschild metric

Definition

Consider a solution of the Einstein field equations sourced by a bounded system of matter with typical mass MM, size rr and internal speeds vv. Assume the system is weakly gravitating and slow: ε2=max⁡(v2/c2, GM/(rc2))≪1\varepsilon^2 = \max\left(v^2/c^2, \, GM/(rc^2)\right) \ll 1, where GG is the gravitational constant and cc the speed of light. The post-Newtonian expansion is the formal expansion of the metric gμνg_{\mu\nu} and of the equations of motion of the matter in powers of 1/c1/c. It is done in the near zone, in coordinates (ct,x)(ct, x) close to Minkowski coordinates. A term is of nn-th post-Newtonian order (nPN) when its size relative to the Newtonian term is ε2n\varepsilon^{2n}. At first post-Newtonian order (1PN), g00g_{00} is computed through order c−4c^{-4}, g0ig_{0i} through order c−3c^{-3} and gijg_{ij} through order c−2c^{-2}, with gμν→ημνg_{\mu\nu} \to \eta_{\mu\nu} as c→∞c \to \infty.

Remarks

At 1PN in the harmonic gauge one finds g00=−1−2Φ/c2−2Φ2/c4+…g_{00} = -1 - 2\Phi/c^2 - 2\Phi^2/c^4 + \dots, where the extra terms of order c−4c^{-4} involve further potentials; Φ\Phi is the Newtonian potential solving Poisson's equation ∇2Φ=4πGρ\nabla^2 \Phi = 4\pi G\rho. The leading order reproduces Newton's law of gravitation. Radiation reaction appears first at 2.5PN order, through odd powers of 1/c1/c. The expansion is valid only in the near zone, at distances small compared with the wavelength of the emitted gravitational waves. Far away it must be matched with a different expansion.

Example: the Schwarzschild metric of mass MM in harmonic coordinates has g00=−(r−m)/(r+m)g_{00} = -(r - m)/(r + m) with m=GM/c2m = GM/c^2. Expanding in 1/c1/c gives g00=−1+2m/r−2m2/r2+O(c−6)g_{00} = -1 + 2m/r - 2m^2/r^2 + O(c^{-6}), which is the 1PN form above with Φ=−GM/r\Phi = -GM/r.

Non-example: the late merger of two black holes has vv close to cc and GM/(rc2)GM/(rc^2) of order one. So ε\varepsilon is not small and the expansion does not apply; numerical relativity is used instead.

Reading path

To understand this page from the ground up, read these 49 pages in order. The foundations come first.

  1. Derivative concept
  2. Differential equation concept
  3. Einstein field equations concept
  4. Euclidean space object
  5. Euclidean vector concept
  6. Function concept
  7. Interval concept
  8. Minkowski space concept
  9. Ordinary differential equation concept
  10. Poisson's equation concept
  11. Ricci curvature concept
  12. Velocity concept
  13. Fréchet derivative definition
  14. Ordinary differential equation definition
  15. Cross product definition
  16. Function of several variables concept
  17. Vector-valued function concept
  18. Lipschitz condition definition
  19. Classical mechanics concept
  20. Schwarzschild metric example
  21. Limit of a function of several variables concept
  22. Momentum concept
  23. Partial derivative concept
  24. Solution of an ordinary differential equation definition
  25. Inertial frame of reference concept
  26. Continuity in R^n definition
  27. Differentiable function of several variables definition
  28. Partial derivative definition
  29. Initial value problem definition
  30. Newton's laws of motion statement
  31. Differentiable on an open set definition
  32. Gradient definition
  33. Higher partial derivatives definition
  34. Picard–Lindelöf hypotheses definition
  35. Iterated derivative definition
  36. Picard operator definition
  37. C^k function characterised by continuous derivatives statement
  38. Integral equation form of an initial value problem statement
  39. An iterate of the Picard operator is a contraction statement
  40. Vector field definition
  41. Picard–Lindelöf theorem statement
  42. Conservative vector field definition
  43. Classical mechanics definition
  44. Momentum definition
  45. Potential energy definition
  46. Angular momentum definition
  47. Central force definition
  48. Newton's law of universal gravitation statement
  49. Post-Newtonian expansion definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Post-Newtonian_expansion
About this page
TagDAX7ZJ352T
URIm2://DAX7ZJ352T
Kinddefinition
WritingW0 stub
MSC83C25, 83C10
DefinesPost-Newtonian expansion
UsesEinstein field equations, Newton's law of universal gravitation, Minkowski space, Poisson's equation, Schwarzschild metric
Changed3 November 2026

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