Math 2.0

Relativity and gravitational theory Definition

Gravitational radiation reaction

Builds on Post-Newtonian expansionQuadrupole formulaGravitational wave

Definition

Consider a slowly moving, weakly self-gravitating source of mass density ρ(t,x)\rho(t, x) and reduced quadrupole moment Qij(t)Q_{ij}(t), as in the quadrupole formula. Consider the post-Newtonian expansion of the equations of motion of the source. The gravitational radiation reaction is the leading part of the force that is odd under time reversal. It appears at order (v/c)5(v/c)^5 relative to the Newtonian force. In Burke–Thorne gauge it is given by the potential ΦRR(t,x)=G5c5d5Qijdt5(t) xixj,\Phi_{RR}(t, x) = \frac{G}{5c^5} \frac{d^5 Q_{ij}}{dt^5}(t)\, x_i x_j, with summation over i,j∈{1,2,3}i, j \in \set{1, 2, 3}, which adds the force density −ρ∇ΦRR-\rho \nabla \Phi_{RR} to the Newtonian equations.

Remarks

The work done by this force on the source is −∫ρ v⋅∇ΦRR d3x=−G5c5Qij(5)Q˙ij-\int \rho\, v \cdot \nabla\Phi_{RR}\, d^3x = -\frac{G}{5c^5} Q^{(5)}_{ij} \dot Q_{ij}, using the continuity equation. After integration by parts in time and averaging over several periods it equals −G5c5⟨Q...ijQ...ij⟩-\frac{G}{5c^5}\langle \dddot Q_{ij} \dddot Q_{ij} \rangle. This is minus the power carried by the gravitational waves, so the energy balance holds.

Example: for two point masses m1,m2m_1, m_2 on a circular orbit of separation aa, energy balance with E=−Gm1m2/(2a)E = -G m_1 m_2/(2a) gives dadt=−645G3m1m2(m1+m2)c5a3.\frac{da}{dt} = -\frac{64}{5} \frac{G^3 m_1 m_2 (m_1 + m_2)}{c^5 a^3}. So a4a^4 decreases linearly in time and the bodies collide after a finite time.

Non-example: the first post-Newtonian corrections to the force, at order (v/c)2(v/c)^2, are even under time reversal. They are conservative, cause effects such as perihelion precession, and are not part of radiation reaction.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Gravitational wave, concept, no evidence levelGravitational waveconceptLinearized gravity, concept, no evidence levelLinearized gravityconceptPost-Newtonian expansion, concept, no evidence levelPost-Newtonian expans…conceptQuadrupole formula, statement, no evidence levelQuadrupole formulastatementGravitational radiation reaction, definition, no evidence levelGravitational radiati…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Gravitational wave concept
  2. Linearized gravity concept
  3. Post-Newtonian expansion concept
  4. Quadrupole formula statement
  5. Gravitational radiation reaction definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Gravitational_wave
About this page
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MSC83C35, 83C10, 83C25
DefinesGravitational radiation reaction
UsesPost-Newtonian expansion, Quadrupole formula, Gravitational wave
Changed3 November 2026

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