Post-Newtonian expansion and radiation reaction

For a nonspinning nearly circular binary define x from total mass and orbital angular frequency. At Newtonian order it equals orbital speed squared over c squared and GM over separation c squared. Defining it through frequency is preferable to treating a coordinate-dependent separation as an invariant observable. One post-Newtonian order represents an additional factor of x relative to the leading conservative result. Half-integer orders can occur in radiation-related quantities.

The conservative binding energy has a known expansion in x and symmetric mass ratio. Its first correction modifies the relation between orbital energy and frequency, rather than adding a measured wave amplitude directly. Coordinate gauges and field redefinitions can change intermediate accelerations or potentials while preserving appropriately defined circular energy-frequency relations. Comparisons between calculations must therefore use matched invariant quantities and mass conventions.

The leading gravitational-wave luminosity scales as the fifth power of x. Combining it with the derivative of binding energy gives slow orbital-frequency evolution. The leading dissipative acceleration is called a relative two-and-a-half-post-Newtonian effect compared with Newtonian acceleration. That label describes its velocity ordering, not a statement that waveform flux begins only at a high relative correction to its own leading radiation formula.

For adiabatic circular inspiral, orbital evolution is slow compared with one orbital period. A convenient local diagnostic is the magnitude of frequency derivative divided by frequency squared. Its leading expression grows strongly with x, so increasing orbital speed erodes the timescale hierarchy. Near plunge, orbit averaging and an expansion around a sequence of stable circular orbits become inadequate even if a truncated expression remains easy to evaluate numerically.

Instantaneous source-mechanical energy loss need not equal the far-zone flux at every time. Near-zone field energy can exchange reversibly with matter, producing additional total-derivative terms often called Schott terms. Orbit-averaged balance eliminates or accounts for such terms in the appropriate setting. Using a secular flux to invent an arbitrary local damping force would fail to specify the gauge, reaction dynamics and angular-momentum balance needed for consistency.

PN corrections include hereditary effects depending on past evolution, such as tails from wave propagation on the source's background curvature. Coefficients and logarithms can make successive terms larger or smaller than a naive power estimate suggests. Waveform phasing accumulates small local errors across many cycles, so convergence should be assessed with invariant diagnostics, comparisons between approximations and other controlled calculations. Numerical relativity and resummation complement the expansion rather than turning every truncated formula into an exact strong-field solution.