Dimensionless Parameters of Compact Binaries

A compact binary is not characterized by one universal expansion parameter. Mass ratio, orbital field strength, eccentricity, spin, body size, internal response, and radiation-reaction time control different approximations. The useful question is therefore not whether the system is “relativistic,” but which independent dimensionless ratios are small, which are held fixed, and which observable is to be predicted.

Required background. The Einstein Equation supplies the nonlinear field law whose solutions are being approximated, and Weak Metric Perturbations and Gauge Freedom explains why coordinate components cannot serve as regime labels by themselves.

Helpful background. Presymplectic Potentials and Currents clarifies invariant generators and conserved quantities, while Black-Hole Perturbation Theory supplies the fixed-background limit used when one mass is much smaller than the other.

Mass ratios and exchange symmetry

Let $m_1\geq m_2\gt 0$ denote the isolated masses of the bodies, and define

$$M=m_1+m_2, \qquad \mu=\frac{m_1m_2}{M}, \qquad q=\frac{m_1}{m_2}\geq1.$$

The symmetric mass ratio and normalized mass difference are

$$\nu=\frac{\mu}{M}=\frac{q}{(1+q)^2}, \qquad \delta=\frac{m_1-m_2}{M}=\sqrt{1-4\nu}.$$

They obey $0\lt \nu\leq1/4$. Equal masses give $(q,\nu,\delta)=(1,1/4,0)$; the small-mass-ratio limit gives $(q,\nu,\delta)\to(\infty,0,1)$. If $\epsilon=m_2/m_1$, then

$$\nu=\frac{\epsilon}{(1+\epsilon)^2} =\epsilon-2\epsilon^2+3\epsilon^3+O(\epsilon^4).$$

Thus $\nu$ and $\epsilon$ agree only at leading order. A coefficient quoted as “first order in mass ratio” is ambiguous unless the expansion variable is named. Exchange-symmetric observables are naturally functions of $\nu$, whereas self-force calculations are often organized in $\epsilon$ with a specified background mass. The mass definitions are given in Blanchet 2024, §3.2.4, Eq. (353).

The masses also need an operational definition. For asymptotically flat conservative calculations, $m_A$ normally means the mass of each isolated body and $M$ is their sum. It is not automatically the ADM energy of the interacting binary, and in a radiating spacetime it is not the time-dependent Bondi mass. Replacing one with another can move terms between nominal perturbative orders.

Orbital strength, velocity, and frequency

For a quasi-circular orbit whose angular frequency $\Omega$ is measured with respect to asymptotic time, define

$$x=(GM\Omega)^{2/3}.$$

Newtonian Kepler motion gives

$$\Omega^2=\frac{GM}{r^3}, \qquad x=\frac{GM}{r}=v^2.$$

Beyond Newtonian order, $r$ and a coordinate velocity $v$ depend on the chosen coordinates, while $\Omega$ and the relation of an invariant energy to $\Omega$ can be compared across gauges. Hence $x$ is the standard circular-orbit PN parameter. The approximation $x\ll1$ expresses weak binding and slow motion, but it does not by itself specify the near-zone expansion, radiative boundary conditions, or mass ratio. The distinct near, wave, and radiation zones are developed in Blanchet 2024, §1.1 and §2.1.

Eccentric bound motion has at least two fundamental frequencies. Periastron advance makes $\Omega_\phi\neq\Omega_r$, so $(GM\Omega_\phi)^{2/3}$ and a parameter built from the radial period are not interchangeable. Generic Kerr motion has three frequencies. A frequency parameter is therefore meaningful only after the orbit family and the time normalization are stated.

For an unbound encounter, the natural inputs live at infinity. With four-velocities $u_1^\mu$ and $u_2^\mu$,

$$\gamma=-u_1\!\cdot u_2\geq1, \qquad j=\frac{J}{GM\mu}, \qquad \widehat b=\frac{b}{GM}$$

describe relative energy, angular momentum, and impact parameter. They are related by the asymptotic momentum and should not all be treated as independent. A post-Minkowskian expansion keeps the asymptotic velocity fixed and expands in $GM/b$; it need not assume $\gamma-1\ll1$.

Eccentricity, spin, and compactness

Newtonian eccentricity is unique, but relativistic eccentricity is definition dependent. PN quasi-Keplerian motion uses distinct radial, time, and angular eccentricities. A turning-point convention may define

$$e_r=\frac{r_{\mathrm a}-r_{\mathrm p}}{r_{\mathrm a}+r_{\mathrm p}}, \qquad p_r=\frac{2r_{\mathrm a}r_{\mathrm p}}{r_{\mathrm a}+r_{\mathrm p}},$$

but the chosen radial coordinate enters both quantities. Frequency ratios, action variables, or explicitly matched turning-point definitions are safer for comparisons.

For each body, a common dimensionless spin is

$$\chi_A=\frac{\lvert S_A\rvert}{Gm_A^2}.$$

For an isolated classical Kerr black hole, $\chi_A\leq1$. This is not a universal bound for arbitrary material bodies, and the spin vector still depends on a representative worldline and spin supplementary condition. Spin-orbit, spin-spin, and spin-induced multipole effects enter at different orders, so “large spin” does not name a unique dynamical correction.

For a material object of radius $R_A$, compactness is

$$\mathcal C_A=\frac{Gm_A}{R_A}.$$

A black hole has no material surface radius, so stellar compactness should not be assigned to it without defining an effective radius. Worldline power counting instead uses ratios such as $R_A/r$ or $R_A/\mathcal R$, where $\mathcal R$ is an external curvature scale.

Response and timescale parameters

If the leading adiabatic quadrupole satisfies

$$Q_{ij}=-\lambda_A\mathcal E_{ij},$$

then, with $G$ explicit, a dimensionless tidal deformability may be defined as

$$\Lambda_A=\frac{G\lambda_A}{(Gm_A)^5}.$$

The leading circular-orbit tidal effect scales schematically as $\Lambda_A x^5$, but dynamical tides introduce $\Omega/\omega_{A,n}$ for each internal mode frequency $\omega_{A,n}$. A small size-to-separation ratio does not justify an instantaneous response close to resonance. Dissipation adds absorption probabilities or phase lags and requires retarded, rather than time-symmetric, boundary conditions. Worldline matching and its scale hierarchy are reviewed in Porto 2016, §§6.5–6.6 and 7.1, pp. 34–40.

For an inspiral, define an orbital time $T_{\mathrm{orb}}$ and a radiation-reaction time $T_{\mathrm{rr}}$. Their ratio

$$\varepsilon_{\mathrm{ad}}=\frac{T_{\mathrm{orb}}}{T_{\mathrm{rr}}}$$

controls adiabatic averaging away from resonances and separatrices. It is conceptually distinct from both $x$ and the mass ratio, even though a particular regime can relate their scalings. Small-mass-ratio inspirals illustrate the distinction sharply: the field expansion, the slow-evolution expansion, and the accumulated-phase expansion must all be tracked Barack and Pound 2019, §§4–5.

A regime is a vector, not a label

A useful specification is a parameter vector

$$\mathcal P= (\nu,x,e,\boldsymbol\chi_1,\boldsymbol\chi_2, \mathcal C_1,\mathcal C_2,\Lambda_1,\Lambda_2, \varepsilon_{\mathrm{ad}},\ldots).$$

PN theory expands primarily in $x$; self-force theory in a mass ratio; PM theory in $GM/b$ at fixed asymptotic energy; worldline EFT in body size over orbital scale; and adiabatic evolution in $T_{\mathrm{orb}}/T_{\mathrm{rr}}$. Overlaps occur when more than one component is small. Those overlaps are valuable because independently derived invariant observables must agree there.

This classification also exposes failure modes. Near a separatrix, radial periods grow and ordinary averaging can fail. Near an internal mode resonance, a nominally high-order tidal term can become enhanced. For ultrarelativistic scattering, the limits $G\to0$ and $\gamma\to\infty$ need not commute. A truncation error inferred from formal power counting alone can therefore be misleading.

Exercises

Show that $0\lt \nu\leq1/4$ for $q\geq1$, and obtain its large-$q$ expansion through $q^{-3}$.

Solution

From $\nu=q/(1+q)^2$,

$$\frac{d\nu}{dq}=\frac{1-q}{(1+q)^3}.$$

The maximum on $q\geq1$ is at $q=1$, where $\nu=1/4$. Setting $\epsilon=q^{-1}$ gives

$$\nu=\epsilon(1+\epsilon)^{-2} =q^{-1}-2q^{-2}+3q^{-3}+O(q^{-4}).$$

A circular binary has $x=0.04$. Estimate $r/(GM)$ and $v$ at Newtonian order, and explain why the estimates are not invariant equalities in the relativistic problem.

Solution

Kepler scaling gives $r/(GM)=x^{-1}=25$ and $v=\sqrt{x}=0.2$. The input $x=(GM\Omega)^{2/3}$ uses the asymptotically normalized orbital frequency, but $r$ and a coordinate velocity depend on gauge beyond leading order. Their numerical values therefore acquire representation-dependent relativistic corrections.

References

• Barack, Leor, and Adam Pound. “Self-Force and Radiation Reaction in General Relativity.” Reports on Progress in Physics 82 (2019): 016904. DOI.
• Blanchet, Luc. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27 (2024): 4. DOI.
• Porto, Rafael A. “The Effective Field Theorist’s Approach to Gravitational Dynamics.” Physics Reports 633 (2016): 1–104. DOI.