Self-Force Information at Finite Mass Ratio

First-order gravitational self-force information constrains a comparable-mass model by fixing its derivative at the test-mass boundary, after both descriptions are written in the same invariant variables. It does not determine the model at finite mass ratio by itself. The disciplined route is invariant self-force datum $\rightarrow$ symmetric mass variable $\rightarrow$ Hamiltonian or effective-one-body (EOB) potential $\rightarrow$ weak-field and geodesic checks. Any completion beyond the known coefficient is an explicit modeling choice to be calibrated and validated separately.

Helpful background. The Einstein Equation supplies the field equations; Weak Metric Perturbations and Gauge Freedom supplies perturbative gauge reasoning; Presymplectic Potentials and Currents supplies invariant Hamiltonian comparisons; Computational Relativity supplies convergence standards; Gravitational Radiation and Waveforms supplies radiative invariants; and Relativistic Two-Body Dynamics supplies the mass-ratio and regime map.

From a small-ratio invariant to a symmetric potential

Use the volume-wide ordering $m_1\geq m_2$ and define the local self-force parameter

$$q=\frac{m_1}{m_2}\geq1, \qquad \epsilon=\frac{m_2}{m_1}=q^{-1}\ll1.$$

Then

$$M=m_1+m_2, \qquad \nu=\frac{m_1m_2}{M^2}=\frac{\epsilon}{(1+\epsilon)^2}.$$

The inverse series is

$$\nu=\epsilon-2\epsilon^2+3\epsilon^3+O(\epsilon^4), \qquad \epsilon=\nu+2\nu^2+5\nu^3+O(\nu^4).$$

Therefore a coefficient linear in $\epsilon$ is also linear in $\nu$, but only if the independent variable and normalization are unchanged. That qualification matters. Self-force calculations around a background of mass $m_1$ naturally use

$$y=(Gm_1\Omega)^{2/3},$$

whereas symmetric two-body descriptions use

$$x=(GM\Omega)^{2/3}=y(1+\epsilon)^{2/3}.$$

Suppose a dimensionless invariant has the small-ratio expansion

$$\mathcal I(y,\epsilon) =\mathcal I_0(y)+\epsilon\,\mathcal I_1(y)+O(\epsilon^2).$$

At fixed $x$, not fixed $y$,

$$y=x(1+\epsilon)^{-2/3} =x-\frac{2}{3}\epsilon x+O(\epsilon^2),$$

and hence

$$\mathcal I(x,\nu) =\mathcal I_0(x) +\nu\left[ \mathcal I_1(x)-\frac{2}{3}x\mathcal I_0'(x) \right] +O(\nu^2).$$

The derivative term is a variable-conversion effect, not new dynamics. Omitting it creates an apparent disagreement with PN theory. Akcay and collaborators state their small-ratio, $\nu$, $x$, and EOB inverse-radius definitions together and use precisely this invariant-frequency bridge in their extraction of the linear-in-$\nu$ EOB potential Akcay et al. 2012, §I.A, pp. 4–5, Eqs. (1)–(2), PDF.

Which invariants constrain which functions?

For nonspinning conservative motion, a standard EOB organization is

$$A(u;\nu)=1-2u+\nu a(u)+\nu^2 a_2(u)+\cdots,$$ $$\bar D(u;\nu)=1+\nu\bar d(u)+O(\nu^2), \qquad Q(u,p_r;\nu)=\nu q_4(u)p_r^4+O(\nu p_r^6,\nu^2),$$

with $u=GM/r_{\mathrm{EOB}}$, dimensionless radial momentum $p_r=P_r/\mu$, and dimensionless $Q=Q_{\rm phys}/\mu^2$ in the squared effective Hamiltonian. The potentials depend on the chosen EOB phase-space gauge, but a fully specified dictionary from an invariant fixes their coefficients in that gauge. The following translations are complementary rather than interchangeable.

Self-force invariant	Finite-ratio information constrained	Essential cross-check
Circular redshift at fixed $\Omega$	The linear-in-$\nu$ circular potential $a(u)$ through the binary first law and EOB energy map	Its small-$u$ series agrees with PN binding-energy coefficients
Periastron ratio $\Omega_\phi/\Omega_r$ for slightly eccentric motion	A combination of $a$, its derivatives, and $\bar d$	It reduces to Schwarzschild geodesic precession as $\nu\to0$
Redshift or radial action through higher powers of eccentricity	Momentum-dependent functions such as $q_4(u)$	Independent self-force codes and the PN expansion agree
Gyroscope precession at fixed orbital frequencies	Linear-mass-ratio gyrogravitomagnetic spin couplings	The zero-spin and weak-field limits commute in the stated domain
Electric and magnetic tidal eigenvalues	Linear-mass-ratio curvature information along the orbit and tidal EOB factors once the body-response convention is fixed	Eigenvalue traces and the geodesic limit are satisfied
Conservative scattering angle $\chi(E,J)$	The linear-in-$\nu$ radial action for unbound motion	Its weak-field expansion agrees with PM and PN scattering

For circular binaries the first law relates variations of the conservative energy and angular momentum to the redshifts. That relation turns a regularized self-force observable into a statement about the Hamiltonian rather than about one gauge-dependent metric component Le Tiec, Blanchet, and Whiting 2012, §§II–IV, especially Eqs. (2.13), (3.8), and (4.1). Combining the first law with the EOB circular-orbit relations yields $a(u)$; Akcay et al. explain this chain and the separate role of periastron data in determining $\bar d(u)$ Akcay et al. 2012, §I, pp. 2–4, PDF.

Slightly eccentric information adds genuinely new phase-space directions. The self-force correction to the radial-to-azimuthal frequency ratio constrains a combination of $a(u)$ and $\bar d(u)$, rather than merely checking the circular redshift Barack, Damour, and Sago 2010, §§II–IV, Eqs. (2.5)–(2.13) and (4.1), PDF. Extending the eccentric expansion further gives access to the momentum-dependent EOB function: two independent numerical codes were used to determine the first-order $q_4$ potential and to compare its weak-field behavior with PN coefficients Akcay and van de Meent 2016, §§II–IV, especially Eqs. (2.24)–(2.26), PDF.

Spin and tidal data obey the same logic but require additional conventions. A spin-precession invariant can determine the linear-mass-ratio part of an EOB gyrogravitomagnetic ratio only after the spin supplementary condition, spin normalization, and orbital-frequency variables are aligned. Bini and Damour carry out that translation for circular motion and compare the resulting high-PN expansion with strong-field data Bini and Damour 2014, §§II–V, PDF. Tidal eigenvalues characterize the external curvature sampled along an orbit; they are not a compact object’s Love numbers. First-order self-force corrections to electric- and magnetic-type eigenvalues provide orbital curvature benchmarks, while a material response coefficient still requires its own matching problem Bini and Geralico 2018, §§II–IV, PDF.

What “finite-mass-ratio information” does and does not mean

Knowing $\mathcal I_1(x)$ exactly over an interval fixes the tangent of $\mathcal I(x,\nu)$ at $\nu=0$:

$$\left.\frac{\partial\mathcal I(x,\nu)}{\partial\nu}\right|_{\nu=0} =\widetilde{\mathcal I}_1(x).$$

It leaves every coefficient of $\nu^2,\nu^3,\ldots$ undetermined. Exchange symmetry does not remove that freedom because any smooth function of $\nu$ is already symmetric under $m_1\leftrightarrow m_2$. A finite-ratio completion may therefore be written as

$$\mathcal I_{\mathrm{model}}(x,\nu) =\mathcal I_0(x)+\nu\widetilde{\mathcal I}_1(x) +\nu^2 C(x,\nu).$$

At equal masses, $\nu=1/4$, so the unknown term is $C(x,1/4)/16$. It need not be small compared with the known first-order term in the strong field. Choosing a Padé form, imposing a pole, or replacing $\nu\widetilde{\mathcal I}_1$ by a nonlinear “symmetric completion” can be useful, but it is resummation or modeling, not a consequence of first-order self-force theory.

The distinction between matching, calibration, and validation is sharp. The self-force coefficient is matched because the finite-ratio model must reproduce its $\nu\to0$ expansion. Unknown pieces of $C$ may be calibrated against numerical-relativity data. A withheld configuration then validates the completed model. Reusing the same datum in all three roles produces circular evidence; the calibration workflow prevents that leakage.

Strong-field nonuniformity

The limits $\nu\to0$ and approach to a singular geodesic structure need not be uniform. In the circular Schwarzschild problem,

$$A(u;\nu)=1-2u+\nu a(u)+O(\nu^2),$$

and the extracted $a(u)$ grows as the light ring $u\to1/3$ is approached. Akcay et al. find $a(u)\simeq0.25(1-3u)^{-1/2}$ in their convention and explain that the small body’s orbital energy also diverges, eventually invalidating fixed-$\nu$ perturbation theory Akcay et al. 2012, §III, Eq. (47), and §VII A–B, pp. 28–30, PDF. Thus one may take $\nu\to0$ at fixed $u\lt1/3$ and then study the coefficient, but one cannot hold a finite $\nu$ and move arbitrarily close to $u=1/3$ while trusting the first-order truncation.

Gauge regularity is another boundary. Detweiler’s circular redshift is invariant under an appropriate class of helically symmetric gauge transformations with the asymptotic time normalization fixed; it is not invariant under every formal gauge vector. Eccentric observables should be labeled by invariant frequencies or actions, not by coordinate eccentricity. Dissipative fluxes require retarded boundary conditions, whereas the conservative dictionaries above use time-symmetric or conservative pieces. Mixing those sectors can imitate a finite-ratio correction.

Solved variable-conversion check

Assume a self-force computation reports

$$\mathcal I(y,\epsilon)=1-\frac{3}{2}y +\epsilon(2y+5y^2)+O(\epsilon^2), \qquad y=(Gm_1\Omega)^{2/3}.$$

Find the coefficient linear in $\nu$ at fixed $x=(GM\Omega)^{2/3}$, and state what the result predicts at equal masses.

Solution

Here $\mathcal I_0(x)=1-3x/2$, $\mathcal I_1(x)=2x+5x^2$, and $\mathcal I_0'(x)=-3/2$. The fixed-$x$ coefficient is therefore

$$\widetilde{\mathcal I}_1(x) =\mathcal I_1(x)-\frac{2}{3}x\mathcal I_0'(x) =3x+5x^2.$$

Thus

$$\mathcal I(x,\nu)=1-\frac{3}{2}x+\nu(3x+5x^2)+O(\nu^2).$$

The test-mass limit correctly returns $1-3x/2$, and the weak-field coefficients can be compared directly with PN theory. At equal masses the known linear contribution is $(3x+5x^2)/4$, but the full answer is not predicted: an unknown term $C(x,1/4)/16$ remains. Treating the displayed expression as exact at $\nu=1/4$ would be an unjustified extrapolation.

The corresponding cross-regime check is to compare the weak-field series with PN theory, the $\nu=0$ term with exact geodesic dynamics, and any proposed finite-$\nu$ completion with genuinely withheld numerical data. The failure-diagnosis page organizes those tests when they disagree.

References

• Akcay, Sarp, and Maarten van de Meent. “Numerical Computation of the EOB Potential q Using Self-Force Results.” Physical Review D 93, 064063 (2016). DOI. Open PDF.
• Akcay, Sarp, Leor Barack, Thibault Damour, and Norichika Sago. “Gravitational Self-Force and the Effective-One-Body Formalism between the Innermost Stable Circular Orbit and the Light Ring.” Physical Review D 86, 104041 (2012). DOI. Open PDF.
• Barack, Leor, Thibault Damour, and Norichika Sago. “Precession Effect of the Gravitational Self-Force in a Schwarzschild Spacetime and the Effective-One-Body Formalism.” Physical Review D 82, 084036 (2010). DOI. Open PDF.
• Bini, Donato, and Andrea Geralico. “Gravitational Self-Force Corrections to Tidal Invariants for Spinning Particles on Circular Orbits in a Schwarzschild Spacetime.” Physical Review D 98, 064040 (2018). DOI. Open PDF.
• Bini, Donato, and Thibault Damour. “Two-Body Gravitational Spin-Orbit Interaction at Linear Order in the Mass Ratio.” Physical Review D 90, 024039 (2014). DOI. Open PDF.
• Le Tiec, Alexandre, Luc Blanchet, and Bernard F. Whiting. “The First Law of Binary Black Hole Mechanics in General Relativity and Post-Newtonian Theory.” Physical Review D 85, 064039 (2012). DOI.