Gauge-Invariant Self-Force Observables

The local gravitational self-force and the coordinate location of a worldline change under a perturbative gauge transformation. Physics is extracted instead from relations among quantities whose operational normalization has been fixed: a redshift as a function of orbital frequency, a ratio of fundamental frequencies, a spin-precession angle, tidal eigenvalues, or a scattering angle at fixed asymptotic energy and angular momentum. These are invariants under a stated class of gauges, not under arbitrary singular transformations or changes of the asymptotic clock.

Required background. PDEs and Green Functions for Gravitation supplies the retarded solution and homogeneous freedom. Helpful background. The Einstein Equation supplies the geometric observables; Weak Metric Perturbations and Gauge Freedom supplies infinitesimal diffeomorphisms; Presymplectic Potentials and Currents supplies asymptotic charges and their variations; Black-Hole Perturbation Theory supplies the regular metric reconstructed in practical gauges.

Green-function bridge. Retarded boundary conditions select an inhomogeneous solution, but one may still add a smooth homogeneous perturbation corresponding to changed physical boundary data or to gauge. The regular field is the difference between the retarded and singular fields and is smooth enough on the worldline for the observable being formed. A gauge vector $\xi^\mu$ changes it by $2\nabla_{(\mu}\xi_{\nu)}$ and moves the coordinate worldline; a valid comparison transforms both and preserves the chosen asymptotic charges and time coordinate.

What “gauge invariant” means here

Let $\varepsilon=m/M\ll1$ denote the small body’s mass divided by the background mass. Write the first-order effective metric as

$$\tilde g_{\mu\nu} =g_{\mu\nu}+\varepsilon h_{\mu\nu}^{\mathrm R}.$$

Under $x^\mu\mapsto x^\mu-\varepsilon\xi^\mu$,

$$h_{\mu\nu}^{\mathrm R} \mapsto h_{\mu\nu}^{\mathrm R} +2\nabla_{(\mu}\xi_{\nu)}.$$

The component $F^r$ at a coordinate radius is therefore not an observable: the gauge transformation changes the metric, the representative worldline, and the label $r$. An invariant is normally a relation such as $U(\Omega)$, not either entry evaluated at fixed coordinate radius.

For a circular-orbit invariant, the admissible class is narrower than “all smooth vectors.” The vector must be sufficiently regular on the worldline, respect helical symmetry, preserve reflection symmetry when used, and approach the identity rapidly enough at infinity to preserve the normalization of $t^\mu$ and $\phi^\mu$. It must also preserve the identification of the background mass. A transformation proportional to $t\,\partial_t$ rescales the asymptotic clock and changes both $U$ and $\Omega$; it is not part of this invariant class until the clock is renormalized. String singularities in a radiation gauge require a parity-regular averaging prescription rather than pointwise use.

This boundary qualification explains why two calculations can be locally gauge related yet initially report different “invariants.” They may use different asymptotic times, mass completions, or one-sided worldline limits. Converting those conventions is part of the comparison.

Circular redshift from the effective metric

Restrict to the conservative circular problem and use proper time $\tau$ of the effective metric $\tilde g$. It differs perturbatively from proper time in the background metric. A dissipative orbit slowly leaves the circular family and does not possess exact helical symmetry.

Let

$$k^\mu=t^\mu+\Omega\phi^\mu$$

be the helical Killing vector and write the circular four-velocity as $u^\mu=Uk^\mu$, where $U=u^t=dt/d\tau$ and $z=U^{-1}=d\tau/dt$. Normalization in the effective metric gives

$$U=\left(-\tilde g_{\mu\nu}k^\mu k^\nu\right)^{-1/2}.$$

At fixed physical frequency $\Omega$, expand $U=U_0+\varepsilon\Delta U$ and define

$$h_{uu}^{\mathrm R} =h_{\mu\nu}^{\mathrm R}u_0^\mu u_0^\nu.$$

Because $g_{\mu\nu}k^\mu k^\nu=-U_0^{-2}$ on the background circular geodesic, varying the normalization yields

$$\Delta U =\frac12U_0h_{uu}^{\mathrm R}.$$

The orbital displacement does not contribute at fixed $\Omega$ because the background circularity condition makes the relevant radial variation stationary. This is why one regularized scalar contraction contains the redshift correction even though individual components and the coordinate radius are gauge dependent.

For a helically symmetric gauge vector, the change in the contraction is

$$\Delta_\xi h_{uu}^{\mathrm R} =2u_0^\mu u_0^\nu\nabla_\mu\xi_\nu =2\frac{d}{d\tau}(u_0^\nu\xi_\nu) -2a_0^\nu\xi_\nu.$$

The background orbit is geodesic, $a_0^\nu=0$, and helical symmetry makes $u_0\cdot\xi$ constant along it. Hence $\Delta_\xi h_{uu}^{\mathrm R}=0$ within the stated gauge class. Detweiler develops $U(\Omega)$ and proves its restricted gauge independence in Detweiler 2008, §§ II–III, especially eqs. (37)–(42) and Appendix B.

A practical trap occurs in Lorenz gauge: the monopole perturbation can approach a nonzero constant in $h_{tt}$ at infinity. One must rescale the Lorenz time to the asymptotically normalized time and transform both $U$ and $\Omega$. Comparing an uncorrected Lorenz $U(\Omega)$ with a radiation-gauge value falsely diagnoses a gauge disagreement.

Periastron advance and two-frequency data

For a stable bound eccentric orbit in the conservative Schwarzschild problem, define the coordinate-time radial period and accumulated azimuth,

$$T_r=\int_0^{\mathcal T_r}u^t\,d\tau, \qquad \Delta\Phi=\int_0^{\mathcal T_r}u^\phi\,d\tau.$$

The fundamental frequencies and periastron factor are

$$\Omega_r=\frac{2\pi}{T_r}, \qquad \Omega_\phi=\frac{\Delta\Phi}{T_r}, \qquad K=\frac{\Omega_\phi}{\Omega_r}.$$

Under a smooth biperiodic gauge transformation that preserves the asymptotic coordinates, endpoint changes cancel over a complete radial cycle. The two frequencies and their ratio are invariant. At finite eccentricity the identity $K=\Omega_\phi/\Omega_r$ supplies no additional information once both frequencies are specified; useful extra data include the averaged redshift at those frequencies. A coordinate description $K(p,e)$ instead requires translating $p,e$ between gauges. For generic inclined Kerr motion, one radial cycle does not generally close the polar motion, so use a full torus or long-time average rather than these single-period integrals.

In the small-eccentricity limit one may write

$$\left(\frac{\Omega_r}{\Omega_\phi}\right)^2 =1-6x+\varepsilon\rho(x)+O(\varepsilon^2), \qquad x=\bigl[G(M+m)\Omega_\phi\bigr]^{2/3}.$$

Here the frequency parameter uses the total mass $M+m$, although the expansion parameter is $\varepsilon=m/M$. This is the convention of the cited periastron result. If one instead uses $y=(GM\Omega_\phi)^{2/3}$, then $x=y(1+\varepsilon)^{2/3}$ and the first-order coefficient becomes $\rho(y)-4y$. A mass-normalization change therefore shifts the coefficient even though the observable is unchanged. For finite eccentricity, the orbit-averaged redshift

$$\langle U\rangle =\frac{T_r}{\mathcal T_r}$$

provides another invariant function of the two frequencies. Barack and Sago derive the conservative periastron correction and the averaged-redshift construction in Barack and Sago 2011, § II.3, eqs. (21)–(22), and §§ III.2–III.5.

An independent-gauge check should calculate the same frequency relation in Lorenz gauge and in a completed Regge–Wheeler or no-string radiation gauge. Agreement must be tested after matching the asymptotic clock, total mass, low multipoles, and side-averaging prescription. A radiation-gauge calculation of circular redshift agreeing with Lorenz-gauge and post-Newtonian results is documented in Shah et al. 2011, §§ II–IV, especially Table I. The same comparison logic applies to $\rho(x)$.

Spin precession and tidal eigenvalues

Give the small body a test spin $S^\mu$ satisfying $S^\mu u_\mu=0$ and parallel transport it in $\tilde g_{\mu\nu}$, with spin small enough that its own force does not change the orbit. In a spatial frame carried around by the helical symmetry, choose positive azimuthal circulation and let $\omega_s$ be the magnitude of the spin’s rotation rate per unit proper time. The orbital rate on the same clock is $u^\phi$, so define

$$\psi=1-\frac{\Psi}{\Phi} =1-\frac{\omega_s}{u^\phi},$$

where $\Psi=\omega_s\mathcal T$ and $\Phi=u^\phi\mathcal T$ over a proper-time interval $\mathcal T$. Report this as $\psi(\Omega)$ with $\Omega=u^\phi/u^t$. The coordinate-time rotation rate is $\omega_s/u^t$, not $\omega_s$. For a Schwarzschild circular geodesic,

$$\psi_0=1-\sqrt{1-3GM/r} =\frac{3GM}{2r}+O\bigl((GM/r)^2\bigr),$$

which recovers weak-field geodetic precession and fixes the meaning of the subtraction from one. The regular-field correction and the invariant’s definition are given in Dolan et al. 2014, § II A–B, eqs. (3)–(7).

Tidal observables probe more derivatives. In an orthonormal frame along the effective-metric orbit, define

$$\mathcal E_{\alpha\beta} =\tilde R_{\alpha\mu\beta\nu}u^\mu u^\nu, \qquad \mathcal B_{\alpha\beta} =\frac12\tilde\epsilon_{\alpha\mu}{}^{\rho\sigma} \tilde R_{\rho\sigma\beta\nu}u^\mu u^\nu.$$

The eigenvalues of the spatial, symmetric trace-free tensors $\mathcal E$ and $\mathcal B$, expressed at fixed $\Omega$, are invariant under the same regular helically symmetric gauge class. They measure curvature rather than coordinate acceleration and expose information not contained in redshift. They also demand a regular field with two controlled derivatives; a puncture adequate for the self-force may be inadequate for tidal eigenvalues. Independent Lorenz- and Regge–Wheeler-gauge calculations of these invariants and their strong-field behavior are given by Dolan et al. 2015, §§ II–IV, especially eqs. (2.7)–(2.31).

Scattering data and asymptotic conditions

For an unbound conservative encounter, define incoming energy $E$, angular momentum $J$, and the total azimuthal deflection,

$$\chi(E,J)=\Delta\phi(E,J)-\pi.$$

The coordinate trajectory is gauge dependent, but $\chi$ is invariant if the gauge vector approaches the same asymptotic Poincaré frame at past and future infinity. A translation or rotation applied differently at the two ends changes the comparison frame and therefore the reported angle. With radiation reaction, incoming and outgoing energies differ; one must report the full dissipative scattering map or define a time-symmetric conservative angle using a declared prescription. The conservative/dissipative split is not interchangeable with simply deleting flux after the calculation.

Scattering observables are especially useful for comparison with post-Minkowskian amplitudes and effective-one-body models because no coordinate radius is needed. Near capture, however, $\chi$ becomes highly sensitive to initial data and perturbative corrections, so an apparently small local self-force can create a large angle shift. A first-order result should state its distance from the critical impact parameter.

The first law as an independent relation

For a conservative circular binary with asymptotically normalized helical symmetry, let $E_{\mathrm{ADM}}$ denote the total conservative ADM energy, distinct from the volume-wide mass sum $M=m_1+m_2$. The point-particle first law is

$$\delta E_{\mathrm{ADM}}-\Omega\,\delta J =z_1\,\delta m_1+z_2\,\delta m_2.$$

It implies the first integral

$$E_{\mathrm{ADM}}-2\Omega J=m_1z_1+m_2z_2.$$

These formulas connect local redshifts to global energy and angular momentum. Le Tiec, Blanchet, and Whiting derive them and their hypotheses in Le Tiec, Blanchet, and Whiting 2012, § II D, eqs. (2.43)–(2.44), and § III A, eqs. (3.7)–(3.8). At first order in the small mass ratio, a computed $z(\Omega)$ determines a corresponding piece of the binding energy. Agreement is a stringent check because redshift and energy are regularized through different local and asymptotic data.

This is not a law for a fully dissipative inspiral at an arbitrary instant. Exact helical symmetry and global asymptotic flatness are incompatible with outgoing radiation; the useful statement applies to the conservative sector or a controlled approximation. Spin, eccentricity, and tail nonlocality require generalized first laws with additional conjugate variables.

Checks and limitations

A credible observable calculation should pass four tests. It must reproduce the geodesic limit as $\varepsilon\to0$. It must agree in two independently implemented gauges after time, mass, completion, and worldline prescriptions are matched. Its redshift and binding-energy data must satisfy the appropriate first-law relation. Finally, its weak-field expansion should agree with independently derived post-Newtonian coefficients over a range where neglected terms are demonstrably smaller.

No single invariant reconstructs the entire dynamics. Circular redshift does not determine eccentric or dissipative motion; periastron data do not fix all spin couplings; tidal eigenvalues require stronger differentiability; and scattering data probe an unbound sector. Near the light ring, separatrix, or capture threshold, first-order coefficients can diverge even while the mass ratio is small, indicating a nonuniform expansion rather than a failure of regularization.

The regular fields entering these quantities can be obtained with Mode-Sum Regularization or Effective-Source and Puncture Methods. At the next perturbative order, defining invariant families also requires the second-order worldline and gauge map described on Second-Order Self-Force and Nonlinear Regularization.

Exercise

Let $\xi^\mu$ be smooth, helically symmetric, asymptotically vanishing, and evaluated on a background circular geodesic. Show that $h_{uu}^{\mathrm R}$ is unchanged by the associated gauge transformation. Then explain why the conclusion fails for $\xi^\mu=\alpha t\,\delta^\mu_t$.

Solution

The transformation gives

$$\Delta_\xi h_{uu}^{\mathrm R} =2u^\mu u^\nu\nabla_\mu\xi_\nu =2\frac{d}{d\tau}(u^\nu\xi_\nu) -2a^\nu\xi_\nu.$$

The background orbit is geodesic, so $a^\nu=0$. Helical symmetry means the scalar $u\cdot\xi$ is constant along an orbit of $k^\mu$, hence its proper-time derivative vanishes. Therefore $\Delta_\xi h_{uu}^{\mathrm R}=0$, and so does the induced change in $\Delta U=\tfrac12U_0h_{uu}^{\mathrm R}$ at fixed $\Omega$.

For $\xi^\mu=\alpha t\,\delta^\mu_t$, the vector is not helically symmetric and does not vanish at infinity. It rescales the asymptotic time coordinate. Consequently $u^t$ and the numerical value of $\Omega=d\phi/dt$ both change. Treating it as an admissible gauge transformation while holding the old clock fixed violates the hypotheses; one must first restore the shared asymptotic normalization.

References

• Barack, Leor, and Norichika Sago. “Beyond the Geodesic Approximation: Conservative Effects of the Gravitational Self-Force in Eccentric Orbits around a Schwarzschild Black Hole.” Physical Review D 83, 084023 (2011). doi:10.1103/PhysRevD.83.084023.
• Detweiler, Steven. “A Consequence of the Gravitational Self-Force for Circular Orbits of the Schwarzschild Geometry.” Physical Review D 77, 124026 (2008). doi:10.1103/PhysRevD.77.124026.
• Dolan, Sam R., Niels Warburton, Abraham I. Harte, Alexandre Le Tiec, Barry Wardell, and Leor Barack. “Gravitational Self-Torque and Spin Precession in Compact Binaries.” Physical Review D 89, 064011 (2014). doi:10.1103/PhysRevD.89.064011.
• Dolan, Sam R., Patrick Nolan, Adrian C. Ottewill, Niels Warburton, and Barry Wardell. “Tidal Invariants for Compact Binaries on Quasi-Circular Orbits.” Physical Review D 91, 023009 (2015). doi:10.1103/PhysRevD.91.023009.
• 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:10.1103/PhysRevD.85.064039.
• Shah, Abhay G., Tobias S. Keidl, John L. Friedman, Dong-Hoon Kim, and Larry R. Price. “Conservative, Gravitational Self-Force for a Particle in Circular Orbit around a Schwarzschild Black Hole in a Radiation Gauge.” Physical Review D 83, 064018 (2011). doi:10.1103/PhysRevD.83.064018.