EOB Potentials and Effective Metrics

The conservative EOB Hamiltonian packages two-body information into a deformed Schwarzschild geometry plus terms that are genuinely nongeodesic. The central objects are radial potentials such as $A$ and $\bar D$, together with a momentum potential $Q$. Their individual coefficients depend on the EOB gauge, whereas circular energies, periastron advance, and scattering angles do not. This page defines one standard nonspinning gauge and shows which invariant data determine which combinations of its potentials.

Required background. Presymplectic Potentials and Currents supplies Hamiltonian flow, action variables, and canonical equivalence.

Helpful background. The Einstein Equation supplies the underlying relativistic dynamics; Weak Metric Perturbations and Gauge Freedom supplies weak-field matching; Black-Hole Perturbation Theory supplies the exact small-mass-ratio limit; and Relativistic Two-Body Dynamics supplies the PN, PM, and self-force regimes being combined.

Effective geometry and mass shell

Let $M=m_1+m_2$, $\mu=m_1m_2/M$, and $\nu=\mu/M$. Use dimensionless inverse radius and canonical momenta

$$u=\frac{GM}{r},\qquad p_r=\frac{P_r}{\mu},\qquad j=\frac{P_\phi}{GM\mu}.$$

For nonspinning motion in the equatorial plane, introduce the effective line element

$$ds_{\rm eff}^2 =-A(u;\nu)\,dt^2 +\frac{dr^2}{A(u;\nu)\bar D(u;\nu)} +r^2d\Omega^2.$$

This equation defines the convention $\bar D=(AB)^{-1}$ if the radial metric coefficient is called $B$. Some papers instead use $D=AB=1/\bar D$; a sign or coefficient comparison is meaningless until that choice is stated. The effective mass shell can be written

$$\widehat H_{\rm eff}^2 =A\left[ 1+j^2u^2+A\bar D\,p_r^2+Q(u,p_r;\nu) \right],$$

where $\widehat H_{\rm eff}=H_{\rm eff}/\mu$. The potential $Q$ contains nongeodesic terms at least quartic in radial momentum. Defining the tortoise momentum $p_{r_*}=A\sqrt{\bar D}\,p_r$ gives the equivalent form

$$\widehat H_{\rm eff}^2 =p_{r_*}^2+A\left[1+j^2u^2+Q\right].$$

The real Hamiltonian is obtained from the EOB energy map, not by identifying $\widehat H_{\rm eff}$ with the real binding energy.

At 3PN order, one commonly used EOB gauge has

$$A(u;\nu) =1-2u+2\nu u^3+a_4\nu u^4+O(u^5),$$ $$\bar D(u;\nu) =1+6\nu u^2+2(26-3\nu)\nu u^3+O(u^4),$$

with

$$a_4=\frac{94}{3}-\frac{41\pi^2}{32}.$$

In the same gauge the leading nongeodesic term is proportional to

$$Q=2\nu(4-3\nu)u^2p_r^4+\cdots.$$

These coefficients follow by canonically matching the 3PN two-body Hamiltonian to the EOB mass shell Damour, Jaranowski, and Schäfer 2000, § IV, Eqs. (4.28)–(4.36), PDF. The ellipses are important: higher PN orders introduce logarithms, additional powers of $p_r$, and, from 4PN onward, conservative hereditary effects that are not globally represented by a simple local metric without an additional prescription.

What is geometry and what is gauge?

Calling $A$ and $\bar D$ “metric potentials” is useful but can mislead. At $\nu=0$,

$$A(u;0)=1-2u,\qquad \bar D(u;0)=1,\qquad Q(u,p_r;0)=0,$$

so the effective dynamics is exactly geodesic motion in Schwarzschild spacetime. At finite $\nu$, however, the effective metric is a representation of two-body dynamics, not a vacuum solution of the Einstein equation. A zero of a resummed $A$, for example, is an effective-coordinate feature; it is not by itself the event horizon of the binary spacetime.

A time-independent canonical transformation generated by $g(\mathbf r,\mathbf p)$ changes the phase-space coordinates and moves terms among $A$, $\bar D$, and $Q$. Use the passive Lie-flow coordinate map $z'=z+\{z,g\}+\tfrac12\{\{z,g\},g\}+\cdots$ for each canonical coordinate $z$. Expressing the same Hamiltonian in the new variables then gives

$$H' = H-\{H,g\}+\frac{1}{2}\{\{H,g\},g\}+\cdots,$$

where the right-hand side is evaluated in the new coordinates. Reversing the generator sign reverses the first bracket, so the coordinate map must always accompany this formula; see canonical transformations. Therefore a coefficient of $p_r^4$ is not an observable in isolation. The transformation must preserve the symplectic form and be accompanied by the corresponding change of variables. Afterward, invariant relations—such as the circular binding energy $E(\Omega)$, periastron advance $K(E,J)$, or conservative scattering angle $\chi(E,J)$—remain unchanged at the retained order. The original EOB derivation makes the distinction between the effective metric and the canonical real-to-effective dictionary explicit Buonanno and Damour 1999, §§ V–VI.

The $Q$ potential is not an optional numerical correction. A purely geodesic effective metric has too few independent functions to reproduce generic 3PN motion while retaining the simple energy map. Choosing a gauge in which $Q$ vanishes for circular motion is valuable because it cleanly separates circular and noncircular information, but a different regular canonical gauge can redistribute the same physics.

How PN, PM, and self-force information enter

The three approximation schemes constrain complementary slices of the potentials.

Post-Newtonian matching expands in $u\sim v^2\ll1$ at arbitrary $\nu$. It determines Taylor coefficients and logarithms order by order. The displayed 3PN series is an example. A finite PN series fixes only the weak-field expansion; it does not uniquely select its strong-field continuation.

Post-Minkowskian matching expands in $G$ at unrestricted asymptotic velocity. The conservative scattering angle $\chi(E,J)$ constrains an energy-dependent radial mass shell. One may choose a “post-Schwarzschild” gauge that places PM information in $Q$, or transform it into other EOB potentials. The invariant statement is equality of scattering data, not equality of gauge-dependent radial coefficients Damour 2016, §§ IV–VI.

Gravitational self-force information expands at fixed strong-field $u$ in $\nu$. Schematically,

$$A=1-2u+\nu a_1(u)+\nu^2a_2(u)+\cdots,$$ $$\bar D=1+\nu \bar d_1(u)+\cdots, \qquad Q=\nu q_1(u,p_r)+\cdots.$$

For exactly circular orbits, $p_r=0$, so redshift and circular binding-energy data chiefly determine $a_1(u)$. The periastron advance in the infinitesimal-eccentricity limit adds information about $\bar d_1$. It does not determine $q_1$: a term quartic in $p_r$ has vanishing second derivatives at $p_r=0$ and cannot enter the linear radial-oscillation frequency. Finite-eccentricity information or scattering is needed for those higher radial-momentum terms. This is why precise circular and infinitesimally eccentric data do not by themselves determine generic-orbit dynamics. Fourth-PN analyses combine analytic PN and self-force information to determine the main radial potential while tracking infrared and hereditary terms Bini and Damour 2013, §§ II–IV.

Circular orbits expose an invariant combination

Set $p_r=0$. Then

$$\widehat H_{\rm eff}^2=A(u)\left(1+j^2u^2\right).$$

A circular orbit is an extremum at fixed $j$, so

$$0=A'(1+j^2u^2)+2A j^2u.$$

Solving for the angular momentum gives

$$j^2(u) =-\frac{A'(u)} {u\left[2A(u)+uA'(u)\right]}.$$

Thus circular conservative dynamics directly probes $A$, while $\bar D$ and $Q$ disappear in this gauge. For the Schwarzschild limit $A=1-2u$,

$$j^2=\frac{1}{u(1-3u)}, \qquad \widehat H_{\rm eff} =\frac{1-2u}{\sqrt{1-3u}}.$$

The minimum of $j^2(u)$ occurs at $u=1/6$, the Schwarzschild innermost stable circular orbit. Combining this effective energy with the nonlinear map, differentiating the real Hamiltonian with respect to $P_\phi$, and eliminating $u$ produces the gauge-invariant relation $E(\Omega)$. At finite $\nu$, the location of an extremum in the chosen EOB radius can shift under a canonical transformation, whereas its predicted energy and frequency are the meaningful comparison.

For small radial perturbations, write $\delta\dot r=H_{P_rP_r}\delta P_r$ and $\delta\dot P_r=-H_{rr}\delta r$, with derivatives of the real Hamiltonian evaluated on the circular orbit. Thus

$$\Omega_r^2=H_{P_rP_r}H_{rr}, \qquad K=\frac{\Omega_\phi}{\Omega_r}.$$

The coefficient $H_{P_rP_r}$ contains $\bar D$, whereas $Q=O(p_r^4)$ drops out. Its contribution first appears in nonlinear radial oscillations. In the Schwarzschild limit the result is $K=(1-6u)^{-1/2}$ on the stable branch $0\lt u\lt1/6$. For finite eccentricity or hyperbolic motion, the radial action can sample all three potentials. These provide independent checks: a model fitted only to circular energy can still fail the periastron advance or scattering angle.

Validity and breakdown

The Taylor forms above require $u\ll1$, momenta within PN power counting, conservative dynamics, and compact objects adequately described as point masses. Resumming the series may extend useful behavior toward merger, but it does not enlarge the information content of the known coefficients. A candidate completion should recover Schwarzschild at $\nu=0$, reproduce every retained PN or PM coefficient when re-expanded, remain real on the intended phase-space domain, and avoid poles before the physical turning point. Spin, tides, absorption, radiation reaction, and merger-ringdown matching introduce additional structures developed elsewhere.

Common pitfalls

Treating every potential as observable. $A$, $\bar D$, and $Q$ are gauge-fixed Hamiltonian functions. Test them through invariant relations and through a fully specified canonical map.

Using circular data to claim generic-orbit control. Circular motion sets $p_r=0$. It cannot determine all radial-momentum terms, so eccentric or scattering information is indispensable.

Equating an effective horizon with a binary horizon. The effective geometry organizes dynamics. Its coordinate features are not automatically geometric features of the full two-black-hole spacetime.

Exercise

Starting from $A=1-2u$, derive the Schwarzschild circular angular momentum and locate the loss of stability using only the EOB circularity condition.

Solution

Here $A'=-2$. Substitution into the general formula gives

$$j^2 =-\frac{-2}{u[2(1-2u)-2u]} =\frac{1}{u(1-3u)}.$$

Differentiate:

$$\frac{d j^2}{du} =-\frac{1-6u}{u^2(1-3u)^2}.$$

The derivative vanishes at $u=1/6$, or $r=6GM$. To distinguish the two branches, differentiate the squared effective potential at fixed $j$ and then use circularity:

$$\left.\frac{\partial^2}{\partial u^2} \left[(1-2u)(1+j^2u^2)\right]\right|_{\rm circ} =2j^2(1-6u).$$

The radial kinetic coefficient is positive outside the horizon, so the branch $0\lt u\lt1/6$ is stable and $1/6\lt u\lt1/3$ is unstable. The denominator of $j^2$ diverges at $u=1/3$, the null circular orbit; the timelike result cannot be continued through it. At $u=1/6$, $\widehat H_{\rm eff}=\sqrt{8/9}$, the test-mass ISCO energy.

References

• Bini, Donato, and Thibault Damour. “Analytical Determination of the Two-Body Gravitational Interaction Potential at the Fourth Post-Newtonian Approximation.” Physical Review D 87, 121501(R) (2013). DOI. Open PDF.
• Buonanno, Alessandra, and Thibault Damour. “Effective One-Body Approach to General Relativistic Two-Body Dynamics.” Physical Review D 59, 084006 (1999). DOI. Open PDF.
• Damour, Thibault, Piotr Jaranowski, and Gerhard Schäfer. “On the Determination of the Last Stable Orbit for Circular General Relativistic Binaries at the Third Post-Newtonian Approximation.” Physical Review D 62, 084011 (2000). DOI. Open PDF.
• Damour, Thibault. “Gravitational Scattering, Post-Minkowskian Approximation and Effective One-Body Theory.” Physical Review D 94, 104015 (2016). DOI. Open PDF.