The EOB Energy Map

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The effective-one-body (EOB) construction does not identify the two-body Hamiltonian with a test-particle Hamiltonian term by term. Its decisive step is a nonlinear dictionary between the total center-of-mass energy of the real binary and the energy of an effective particle of mass $\mu$ . That dictionary preserves the relativistic scattering energy, reproduces the post-Newtonian (PN) spectrum, and has the correct test-mass limit. This page derives the map for the conservative, nonspinning problem; spin and dissipation are added on later pages without changing the basic distinction between real and effective energies.

Required background. Presymplectic Potentials and Currents supplies Hamiltonian flow, action variables, and the role of canonical transformations.

Helpful background. The Einstein Equation fixes the relativistic field dynamics behind the Hamiltonian; Weak Metric Perturbations and Gauge Freedom supplies the weak-field expansion; Black-Hole Perturbation Theory supplies the exact test-body benchmark; and Relativistic Two-Body Dynamics places EOB among PN, post-Minkowskian, and self-force regimes.

The relativistic energy dictionary

Let

M=m_1+m_2,\qquad \mu=\frac{m_1m_2}{M},\qquad \nu=\frac{\mu}{M}=\frac{m_1m_2}{M^2},

with $0\lt \nu\leq 1/4$ . In the center-of-mass frame, denote the real total conservative energy, including both rest masses, by $H_{\rm real}$ . The effective Hamiltonian $H_{\rm eff}$ includes the effective particle’s rest energy $\mu$ . It is useful to define

h\equiv \frac{H_{\rm real}}{M}, \qquad \widehat H_{\rm eff}\equiv\frac{H_{\rm eff}}{\mu}.

The EOB energy map is

h^2=1+2\nu\left(\widehat H_{\rm eff}-1\right),

or, equivalently,

H_{\rm real} =M\sqrt{1+2\nu\left(\widehat H_{\rm eff}-1\right)}.

This relation emerged from matching the real and effective action variables and PN energy levels in the original construction Buonanno and Damour 1999, §§ II–IV. Its especially transparent scattering interpretation begins with asymptotic four-momenta. With the site’s mostly-plus metric, $p_i^2=-m_i^2$ , and

s\equiv-(p_1+p_2)^2=H_{\rm real}^2.

The invariant relative Lorentz factor is

\gamma\equiv-\frac{p_1\cdot p_2}{m_1m_2} =\frac{s-m_1^2-m_2^2}{2m_1m_2}.

Substituting $s=M^2h^2$ and $m_1m_2=\nu M^2$ gives

\gamma =1+\frac{h^2-1}{2\nu}.

Choosing $\widehat H_{\rm eff}=\gamma$ for an asymptotic scattering state yields the same map as the PN spectral matching and makes exchange symmetry explicit. This is a dynamical dictionary motivated by invariant kinematics, not a consequence of the definition of $s$ alone: the effective Hamiltonian must still reproduce the real scattering angle at the mapped energy and the same angular momentum. See Buonanno and Damour 1999, § IV, Eqs. (4.25)–(4.26), PDF and Damour 2016, §§ II–III, Eqs. (5)–(19).

From effective energy to binding energy

For orbital dynamics one usually evolves the binding Hamiltonian rather than the total rest energy. Define

H_{\rm EOB}\equiv H_{\rm real}-M, \qquad \widehat H_{\rm EOB}\equiv\frac{H_{\rm EOB}}{\mu}.

The energy map then becomes

\widehat H_{\rm EOB} =\frac{1}{\nu} \left[ \sqrt{1+2\nu\left(\widehat H_{\rm eff}-1\right)}-1 \right].

Writing $z=\widehat H_{\rm eff}-1$ , its small-binding-energy expansion is

\widehat H_{\rm EOB} =z-\frac{\nu}{2}z^2+\frac{\nu^2}{2}z^3 -\frac{5\nu^3}{8}z^4+\cdots.

The first term is the ordinary reduced one-body energy; the nonlinear terms are finite-mass-ratio corrections fixed by the map. They must not be confused with corrections to the effective metric itself. At any chosen PN order, information is distributed between this universal composition rule, the effective potentials, and momentum-dependent terms. Canonical transformations can rearrange the latter two, but they cannot alter gauge-invariant energy–frequency or scattering-angle relations.

Why not use the identity $H_{\rm eff}-\mu=H_{\rm real}-M$ ? At 2PN order that naive identification overconstrains a purely energy-independent effective metric if one also requires the effective mass to equal $\mu$ and the linearized metric to be Schwarzschild with mass $M$ . Allowing a nonlinear energy relation resolves the mismatch. With $e=(H_{\rm real}-M)/\mu$ , the original matching fixes the real-to-effective relation

\widehat H_{\rm eff}-1=e+\frac{\nu}{2}e^2

through that order: the coefficient of $e^2$ is $\nu/2$ and that of $e^3$ vanishes Buonanno and Damour 1999, § IV, Eqs. (4.23)–(4.25), PDF. Inverting this polynomial produces the square-root map and its negative quadratic coefficient in $z$ above. Keeping the direction of the map explicit avoids confusing those opposite signs.

Hamiltonian composition and equations of motion

The map is a composition of functions, not an additional equation of motion. Once an effective Hamiltonian $\widehat H_{\rm eff}(\mathbf r,\mathbf p)$ has been selected, the conservative real Hamiltonian is obtained by the square root above. Hamilton’s equations use derivatives of the real binding Hamiltonian:

\dot r^i=\frac{\partial H_{\rm EOB}}{\partial p_i}, \qquad \dot p_i=-\frac{\partial H_{\rm EOB}}{\partial r^i}.

The chain rule supplies a common energy-dependent factor,

\frac{\partial H_{\rm EOB}}{\partial H_{\rm eff}} =\frac{1}{h}.

Consequently, effective and real trajectories have the same phase-space level sets in the conservative problem, while their Hamiltonian time parametrizations differ by the factor $1/h$ . This observation is useful when translating an effective radial action into a real orbital frequency or scattering angle. For scattering, the angle is invariant under this monotonic energy relabeling when the same physical $H_{\rm real}$ and angular momentum are held fixed; the explicit PM correspondence is developed in Damour 2016, §§ III–V, PDF.

The boundary conditions also matter. The conservative map presumes an asymptotically flat two-body problem with well-defined incoming or bound-state energy and with radiation reaction omitted. For a dissipative inspiral, $H_{\rm EOB}$ changes along the orbit, but the instantaneous conservative Hamiltonian still uses the same algebraic map while a separate force supplies the loss. Incoming-radiation boundary data, nonlocal hereditary Hamiltonians, matter effects, or modified gravity may require extra state variables or a revised dictionary; the square root is not by itself a proof that such systems share the same effective description.

Three invariant checks

Newtonian limit. In a weak field with small momentum,

z=\widehat H_{\rm eff}-1 =\frac{\mathbf p^2}{2\mu^2}-\frac{GM}{r} +O(v^4).

Because $z=O(v^2)$ , the nonlinear term $-\nu z^2/2$ begins one PN order later. Hence $H_{\rm EOB}/\mu=z+O(v^4)$ , giving the reduced Newtonian Hamiltonian.

Test-mass limit. Hold $M$ and the background geometry fixed while $\nu\to0$ . Expanding the square root gives

H_{\rm real}=M+\mu\left(\widehat H_{\rm eff}-1\right)+O(\mu\nu),

so the binary binding energy becomes the energy of a test body in the background. The map introduces no spurious finite correction in this limit.

Weak-field circular check. For a Schwarzschild effective geometry, a circular geodesic has

\widehat H_{\rm eff} =\frac{1-2u}{\sqrt{1-3u}} =1-\frac{u}{2}+\frac{3u^2}{8}+O(u^3), \qquad u=\frac{GM}{r}.

Inserting this into the energy map yields

\widehat H_{\rm EOB} =-\frac{u}{2}+\frac{3-\nu}{8}u^2+O(u^3).

The $-\nu u^2/8$ term comes from the energy map rather than from deforming Schwarzschild. To finish the invariant check, use the chain-rule factor $1/h$ : here $GM\Omega=u^{3/2}/h$ , so

x=(GM\Omega)^{2/3}=u h^{-2/3} =u+\frac{\nu}{3}u^2+O(u^3).

Inverting gives $u=x-\nu x^2/3+O(x^3)$ , and hence

\widehat H_{\rm EOB}(x) =-\frac{x}{2}+\left(\frac38+\frac{\nu}{24}\right)x^2 +O(x^3).

This is the 1PN circular binding energy. The first deformation $A-(1-2u)=2\nu u^3+\cdots$ enters only at the next order. The calculation shows why comparing coefficients at an untransformed coordinate radius would give the wrong mass-ratio dependence.

Common pitfalls

Dropping rest masses inconsistently. The square-root formula maps total energies. If $M$ and $\mu$ are subtracted on only one side, the Newtonian and test-mass limits appear to fail.

Calling the map an exact solution. The algebraic dictionary is structurally powerful, but its predictive content depends on the PN, PM, self-force, or calibrated information inserted into $\widehat H_{\rm eff}$ . A finite-order effective Hamiltonian still has a finite domain of accuracy.

Comparing gauge-dependent radii. Canonical coordinates may differ between PN and EOB descriptions. Compare binding energy versus frequency, periastron advance, redshift invariants, or a scattering angle rather than bare coefficients at an untransformed radius.

Exercise

Suppose an effective state has $\widehat H_{\rm eff}=1+z$ , where $\lvert z\rvert\ll1$ . Derive the real reduced binding energy through $z^3$ , and show directly that the equal-mass and test-mass limits behave regularly.

Solution

Using $H_{\rm EOB}/\mu=\nu^{-1}[\sqrt{1+2\nu z}-1]$ and the binomial series,

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

Therefore

\frac{H_{\rm EOB}}{\mu} =z-\frac{\nu}{2}z^2+\frac{\nu^2}{2}z^3+O(z^4).

At equal masses, $\nu=1/4$ , the result is $z-z^2/8+z^3/32+\cdots$ , with no singular coefficient. In the test-mass limit $\nu\to0$ , every nonlinear correction vanishes and $H_{\rm EOB}/\mu\to z$ . The apparent $1/\nu$ in the exact expression is removable because the numerator is itself $O(\nu)$ .

References

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. “Gravitational Scattering, Post-Minkowskian Approximation and Effective One-Body Theory.” Physical Review D 94, 104015 (2016). DOI. Open PDF.