Radiation-Reaction Potentials and Effective Actions

Retarded boundary conditions make radiation reaction irreversible, but an ordinary action varied with one history produces time-symmetric equations. The resolution is to formulate an initial-value problem with doubled histories. Integrating out the radiation field on a closed time path yields a causal in-in effective action; varying its difference field produces the local and hereditary reaction force.

Helpful background. The Einstein Equation fixes the bulk action, Weak Metric Perturbations and Gauge Freedom supplies radiative fields, Presymplectic Potentials and Currents supplies variational structure, Partial Differential Equations and Green Functions supplies retarded solutions, Gravitational Radiation and Waveforms supplies flux normalization, and Relativistic Two-Body Dynamics supplies source multipoles.

Why one history is insufficient

For a linear field $\phi$ coupled to a source $J$,

$$S[\phi,J] =\frac12\int \phi D\phi+\int J\phi.$$

In this sign convention $D\phi=-J$ and the retarded response kernel is defined by $DG_R=-\delta$, so that $\phi=G_RJ$. Keeping this operator/source sign together with the kernel prevents an overall-sign convention from being mistaken for a physical disagreement.

Eliminating $\phi$ in a conventional in-out action gives a symmetric kernel: the variation cannot select a purely retarded response because mixed second functional derivatives commute. A retarded equation is an initial-value statement, whereas the usual stationary action treats endpoint data symmetrically. The failure of the regular in-out action and the doubled initial-value construction are shown explicitly in Galley and Tiglio 2009, §§ II.C–III.C, pp. 7–11, especially Eqs. (2.32) and (3.17)–(3.33).

Double the histories, $(\phi_1,J_1)$ and $(\phi_2,J_2)$, and define

$$J_+=\frac12(J_1+J_2), \qquad J_-=J_1-J_2.$$

After the field is integrated out, the classical influence action contains

$$S_{\mathrm{IF}} =\int d^4x\,d^4x'\, J_-(x)G_R(x,x')J_+(x') +\frac{i}{2} \int d^4x\,d^4x'\, J_-(x)G_H(x,x')J_-(x').$$

$G_R$ is retarded. Here the state or noise kernel is the half-anticommutator

$$G_H(x,x') =\frac12\left\langle\left\{\phi(x),\phi(x')\right\}\right\rangle,$$

which fixes the coefficient $i/2$ above. Galley and Tiglio instead denote the full anticommutator by $D_H=2G_H$; their corresponding normalization is given in Galley and Tiglio 2009, § III.C, p. 11, Eqs. (3.28)–(3.33), and Appendix C, pp. 18–19, Eqs. (C5), (C9), and (C11)–(C12). Deterministic classical reaction follows from

$$\left. \frac{\delta S_{\mathrm{eff}}}{\delta x_-} \right|_{x_-=0}=0.$$

The physical limit is taken only after variation. Setting the two histories equal earlier erases the dissipative term.

The leading quadrupole reaction

For a slow compact source, use the symmetric trace-free Newtonian mass quadrupole $I_{ij}=\sum_A m_A(x_A^i x_A^j-\delta_{ij}r_A^2/3)$ in the center-of-mass frame. A parenthetical superscript denotes an ordinary time derivative, $I_{ij}^{(n)}=d^nI_{ij}/dt^n$. Define $I_+=(I_1+I_2)/2$ and $I_-=I_1-I_2$, with the same history convention used above. Expanding the retarded kernel in the near zone produces the leading local influence term

$$S_{\mathrm{RR}} =-\frac{G}{5} \int dt\, I_-^{ij}(t)I_{+\,ij}^{(5)}(t),$$

in the common STF quadrupole normalization and with $c=1$. For these definitions the minus sign is fixed: varying with respect to the difference coordinates yields the leading Burke–Thorne acceleration

$$a_{A,\mathrm{RR}}^i =-\frac{2G}{5}x_A^j I_{ij}^{(5)}.$$

The primary closed-time-path derivation is Galley and Tiglio 2009, § IV.B, pp. 14–15, Eqs. (4.20)–(4.24); the equivalent compact multipole form is Galley et al. 2016, § 2, pp. 5–6, Eqs. (2.8)–(2.16). Defining $I_-=I_2-I_1$ would reverse the displayed action sign but not the physical acceleration after the variational convention is translated. The force is odd under time reversal: replacing retarded by advanced boundary conditions reverses its dissipative part.

Contracting the force with velocities gives $\dot E_{\mathrm{mech}}=-(G/5)I_{ij}^{(1)}I_{ij}^{(5)}$. The factor of two is absorbed because differentiating the symmetric quadrupole produces two identical position–velocity terms. Two integrations by parts are encoded in the exact identity

$$I_{ij}^{(1)}I_{ij}^{(5)} =I_{ij}^{(3)}I_{ij}^{(3)} +\frac{d}{dt}\left(I_{ij}^{(1)}I_{ij}^{(4)} -I_{ij}^{(2)}I_{ij}^{(3)}\right).$$

Thus this representative has $E_{\mathrm{Schott}}=(G/5)(I_{ij}^{(1)}I_{ij}^{(4)}-I_{ij}^{(2)}I_{ij}^{(3)})$ and obeys

$$\dot E_{\mathrm{mech}} =-\frac{G}{5} I_{ij}^{(3)}I_{ij}^{(3)} -\dot E_{\mathrm{Schott}}.$$

The first term is minus the leading quadrupole flux; the second depends on near-zone variables. Averaging a periodic source eliminates the total derivative and reproduces

$$\left\langle\dot E_{\mathrm{mech}}\right\rangle =-\frac{G}{5} \left\langle I_{ij}^{(3)}I_{ij}^{(3)} \right\rangle.$$

This power check fixes the sign and the fifth derivative in the effective action. Galley et al. derive the Schott shift and the nonpositive averaged quadrupole balance in the same Galley et al. 2016, § 2, pp. 5–6, Eqs. (2.11)–(2.16). The in-in EFT derivation and its extensions are reviewed in Porto 2016, §§7.3–7.7, pp. 52–68 and Levi 2020, §5.1, pp. 42–54.

Potentials, actions, and equivalence

A near-zone radiation-reaction potential and an in-in action can yield different-looking accelerations. Contact transformations and total derivatives move terms between the force and Schott energy. They are equivalent if the transformed equations give the same invariant secular evolution and horizon-plus-infinity balance.

The local derivative expansion assumes source variation slow compared with the radiation kernel’s short scales. Curvature backscatter produces nonanalytic frequency dependence and a hereditary tail that cannot be represented by finitely many time derivatives. Internal absorption likewise requires matched retarded worldline correlators rather than a real local coefficient.

The physical prescription must state:

1. retarded, advanced, or time-symmetric boundary conditions;
2. in-in versus in-out histories;
3. the multipole normalization and subtraction scheme;
4. which conservative time-symmetric part is retained;
5. the Schott convention used in balance.

Without these declarations, a sign comparison between two reaction potentials is ambiguous.

A numerical check should evaluate the work done by the derived force and the independently computed radiative flux on the same periodic orbit. Their difference must be a resolved total derivative whose cycle average converges to zero. Replacing $G_R$ by $G_A$ should reverse the dissipative power while leaving the time-symmetric conservative part unchanged.

Exercises

Show that the deterministic in-in term vanishes before variation in the physical limit but still affects the equations.

Solution

The term is proportional to $J_-$, so setting $J_-=0$ makes the action value zero. Its derivative with respect to $J_-$ is

$$\frac{\delta S_{\mathrm{IF}}}{\delta J_-(x)} =\int d^4x'\,G_R(x,x')J_+(x'),$$

which remains nonzero when the physical limit is taken afterward. Variation must therefore precede identifying the histories.

Why is the averaged quadrupole power nonpositive for the orbit?

Solution

The contraction $I_{ij}^{(3)}I_{ij}^{(3)}$ is a sum of real squares in the source frame and is nonnegative. With the stated flux sign, its coefficient in $\langle\dot E_{\mathrm{mech}}\rangle$ is negative. The averaged Schott derivative vanishes for periodic motion.

References

• Galley, Chad R., Adam K. Leibovich, Rafael A. Porto, and Andreas Ross. “The Tail Effect in Gravitational Radiation-Reaction: Time Non-Locality and Renormalization Group Evolution.” Physical Review D 93 (2016): 124010. DOI.
• Galley, Chad R., and Manuel Tiglio. “Radiation Reaction and Gravitational Waves in the Effective Field Theory Approach.” Physical Review D 79 (2009): 124027. DOI.
• Levi, Michèle. “Effective Field Theories of Post-Newtonian Gravity: A Comprehensive Review.” Reports on Progress in Physics 83 (2020): 075901. DOI.
• Porto, Rafael A. “The Effective Field Theorist’s Approach to Gravitational Dynamics.” Physics Reports 633 (2016): 1–104. DOI.