Dissipative Worldline Degrees of Freedom

A local worldline action with real Wilson coefficients describes reversible finite-size response, but it cannot by itself absorb energy. Horizon heating, viscosity, and internal-mode damping require additional worldline degrees of freedom whose retarded correlation functions encode what the long-distance theory cannot resolve. The result is still an effective theory: the microscopic interior is replaced by causal response data matched at frequencies well below the inverse body size.

Required background. The Gibbons-Hawking-York Boundary Term supplies the gravitational variational setting, and Weak Metric Perturbations and Gauge Freedom supplies linear response and gauge translation.

Helpful background. The Einstein Equation fixes the bulk dynamics, Presymplectic Potentials and Currents clarifies fluxes, and Black-Hole Perturbation Theory supplies the absorption data used in matching.

Internal operators on the worldline

Let $e^\mu{}_a(\tau)$ be a body-fixed orthonormal frame and let $E_{ab}$ and $B_{ab}$ denote the electric- and magnetic-type tidal tensors in that frame. A nonspinning body’s leading quadrupolar dissipative couplings may be written schematically as

$$S_{\mathrm{int}} =-\frac12\int d\tau \left[ Q_E^{ab}(\tau)E_{ab}(\tau) +Q_B^{ab}(\tau)B_{ab}(\tau) \right].$$

The symmetric trace-free operators $Q_E^{ab}$ and $Q_B^{ab}$ are not classical multipole coefficients assigned in advance. They represent unresolved internal degrees of freedom. Their correlation functions are matched to the object’s absorption probabilities or retarded response. Rotational invariance in the object’s rest frame reduces the two-point function to a scalar response times the symmetric trace-free projector,

$$\chi_{R,E}^{ab,cd}(\omega) =\mathcal P_{\mathrm{STF}}^{ab,cd}\,\chi_{R,E}(\omega),$$

and similarly in the magnetic sector. Spin, anisotropy, or a material preferred direction enlarges this tensor basis.

It is useful to absorb the factor in the interaction into a generalized force $F_{ab}=-E_{ab}/2$. With the operational convention

$$\langle Q_E^{ab}(\omega)\rangle =\chi_{R,E}^{ab,cd}(\omega)F_{cd}(\omega),$$

causality requires $\chi_R(\tau)=0$ for $\tau\lt 0$. Consequently $\chi_R(\omega)$ is analytic in the upper half of the complex $\omega$-plane and its real and imaginary parts obey dispersion relations. The real, even part controls conservative response; the imaginary, odd part controls absorption. Changing the sign used to define $F_{ab}$ changes the displayed response sign but not any absorbed power.

With the induced-multipole convention $Q_E^{ab}=-\lambda_E^{\rm ret}E^{ab}$ used in the tides chapter, the normalizations above imply

$$\lambda_E^{\rm ret}=\frac12\chi_{R,E}.$$

This factor is conventional, but it must accompany any numerical response or absorption coefficient moved between the two descriptions.

The microscopic retarded correlator makes the causal sign concrete. With $H_{\rm int}=-Q^{ab}F_{ab}$ and $\hbar$ restored only in this formula,

$$\chi_R^{ab,cd}(\tau) =\frac{i}{\hbar}\theta(\tau) \left\langle[Q^{ab}(\tau),Q^{cd}(0)]\right\rangle.$$

The expectation value is taken in the body’s specified stationary state. For a single oscillator of mass $m_Q$ and frequency $\omega_0$, this gives $\chi_R(\tau)=\theta(\tau)\sin(\omega_0\tau)/(m_Q\omega_0)$ and hence $\chi_R(\omega)=1/[m_Q(\omega_0^2-(\omega+i0)^2)]$. Its poles approach the real axis from below, checking both the Fourier convention and the response sign. At positive frequency a passive ground state has positive spectral weight; an excited or rotating state requires the corresponding state and energy balance to be examined.

Why a doubled history is necessary

An ordinary in-out effective action computes time-ordered amplitudes and does not impose a causal initial-value problem. Dissipative orbital dynamics instead uses a closed-time-path, or in-in, description. Duplicate every field, $E_1$ and $E_2$, and form

$$E_+=\frac12(E_1+E_2), \qquad E_-=E_1-E_2.$$

After the internal operators are integrated out, the quadratic influence action has the universal form

$$S_{\mathrm{IF}} =\int\frac{d\omega}{2\pi} \left[ E_-(-\omega)\,\mathcal G_R(\omega)\,E_+(\omega) +\frac{i}{2} E_-(-\omega)\,\mathcal N(\omega)\,E_-(\omega) \right],$$

with tensor indices and normalization factors suppressed. Varying with respect to the difference field and then taking the physical limit $E_-=0$ produces the retarded force. The positive noise kernel $\mathcal N$ describes fluctuations. In thermal equilibrium it is related to the absorptive part of $\mathcal G_R$ by a fluctuation–dissipation relation; away from equilibrium, response and noise must be matched independently.

This structure separates three notions that are often conflated. A real local operator such as $C_EE_{ab}E^{ab}$ is conservative. The imaginary part of a retarded correlator describes deterministic absorption. The noise kernel describes fluctuations of the unresolved degrees of freedom. None can be reconstructed from either of the others without additional state information.

The in-in construction and its application to radiation reaction and absorption are reviewed in Porto 2016, §§7.5–7.8, pp. 60–68.

Spectral positivity and absorbed power

Consider one component driven as $F(\tau)=\operatorname{Re}(F_0e^{-i\omega\tau})$. The cycle-averaged power delivered to a passive internal system is

$$\overline P_{\mathrm{abs}} =\frac{\omega}{2} \operatorname{Im}\chi_R(\omega)\, \lvert F_0\rvert^2.$$

Thus passivity implies

$$\omega\,\operatorname{Im}\chi_R(\omega)\ge 0.$$

This is the worldline version of the optical-theorem sign: a positive absorption probability gives a nonnegative spectral density. It is a stronger check than merely finding an imaginary term. One must also verify the Fourier convention, the sign of the interaction, and which system’s energy is being measured.

For a stationary rotating body, the relevant frequency for a mode with azimuthal number $m$ is the co-rotating frequency $\widetilde\omega=\omega-m\Omega$. A black-hole horizon is passive with respect to its generator, so positivity applies to $\widetilde\omega\,\operatorname{Im}\chi_R$. A superradiant mode has $\widetilde\omega\lt 0$ and can carry negative Killing-energy flux through the horizon while increasing the outgoing flux. This is not a violation of causality or of the appropriately defined spectral condition.

Low-frequency black-hole matching

For a four-dimensional, nonrotating black hole in general relativity, matching the quadrupolar correlator to the long-wavelength graviton absorption cross section gives the scalings

$$\operatorname{Im}\chi_{R,E}(\omega) \sim \frac{(GM)^6}{G}\,\omega, \qquad \sigma_{\mathrm{abs}}(\omega) \sim (GM)^6\omega^4,$$

up to convention-dependent numerical and polarization factors. The dimensions provide an immediate check: $\chi_E$ has dimensions $\text{length}^5/G$, while $\sigma_{\mathrm{abs}}$ has dimensions of area. The $\omega^4$ suppression also explains why horizon absorption by nonspinning bodies first enters a quasicircular inspiral much later than the leading flux to infinity.

These scalings are derived by matching worldline two-point functions to the black-hole absorption cross section in Goldberger and Rothstein 2006, §§II–III. The numerical coefficient must be imported with the same graviton normalization, polarization sum, and definition of $E_{ab}$. For neutron stars or other material bodies, viscosities and internal modes generate different low-frequency powers and resonant structure; a black-hole spectral function cannot be reused.

Domain and checks

The correlator description assumes that external wavelengths are long compared with the object and that the retained worldline operators span the relevant symmetry channels. A local derivative expansion further requires frequencies far from unresolved narrow resonances. Near a mode pole, that mode should be promoted to an explicit worldline field. Near merger, the separation between body, orbit, and radiation scales can fail altogether.

A reliable calculation checks:

1. analyticity of the retarded response in the upper half-plane;
2. the optical-theorem or passivity sign in the appropriate rotating frame;
3. dimensional and parity properties of every spectral function;
4. agreement with an independently computed absorption cross section;
5. cancellation of arbitrary matching scales in the final orbital observable.

The object-specific horizon or material perturbation problem belongs to Black-Hole Perturbation Theory or Relativistic Matter and Compact Objects. This page’s worldline theory begins after that response has been matched.

Exercises

Show directly that a damped oscillator has the absorptive sign required for a passive worldline degree of freedom.

Solution

For

$$\chi_R(\omega) =\frac{1}{\omega_0^2-\omega^2-2i\gamma\omega}, \qquad \gamma\gt 0,$$

one finds

$$\operatorname{Im}\chi_R(\omega) =\frac{2\gamma\omega} {(\omega_0^2-\omega^2)^2+4\gamma^2\omega^2}.$$

Therefore $\omega\operatorname{Im}\chi_R(\omega)\ge0$ for every real $\omega$. Its poles lie in the lower half-plane, which independently confirms retarded decay rather than exponentially growing response.

Use dimensions to recover the frequency dependence of the Schwarzschild quadrupolar spectral function from the low-frequency cross-section scaling.

Solution

The response coefficient has dimensions $[\chi_E]=\text{length}^5/G$. The only body scale is $GM$, and the absorptive part must be odd in $\omega$, so its leading analytic behavior is

$$\operatorname{Im}\chi_E\propto \frac{(GM)^6\omega}{G}.$$

The optical-theorem relation supplies three further powers of frequency and one power of $G$, giving

$$\sigma_{\mathrm{abs}} \propto G\omega^3\operatorname{Im}\chi_E \propto(GM)^6\omega^4,$$

an area as required.

References

• Goldberger, Walter D., and Ira Z. Rothstein. “Dissipative Effects in the Worldline Approach to Black Hole Dynamics.” Physical Review D 73 (2006): 104030. DOI.
• Porto, Rafael A. “The Effective Field Theorist’s Approach to Gravitational Dynamics.” Physics Reports 633 (2016): 1–104. DOI.