Extended Homogeneous Solutions (EHS) in Perturbation

• Extended Homogeneous Solutions (EHS) are a method where frequency–domain homogeneous solutions are extended into singular source regions to yield rapidly convergent time–domain fields.
• The approach constructs separate 'in' and 'up' solutions that are stitched together using jump conditions to accurately model gravitational perturbations in black hole spacetimes.
• EHS overcomes traditional Gibbs oscillations with exponential convergence, enabling high-precision waveform modeling and reliable self-force regularization.

Extended Homogeneous Solutions (EHS) are a class of solution representations developed for physical, geometric, and partial differential equation problems in which the source or boundary conditions exhibit singular or distributional structure. In particular, EHS methodologies have achieved prominence in gravitational perturbation theory for modeling fields sourced by point particles in black hole backgrounds. The defining feature of EHS approaches is the explicit extension of frequency–domain homogeneous solutions of linearized equations into regions containing the singular source, enabling rapid time–domain reconstruction and sharply improved control of singularities.

1. Motivation and Conceptual Framework

The motivation for Extended Homogeneous Solutions stems from limitations in standard Fourier synthesis applied to linear PDEs and wave equations with distributional sources. In frequency–domain calculations (especially for self-force computations in general relativity), the presence of Dirac δ and ∂δ source terms leads to inhomogeneous equations for Fourier-harmonic modes. Attempts to reconstruct the time–domain fields in the source region through summation of these inhomogeneous modes suffer from algebraic (slow) convergence and the Gibbs phenomenon, manifesting as persistent overshoots and oscillations near singularities. Classical examples include the Regge–Wheeler and Zerilli equations sourced by particles in black hole spacetimes.

The EHS methodology, originally developed for self-force problems by Barack, Ori, and Sago and subsequently generalized for gravitational perturbations (Hopper et al., 2010), prescribes the following solution procedure:

• Construct both "in" and "up" homogeneous solutions—those satisfying boundary conditions at the black hole horizon and at spatial infinity, respectively—for each Fourier-harmonic mode of the (typically hyperbolic) master equation.
• Assemble two partial time–domain fields, $\Psi^+(t, r)$ and $\Psi^-(t, r)$, as summations (over the Fourier index $n$) of these homogeneous spatial modes modulated by frequency exponentials.
• Define the total solution $\Psi(t, r)$ as a partitioned function:

$$\Psi(t, r) = \Psi^+(t, r) \Theta(r - r_p(t)) + \Psi^-(t, r) \Theta(r_p(t) - r),$$

where $r_p(t)$ is the instantaneous particle position, and $\Theta$ is the Heaviside step function.

• Enforce the appropriate jump conditions derived from the singular source terms, ensuring well-posedness of the solution as a distribution (typically of class $C^{-1}$).

This approach ensures that the partial sums in the mode decomposition converge exponentially fast, even at the location $r = r_p(t)$ of the distributional source—a dramatic improvement over standard techniques.

2. Frequency–Domain EHS in Black Hole Perturbation Theory

In Schwarzschild black hole backgrounds, the EHS method is employed within the Regge–Wheeler–Zerilli (RWZ) formalism. Here, the field equations for even- and odd-parity master functions $\Psi(t, r)$ are

$$\left[ -\frac{\partial^2}{\partial t^2} + \frac{\partial^2}{\partial r_*^2} - V_\ell(r) \right] \Psi_{\ell m}(t, r) = S_{\ell m}(t, r),$$

where $r_*$ is the tortoise coordinate, $V_\ell(r)$ is the potential (Zerilli or Regge–Wheeler), and $S_{\ell m}(t, r)$ encodes the singular particle source—including both delta and derivative-of-delta functions.

The EHS procedure in this setting involves:

• Computing "in" and "up" homogeneous solutions $R^\pm_{\ell mn}(r)$ for each frequency $\omega_{mn}$, with boundary conditions:
  • $R^+_{\ell mn}(r) \sim \exp(i\omega_{mn} r_*)$ as $r_* \to +\infty$,
  • $R^-_{\ell mn}(r) \sim \exp(-i\omega_{mn} r_*)$ as $r_* \to -\infty$.
• Forming $$\Psi^+_{\ell m}(t, r) = \sum_n R^+_{\ell mn}(r) e^{-i\omega_{mn} t}$$ and analogously for $\Psi^-$.
• Stitching the complete field from these via the particle-position-based partition.
• Implementing the jump conditions for $\Psi$ and its first derivative (obtained by integrating the master equation in a small region around $r = r_p(t)$), ensuring that the delta-function and its derivative in the source are correctly reflected in the solution.

Numerically, the method requires high-precision integration of ODEs for $R^\pm_{\ell mn}(r)$, careful treatment of delta-prime contributions, and efficient Fourier-mode summation. The resulting time–domain field enables accurate extraction of physical observables, such as energy and angular momentum fluxes at the horizon and infinity, with relative errors at the level of $10^{-9}$ to $10^{-12}$, and is critical for high-accuracy gravitational waveform modeling.

3. Advantages: Convergence and Weak Solution Structure

The primary advantage of EHS over traditional mode-sum approaches is exponential convergence of the time–domain partial sum, regardless of whether $r$ coincides with $r_p(t)$. This is in sharp contrast with the algebraic convergence and large Gibbs oscillations encountered with inhomogeneous Fourier synthesis in the presence of singular sources.

More precisely, if the mode sum for the inhomogeneous solution converges as $N^{-\alpha}$ (with $\alpha$ determined by the order of the source singularity), the EHS sum converges as $\exp(-\gamma N)$ for some positive $\gamma$, enabling far fewer terms to be summed for a given level of precision.

Mathematically, the resulting $\Psi(t, r)$ is a weak solution: it may be only of class $C^{-1}$ (i.e., discontinuous across $r = r_p(t)$), with possible embedded distributional (delta) terms in derivatives. This structure is not merely an artifact; it precisely mirrors the physical singularity structure required by the particle source and is directly exploited in regularization strategies for self-force computations.

4. Implementation Strategies and Practical Steps

Implementing EHS in practice, particularly for numerical codes, involves:

1. Orbital specification: Compute geodesic parameters (like semi-latus rectum $p$, eccentricity $e$) and associated frequencies $(\Omega_r, \Omega_\varphi)$.
2. Boundary-value ODEs: For each mode $(\ell, m, n)$, integrate the homogeneous ODEs for $R^\pm_{\ell mn}(r)$ from boundaries to a matching point. Use analytic expansions at boundaries to deal with stiffness.
3. Normalization: Compute the normalization coefficients (source integrals) for each frequency, handling $\delta$ and $\delta'$ terms by switching to integration by parts at turning points of the radial motion.
4. Synthesis: Sum over $n$ the time–domain reconstructed modes, forming $\Psi^+(t, r)$ and $\Psi^-(t, r)$.
5. Jump conditions: Evaluate the explicit formulas for jump conditions to correctly piece together the global solution and ensure the solution matches the distributional structure of the source.
6. Metric reconstruction: Use analytic relations to recover metric perturbation amplitudes from the computed master functions—this may involve derivatives that preserve or alter the singularity structure (delta terms in $h_{rr}$, $h_{tt}$, etc.).

Practical challenges include mitigating round-off errors in large-$n$ sums, maintaining numerical stability at the radial turning points, and implementing the distributional regularization required at the particle location.

5. Metric Reconstruction and Distributional Content

An essential component of the EHS approach is the accurate reconstruction of the metric perturbation amplitudes ("MP amplitudes") from the master function. In the Regge–Wheeler gauge, the even- and odd-parity metric components are expressed as combinations of the master function and its derivatives. For example, for the even-parity $h_{rr}$ component,

$$h_{rr}^S(t) = r_p [\!\![ \partial_r \Psi ]\!\!]_p + (r_p A_p)[\!\![\Psi]\!\!]_p,$$

where $[\!\![ \cdot ]\!\!]_p$ denotes the jump across the particle, and $A_p$ is a function of $r_p$. The smooth and singular components are treated separately:

• Some amplitudes (such as the sphere amplitude $K$) remain only jump-discontinuous without embedded delta-functions.
• Others (primarily $h_{tt}$, $h_{tr}$, $h_{rr}$) acquire explicit, time-dependent delta-distributions at the instantaneous location of the particle.

The correct identification and computation of these singular contributions is crucial for self-force regularization procedures, as the singular part directly influences the subtraction of local (Detweiler–Whiting) fields in regularization schemes.

6. Significance and Broader Implications

The EHS methodology is now central in frequency-domain self-force and waveform computations for systems including generic orbits, eccentric binaries, and EMRI modeling in the black hole perturbation framework. The capability of EHS to deliver exponentially convergent, well-controlled weak solutions with correct singularity structure allows for:

• Precision modeling of gravitational radiation for highly relativistic sources—even in regimes of large eccentricities.
• Direct input for mode-sum or other self-force regularization methods, as metric and its derivatives at the worldline are accurately and controllably extracted.
• Stable and reliable use in future extensions, including non–bound orbits, multiparticle interaction scenarios, and explorations beyond Schwarzschild backgrounds (with suitable generalizations).

A possible limitation, noted in subsequent research (Whittall et al., 23 Sep 2025), is the degradation of EHS convergence for extreme eccentricities or in unbound (scattering) orbits, where large numerical cancellations hinder summation efficiency. This has motivated alternative projections (such as Gegenbauer reconstruction) to address these challenges.

7. Conclusion

The method of Extended Homogeneous Solutions represents a major advance in the computation of fields sourced by singular distributions in linearized gravitational (and other classical) field theories. By leveraging the analyticity and exponential convergence properties of frequency–domain homogeneous solutions stitched together at the singularity, EHS circumvents both the practical and conceptual limitations of traditional Fourier synthesis. Its rigorous enforcement of distributional jump conditions and its seamless compatibility with gauge-invariant master equation formalisms make it an indispensable tool in high-precision numerical relativity and gravitational self-force research (Hopper et al., 2010).

The EHS framework also provides a paradigm for constructing singular solutions in other settings where distributional sources and rapid convergence are needed, and underpins ongoing developments in black hole perturbation theory, waveform modeling, and the mathematical analysis of wave equations with singularities.