Second-Order Gravitational Tails

• Second-order gravitational tails are nonlinear radiative components in general relativity produced from quadratic metric perturbations, leading to distinctive power-law decays.
• They emerge from the self-coupling of linear quasinormal modes, resulting in sum and difference frequency tones that can dominate late-time black hole signals.
• Their unique, long-lived signatures enhance gravitational-wave template modeling and provide a robust test of general relativity's nonlinear dynamics.

Second-order gravitational tails are nonlinear radiative components in general relativity arising from second-order (quadratic) metric perturbations. Unlike the well-known linear “Price tails” of black hole perturbation theory—which originate from first-order (linear) effects and decay steeply at late times—second-order gravitational tails result from the nonlinear self-coupling of gravitational perturbations. These effects are significant both conceptually and quantitatively: they appear as additional, long-lived signals in the late-time behavior of compact-object mergers, can dominate over linear tails at sufficiently late times, and provide unique observational signatures of general relativity’s nonlinearity in the strong-field regime.

1. Nonlinear Perturbation Theory and Second-Order Tails

In the perturbative framework, the metric is expanded as $$g_{ab} = g_{ab}^{(0)} + h^{(1)}_{ab} + h^{(2)}_{ab} + \ldots$$, with $h^{(1)}_{ab}$ the linear perturbation and $h^{(2)}_{ab}$ the second-order part. The field equation for $h^{(2)}$ generically takes the form

$$\left[ -\partial_t^2 + \partial_x^2 - V(x) \right] \phi^{(2)}(t,x) = S^{(2)}(t,x),$$

where the source term $S^{(2)}$ is quadratic in the first-order field:

$S^{(2)}(t,x) = F(x)\left[\phi^{(1)}(t,x)\right]^2,$

with $F(x) \propto (|x|+1)^{-2}$ for a typical choice of potential (0803.0501).

The first-order solution is typically dominated by a set of exponentially damped quasinormal modes (QNMs):

$\phi^{(1)}(t,x) \sim \sum_n C_n e^{s_n (t - x)},$

where $s_n$ are the complex frequencies of the QNMs. The key point is that when this $\phi^{(1)}$ is squared, the second-order source produces new frequency components—specifically, “sum” and “difference” tones:

$C_n C_m e^{(s_n + s_m)(t-x)},$

generating frequencies $\Omega_\text{sum} = \Omega_1+\Omega_2$, $\Omega_\text{diff} = |\Omega_1-\Omega_2|$, and for self-coupled terms, $2s_0 = 2\gamma_0 - 2i\Omega_0$ and $s_0 + s_0^* = 2\gamma_0$.

2. Characterization and Origin of the Second-Order Power-Law Tail

A central discovery is the existence of a genuinely new, nonoscillatory power-law tail in the second-order field, absent in linear (first-order) theory (0803.0501). At late times, away from the central region, this tail can be written as

$$\phi^{(2)}_\mathrm{T}(t,x) \simeq H\left[\frac{1}{t-x+2} - \frac{1}{t+x+2}\right],$$

where $H$ encodes the excitation amplitude. For $x\gg1$, the decay transitions from $\sim t^{-1}$ ($x < t \lesssim 2x$) to $\sim t^{-2}$ as $t$ increases further. This decay law is strikingly slower than that of the linear (Price) tail, which for a multipole $\ell$ is $t^{-2\ell-3}$. The slower decay allows the second-order tail to eventually dominate the late-time waveform.

The physical mechanism is inherently nonlinear: it results from evaluating the Green's function on the “edge” of the first-order source distribution, where the phase is nearly stationary, yielding amplification of nonoscillatory contributions. This is not a remnant of the branch-cut structure that generates the first-order tail but a consequence of the quadratic source structure.

3. Mathematical Structure: Second-Order QNMs and Nonlinear Tail

The following table summarizes the mathematical structure of the contributions:

Phenomenon	Frequency Structure	Decay Law
First-order QNMs	$s_n = \gamma_n - i\Omega_n$	$e^{\gamma_n t}$
Sum/difference tones	$2s_0,\, s_0 + s_0^*$, $s_n + s_m$	$\sim e^{(s_n+s_m)t}$
Second-order tail	Nonoscillatory; generated at source edge	$\sim t^{-1},\, t^{-2}$

The explicit second-order source, using $F(x)\propto (|x|+1)^{-2}$,

$S^{(2)}(t, x) = F(x) [\phi^{(1)}(t, x)]^2,$

acts as the driving term for both the shifted QNMs and the slowly decaying tail.

4. Implications for Black Hole Ringdown and Gravitational Waveforms

Second-order gravitational tails have several direct implications for the modeling and interpretation of black hole ringdown signals:

• Dominance of Second-Order Tail at Late Times: Although the first-order (linear) signal will appear largest at early and intermediate times, the slower decay of the second-order tail allows it to eventually surpass the amplitude of first-order modes (0803.0501). This is especially relevant in the context of binary black hole coalescences, where the ringdown period is energetically significant.
• Additional Spectral Content: The presence of “sum” and “difference” QNM tones provides a clean signature of nonlinearity. Observing frequencies at $2\Omega_0$ or at $|\Omega_1-\Omega_2|$ in the post-merger spectrum—frequencies not present in linear theory—is a distinctive marker.
• Tail Origin and Geometry Dependence: The second-order power-law tail is primarily determined by the spatial asymptotics of the source term, not by the detailed structure of the potential. As such, it is less sensitive to the background spacetime geometry.
• Precision Template Construction: Waveform templates that neglect these effects may underfit or misestimate key ringdown features, particularly at late times, and may therefore yield biased parameter inferences for mass and spin. Incorporating nonlinear tails in template construction improves the fidelity of the inferred remnant properties and strengthens general relativity tests.

5. Observational and Theoretical Consequences

Detection of second-order gravitational tails would constitute a test of the Einstein equations’ genuinely nonlinear regime. Observable features include:

• Late-Time Enhancement: The amplitude of the tail may become measurable in high-SNR post-merger datasets, opening access to strong-field diagnostics.
• Spectral Nonlinearity: Unambiguous identification of nonlinear QNM combinations (sum/difference tones).
• Data Analysis Applications: The tail’s unique decay profile offers a handle to suppress or identify spurious late-time signals and to refine event selection.
• General Relativity Tests: Quantitative agreement between predicted and observed tail properties constrains deviations from Einstein’s equations, particularly in the nonlinear sector.

6. Practical Recommendations and Further Directions

Implementation in gravitational-wave data analysis should proceed with the following considerations:

• Template Extensions: Include nonlinear QNM frequencies and power-law tail contributions in ringdown models for binary black hole analysis.
• Parameter Extraction: Use summed and difference QNM frequencies to break degeneracies in the estimation of black hole mass, spin, and final state properties.
• Tail Amplitude Modeling: Since the amplitude $H$ of the second-order tail depends on the specifics of the source truncation and nonlinear coupling, detailed modeling or numerical calibration (e.g., from simulations of merger scenarios) is necessary.
• Generalization to Astrophysical Initial Data: Extending the analysis to head-on collisions and realistic mergers (using, for example, Misner initial data) to validate the universality of the second-order tail.

7. Summary Table: First vs. Second-Order Tails

Order	Decay Law	Frequency Content	Origin	Dominance at Late Times
First (linear)	$t^{-2\ell-3}$	QNM complex frequencies	Branch-cut of Green's	No
Second (nonlinear)	$t^{-1}$, $t^{-2}$ (generic)	Sum/difference QNM combinations; nonoscillatory tail	Edge/truncation of first-order field via quadratic source	Yes

8. Outlook

Second-order gravitational tails are essential components in the theoretical and observational paper of strong-field general relativity. Their detection would confirm the nonlinear nature of spacetime dynamics and significantly enrich the scientific potential of gravitational-wave astronomy, enabling finer probes of black hole ringdown and deeper tests of gravitational dynamics in the strong-field, nonlinear regime (0803.0501).