On the Fourier analysis of the Einstein-Klein-Gordon system: Growth and Decay of the Fourier constants (2004.11049v3)

Abstract: We consider the $(1 + 3)$-dimensional Einstein equations with negative cosmological constant coupled to a spherically-symmetric, massless scalar field and study perturbations around the Anti-de Sitter spacetime. We derive the resonant systems, pick out vanishing secular terms and discuss issues related to small divisors. Most importantly, we rigorously establish (sharp, in most of the cases) asymptotic behaviour for all the interaction coefficients. The latter is based on uniform estimates for the eigenfunctions associated to the linearized operator as well as on some oscillatory integrals.

• The paper rigorously establishes asymptotic bounds for Fourier constants, clarifying resonant and non-resonant dynamics in the Einstein-Klein-Gordon system on AdS space.
• It employs detailed perturbative expansions and eigenfunction analysis to quantify mode couplings and manage secular growth in spherically symmetric configurations.
• The derived estimates inform both theoretical insights into AdS stability and practical strategies for accurate numerical simulations in gravitational dynamics.

This paper (2004.11049) initiates a rigorous paper of the dynamics in Fourier space of spherically symmetric solutions to the coupled Einstein-Klein-Gordon system with a negative cosmological constant ($\Lambda < 0$) around the Anti-de Sitter (AdS) spacetime. The primary goal is to understand the asymptotic behavior (growth and decay rates) of the Fourier constants that arise from perturbative expansions of the system. This is motivated by the AdS instability conjecture and recent numerical work suggesting the existence of time-periodic solutions for specific initial data, which relies on the ability to cancel secular terms in perturbative expansions.

The system is described by the Einstein field equations coupled to a massless scalar field $\phi$, specialized to spherical symmetry in $(1+3)$ dimensions using a specific metric ansatz: $$g(t,x,\omega) = \frac{1}{\cos ^2 (x)} \left( - A(t,x)e^{-2\delta(t,x)} dt^2 + A^{-1}(t,x) dx^2+ \sin ^2 (x) d \omega ^2 \right)$$, where $x \in [0, \pi/2)$ represents the spatial radial coordinate in compactified coordinates. The dynamics of $\phi$ is governed by the wave equation $\Box_g \phi = 0$, and the metric functions $A$ and $\delta$ are determined by the energy-momentum tensor of the scalar field through the Einstein equations.

The linearized operator governing perturbations around the AdS solution $(A=1, \delta=0, \phi=0)$ is given by $$L[f](x) = -\frac{1}{\tan^2(x)} \partial_x (\tan^2(x) \partial_x f(x))$$. The eigenfunctions $\{e_j(x)\}_{j=0}^\infty$ of this operator with Dirichlet boundary conditions at $x=\pi/2$ are related to weighted Jacobi polynomials. The corresponding eigenvalues are $\omega_j^2 = (3+2j)^2$, with frequencies $\omega_j = 3+2j$. These eigenfunctions form an orthogonal basis for the weighted $L^2$ space with inner product $(f|g) = \int_0^{\pi/2} f(x)g(x) \tan^2(x) dx$.

The paper investigates two types of perturbative expansions of the fields $$(\Phi = \partial_x \phi, \Pi = A e^{-\delta} \partial_t \phi, A, \delta)$$ around the AdS background $(0, 0, 1, 0)$ in powers of a small parameter $\epsilon$.

Perturbation 1: Infinite Series Expansion

In this approach, all fields are expanded as infinite power series in $\epsilon$, and time is rescaled by a frequency $\Omega_\gamma$ that also has an expansion in $\epsilon$ (starting with the frequency of a dominant mode $\omega_\gamma$): $$\Phi = \sum_{\lambda=0}^\infty \psi_{2\lambda+1} \epsilon^{2\lambda+1}$$, $$\Pi = \sum_{\lambda=0}^\infty \sigma_{2\lambda+1} \epsilon^{2\lambda+1}$$, $$A e^{-\delta} = \sum_{\lambda=0}^\infty \xi_{2\lambda} \epsilon^{2\lambda}$$, $$e^{-\delta} = \sum_{\lambda=0}^\infty \zeta_{2\lambda} \epsilon^{2\lambda}$$, $\tau = \Omega_\gamma t$. Substituting these expansions into the field equations and collecting terms of the same order in $\epsilon$ yields a hierarchical system of partial differential equations for $$(\psi_{2\lambda+1}, \sigma_{2\lambda+1}, \xi_{2\lambda}, \zeta_{2\lambda})$$. These solutions are then expanded in the spatial eigenfunctions $e_j(x)$ and their derivatives $\frac{e_j'(x)}{\omega_j}$, leading to an infinite system of coupled ordinary differential equations (ODEs) for the expansion coefficients (Fourier coefficients) in time $\tau$.

The Fourier constants in this approach are defined by integrals involving products of eigenfunctions and powers of $\tan x$:

• $$C_{ij}^{(m)} := \int_0^{\pi/2} e_i(x) e_j(x) e_m(x) \tan^2(x) dx$$
• $$\overline{C}_{ij}^{(m)} := \int_0^{\pi/2} e_i(x) \frac{e_j'(x)}{\omega_j} \frac{e_m'(x)}{\omega_m} \tan^2(x) dx$$
• $$\overline{A}_{ijl}^{(m)} := \int_0^{\pi/2} \frac{e_i'(x)}{\omega_i} \frac{e_j'(x)}{\omega_j} e_l(x) \frac{e_m'(x)}{\omega_m} \frac{\sin^3(x)}{\cos(x)} dx$$
• $$\overline{B}_{ijl}^{(m)} := \int_0^{\pi/2} e_i(x) e_j(x) e_l(x) \frac{e_m'(x)}{\omega_m} \frac{\sin^3(x)}{\cos(x)} dx$$
• $\widetilde{A}_{ijl}^{(m)}$ and $\widetilde{B}_{ijl}^{(m)}$ involve integrals of products of eigenfunctions and integrals of eigenfunctions, weighted by powers of $\tan x$.

The time evolution of the Fourier coefficients $f_{2\lambda+1}^{(m)}(\tau)$ (for $\psi_{2\lambda+1}$) is governed by forced harmonic oscillator equations. Secular terms (terms growing linearly with $\tau$) appear when the forcing term contains a frequency that matches the natural frequency $\omega_m/\omega_\gamma$. The numerical work [PhysRevLett111051102] suggested these secular terms could be canceled by appropriately choosing initial data and the frequency shifts $\omega_{\gamma,2\lambda}$. The validity of this procedure depends crucially on the properties of the Fourier constants.

Perturbation 2: Finite Sum with Error

This approach considers a finite-order expansion (up to $\epsilon^3$ for $\Phi, \Pi$ and $\epsilon^4$ for $A, e^{-\delta}$) plus error terms: $\Phi = \Phi_1 \epsilon + \Psi \epsilon^3$, $\Pi = \Pi_1 \epsilon + \Sigma \epsilon^3$, $A = 1 - A_2 \epsilon^2 - B \epsilon^4$, $$e^{-\delta} = 1 - \delta_2 \epsilon^2 - \Theta \epsilon^4$$. The linear terms $(\Phi_1, \Pi_1)$ and second-order terms $(A_2, \delta_2)$ are solved explicitly based on a dominant mode (chosen as the $j=0$ mode for simplicity). The resulting expressions for $A_2$ and $\delta_2$ involve spatial functions $\Gamma_1, \Gamma_2, \Gamma_3, \Gamma_4$ which are explicit integrals of products of the $j=0$ eigenfunctions and their derivatives. The goal is then to paper the system for the error terms $(\Psi, \Sigma, B, \Theta)$.

Expanding the error terms in eigenfunctions leads to a system of coupled ODEs for their Fourier coefficients. The Fourier constants in this approach involve integrals with the functions $\Gamma_a(x)$ as additional weights, along with products of eigenfunctions:

• $$\mathbb{C}_{abij}, \mathbb{D}_{abij}, \mathbb{E}_{1423ij}$$ (integrals involving $\Gamma_a, \Gamma_b$ products and two eigenfunctions, weighted by $\tan^2 x$).
• $\mathbb{G}_{jki}$ (same as $C_{ijk}$), $\mathbb{F}_{ajki}$ (integrals involving $\Gamma_a$ and three eigenfunctions, weighted by $\tan^2 x$).
• $$\overline{\mathbb{C}}_{abij}, \overline{\mathbb{D}}_{abij}, \overline{\mathbb{E}}_{1423ij}, \overline{\mathbb{G}}_{kji}, \overline{\mathbb{F}}_{akji}, \overline{\mathbb{H}}_{kjli}$$ (similar integrals but with derivatives of eigenfunctions).
• $$\mathbb{J}_{jki}, \mathbb{I}_{jki}, \mathbb{P}_{ajki}, \mathbb{Q}_{ajki}, \mathbb{R}_{klji}, \mathbb{S}_{jkli}$$ (integrals involving up to four eigenfunctions/derivatives, weighted by $\frac{\sin^3 x}{\cos x}$).
• $$\overline{\mathbb{J}}_{jki}, \overline{\mathbb{I}}_{jki}, \overline{\mathbb{P}}_{ajki}, \overline{\mathbb{Q}}_{ajki}, \overline{\mathbb{R}}_{klji}, \overline{\mathbb{S}}_{jkli}$$ (integrals involving up to three eigenfunctions/derivatives and an integral of an eigenfunction, weighted by $\tan^2 x$).

This approach also reveals sources of secular terms and potential small divisors in the ODEs for the error term coefficients, related to frequencies like $\omega_i/\omega_0$, $\omega_i/3\omega_0$, $\omega_i/5\omega_0$. The paper shows how specific choices of initial data and the frequency shift $\theta_0$ (which corresponds to $\omega_{\gamma,2}$ in the first approach) can cancel some of these secular terms.

Main Results on Growth and Decay

The core contribution of the paper is the rigorous establishment of asymptotic bounds for these various Fourier constants as the mode indices go to infinity. These bounds quantify the strength of the coupling between different modes. The analysis relies on:

• Detailed asymptotic formulas for the eigenfunctions $e_j(x)$ and $\frac{e_j'(x)}{\omega_j}$ for large $j$.
• Asymptotic analysis of oscillatory integrals using integration by parts (Lemma \ref{byparts1}, \ref{byparts2}).
• Dirichlet-Kernel-type identities for trigonometric sums.
• $L^\infty$ bounds for eigenfunctions and related quantities.
• Holder's inequality.

The bounds presented distinguish between two cases for the modes involved (e.g., $i, j, m$ for $C_{ij}^{(m)}$):

1. Resonant Case: When linear combinations of the mode frequencies $\omega_i \pm \omega_j \pm \omega_m$ (or more for higher-order constants) do not go to infinity. This corresponds to near-resonance conditions between the modes. In these cases, the constants generally show slower decay or even growth with respect to the mode indices. For example, $C_{ij}^{(m)}$ and $\overline{C}_{ij}^{(m)}$ are $\mathcal{O}(\omega_i)$, $\overline{A}_{ijl}^{(m)}$ and $\overline{B}_{ijl}^{(m)}$ are $\mathcal{O}(\omega_l)$.
2. Non-Resonant Case: When all linear combinations of the mode frequencies do go to infinity. In these cases, the constants decay rapidly with increasing mode indices, often like a negative power of the frequency combinations ($\sim 1/(\dots)^N$ for some large $N$). The specific decay rate depends on the constant and is determined by the leading non-zero term in the asymptotic expansion of the relevant integral using integration by parts. For example, in the non-resonant case, $C_{ij}^{(m)}$ and $\overline{C}_{ij}^{(m)}$ are $$\sum_\pm \mathcal{O}(1/(\omega_i \pm \omega_j \pm \omega_m)^2)$$.

The paper provides detailed tables listing the asymptotic behavior for the various groups of Fourier constants defined in both perturbation approaches. For constants involving functions $\Gamma_a(x)$, the decay rate in the non-resonant case depends on the order of the first non-zero derivative of the function $\Gamma_a(x)$ (or products/ratios involving it) at the boundary $x=\pi/2$ in the integration by parts formula. The analysis shows that some constants decay polynomially fast with increasing mode numbers, while others may grow in resonant scenarios.

Practical Implications

These rigorous bounds on the Fourier constants are essential for:

• Analyzing AdS Stability: Understanding the coupling strengths between modes helps determine if energy can cascade to high frequencies, potentially leading to blow-up or black hole formation, as conjectured for AdS instability.
• Constructing Time-Periodic Solutions: The decay rates inform whether the infinite sums involved in the perturbative approach converge and how quickly. The behavior in resonant cases is particularly important for identifying potentially problematic terms (secular terms) and devising strategies (like tuning initial data or frequency shifts) to cancel them, following the ideas from KAM theory and numerical simulations like those in [PhysRevLett111051102].
• Numerical Simulations: The asymptotic behavior suggests which modes are dominant and how many modes need to be included in numerical truncations for accurate long-time simulations of the system. Faster decay implies fewer modes are needed for a given accuracy.
• Small Divisors Problem: The explicit dependence of bounds on frequency differences highlights the small divisors issue, which is central to understanding the potential regularity or instability of solutions in systems with infinite-dimensional phase spaces.

The paper also provides explicit closed formulas for several Fourier constants involving the $j=0$ mode (Appendix B), which are valuable for low-order perturbative calculations and validation of numerical results.

In summary, this paper provides a detailed mathematical analysis of the spatial coupling coefficients in the spherically symmetric Einstein-Klein-Klein-Gordon system around AdS, establishing rigorous bounds on their asymptotic behavior. These results provide crucial quantitative information for the ongoing efforts to understand the non-linear dynamics and potential instability or existence of periodic solutions in Anti-de Sitter spacetime, particularly in the context of resonant phenomena and perturbative methods.