[Paper Review] Linear Stability of Higher Dimensional Schwarzschild Spacetimes: Decay of Master Quantities

Pei‐Ken Hung, Jordan Keller+1 more|arXiv (Cornell University)|Sep 13, 2018

Black Holes and Theoretical Physics27 references4 citations

TL;DR

This paper establishes linear stability of higher-dimensional Schwarzschild-Tangherlini black holes by decomposing linearized gravitational perturbations via spacetime Hodge decomposition into scalar, co-vector, and two-tensor components. It introduces gauge-invariant master quantities satisfying Regge-Wheeler-type wave equations, proving uniform boundedness and, in six or fewer dimensions, uniform $L^2$-decay for all modes, extending mode stability to uniform estimates and confirming decay for low-frequency perturbations.

ABSTRACT

In this paper, we study solutions to the linearized vacuum Einstein equations centered at higher-dimensional Schwarzschild met- rics. We employ Hodge decomposition to split solutions into scalar, co-vector, and two-tensor pieces; the first two portions respectively cor- respond to the closed and co-closed, or polar and axial, solutions in the case of four spacetime dimensions, while the two-tensor portion is a new feature in the higher-dimensional setting. Rephrasing earlier work of Kodama-Ishibashi-Seto in the language of our Hodge decomposition, we produce decoupled gauge-invariant master quantities satisfying Regge- Wheeler type wave equations in each of the three portions. The scalar and co-vector quantities respectively generalize the Moncrief-Zerilli and Regge-Wheeler quantities found in the setting of four spacetime dimen- sions; beyond these quantities, we further discover a higher-dimensional analog of the Cunningham-Moncrief-Price quantity in the co-vector por- tion. In the analysis of the master quantities, we strengthen the mode stability result of Kodama-Ishibashi to a uniform boundedness estimate in all dimensions; further, we prove decay estimates in the case of six or fewer spacetime dimensions. Finally, we provide a rigorous argument that linearized solutions of low angular frequency are decomposable as a sum of pure gauge solution and linearized Myers-Perry solution, the lat- ter solutions generalizing the linearized Kerr solutions in four spacetime dimensions.

Motivation & Objective

• To establish linear stability of higher-dimensional Schwarzschild-Tangherlini black holes under small perturbations.
• To generalize four-dimensional gravitational perturbation theory (Moncrief-Zerilli, Regge-Wheeler) to higher dimensions using Hodge decomposition.
• To prove uniform boundedness and decay estimates for gauge-invariant master quantities in all dimensions, with stronger decay in six or fewer dimensions.
• To rigorously show that low-frequency linearized solutions decompose into pure gauge and linearized Myers-Perry solutions.

Proposed method

• Applying spacetime Hodge decomposition to split symmetric two-tensors into scalar, co-vector, and two-tensor components based on spherical symmetry.
• Constructing gauge-invariant master quantities—$Q^{(+)}_{ ext{scalar}}$, $Q^{(-)}_{ ext{co-vector}}$, $S_{ ext{co-vector}}$, and $ ilde{h}_{ ext{two-tensor}}$—that satisfy decoupled Regge-Wheeler-type wave equations.
• Using Hardy estimates and red-shift/Morawetz-type estimates to strengthen Kodama-Ishibashi's mode stability to uniform boundedness across all spacetime dimensions.
• Proving uniform $L^2$-decay in six or fewer spacetime dimensions via energy estimates involving higher-order derivatives and the Killing vector field $ ilde{K}$.
• Summing mode-by-mode estimates over angular harmonics using $L^2(S^n)$-convergence of potentials to obtain global boundedness and decay for total master quantities.
• Employing commutation with angular Killing fields and Sobolev embeddings to derive pointwise estimates from $L^2$-based energy bounds.

Experimental results

Research questions

• RQ1 Can the linearized Einstein equations about higher-dimensional Schwarzschild spacetimes be decoupled into gauge-invariant master quantities satisfying wave equations?
• RQ2 Do these master quantities satisfy uniform boundedness and decay estimates across all spacetime dimensions?
• RQ3 In which dimensions does uniform $L^2$-decay of the master quantities hold?
• RQ4 Can low-frequency linearized solutions be decomposed into pure gauge and linearized Myers-Perry components?
• RQ5 How do the higher-dimensional generalizations of Moncrief-Zerilli and Cunningham-Moncrief-Price quantities behave under energy estimates?

Key findings

• The scalar portion of the perturbation gives rise to a master quantity $Q^{(+)}_{ ext{total}}$ satisfying uniform boundedness: $\check{E}^{N}_{Q^{(+)}}(\Sigma_{\tau}) \lesssim \check{E}^{N}_{Q^{(+)}}(\Sigma_{0})$ in all spacetime dimensions.
• In six or fewer spacetime dimensions, the total scalar master quantity $Q^{(+)}_{\text{total}}$ satisfies uniform $L^2$-decay: $\check{E}^{N}_{Q^{(+)}}(\Sigma_{\tau}) \lesssim \frac{I_{Q^{(+)}}(\Sigma_{0})}{\tau^{2}}$.
• The two-tensor and co-vector portions also satisfy uniform boundedness estimates across all dimensions, with decay in six or fewer dimensions.
• The authors identify a higher-dimensional analog of the Cunningham-Moncrief-Price quantity in the co-vector sector, extending four-dimensional results.
• The analysis confirms that low-frequency linearized solutions decompose into pure gauge and linearized Myers-Perry solutions, generalizing the four-dimensional linearized Kerr result.
• All estimates are uniform in angular mode numbers $\ell$ and $m_{s}(n,\ell)$, enabling summation over harmonics via $L^2(S^n)$-convergence of potentials.