[Paper Review] Localized energy estimates on Myers-Perry space-times

Parul Laul, Jason Metcalfe+2 more|arXiv (Cornell University)|Jan 2, 2014

Black Holes and Theoretical Physics44 references3 citations

TL;DR

This paper establishes localized energy estimates for the wave equation on (1+4)-dimensional Myers-Perry black hole spacetimes with small angular momenta, adapting a pseudodifferential multiplier method originally developed for Schwarzschild spacetimes. The key contribution is a robust, loss-free localized energy estimate despite the presence of trapped null geodesics at the photon sphere, achieved by constructing a microlocal parametrix and carefully handling the degeneracy of the metric near the trapped set using a refined symbol calculus and frequency localization.

ABSTRACT

Localized energy estimates for the wave equation have been increasingly used to prove various other dispersive estimates. This article focuses on proving such localized energy estimates on $(1+4)$-dimensional Myers-Perry black hole backgrounds with small angular momenta. The Myers-Perry space-times are generalizations of higher dimensional Kerr backgrounds where additional planes of rotation are availabile while still maintaining axial symmetry. Once it is determined that all trapped geodesics have constant $r$, the method developed by Tataru and the fourth author, which perturbs off of the Schwarzschild case by using a pseudodifferential multiplier, can be adapted.

Motivation & Objective

• To extend localized energy estimates—previously established on Minkowski and Schwarzschild spacetimes—to higher-dimensional rotating black holes with axial symmetry.
• To address the challenge of trapping at the photon sphere in (1+4)-dimensional Myers-Perry spacetimes, where standard energy estimates fail due to degeneracy.
• To develop a robust microlocal framework that avoids losses in the energy estimate despite the presence of trapped geodesics.
• To adapt the pseudodifferential multiplier method of Tataru and Tohaneanu to the Myers-Perry background, which features additional rotational planes and non-trivial angular momentum.

Proposed method

• Adapts the positive commutator method with a pseudodifferential multiplier to the Myers-Perry metric, using a symbol $ s $ that vanishes on the trapped set and is constructed via the Malgrange preparation theorem to handle non-polynomial dependence on $ \tau $.
• Introduces a microlocal parametrix construction by defining a symbol $ s $ that captures the behavior near the photon sphere, ensuring the Poisson bracket $ \{ \rho^2 p, s \} $ is non-negative and controls the error terms.
• Uses a frequency-localized cutoff $ \chi_{>1} $ to truncate low frequencies and a spatial cutoff $ \chi $ supported near $ r_{ps} $ to localize the analysis near the trapped set.
• Constructs the operator $ S = \chi s^w \chi $ and a modified error operator $ E = \chi e^w \chi - e^w_{\text{aux}} D_t $ to cancel the $ D_t^3 $ term in the commutator expansion, ensuring the energy estimate holds.
• Employs a sum-of-squares decomposition of the symbol $ r^2 q^S $ in terms of angular derivatives and radial components, using $ \lambda_i^2 $ for the spherical Laplacian and $ \xi^2 $ for radial derivatives.
• Applies a partition of unity in $ r $, $ 1 = \chi^2 + \chi_o^2 $, to separate the analysis into trapped and non-trapped regions, enabling uniform control of the error terms.

Experimental results

Research questions

• RQ1 Can localized energy estimates be established on (1+4)-dimensional Myers-Perry spacetimes with small angular momenta, despite the presence of trapped null geodesics?
• RQ2 How can the pseudodifferential multiplier method be adapted to higher-dimensional rotating black holes with multiple rotational parameters?
• RQ3 What modifications are required to the standard energy estimate framework to eliminate losses when trapping occurs at the photon sphere?
• RQ4 Can the degeneracy of the metric near the trapped set be controlled using microlocal techniques and symbol calculus?
• RQ5 Is it possible to construct a symbol $ s $ that vanishes on the trapped set and yields a positive definite commutator, even when $ s $ is not a polynomial in $ \tau $?

Key findings

• The paper establishes a localized energy estimate of the form $ \|u'\|_{L^\infty_t L^2_x} + \|u\|_{LE^1} \lesssim \|u'(0,\cdot)\|_{L^2} + \|\Box u\|_{LE^{*} + L^1_t L^2_x} $ on (1+4)-dimensional Myers-Perry spacetimes with small angular momenta.
• The estimate is loss-free, even in the presence of trapping at the photon sphere, by constructing a symbol $ s $ that vanishes on the trapped set and using a microlocal parametrix to control error terms.
• The method relies on a refined symbol calculus that handles the non-polynomial dependence of the symbol on $ \tau $, using the Malgrange preparation theorem to reduce the problem to a polynomial-like structure.
• The construction of the error operator $ E $ ensures cancellation of the $ D_t^3 $ term in the commutator, which is essential for controlling the energy growth.
• The sum-of-squares decomposition of the symbol $ r^2 q^S $ allows uniform control of angular and radial derivatives, with $ \nu(r) \in (0,1) $ ensuring positivity near the trapped set.
• The final estimate holds for small enough $ \epsilon_0 $, confirming the robustness of the method under small perturbations of the Schwarzschild background.