Pseudospectra of holographic diffusion: gauge fields breaking free from the master scalar

We study pseudospectra of quasinormal frequencies and complex linear momenta of a U(1) gauge field in a Schwarzschild black branes in Anti-de Sitter. We present a novel approach for computing the pseudospectra which uses directly the gauge field variables and contrast it to a conventional master scalar field approach. Upon clarifying a subtlety in the energy norm of the master scalar we show that the pseudospectra of both approaches conincide. In the hydrodynamic regime we find that the hydrodynamic quasinormal frequency, the diffusive mode, is spectrally stable to a very good approximation. On the other hand hydrodynamic complex linear momenta show enhanced spectral instability as a consequence of a pole-collision at zero frequency.

💡 Research Summary

The paper investigates the ε‑pseudospectrum of quasinormal frequencies (QNFs) and complex linear momenta (CLMs) for a U(1) gauge field propagating on Schwarzschild‑Anti‑de Sitter (SAdS) black branes. The authors introduce a novel computational scheme that works directly with the gauge‑field components (the “GF” approach) and compare it to the traditional master‑scalar (MS) method, where the longitudinal sector of the gauge field is reduced to an effective scalar equation.

A central theme is the role of the norm in defining a pseudospectrum. For non‑self‑adjoint operators, eigenvalues alone do not guarantee spectral stability; small perturbations can cause large eigenvalue shifts. The ε‑pseudospectrum quantifies the maximal displacement of eigenvalues under perturbations of size ε, but its shape depends on the chosen operator norm. Physically, the natural norm is the energy norm of the perturbation. In the MS framework this norm is constructed from the energy of the effective scalar on a constant‑time slice. However, the authors point out a subtlety: the scalar energy acquires a boundary term at the AdS boundary, which can spoil positive‑definiteness and thus invalidate the norm for certain function spaces.

To avoid this issue, the GF approach defines the norm directly from the Maxwell energy density, eliminating any ambiguous boundary contribution. By dimensional reduction to an effective three‑dimensional spacetime and invoking Hodge duality, the authors show that the MS scalar and the GF gauge field are mathematically equivalent, provided the boundary term is handled correctly. Consequently, both frameworks yield identical pseudospectra when the energy norm is properly defined.

The numerical implementation discretizes the linearized equations using pseudospectral collocation, leading to a generalized eigenvalue problem for a large but finite matrix. The authors compute ε‑pseudospectra for several backgrounds: 5‑dimensional and 4‑dimensional SAdS black branes, as well as an AdS₄ black hole. Convergence tests (Appendices C–D) confirm that the results are robust against grid refinement.

Physical findings can be summarized as follows:

1. Hydrodynamic quasinormal mode (diffusion) – The QNF ω = −i D k² (with diffusion constant D) remains spectrally stable in the low‑momentum regime. The ε‑pseudospectrum stays tightly clustered around the exact eigenvalue, indicating that small perturbations of the operator do not move the diffusion pole appreciably. This matches the expectation that hydrodynamics, being a universal low‑energy description, should be robust.

2. Complex linear momenta (CLMs) – When the frequency is held real and one solves for complex spatial momentum, the two hydrodynamic CLMs behave as k ≈ ±√(i ω) near ω = 0. As ω→0 the two branches collide at the origin, forming an exceptional point. Exceptional points are known to amplify non‑normal effects, and indeed the ε‑pseudospectrum inflates dramatically, signalling strong spectral instability. Thus, while the diffusion QNF is stable, its CLM counterpart is highly sensitive to perturbations.

3. Agreement between GF and MS – After correcting the MS energy norm for the boundary term, the pseudospectra obtained from the scalar and gauge‑field formulations coincide. This validates the Hodge‑duality argument and demonstrates that the GF method can serve as a more universal tool, especially in situations where a master scalar is not readily available (e.g., higher‑spin fields or coupled gauge‑gravity systems).

4. General observations – Non‑hydrodynamic QNFs are generically unstable, consistent with earlier work on asymptotically flat black holes. The CLM pseudospectra, however, converge nicely and are generally more stable than QNFs, except in the hydrodynamic limit where the exceptional‑point mechanism dominates.

The authors conclude that the choice of norm is crucial for a physically meaningful pseudospectrum, and that the gauge‑field based approach offers a clean, extensible framework. Their results have implications for holographic transport, the reliability of hydrodynamic approximations, and the interpretation of pole collisions in the complex momentum plane. Future directions include extending the analysis to non‑linear gauge sectors, mixed gauge‑gravity perturbations, and exploring the impact of spectral instability on chaotic dynamics and strong cosmic censorship in AdS spacetimes.