Sourced Carrollian Fluids Dual to Black Hole Horizons

TL;DR

This work establishes a dynamical Carrollian-fluid dual for a black-hole event horizon perturbed by a bulk massless scalar field, using a stretched Carrollian geometry framework to derive horizon equations that mirror Carrollian hydrodynamics with a driving source. Implementing a perturbative expansion around Schwarzschild, the authors recast the horizon dynamics in spin-weighted variables, and solve numerically for a small scalar-field amplitude up to second order, revealing how the horizon’s expansion and Hájíček momentum relax as the horizon approaches a non-expanding state. The results show the emergence of quadratic quasinormal modes, a close relation between horizon area growth and bulk energy flux, and a consistent angular-momentum transfer between the horizon and the scalar field; equilibration of the Carrollian fluid is tied to the horizon’s relaxation. Overall, the study strengthens the Carrollian-fluid/horizon duality in nonvacuum, dynamical settings and highlights nonlinear features that warrant further exploration in fully dynamical or more general null-surface contexts.

Abstract

The (degenerate) geometry of event horizons is linked to Carrollian fluids. We investigate the behavior of event horizons via a perturbative coupling to a massless scalar field, making connections to Carrollian hydrodynamics with a driving source, and discuss the fluid equilibration in tandem with the horizon's relaxation to equilibrium. We observe that after the perturbation dies off, the Carrollian fluid energy and momentum densities approach equilibrium as the horizon asymptotically becomes non-expanding. We connect the equilibration of the Carrollian fluid dual to the black hole horizon through the expansion of its background geometry.

Figures (7)

• Figure 1: Time profile of different $(\ell,m)$ modes on the horizon. Peak amplitudes, $\mathcal{A}_{\ell m}$, are set to satisfy: 1 = $\mathcal{A}_{00} = 2 \mathcal{A}_{11} = - 2 \mathcal{A}_{1-1} = 4 \mathcal{A}_{20}$. The negative amplitude is needed for the $\left( \ell,m \right) =(1,-1)$ to ensure the scalar field is real. See Appendix \ref{['appendix:scalarfield']} for details on the construction of this profile.
• Figure 2: Horizon 2-sphere embedding at different times. Radial coordinate is scaled with $\sqrt{\det{q_{AB}}}$ to illustrate the deformations of the surface. The peak of the scalar field is reached at $v = 0$. The color map refers to the value of the null expansion in $\mathcal{O}\left(\epsilon^2\right)$, $\vartheta\left(\theta,\phi\right)$, evaluated on a grid of angles on the 2-sphere. A cut-off value of $10^{-16}$ is used in the color map. That is, $\vartheta\left(\theta,\phi\right)$ is set to 0 for any cell on the grid on which the value of null expansion is less than $10^{-16}$. The perturbative parameter is set to satisfy $\epsilon^2 = 3 \times 10^{-8}$, which is chosen such that the deformations on the metric are visible in the embedding plot at early times.
• Figure 3: Real and imaginary parts of the quasinormal mode (QNM) frequencies extracted from the evolution of the null expansion in $\mathcal{O}\left(\epsilon^2\right)$. Grey squares denote the input QNM frequencies of the scalar field source. Note that QNM frequencies of $\vartheta_{\ell m}$ that appear multiple times are generated by different parent modes. Parent modes are written near the markers in the same color as the markers.
• Figure 4: Difference between the final black hole area at time $v_f$ and the area at time $v$, computed from the null expansion (solid orange curve) and the energy flux (dashed blue curve) versus time. The perturbative parameter is the same as in Fig. \ref{['fig:snapshots']}, which satisfies $\epsilon^2 = 3 \times 10^{-8}$.
• Figure 5: Angular momentum of the horizon computed by integrating the Hájíček field (solid orange curve) and integrating the angular momentum flux due to the scalar field (dashed blue curve). The perturbative parameter is the same as in Fig. \ref{['fig:snapshots']}, which satisfies $\epsilon^2 = 3 \times 10^{-8}$.