Black Hole Tomography: Unveiling Black Hole Ringdown via Gravitational Wave Observations (2501.08964v2)

Abstract: During the post-merger regime of a binary black hole merger, the gravitational wave signal consists of a superposition of quasi-normal modes (QNMs) of the remnant black hole. It has been observed empirically, primarily through numerical simulations and heuristic arguments, that the infalling radiation at the horizon is also composed of a superposition of QNMs. In this paper, we provide an analytic explanation for this observation in the perturbative regime. Our analysis is based on a characteristic initial value formulation where data is prescribed on the horizon (modeled as a perturbed isolated horizon), and on a transversal null-hypersurface which registers the outgoing radiation. This allows us to reformulate the traditional QNM problem in a fully 4-dimensional setting. Using a mode-decomposition, we demonstrate that the radiation modes crossing $\mathcal{H}$ are highly correlated with the outgoing modes crossing $\mathcal{I}$, and provide explicit expressions linking $\widetilde{\Psi}_0$ at the horizon with $\widetilde{\Psi}_4$ at null infinity.

• The paper develops an analytical framework for black hole tomography, showing how infalling and outgoing gravitational waves share quasi-normal mode characteristics.
• Quasi-normal mode frequencies are deduced by demanding analytic and stable solutions that naturally satisfy physical boundary conditions at the horizon and infinity.
• The paper links near-horizon black hole physics to far-field GW observations, providing a theoretical basis for empirical tomography.

Black Hole Tomography: Unveiling Black Hole Ringdown via Gravitational Wave Observations

The paper of black holes through gravitational wave observations reveals intricate aspects of their post-merger dynamics. The paper "Black Hole Tomography: Unveiling Black Hole Ringdown via Gravitational Wave Observations" investigates the theoretical underpinnings of such phenomena using perturbative techniques. It primarily addresses the question of why the infalling radiation onto a black hole bears the same quasi-normal mode (QNM) characteristics as the outgoing gravitational waves observed in the far-field region. This work provides an analytical framework supporting gravitational wave tomography, making it possible to infer the near-horizon physics from observations at null infinity.

Perturbative Framework and Initial Conditions

In the context of binary mergers, the research explores the ringdown phase where gravitational waves emanate as damped sinusoids governed by QNMs. These modes emerge from the dynamics of black hole perturbations, which converge to specific discrete complex frequencies known as QNM frequencies. The paper employs a perturbative regime wherein the gravitational radiation influences the black hole horizon. By establishing a characteristic initial value problem that prescribes data on the black hole horizon (modeled as a perturbed isolated horizon) and outgoing null surfaces, this research provides a coherent reformulation of the QNM problem.

Analytic Solutions and Boundary Conditions

A significant achievement of this analysis lies in extending the traditional QNM formulation to a four-dimensional setting, using the Newman-Penrose formalism. The paper shows that QNM frequencies can naturally be deduced by demanding analytic and stable solutions. Unlike the conventional approach where the absence of incoming radiation is a priori imposed, the present framework inherently satisfies such conditions by favoring solutions that correspond to ingoing modes at the horizon and outgoing modes at infinity.

Quasi-Normal Modes and Stability Criteria

The analysis demonstrates that the frequencies encountered are consistent with those derived through other methods within the field of general relativity, highlighting their independence from initial data choices. The framework additionally brings to light how the minimal set of boundary conditions—pertaining to analyticity and causality—leads to a clean extraction of QNMs without invoking additional assumptions. Through this novel formulation, the distinct phases encapsulated by QNMs are shown to be seamlessly connected to the physical boundary and stability criteria.

Implications and Future Directions

This research holds relevance not just for understanding black hole ringdown, but also for future gravitational wave astronomy related to extreme mass-ratio inspirals (EMRIs) and more generalized black holes encompassing arbitrary spins. The explicit relationship established between the infalling and outgoing radiation paves the way for potentially observing horizon dynamics through gravitational wave data, thereby marking a meaningful step towards empirical black hole tomography. The implications extend to understanding the nonlinearities in gravitational waveforms and enhancing waveform models used for detection analysis.

Looking forward, the work suggests a reevaluation of black hole stability, especially around dynamical horizons, through second-order perturbations that might capture more intricate details such as horizon area change. Moreover, the theoretical insights gained here invite further studies into the full metric reconstruction of perturbed black holes, offering a more comprehensive view of black hole dynamics post-coalescence.

In conclusion, this paper lays a robust theoretical foundation for interpreting gravitational wave signals through the lens of isolated and perturbed horizon formalism, establishing an analytical basis for probing the near-horizon region of black holes via observations made at cosmic distances. This approach not only reconciles theoretical predictions with computational findings but also enhances our understanding of black hole physics in the strongest gravitational regimes.