Black Hole Formation from the Collision of Plane-Fronted Gravitational Waves

Abstract: This paper introduces a new effort to study the collision of plane-fronted gravitational waves in four dimensional, asymptotically flat spacetime, using numerical solutions of the Einstein equations. The pure vacuum problem requires singular, Aichelburg-Sexl type sources to achieve finite energy solutions, which are problematic to treat both mathematically and numerically. Instead then, we use null (massless) particles to source non-trivial geometry within the initial wave fronts. The main purposes of this paper are to (a) motivate the problem, (b) introduce methods for numerically solving the Einstein equations coupled to distributions of collisionless massless or massive particles, and (c) present a first result on the formation of black holes in the head-on collision of axisymmetric distributions of null particles. Regarding the last-named, initial conditions are chosen so that a black hole forms promptly, with essentially no matter escaping the collision. This can be interpreted as approaching the ultra-relativistic collision problem from within an infinite boost limit, but where the matter distribution is spread out, and thus non-singular. We find results that are consistent with earlier perturbative calculations of the collision of Aichelburg-Sexl singularities, as well as numerical studies of the high-speed collision of boson stars, black holes, and fluid stars: a black hole is formed containing most of the energy of the spacetime, with the remaining $15\pm1\%$ of the initial energy radiated away as gravitational waves. The methods developed here could be relevant for other problems in strong field gravity and cosmology that involve particle distributions of matter.

• The paper demonstrates that a black hole forms from the collision of plane-fronted gravitational waves, capturing about 85% of the initial energy.
• It employs a novel numerical approach using collisionless massless particles to overcome challenges posed by singularities in strong-field gravity.
• Results align with perturbative predictions and offer significant insights for gravitational wave astronomy and cosmic censorship studies.

Overview of Black Hole Formation from Gravitational Wave Collisions

The research article by Pretorius and East focuses on the intriguing problem of black hole formation resulting from the collision of plane-fronted gravitational waves in asymptotically flat spacetime. This investigation leverages numerical solutions to the Einstein equations, representing a significant methodological shift from conventional analytical approaches to a problem first explored by Penrose and Khan nearly half a century ago.

Problem and Approach

This study explores a facet of general relativity that starkly contrasts with the linear dynamics inherent in Newtonian gravity — the non-linear nature of Einstein's field equations, which prominently manifest in strong-field regimes. The motivation hinges on the quest to understand ultra-relativistic scattering dynamics, a less-explored terrain of the vast landscape that strong-field gravity encompasses.

To navigate the mathematical and numerical challenges associated with singular Aichelburg-Sexl type sources — typically employed in the vacuum problem to simulate plane-fronted gravitational waves — the authors utilize distributions of collisionless massless particles to source non-trivial geometry within the initial wave fronts. This adoption mitigates difficulties in handling singularities, enabling a rigorous exploration of black hole formation in head-on collisions of ultra-relativistic setups.

Key Findings and Numerical Results

The numerical results presented in the article are consistent with earlier perturbative predictions and previous high-speed collision studies involving boson stars and fluid stars. Notably, a black hole is formed containing approximately 85% of the initial system energy, with the remainder, around 15% ± 1%, radiated away as gravitational waves — a finding of practical significance for gravitational wave astronomy, especially in probing such events' radiative properties.

Moreover, the research underscores the beamed nature of the gravitational radiation emitted during these collisions, positing an alignment with expected behaviors derived from perturbative analyses of large impact parameters. The shock wave characteristics, notably, propagate outward from the collision site, forming concentric patterns around the resulting black hole.

Theoretical Implications and Future Directions

The study's results advance our understanding of the strong-field dynamics in scenarios lacking timelike compact objects or symmetries. Importantly, the methodologies and insights gained here could propel new questions regarding the universality of black hole formation thresholds across different matter models and dimensions. This work also supplies a lever for future study into gravitational interactions at the Planck scale, particularly if we assume scenarios with lowered dimensional thresholds for quantum gravitational effects.

In terms of theoretical implications, these findings may further inform the ongoing discourse on cosmic censorship, critical phenomena at the thresholds of singularity formation, and the broad classification of event horizon solutions across complex spacetimes, including those extending into higher dimensions.

The particle code developed here, with its capacity to simulate the Einstein-collisionless particle system, holds promise for future investigations into other complex problems within strong-field gravity and cosmology. Importantly, addressing existing computational challenges—such as managing particle number for precision and devising robust gauge conditions—will be crucial for the extensive exploration of these strong-field phenomena.

Given these advancements, this work not only contributes to the foundational understanding of gravitational wave dynamics but also provides a versatile platform for further theoretical and numerical inquiries into high-energy gravitational phenomena.