High-Energy Collisions and Horizon Destruction

Near an extremal rotating horizon, the center-of-mass energy of two idealized particles can become arbitrarily large when one trajectory is tuned to a critical angular momentum. This is a local kinematic statement. It neither guarantees that such a particle is prepared from infinity nor that high-energy debris escapes with comparable energy, and it does not establish destruction of the horizon once backreaction is included.

Helpful background. Horizon generators and extremality build on Black-Hole Geometry and Mechanics; near-horizon modes use Black-Hole Perturbation Theory; conserved particle charges come from Formulations of General Relativity; escape is a question in Causality and Global Structure; and nonlinear tests require Computational Relativity.

Local collision energy

For particles with four-momenta $p_1^a$ and $p_2^a$, the invariant collision energy is

$$E_{\rm cm}^2=-(p_1+p_2)^2 =m_1^2+m_2^2-2p_1\cdot p_2.$$

In extremal Kerr, a particle with critical relation $E-\Omega_H L=0$ becomes asymptotically tangent to the horizon generator. Its relative boost against a generic infaller can diverge as the collision radius approaches the horizon. Bañados, Silk, and West display the resulting near-horizon divergence for equatorial geodesics Bañados, Silk, and West 2009, Eqs. (11)–(14).

The limiting procedure matters. Exactly critical trajectories can require infinite proper or coordinate preparation, near-extremal holes cut off the redshift hierarchy, and generic particles from infinity may encounter a turning point. Finite size, gravitational radiation, self-force, and the particles’ own center-of-mass energy eventually invalidate the fixed-background approximation.

Energy that reaches infinity

A large $E_{\rm cm}$ is measured locally. Debris climbing out of the near-horizon region is redshifted, and much of its allowed momentum cone points into the hole. A collisional Penrose process can extract rotational energy when a negative Killing-energy product falls through the horizon, but efficiency depends on the preparation channel, collision radius, masses, and allowed outgoing trajectory. Schnittman’s review separates local collision energy from energy-at-infinity bounds across these scenarios Schnittman 2018, §§ 2–4.

For a stationary Kerr background, define each conserved Killing energy by $E_i=-t_ap_i^a$, with $t^a$ normalized at infinity. A two-to-two collision obeys $E_1+E_2=E_3+E_4$, but the escaping efficiency

$$\eta=\frac{E_3}{E_1+E_2}$$

requires particle $3$ to reach infinity. The condition $\eta\gt1$ means particle $4$ carries negative Killing energy into the hole; it does not mean its locally measured energy is negative. Nor is $\eta$ a function of $E_{\rm cm}$ alone. A particle that turns around outside the horizon before colliding can open escape channels absent when both incident particles are ingoing. Schnittman checks these trajectory and reaction constraints explicitly in Schnittman 2014, Eqs. (1)–(3) and Figs. 1–3, pp. 1–3, PDF. Such preparation choices must accompany any stated efficiency bound.

Horizon destruction would require more: the absorbed stress-energy and angular momentum, the escaping flux, and the perturbed geometry must evolve to a globally visible singularity or to data with no regular horizon. Treating debris as test particles while claiming an order-one change to the black hole is internally inconsistent.

Collision results through 2026-08-12

• Established within the model: the invariant collision energy is unbounded in a singular extremal, critical-geodesic limit.
• Model-dependent: maximal energy at infinity and preparation probability depend on the particle model, spin, inclination, and near-extremality parameter.
• Competing interpretation: one view emphasizes the horizon as a natural accelerator; another emphasizes redshift, fine tuning, and backreaction as operational cutoffs. They differ in the observable being bounded, not in the algebra for $E_{\rm cm}$.
• Unresolved: no controlled calculation in classical general relativity has turned this kinematic divergence into a generic destruction of a regular horizon.

Distinguish fixed-background particles from self-consistent Einstein–matter initial data, and local energy from energy escaping to infinity. This discussion covers results through 2026-08-12. A collision whose products all cross the horizon or escape with finite Killing energy disproves the inference “unbounded local energy implies an observable unbounded output.” A stronger conclusion requires a convergent self-gravitating evolution resolving both the near-horizon collision and future null infinity, or a theorem controlling the relevant nonlinear limit.

Check your understanding

Suppose $E_{\rm cm}$ grows without bound along a sequence of collision radii, while the escaping product has conserved Killing energy $E_\infty\le5m$. Does the sequence produce unbounded energy for a distant observer?

Solution

No. $E_{\rm cm}$ is the energy in the local center-of-mass frame; $E_\infty$ is the conserved quantity relevant to the distant stationary observer. Near-horizon redshift and the escape cone can keep $E_\infty$ finite even as the local relative boost diverges. One must not substitute one observable for the other.

References

• Bañados, Máximo, Joseph Silk, and Stephen M. West. “Kerr Black Holes as Particle Accelerators to Arbitrarily High Energy.” Physical Review Letters 103 (2009): 111102. doi:10.1103/PhysRevLett.103.111102.
• Schnittman, Jeremy D. “Revised Upper Limit to Energy Extraction from a Kerr Black Hole.” Physical Review Letters 113 (2014): 261102. doi:10.1103/PhysRevLett.113.261102.
• Schnittman, Jeremy D. “The Collisional Penrose Process.” General Relativity and Gravitation 50 (2018): 77. doi:10.1007/s10714-018-2373-5.