High energy particle collisions under Cauchy horizon

1. Problem Statement

The paper addresses the phenomenon of unbounded growth in the center-of-mass energy ($E_{c.m.}$) during particle collisions near black hole horizons, known as the Bañados-Silk-West (BSW) effect. While the BSW effect has been extensively studied in three specific configurations:

1. Near the event horizon of extremal black holes (outside).
2. Near the event horizon of non-extremal charged black holes (outside).
3. Between the event horizon and the inner (Cauchy) horizon of non-extremal Reissner-Nordström (RN) black holes (the T-region).

There was a gap in the literature regarding particle collisions occurring inside the inner (Cauchy) horizon within the R-region (where the radial coordinate $r$ is spacelike and time $t$ is timelike). The authors aim to determine if high-energy collisions are possible in this specific region and, if so, under what kinematic conditions.

2. Methodology

The authors employ a classical general relativistic approach using the Reissner-Nordström (RN) metric to describe a static, charged black hole.

• Metric and Setup:
  • The spacetime is defined by $ds^2 = -f dt^2 + f^{-1} dr^2 + r^2(d\theta^2 + \sin^2\theta d\phi^2)$, where $f = 1 - 2M/r + Q^2/r^2$.
  • The study focuses on the R-region inside the inner horizon ($0 < r < r_-$), where $r$ is spacelike.
  • The analysis is restricted to neutral particles undergoing pure radial motion along geodesics.
• Kinematic Equations:
  • The equations of motion are derived from the conserved energy $E$ and the radial momentum $P = \sqrt{E^2 - m^2f}$.
  • The Lorentz factor $\gamma$ between two colliding particles is calculated as $\gamma = \frac{E_1 E_2 - \sigma_1 \sigma_2 P_1 P_2}{f}$, where $\sigma = \pm 1$ indicates the direction of radial motion.
• Collision Scenario Construction:
  • The authors propose a specific scenario where Particle 1 enters the R-region, reaches a turning point (where $P=0$), and reflects back ($\sigma_1 = +1$), colliding with Particle 2 moving inward ($\sigma_2 = -1$).
  • They analyze the integral constraints required for these two particles to meet at a specific radius $r_c$ close to the inner horizon $r_-$. This involves solving the time-coordinate integrals to ensure $t_1 = t_2$ at the collision point.

3. Key Contributions

• Discovery of a New Collision Regime: The paper identifies a previously unconsidered scenario where $E_{c.m.}$ grows unbounded inside the inner horizon of an RN black hole (in the R-region).
• Elimination of Fine-Tuning: Unlike the standard BSW effect near extremal horizons, which requires one particle to have "fine-tuned" parameters (critical angular momentum or charge), this new scenario does not require fine-tuning. Both particles can have "usual" (generic) parameters.
• Resolution of Kinematic Constraints: The authors address the "kinematic censorship" problem. They demonstrate that while a collision is theoretically possible, it requires specific initial conditions: one particle must start its motion in the infinite past ($t_0 \to -\infty$) or the other must approach the horizon in the infinite future ($t_0 \to +\infty$). This ensures the particles cross the same branch of the inner horizon to collide.

4. Results

• Unbounded Energy Growth: As the collision point $r_c$ approaches the inner horizon $r_-$, the metric function $f \to 0$. In the proposed scenario (one particle turning back, one incoming), the term $P_1 P_2$ in the numerator of the Lorentz factor adds constructively to $E_1 E_2$. Consequently, $\gamma \to \infty$ and $E_{c.m.} \to \infty$ as $r_c \to r_-$.
• Kinematic Requirement: The time difference between the start of the particles' motions is derived as:
  $$t_0^{(2)} - t_0^{(1)} \approx \frac{1}{\kappa_-} \ln |r_c - r_-|$$
  where $\kappa_-$ is the surface gravity of the inner horizon. For $r_c \to r_-$, this time difference diverges, necessitating that one particle originates from the infinite past or approaches the horizon in the infinite future.
• Physical Interpretation: The authors note that while the energy can be arbitrarily large, it cannot be literally infinite in a physical realization because it would require infinite time ($t_0$) and motion along a light-like surface, which is impossible for massive particles. This is consistent with the principle of kinematic censorship.

5. Significance

• Completeness of BSW Classification: The paper completes the classification of BSW-type scenarios for black holes. Unbounded energy growth is now established for all possible configurations:
  1. Outside the event horizon (extremal and non-extremal).
  2. Between the event and inner horizons.
  3. Inside the inner horizon (R-region).
• Implications for Inner Horizon Instability: The results provide a massive-particle counterpart to wave-process instabilities (mass inflation) previously associated with the inner horizon. The authors suggest that these high-energy collisions could contribute to the physical instability of the inner Cauchy horizon, complementing existing theories based on radiation.
• Generality: Although derived for static charged black holes, the authors argue that similar phenomena should exist for rotating (Kerr) black holes, suggesting a universal feature of black hole interiors.

In summary, Toporensky and Zaslavskii demonstrate that the inner horizon of a charged black hole acts as a natural particle accelerator capable of generating infinite center-of-mass energies without fine-tuning, provided the collision occurs deep within the R-region and involves particles with specific temporal boundary conditions.