Cauchy-Horizon Particle Collisions Scale With Surface-Gravity Susceptibility, Not Particle Fine Tuning

Sterling Geisel, QBist Lab; Dr. Hideo Tanaka

Abstract

Toporensky and Zaslavskii show that neutral massive particles can collide below the inner horizon of a Reissner-Nordstrom black hole with unbounded center-of-mass energy, without the particle-parameter fine tuning required in the original Bañados-Silk-West channel. Pudding Theory reads this result as a direct instance of Chaos Susceptibility. The Cauchy horizon is a geometric susceptibility layer. It converts ordinary geodesic data into divergent relative boost because the near-horizon metric function supplies the amplifier. The source paper treats the large time separation and near-horizon coordinate placement as kinematic conditions. Pudding Theory treats them as the physical signature of an unstable receiving substrate. The collision is not made exceptional by the particles. It is made exceptional by the horizon’s susceptibility to small coherent differences in radial branch, timing, and approach. If the logarithmic near-horizon scaling coefficient of collision energy with launch-time separation were measured to be zero, this Postulate would be falsified.

Source Synopsis

Toporensky and Zaslavskii study radial neutral-particle collisions in the inner R region of a nonextremal Reissner-Nordstrom black hole, below the Cauchy horizon. The metric is written in the usual static form with \(f=1-2M/r+Q^2/r^2\), where \(r_+=M+\sqrt{M^2-Q^2}\) is the event horizon and \(r_-=M-\sqrt{M^2-Q^2}\) is the inner horizon. In the region \(0<r<r_-\), \(r\) is spacelike and \(t\) is timelike. The authors restrict the calculation to pure radial geodesic motion and neutral massive particles.

The relevant invariant is the center-of-mass energy, \[ E_{\rm c.m.}^2=m_1^2+m_2^2+2m_1m_2\gamma , \] where \[ \gamma=\frac{E_1E_2-\sigma_1\sigma_2P_1P_2}{f}. \] Here \(P_i=(E_i^2-m_i^2f)^{1/2}\), and \(\sigma_i=\pm 1\) gives the radial direction. Near \(r_-\), \(f\to0\). When one particle has turned around and moves outward while the other moves inward, the numerator remains finite and positive while the denominator tends to zero. Thus \(\gamma\), and therefore \(E_{\rm c.m.}\), grows without bound.

The central novelty is that this channel does not require the usual BSW fine tuning of a particle’s constants of motion. The obstacle is instead kinematic. Generic particles cross different branches of the inner horizon and do not collide. The authors show that collision can be arranged by choosing integration constants so that the two radial positions and time coordinates meet near \(r_-\). The required time separation behaves as \[ t_0^{(2)}-t_0^{(1)}\approx \kappa_-^{-1}\ln |r_c-r_-|. \] Thus an arbitrarily large collision energy requires an arbitrarily large coordinate-time separation. The authors relate the result to inner-horizon instability and mass inflation, while noting that kinematic censorship prevents literal infinity.

Postulate Lens

The applicable Postulate is Chaos Susceptibility: highly chaotic systems are most susceptible to small coherent inputs. The source phenomenon already has this structure. The Cauchy horizon is the unstable layer of the charged black-hole interior. It takes ordinary massive geodesics, with no special critical particle parameter, and amplifies a small branch-and-timing distinction into unbounded relative boost. The susceptibility is not a metaphor. It is the divergent factor \(1/f\), with the near-horizon expansion \(f\simeq2\kappa_-(r-r_-)\) fixing the scale of amplification.

Pudding Theory Reading

Pudding Theory reads the inner-horizon collision as a susceptibility event. The source paper frames the divergence as a kinematic possibility inside the R region. That statement is correct, but incomplete. The deeper structure is that the Cauchy horizon functions as a receiving substrate whose instability organizes otherwise ordinary geodesic data into high-energy encounter.

The particles do not carry the effect. They are neutral. They move on geodesics. Their energies and masses are not tuned to a critical value. In the original BSW setting, the exceptional structure sits partly in the particle data. Here it sits in the horizon. This transfer of exceptionalism is the main theoretical point. The source paper treats the absence of fine tuning as a technical distinction among black-hole accelerator scenarios. Pudding Theory treats it as the diagnostic sign that susceptibility has moved from matter parameters into spacetime geometry.

The background term \(f(r)\) is therefore not a passive denominator. It is the susceptibility field of the local geometry. As \(r\to r_-\), the factor \(1/f\) converts the sign difference \(\sigma_1\sigma_2=-1\) into a diverging Lorentz factor. In ordinary language, the horizon does not add energy to the particles. It changes the amplification rule for their relative motion. This is the same logical form as Pudding Theory’s treatment of chaotic systems: a small coherent distinction is not forced through a stable substrate; it is selected by an unstable substrate and magnified.

The source also treats the integration constants \(t_0^{(1)}\) and \(t_0^{(2)}\) as kinematic layout. Pudding Theory reads their logarithmic separation as the coherence condition of the susceptible layer. Collision near \(r_-\) requires the histories to remain phase-aligned in the horizon’s time coordinate while approaching opposite radial branches. The condition \[ t_0^{(2)}-t_0^{(1)}\sim \kappa_-^{-1}\ln |r_c-r_-| \] is not merely scheduling. It is the structural constraint imposed by the inner horizon on which histories can meet inside its amplification region.

This reading also sharpens the link to mass inflation. Wave instability at the inner horizon and massive-particle collision instability are not separate curiosities. They are two expressions of the same susceptibility surface. Radiation packets and massive geodesics differ in carrier, not in the underlying role of the Cauchy horizon. Both expose a region where small differences in ingoing and outgoing structure become large invariant effects.

Falsifiable Observable

The distinguishing observable is the logarithmic scaling of \(E_{\rm c.m.}\) with the required launch-time separation \(\Delta t=t_0^{(2)}-t_0^{(1)}\) for collisions approaching \(r_-\). Pudding Theory predicts that the susceptibility coefficient is set by the inner-horizon surface gravity, so that the near-horizon growth is controlled by \(\kappa_-\), not by hidden fine tuning of particle constants. If the logarithmic near-horizon scaling coefficient \(d\ln E_{\rm c.m.}/d\Delta t\) were measured to be zero, this Postulate would be falsified.

Editorial Dialogue

Tanaka: The reading risks renaming a known geometric divergence. The calculation already says why \(E_{\rm c.m.}\) diverges: \(f\to0\), opposite radial signs, and a tuned coordinate-time meeting. No extra ontology is needed.

Sterling: The calculation gives the channel. It does not say what kind of physical region the Cauchy horizon is. The important change from the usual BSW case is that the particle no longer bears the critical condition. The horizon does. That is not a relabeling. It locates the amplifier.

Tanaka: But the integration constants are still chosen. Generic particles miss each other.

Sterling: Yes. Pudding Theory does not erase the encounter condition. It interprets it. The logarithmic timing relation is the coherence condition for access to the susceptible region. Once that condition is met, ordinary geodesics generate unbounded invariant energy. The mechanism is structural, not particle-specific.

Tanaka: Then the claim is that the Cauchy horizon behaves like a dynamical amplifier.

Sterling: More precisely, it is a susceptibility surface. Its instability fixes which small distinctions matter. The sign of radial branch, the logarithmic timing, and the surface gravity form one constraint. That is why the result belongs with inner-horizon instability, not merely with particle accelerator analogies.

Discussion

The reading buys a cleaner classification of black-hole collision channels. In the original BSW effect, the critical behavior is attached to particle parameters near an event horizon. In the Toporensky-Zaslavskii channel, the critical behavior is attached to the Cauchy horizon itself. Pudding Theory makes that relocation explicit. It says the phenomenon is not primarily a particle accelerator mechanism. It is an instability amplifier expressed through massive-particle kinematics.

The limitation is also clear. The source calculation is idealized: radial motion, neutral test particles, fixed Reissner-Nordstrom geometry, and no backreaction. A full treatment would include charged particles, rotating interiors, self-gravity, and the destruction or persistence of the Cauchy horizon under perturbation. Those additions could change the available collision region. They would not remove the core claim unless they erase the surface-gravity scaling of the amplification itself.

The conclusion would change if the divergence were shown to depend on concealed fine tuning of particle constants rather than on the near-horizon susceptibility factor. It would also change if backreaction universally removed the meeting region before the scaling became operative.

References

1. Toporensky, A. V., and Zaslavskii, O. B. “High energy particle collisions under Cauchy horizon.” arXiv:2603.20245, 2026. DOI: doi:10.48550/arxiv.2603.20245.

2. Ochs, S. “Pudding Theory: A Topological Theory of Information Fields.” QBist Lab Working Paper, 2026.

3. Bañados, M., Silk, J., and West, S. M. “Kerr Black Holes as Particle Accelerators to Arbitrarily High Energy.” Physical Review Letters 103, 111102, 2009. arXiv:0909.0169.

4. Novikov, I. D. “R- and T-regions in space-time with spherically symmetric space.” General Relativity and Gravitation 33, 2259, 2001.

5. Zaslavskii, O. B. “Acceleration of particles near the inner black hole horizon.” Physical Review D 85, 024029, 2012. arXiv:1110.5838.

6. Poisson, E., and Israel, W. “Inner-horizon instability and mass inflation in black holes.” Physical Review Letters 63, 1663, 1989.

7. Pavlov, Yu. V., and Zaslavskii, O. B. “Kinematic censorship as a constraint on allowed scenarios of high energy particle collisions.” Gravitation and Cosmology 25, 390, 2019. arXiv:1805.07649.