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A Quantum Weyl Conjecture

Ivo Sachs, Marc Schneider

Abstract

We perform a quantum probing of colliding plane-wave space-times. In particular, we consider the Khan-Penrose and the Ferrari-Ibáñez solutions, which admit a strong and a weak singularity after the two waves collide. While we find that, like Schwarzschild, for the Khan-Penrose solution the singularity cannot be probed by quantum field theory, the Ferrari-Ibáñez singularity can be traversed. Our results culminate in a quantum Weyl conjecture: The significant geometric property to classify space-times with respect to quantum probes is given by the Coulomb part of the Weyl tensor. We then use this conjecture to sketch a possible backreaction scenario for plane waves.

A Quantum Weyl Conjecture

Abstract

We perform a quantum probing of colliding plane-wave space-times. In particular, we consider the Khan-Penrose and the Ferrari-Ibáñez solutions, which admit a strong and a weak singularity after the two waves collide. While we find that, like Schwarzschild, for the Khan-Penrose solution the singularity cannot be probed by quantum field theory, the Ferrari-Ibáñez singularity can be traversed. Our results culminate in a quantum Weyl conjecture: The significant geometric property to classify space-times with respect to quantum probes is given by the Coulomb part of the Weyl tensor. We then use this conjecture to sketch a possible backreaction scenario for plane waves.
Paper Structure (14 sections, 68 equations, 3 figures)

This paper contains 14 sections, 68 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The geometry of colliding plane waves
  3. Plane-wave space-time
  4. Colliding plane waves
  5. Khan-Penrose space-time
  6. Khan-Penrose space-time: geometry
  7. Khan-Penrose space-time: quantum probing
  8. Weakly singular colliding wave solutions
  9. Ferrari-Ibáñez space-time: geometry
  10. Ferrari-Ibáñez space-time: quantum probing
  11. Classification and consequences
  12. The $\Psi_2$-conjecture
  13. Backreaction
  14. Discussion

Figures (3)

  • Figure 1: Penrose diagram of the Khan-Penrose space-time showing two colliding waves at $U=0$ and $V=0$ that cut the space-time into four portions. In the orange shaded, focusing regions, null-geodesics that have crossed the wave are focused at the focal plane $V=1$ in II, or $U=1$ in III. These mark the folding singularities. The purple shaded region is curved and admits a space-like curvature singularity at $U^2+V^2=1$.
  • Figure 2: Penrose diagram of the Ferrari-Ibáñez space-time showing two colliding waves at $U=0$ and $V=0$ that cut the space-time into four portions. In the orange shaded, focusing regions, null-geodesics that have crossed the wave are focused at the focal plane $V=\frac{\pi}{2}$ in II, or $U=\frac{\pi}{2}$ in III. The purple shaded region is curved and admits a Cauchy horizon at $U+V=\frac{\pi}{2}$.
  • Figure 3: The null-field wave packet $\varphi(x)$ (depicted as the orange bar) is radiated across an impulsive plane-wave $H(V)$ into the region II. After crossing the plane-wave, the field scatters and creates a non-vanishing, actually diverging, $\Psi_2$-Weyl scalar through backreaction. The folding singularity in II transforms into a spacelike singularity and, therefore, into a Khan-Penrose space-time. The focusing that occurs through the wave packet itself may lead to a folding singularity at some $U_0\gg1$.

Theorems & Definitions (1)

  • Conjecture : Weyl Quantum Conjecture