Causality in the maximally extended extreme Reissner--Nordström spacetime with identifications

Technical Summary: Causality in the Maximally Extended Extreme Reissner–Nordström Spacetime with Identifications

Problem Statement
This paper investigates whether the identification of asymptotically flat regions (AFRs) in the maximally extended extreme Reissner–Nordström (RN) spacetime ($e^2 = m^2$) leads to violations of causality. Specifically, the study asks if an observer in an AFR can send messages to their own past using timelike or null geodesics. This inquiry follows previous work (Ref. [1]) which demonstrated that such causality breaches do occur in the non-extreme RN metric ($e^2 < m^2$) when AFRs are identified, allowing radial timelike geodesics to return to the emitter's past. The current work seeks to determine if the extreme case ($e^2 = m^2$) exhibits similar behavior or if the causal structure prevents such loops.

Methodology
The author employs a combination of analytical derivation and numerical integration:

1. Metric Derivation: The maximal extension of the extreme RN metric is re-derived using curvature coordinates transformed into null coordinates ($p, q$) and subsequently into Kruskal-like coordinates ($P, Q$) and spacetime coordinates ($U, V$). In this specific coordinate system, the singularity at $r=0$ is represented as a straight line ($U=-1$), and the event horizon at $r=m$ corresponds to specific null boundaries.
2. Geodesic Equations: The equations of motion for timelike and null geodesics are analyzed. First integrals involving energy ($\Gamma$) and angular momentum ($J_0$) are utilized. The behavior of radial and non-radial geodesics is examined, particularly focusing on turning points (TPs) where the radial velocity vanishes.
3. Numerical Integration: Specific geodesic trajectories are numerically integrated starting from points in Sector I (an asymptotically flat region). The paths are tracked as they cross the horizon into Sector II and approach the "future copy" of the initial point in Sector I' (created by identifying the AFRs).
4. Causal Analysis: The positions of the turning points of these geodesics are compared against the past light cones of the future copies of the emission points. If a turning point lies to the future of the past light cone of the copy, the geodesic cannot return to the emitter's past, preserving causality.

Key Contributions and Results

• Geometric Structure: The paper establishes that in the extreme RN metric, the image of the singularity ($r=0$) in the $(U, V)$ plane is a straight line ($U=-1$), distinct from the $e^2 < m^2$ case where the singularity's shape depends on mass and charge parameters. The horizon ($r=m$) is shown to be tangent to timelike and non-radial null geodesics in the $(U, V)$ coordinates.
• Radial Geodesics:
  • Ingoing Radial Null Geodesics: These hit the singularity at $r=0$ and terminate there; they do not traverse to other regions.
  • Radial Timelike Geodesics: Numerical examples show that these geodesics approach the horizon tangentially. Their turning points (where they reverse direction) lie strictly to the future of the past light cone of the emitter's future copy. Consequently, they cannot return to the emitter's past.
• Non-Radial Geodesics:
  • Null Geodesics: Analytical solutions for turning points reveal that for non-radial null geodesics, one turning point always exists inside the horizon. For sufficiently large angular momentum ($|J_0|$), two additional turning points exist outside the horizon. Geodesics with large $|J_0|$ can oscillate between the inner and outer turning points, crossing the horizon in each cycle.
  • Numerical Verification: Numerical integrations of non-radial timelike and null geodesics crossing the horizon confirm that their turning points also lie to the future of the past light cones of the future copies of their emission points.
• Causality Conclusion: Unlike the $e^2 < m^2$ case, the identification of asymptotically flat regions in the extreme RN spacetime does not lead to causality breaches for the tested geodesics. Messages sent via these geodesics cannot reach the causal past of the emitter's future copy.

Significance and Limitations
The primary significance of this work is the demonstration, through numerical examples, that the extreme RN spacetime ($e^2 = m^2$) is causally stable under the identification of its asymptotically flat regions, in contrast to the non-extreme case. The paper explicitly states that while the numerical evidence is robust, a formal mathematical proof of this thesis (i.e., proving that all such geodesics behave this way, not just the examples provided) remains an open problem. The author notes that non-existence of causality violations cannot be proven by examples alone, and the derivation of a rigorous analytical proof is a desired future step.

The paper does not propose new experimental applications or speculate on broader physical implications beyond the specific causal structure of this spacetime model. It serves as a rigorous continuation of previous work on RN metric extensions, clarifying the distinct causal properties of the extreme limit.