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

TL;DR

The work investigates whether identifying asymptotically flat regions in the maximally extended extreme Reissner–Nordström spacetime creates causal loops via timelike or null geodesics. By re-deriving the maximal extension in suitable coordinates and analyzing both radial and nonradial geodesics, the study finds that turning points of these geodesics occur in the future relative to the emitter’s past light cone, preventing messages to the past of the emitter in the extreme case, in contrast to the non-extreme scenario. While numerical evidence supports no acausality, a general formal proof is still sought. The analysis also clarifies horizon tangency properties and the role of angular momentum in generating additional turning points outside the horizon. Overall, identifications do not evidently breach causality for the extreme RN spacetime within the explored parameter space, highlighting a distinction from the $e^2<m^2$ case.

Abstract

In continuation of the similarly titled paper on the $e^2 < m^2$ Reissner -- Nordström (RN) metric (arXiv 2409.03786), in this paper it was verified whether it is possible to send (by means of timelike and null geodesics) messages to one's own past in the maximally extended {\it extreme} ($e^2 = m^2$) RN spacetime with the asymptotically flat regions being identified. Numerical examples show that timelike and nonradial null geodesics originating outside the horizon have their turning points to the future of the past light cone of the future copy of the emitter. This means that they cannot reach the causal past of the emitter's future copy. Ingoing radial null geodesics hit the singularity at $r = 0$ and stop there. So, unlike in the $e^2 < m^2$ case, identification of the asymptotically flat regions does not lead to causality breaches. A formal mathematical proof of this thesis (as opposed to the numerical examples given in this paper) is still lacking and desired.

Figures (10)

• Figure 1: The graph of the function $\zeta(r)$ defined by (\ref{['2.2']}), with $m = 0.95$ (the same value as in Ref. Kras2025).
• Figure 2: The subsets of the $(U, V)$ coordinate plane corresponding to $r \leq m$ (left panel) and $r \geq m$ (right panel). The vertical hyperbolae are the $r =$ constant lines, they are timelike. They degenerate to the pairs of straight segments (which are null) in the limits $r \to m$ and $r \to \infty$. The horizontal hyperbolae are the $t =$ constant lines.
• Figure 3: The maximal extension of the extreme ($e^2 = m^2$) R--N metric. The thin straight segments are the images of the null infinities, where $r \to \infty$. The hyperbola arcs are the timelike $r =$ constant $\neq m$ lines. The thick straight segments are the spurious singularities (event horizons) at $r = m$. The hatched straight line is the true singularity at $r = 0$; it coincides with the ${\cal U} = -1$ coordinate line and is timelike. Just as in the $e^2 < m^2$ case, we can identify sectors I and I$'$.
• Figure 4: Right panel: Three future-directed radial timelike geodesics emitted at points E1 and E3 in sector I of Fig. \ref{['rnextmax']}. The G1a has $\Gamma = 1.1$, G2a has $\Gamma = 3.0$ (i.e. larger energy in the Newtonian limit), G3a is 'elliptic' and has $\Gamma = 0.5$. R1 is a radial ray emitted at E1; it hits the singularity at $r = 0$ in a finite interval of the affine parameter. The crosses at the upper ends of G1a, G2a and G3a mark the first points on them at which $r < m$. Left panel: The continuation of G1a, G2a and G3a into sector II of Fig. \ref{['rnextmax']}. Their endpoints, marked with dots, are at the turning points. LC1 is the radial generator of the past light cone of E1$'$ -- the copy of E1 in sector I$'$ of Fig. \ref{['rnextmax']}, LC3 is the analogue of LC1 for point E3$'$. See the text for more explanation. Upper inset: An enlarged image of the area around the upper endpoints of G1a and G2a.
• Figure 5: The two lower panels of Fig. \ref{['timgeosm']} put into their correct relative positions.