Closed Timelike Curves Induced by a Buchdahl-inspired Vacuum Spacetime in $R^2$ Gravity

The recently obtained $\textit{special}$ Buchdahl-inspired metric [Phys. Rev. D 107, 104008 (2023)] describes asymptotically flat spacetimes in pure Ricci-squared gravity. The metric depends on a new (Buchdahl) parameter $\tilde{k}$ of higher-derivative characteristic, and reduces to the Schwarzschild metric, for $\tilde{k}=0$. For the case $\tilde{k}\in(-1,0)$, it was shown that it describes a traversable Morris-Thorne-Buchdahl (MTB) wormhole [Eur. Phys. J. C 83, 626 (2023)], where the weak energy condition is formally violated. In this paper, we briefly review the $\textit{special}$ Buchdahl-inspired metric, with focuses on the construction of $\zeta-$Kruskal-Szekeres (KS) diagram and the situation for a wormhole to emerge. Interestingly, the MTB wormhole structure appears to permit the formation of closed timelike curves (CTCs). More specifically, a CTC straddles the throat, comprising of two segments positioned in opposite quadrants of the $\zeta-$KS diagram. The closed timelike loop thus passes through the wormhole throat twice, causing $\textit{two}$ reversals in the time direction experienced by the (timelike) traveller on the CTC. The key to constructing a CTC lies in identifying any given pair of antipodal points $(T,X)$ and $(-T,-X)$ $\textit{on the wormhole throat}$ in the $\zeta-$KS diagram as corresponding to the same spacetime event. It is interesting to note that the Campanelli-Lousto metric in Brans-Dicke gravity is known to support two-way traversable wormholes, and the formation of the CTCs presented herein is equally applicable to the Campanelli-Lousto solution.