[Paper Review] Recurrence metrics and the physics of closed time-like curves

Moninder Singh Modgil, Deshdeep Sahdev|ArXiv.org|Jul 16, 2001

Cosmology and Gravitation Theories5 references3 citations

TL;DR

This paper proposes a cosmological model with a closed time dimension (S¹ topology) where all geodesics are closed time-like curves (CTCs), leading to universal recurrence and periodic reversal of proper time. It shows that such a universe avoids causality paradoxes by ensuring all systems, including observers, evolve unitarily and reversibly, with entropy decreasing during backward proper time evolution, rendering paradoxes like the grandfather paradox physically incoherent due to state reversion and memory erasure.

ABSTRACT

We investigate vacuum solutions of Einstein's equation for a universe with an S^1 topology of time. Such a universe behaves like a time-machine and has geodesics which coincide with closed time-like curves (CTCs). A system evolving along a CTC experiences the Loschmidt velocity reversion and undergoes a recurrence commensurate with the universal period. We indicate why this universe is free of the causality paradoxes, usually associated with CTCs.

Motivation & Objective

• To investigate the physical implications of a universe with a closed time dimension (S¹ topology), where time is periodic and geodesics form closed time-like curves (CTCs).
• To resolve the longstanding causality paradoxes associated with CTCs by showing that unitary evolution and state recurrence prevent logical inconsistencies.
• To demonstrate that in such a universe, all systems, including observers, undergo periodic reversals in proper time, leading to recurrence and entropy decrease during backward evolution.
• To argue that the psychological sense of time is preserved despite time reversals, due to memory erasure and the observer's inability to retain information from higher-entropy states.
• To provide a classical framework for understanding time machines and CTCs that avoids the need for external forces or causal isolation, by making the entire universe a time-machine.

Proposed method

• Construct a vacuum solution of Einstein's equations for a universe with S¹ topology in time, analogous to Minkowski space but with periodic time.
• Apply time-closure constraints to ensure all geodesics are periodic, leading to recurrence of initial Cauchy data after a universal period T.
• Analyze the behavior of proper time along CTCs, showing it is periodic and smooth, with turning points where proper time evolves backward relative to coordinate time.
• Use the Zollfrei metric structure (as in S³×S¹) to ensure all null geodesics close periodically, enabling universal recurrence of physical states.
• Model the evolution of irreversible processes (e.g., entropy increase) and show that during backward proper time evolution, entropy decreases and memories are erased.
• Demonstrate that observers cannot access times prior to their own birth due to state reversion, thereby resolving the grandfather paradox via consistent state tracking.

Experimental results

Research questions

• RQ1 How does a universe with a closed time dimension (S¹) avoid causality paradoxes typically associated with closed time-like curves (CTCs)?
• RQ2 What are the implications of periodic proper time evolution along CTCs for the arrow of time and entropy change?
• RQ3 Why is the psychological sense of time preserved in a universe where proper time reverses periodically?
• RQ4 Can unitary evolution be maintained in a time-machine universe without requiring causal isolation or external forces?
• RQ5 How does memory erasure during time reversals prevent observers from retaining information about higher-entropy states, making entropy decrease unobservable?

Key findings

• In a universe with S¹-time, all geodesics are closed time-like curves (CTCs), and every system, including observers, undergoes periodic recurrence with a universal period T.
• Proper time along CTCs is periodic and smooth, requiring an even number of turning points per cycle, each corresponding to a reversal in the direction of proper time evolution.
• During backward proper time evolution, entropy decreases, and all systems, including memory, revert to previously occupied states, erasing information about the recent past.
• The grandfather paradox is resolved because an observer moving along a CTC only revisits their own past and cannot access times prior to their birth, as their state continuously reverts to earlier configurations.
• The psychological sense of time persists because observers only retain memories of their conventional past, which shrinks with each cycle due to continuous erasure of recent information.
• The model avoids causality violations by ensuring that all physical processes, including measurement and memory, are unitary and consistent with recurrence, making non-unitary evolution and paradoxes unobservable.