[Paper Review] Thermodynamics of Time Machines

Michael Devin|arXiv (Cornell University)|Feb 8, 2013

Advanced Thermodynamics and Statistical Mechanics9 references3 citations

TL;DR

This paper develops a thermodynamic framework for time machines using noisy, nonunitary evolution to resolve causal paradoxes like the grandfather paradox. By introducing a noise parameter $k$ that weights consistent histories, it shows how post-selected quantum ensembles and time machine models are formally dual, enabling consistent predictions and linking time travel to entropy and work limits.

ABSTRACT

In this note, a brief review of the consistent state approach to systems containing closed timelike curves or similar devices is given, and applied to the well known thermodynamic problem of Maxwell's demon. The 'third party paradox' for acausal systems is defined and applied to CTC censorship and black hole evaporation. Some traditional arguments for chronology protection are re-examined.

Motivation & Objective

• To resolve causal paradoxes in time machine models using a noise-based consistency mechanism.
• To apply thermodynamic principles—especially entropy and work—to systems with closed timelike curves (CTCs).
• To establish a formal duality between time machines and post-selected quantum ensembles, enabling consistent predictions.
• To examine the implications of chronology protection and information loss in black hole physics via this framework.
• To explore the feasibility of time travel and 'tourist' paradoxes in quantum mechanical models.

Proposed method

• Introduces a noise parameter $k$ to model bit errors in time machine channels, ensuring non-zero probability for all consistent histories.
• Uses a weight function $\omega(\psi_{\text{in}}, \psi_{\text{out}})$ to assign probabilities to in/out state pairs, with $\omega > 0$ for at least one $\psi$ to avoid the grandfather paradox.
• Applies the Brouwer fixed point theorem to the Bloch sphere to prove existence of invariant states under unitary evolution with intrinsic phase noise.
• Derives effective error rate $k$ by integrating over all qubit states, relating classical noise to quantum time machine behavior.
• Models time machine evolution as a post-selected ensemble, where measurement outcomes are weighted by noise and unitary evolution.
• Uses a composite ensemble model where $2k$ fraction of uncorrelated in/out states are admitted, enabling finite normalization even when post-selection would otherwise fail.

Experimental results

Research questions

• RQ1 How can causal paradoxes in time machines be resolved without violating quantum mechanics or unitarity?
• RQ2 What thermodynamic constraints limit the operation of time machines, particularly in terms of error rates and entropy?
• RQ3 Can time machines be formally equivalent to post-selected quantum ensembles, and what are the implications for predictability?
• RQ4 How does the presence of noise in time machine channels affect the consistency and probability of closed timelike curve histories?
• RQ5 What is the role of chronology protection in preventing macroscopic time travel, and can it be statistical rather than absolute?

Key findings

• The grandfather paradox is resolved by introducing a noise parameter $k$, ensuring non-zero probability for all consistent histories.
• A time machine with a noise rate $k$ produces a mixed output state $\rho_{\text{out}}(A)$ that depends on the external evolution $A$, with explicit dependence on $k$ and normalization factors.
• The effective error rate $k$ can be derived from the weight function by integrating over all qubit states, yielding a measurable parameter for quantum time machines.
• For a single qubit, the probability of measuring $|+\rangle$ after unitary $A$ and noise is given by a normalized sum over four possible histories, with $k$ suppressing unphysical outcomes.
• The model shows that post-selection with noise avoids the divergence of sample size in post-selected ensembles, making predictions finite and well-defined.
• A lower bound on the error rate of approximately 7% is suggested from $S = \ln 2$, linking time machine operation to black hole entropy and the Maldacena mechanism.