[Paper Review] Loschmidt's paradox, entropy and the topology of spacetime

Moninder Singh Modgil|ArXiv.org|Jul 17, 2009

Black Holes and Theoretical Physics11 references3 citations

TL;DR

This paper investigates the topology of spacetime as a resolution to Loschmidt's paradox by proposing a time-periodic universe with $S^1$ topology, where entropy increases and decreases symmetrically over a cycle. Using Fourier expansions of physical variables and enforcing periodic boundary conditions, it derives infinite linear systems constraining dynamical laws, showing that only consistent, recurrent dynamics can exist in such a universe.

ABSTRACT

The issue of the "eternal return" is examined from the perspective of the topology of spacetime. Constraints on dynamical laws for the periodic evolution of a system or universe are highlighted. Using a Fourier series expansion, an infinite set of simultaneous linear equations is formulated for the periodic evolution of a system/universe.

Motivation & Objective

• To resolve Loschmidt's paradox—why irreversible entropy increase arises from time-symmetric dynamics—by examining spacetime topology.
• To investigate whether a universe with compact time topology ($S^1$) can support exact, deterministic recurrence of physical states.
• To derive constraints on dynamical laws in a time-periodic universe through Fourier series expansions of physical variables.
• To distinguish between mere recurrence of initial conditions and identical evolution across cycles, emphasizing the need for periodicity in all time derivatives.
• To assess the physical viability of such a model using criteria of falsifiability and experimental feasibility, including IR/UV cutoffs in measurable modes.

Proposed method

• Formulates a recurrence metric with line element $ds^2 = \left(\frac{\pi}{T}\right)^2 \cos^2\left(\frac{\pi t}{T}\right) dt^2 - dx_1^2 - dx_2^2 - dx_3^2$, modeling a time-periodic universe with period $T$.
• Expands any physical variable $A(t)$ and its derivatives $A^{(n)}(t)$ in a Fourier series over the interval $t \in [0, T]$, yielding infinite series in $\cos(2\pi m t / T)$ and $\sin(2\pi m t / T)$.
• Derives an infinite system of linear equations linking initial conditions $A(0)$, $A^{(n)}(0)$ to Fourier coefficients $C_m^1$, $C_m^2$, with coefficients involving $\left(\frac{2\pi m}{T}\right)^n$.
• Imposes periodic boundary conditions $A(t) = A(t+T)$ and $A^{(n)}(t) = A^{(n)}(t+T)$ for all $n$, ensuring identical evolution across cycles.
• Analyzes the solvability of the resulting infinite matrix system using known results from infinite linear algebra and sequence spaces.
• Introduces IR and UV cutoffs for measurable Fourier modes due to experimental and quantum limitations, reducing the infinite system to a finite one for practical analysis.

Experimental results

Research questions

• RQ1 Can a universe with $S^1$ topology of time exhibit exact recurrence of physical states, including all time derivatives?
• RQ2 How do the constraints of periodicity in all time derivatives affect the form of dynamical laws in such a universe?
• RQ3 What is the relationship between the entropy function $S(t) \sim \sin(\pi t / T)$ and the reversal of the thermodynamic arrow of time at $t = \pm T/2$?
• RQ4 How does the $S^1$ time topology differ from asymptotic Poincaré recurrence and from FRW oscillating models with $k=1$?
• RQ5 What are the implications of the infinite system of linear equations for the existence and uniqueness of solutions in a time-periodic universe?

Key findings

• The recurrence metric $ds^2 = \left(\frac{\pi}{T}\right)^2 \cos^2\left(\frac{\pi t}{T}\right) dt^2 - dx_1^2 - dx_2^2 - dx_3^2$ yields sinusoidal particle trajectories with period $T$, ensuring exact recurrence for non-interacting particles.
• The entropy function $S(t) \sim \sin(\pi t / T)$ increases for $t \in [-T/2, T/2]$ and decreases for $t \in (T, -T/2) \cup [T/2, T)$, with $dS/dt = 0$ at $t = \pm T/2$, indicating reversal of the thermodynamic arrow of time.
• Morse theory implies that the entropy function must have an even number of critical points on a closed time loop, consistent with the observed symmetric increase and decrease of entropy.
• The condition $q(t) = q(t+T)$, ensuring identical evolution across cycles, requires periodicity in all derivatives $q^{(n)}(t)$, not just position and velocity.
• The infinite system of linear equations derived from Fourier expansions of $A(t)$ and $A^{(n)}(t)$ imposes strict constraints on dynamical laws, with solvability depending on the properties of the infinite matrix $a_{nm}$.
• In the limit $T \to \infty$, all time derivatives $A^{(n)}(t) \to 0$, implying a frozen, non-dynamic universe, which is inconsistent with observed dynamics.