Topological Entropy: A New Principle from Worldline Non-Injectivity

Topological Entropy: A New Principle from Worldline Non-Injectivity We introduce a novel concept of entropy — **topological entropy** — that emerges from the kinematic condition of worldline non-injectivity: when a single ultra-relativistic worldline (Lorentz factor γ > γ_crit) intersects a constant-time hypersurface Σ_t in N > 1 distinct spatial points, the same physical system appears simultaneously on N topological sheets. In standard thermodynamics, entropy is a single-sheet quantity that obeys the second law. We show that on any single sheet, entropy can decrease in a controlled manner, as illustrated by a relativistic mirror experiment: incoherent thermal radiation reflected from a mirror moving at γ > γ_crit becomes a coherent pure state, with von Neumann entropy decreasing from S_inc > 0 to 0. This apparent violation of the second law is not a violation. It is compensated by two contributions invisible from a single-sheet perspective: 1. **Entropy cost of non-injectivity:** maintaining γ > γ_crit requires acceleration, which dissipates energy E_acc into the environment, increasing environmental entropy by at least E_acc/T (relativistic analogue of Landauer's principle). 2. **Entropy on other sheets:** the original mixed-state information is not destroyed but redistributed across the remaining N−1 sheets, each carrying part of the original entropy. The total entropy over all sheets plus environment never decreases. This leads to the **Topological Entropy Principle**: \[Sₜₒₚ = 1/N(ε) ∑ᵢ₌₁N(ε) S_iᵗᵒᵗᵃˡ, d/dt Sₜₒₚ ≥ 0,\] which reduces to the classical second law for N=1 (injective worldline). The principle is not a new postulate. It follows from the same **Topological Emergence Identity** \( N(ε)·εᵈ⁻² = O(1) \) that regularises UV divergences in holographic entanglement entropy (Ryu–Takayanagi), Coulomb self-energy in classical electrodynamics, and gravitational singularities. Consequences derived without additional assumptions: - **Arrow of time:** the direction of increasing N (deepening non-injectivity) is the thermodynamic arrow of time.- **Resolution of Maxwell's demon:** a demon operating a relativistic mirror cannot decrease total topological entropy because its own memory is distributed across all N sheets.- **Generalised second law for black holes:** the apparent decrease of Bekenstein–Hawking entropy during Hawking evaporation is compensated by entropy on other sheets, providing a topological resolution of the black hole information paradox. The paper is self-contained, derives all results from first principles (non-injectivity, extended Lorentz transformations, ontological identity), and connects to the broader TPST–DGQ framework (holographic gravity, quantum mechanics, quantum computation).