Special Relativity as an Emergent Structure in a Timeless Euclidean Model

A model is considered in which the observable spacetime structure emerges from a real scalar field satisfying the Laplace equation in a four-dimensional Euclidean space without distinguished time or directions. The observer is described as a localized configuration of the same field on the hypersurfaces of a foliation; _events_ are defined as local detector activations specified by a functional of a finite number of mode-decomposition coefficients of the field and the observer’s parameters. It is shown that the choice of foliation gives rise to inertial reference frames, and that a consistent reconstruction under transitions between them is possible without introducing a global set of events—solely on the basis of the observer’s operational description. The model implies that the event structures of different inertial frames may differ, so that no global event space exists. It is proven that within the model it is impossible to transmit information about an event absent in a given frame but present in another. This leads to the distinction between two types of transformations. The first, _direct transformations_, describe the actual rearrangement of the event structure under a change of the inertial frame. The second, _observable transformations_, represent the operational re-description performed by an observer within their own frame, based on a hypothetical assumption of a global event set. The invariance of all foliations, resulting from the full \(O(4)\) symmetry of the Laplace equation, together with the justified existence of a finite maximal propagation speed \(v_{\max}\), leads to observable transformations of Lorentz type with invariant \(v_{\max}\). Thus, both postulates of special relativity are reproduced, and the causal structure emerges as a cone \(\|\Delta \mathbf r\| = v_{\max} |\Delta t|\) within each frame. The results demonstrate that special relativity can emerge within a strictly Euclidean, timeless model.