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If textbook Lorentz invariance is actually a property of the equations describing a sector of the excitations of vacuum above some critical distance scale, several sectors of matter with different critical speeds in vacuum can coexist and an absolute rest frame (the vacuum rest frame) may exist without contradicting the apparent Lorentz invariance felt by "ordinary particles (particles with critical speed in vacuum equal to c, the speed of light). Sectorial Lorentz invariance, relfected by the fact that all particles of a given dynamical sector have the same critical speed in vacuum, will then be an expression of a fundamental sectorial symmetry (e.g. preonic grand unification or extended supersymmetry) protecting a parameter of the equations of motion. Furthermore, the sectorial Lorentz symmetry may be only a low-energy limit, in the same way as the relation w (frequency) = c_s (speed of sound) k (wave vector) holds for low-energy phonons in a crystal. We show that, in this context, phenomena such as the absence of Greisen-Zatsepin-Kuzmin cutoff and the stability of unstable particles at very high energy are basic properties of a wide class of noncausal models where local Lorentz invariance is broken, introducing a fundamental length. Then, observable phenomena are produced at the wavelength scale of the highest-energy cosmic rays or even below this energy, but Lorentz symmetry violation remains invisible to standard low-energy tests. We discuss possible theoretical, phenomenological, experimental and cosmological implications of this new approach to matter and space-time, as well as prospects for future developments.