  If textbook Lorentz invariance is actually a property of the equations
describing a sector of matter 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, possibly related to the local rest
frame of the expanding Universe) 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). The real geometry of space-time
will then be different from standard Lorentz invariance, and the Poincare
relativity principle will be a local (in space and time), approximate sectorial
property. It seems natural to assume that particles with critical speed in
vacuum different from c are superluminal. We illustrate such a scenario using
as an example a spinorial space-time where the modulus of the spinor,
associated to the time variable, is the size of an expanding Universe. Several
properties of superluminal particles, and of matter without a universal
relativity principle, are discussed in view of experimental applications. If
the vacuum rest frame is close to that suggested by the cosmic microwave
background, experimental searches for superluminal particles on earth should
mainly contemplate a laboratory speed range around 10E3 c , even for very high
energy superluminal cosmic rays. The detectability of several consequences of
the new scenario is briefly discussed.
