Time as an Imaginary Fourth Dimension of Space is Incoherent

Abstract

IT may be profitable in relation to current paradox to inquire very briefly into the relation of time to space. Consider two adjacent localities, one around a centre P and the other around a centre Q. The familiar conical diagram asserts that for any sub-locality that is within a certain cone centring on P the relative time is real, while outside that cone it is pure imaginary; and likewise for a cone centring on Q. These cones intersect: and there are regions R, inside one of them and outside the other, for which time is real in relation to P and imaginary in relation to Q, or, in current language, the region around R is time-like in relation to P but space-like in relation to Q. This is something far more complex than the properties of a simple continuum such as a practicable space.