A nonzero vector in -dimensional Lorentzian space is said to be negative timelike
if it has imaginary (Lorentzian) norm and if its first component
is negative. Symbolically, is negative timelike if both

and

hold. Note that equation (6) above expresses the imaginary norm condition by saying, equivalently, that the vector has a negative squared norm.

Misner, C. W.; Thorne, K. S.; and Wheeler, J. A. Gravitation.
San Francisco, CA: W. H. Freeman, p. 53, 1973.Ratcliffe,
J. G. Foundations
of Hyperbolic Manifolds. New York: Springer-Verlag, 2006.