Minkowski vector space

Definition

In this collection, Minkowski vector space is $\mathbb R^{1,3}=\mathbb R^4$ with coordinates $(t,x,y,z)$ , symmetric bilinear form

\eta(u,v)=-u_0v_0+u_1v_1+u_2v_2+u_3v_3,

and associated quadratic form

q(v)=\eta(v,v)=-t^2+x^2+y^2+z^2.

Thus the matrix of $\eta$ is $\operatorname{diag}(-1,1,1,1)$ ; throughout this collection, the signature notation $(1,3)$ means one negative and three positive directions.

Causal types

A nonzero vector $v$ is timelike, null (or lightlike), or spacelike according as $q(v)<0$ , $q(v)=0$ , or $q(v)>0$ . The timelike cone has two connected components. Choosing one as the future cone is a time orientation.

Sources that list positive directions first call the displayed form's signature $(3,1)$ . Other sources reverse the overall sign and use $t^2-x^2-y^2-z^2$ ; in this collection's negative-first ordering, that opposite form has signature $(3,1)$ . Formulas in linked knowls use the displayed $(-+++)$ form, so the Hermitian matrix model satisfies $\det X(v)=-q(v)$ , not $\det X(v)=q(v)$ .

References

Barrett O'Neill, Semi-Riemannian Geometry With Applications to Relativity, Academic Press, 1983, Chapter 5. Publisher record.
Gregory L. Naber, The Geometry of Minkowski Spacetime, 2nd ed., Springer, 2012, Chapter 1. Publisher record.