Classification of 4-Vectors

Three different types of 4-vectors exist:

$$\textrm{if}\quad a_\mu a^\mu < 0, \quad a^\mu\ \textrm{is a space-like vector},$$ (2.15)

$$\textrm{if}\quad a_\mu a^\mu = 0, \quad a^\mu\ \textrm{is a null vector},$$ (2.16)

$$\textrm{if}\quad a_\mu a^\mu > 0, \quad a^\mu\ \textrm{is a time-like vector}.$$ (2.17)

For a time-like vector, $a^0>0$ means the vector points towards the future; $a^0<0$ means the vector points towards the past.