3-Vector, 4-Vector, and Scalar Product

The three space components of a contravariant 4-vector, $a^\mu$, form a 3-vector.

$$a^\mu\equiv(a^0,a^1,a^2,a^3)\equiv(a^0,\vec{a}) \Rightarrow \vec{a}\equiv(a_x,a_y,a_z) ,$$ (2.12)

where $a^1=a_x,a^2=a_y,a^3=a_z$.

The scalar product of a 3-vector is defined as

$$a\equiv(\vec{a}\cdot\vec{a})^{1/2}\equiv(a_x^2+a_y^2+a_z^2)^{1/2} .$$ (2.13)

Sometimes we omit the index and denote a 4-vector, $a^\mu$, by just $a$. We can thus write the scalar product of a 4-vector as

$$a\cdot b=a_\mu b^\mu = a^\mu b_\mu=a^\mu g_{\mu\nu}b^\nu=a^0b^0-\vec{a}\cdot\vec{b} .$$ (2.14)

This is often taken as the definition of the metric tensor, $g_{\mu\nu}$. We notice that $a\cdot b$ can be positive, negative, or zero.