In space-time, vectors that have four components are called four-vectors. A four-vector is an object that has a single time-like component and three space-like components. Four-vectors are made from a time-like part and a space-like part and are displayed in italic script. Therefore, their position in space-time is written as x, where x=(t,x ⃗ ). The components of four-vectors will be given a Greek index; for example, x^μ, where μ = 0, 1, 2, 3. We say that the zeroth component, x^0, is time liketime-like.
Some examples of four-vectors include
• the energy-momentum four-vector, p = (E, p),
• the current density four-vector, j = (ρ, j), and
• the vector potential four-vector, A = (V, A).
The four-dimensional derivative operators ∂_μ and ∂^μ are also combinations of a time-like part and a space-like part.
∂_μ= ∂/(∂x^μ )=(∂/∂t,Δ)=(∂/∂t,∂/∂x,∂/∂y,∂/∂z)
∂_μ= ∂/(∂x_μ )=(∂/∂t,-Δ)=(∂/∂t,-∂/∂x,-∂/∂y,-∂/∂z)
In general, the four-vector inner product is
a.b=a^μ b_μ=a_μ b^μ=g_μv a^μ b^v=b^0 a^0-ab,
where the metric tensor g_μv is given by
g_μv=(■(1&0&■(0&0)@0&-1&■(0&0)@■(0@0)&■(0@0)&■(■(-1@0)&■(0@-1)))).
The upstairs and downstairs vectors are related by the metric tensor via
A_μ=g_μv A^v,
so we can lower or raise an index by inserting