Four-vector - Aetherwisp

A four-vector or 4-vector is a vector in four dimensions used to describe dynamics in spacetime. Three coordinates are spatial, while the fourth is temporal.

a_{\mu}=(a_{0},a_{1},a_{2},a_{3})=(a_{0},\mathbf{a})

where $a_{0}$ is the time coordinate and $\mathbf{a}=(a_{1},a_{2},a_{3})$ are the spatial coordinates.

A note on notation

In relativity, since indexes are ever-present, we use a specific convention to denote what the indexes mean. This is known as Einstein notation. For four-vectors specifically, we identify two kinds of four-vectors: covariant and contravariant. Covariant four-vectors have their index written at the bottom, like $a_{\mu}=(a_{0},a_{1},a_{2},a_{3})$ . Contravariant four-vectors have theirs written at the top, like $a^{\mu}=(a^{0},a^{1},a^{2},a^{3})$ . The difference between the two is the time coordinate, as $a_{0}=-a^{0}$ . The space coordinates are unchanged. More formally, covariant and contravariant four-vectors are bound by the metric tensor $g_{\mu \nu}$ by $a_{\mu}=g_{\mu \nu}a^{\nu}$ .

Another convention is the lettering used. Four-vectors are generally indexed by Greek letters, typically $\mu$ and $\nu$ , which go from $0$ to $3$ . The spatial coordinates specifically, however, use Latin letters, typically $i$ and $j$ , which go from $1$ to $3$ . If you prefer, Latin letters are used for space, whereas Greek letters are used for spacetime.

Product#

As vectors, the definition of product needs to be approached with care. In spacetime, the product between two four-vectors $a$ and $b$ is defined as

a\cdot b=-a_{0}b_{0}+a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}=-a_{0}b_{0}+\mathbf{a}\cdot \mathbf{b}

This is similar to the three-dimensional Scalar product, but in four-dimensions and with the sign of the first product being minus instead of plus. This is expressed in terms of the metric tensor $g_{\mu \nu}$ as

a\cdot b=a_{\mu}b_{\nu}g^{\mu \nu}

This product is a relativistic invariant.