Position Four-Vector

The Contravariant Four-Vector arising in special and general relativity ,

$$x^\mu =\left[{\matrix{x^0\cr x^1\cr x^2\cr x^3\cr}}\right]\equiv\left[{\matrix{ct\cr x\cr y\cr z\cr}}\right],$$

where $c$ is the speed of light and $t$ is time. Multiplication of two four-vectors gives the spacetime interval

$$I = g_{\mu\nu}x^\mu x^\nu=(x^0)^2-(x^1)^2-(x^2)^2-(x^3)^2$$

$$= (ct)^2-(x^1)^2-(x^2)^2-(x^3)^2$$

See also Four-Vector, Lorentz Transformation, Quaternion