Spacetime algebra

In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl_1,3(R), or equivalently the geometric algebra G₄ = G(M⁴), which can be particularly closely associated with the geometry of special relativity and relativistic spacetime.

It is a vector space allowing not just vectors, but also bivectors (directed quantities associated with particular planes, such as areas, or rotations) or multivectors (quantities associated with particular hyper-volumes) to be combined, as well as rotated, reflected, or Lorentz boosted. It is also the natural parent algebra of spinors in special relativity. These properties allow many of the most important equations in physics to be expressed in particularly simple forms, and can be very helpful towards a more geometric understanding of their meanings.

The spacetime algebra is built up from combinations of one time-like basis vector ${\displaystyle \gamma _{0}}$ and three orthogonal space-like vectors, ${\displaystyle \{\gamma _{1},\gamma _{2},\gamma _{3}\}}$, under the multiplication rule

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