Spacetime Algebra - Spacetime Split

Spacetime Split

Spacetime split – examples:

with γ the Lorentz factor

In spacetime algebra, a spacetime split is a projection from 4D space into (3+1)D space with a chosen reference frame by means of the following two operations:

a collapse of the chosen time axis, yielding a 3D space spanned by bivectors, and
a projection of the 4D space onto the chosen time axis, yielding a 1D space of scalars.

This is achieved by pre or post multiplication by the timelike basis vector, which serves to split a four vector into a scalar timelike and a bivector spacelike component. With we have

$\begin{align}x \gamma_0 &= x^0 + x^k \gamma_k \gamma_0 \\ \gamma_0 x &= x^0 - x^k \gamma_k \gamma_0 \end{align}$

As these bivectors square to unity, they serve as a spatial basis. Utilizing the Pauli matrix notation, these are written . Spatial vectors in STA are denoted in boldface; then with the -spacetime split and its reverse are:

$\begin{align}x \gamma_0 &= x^0 + x^k \sigma_k = x^0 + \mathbf{x} \\ \gamma_0 x &= x^0 - x^k \sigma_k = x^0 - \mathbf{x} \end{align}$

Read more about this topic: Spacetime Algebra

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