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

Famous quotes containing the word split:

“The Federated Republic of Europe—the United States of Europe—that is what must be. National autonomy no longer suffices. Economic evolution demands the abolition of national frontiers. If Europe is to remain split into national groups, then Imperialism will recommence its work. Only a Federated Republic of Europe can give peace to the world.”
—Leon Trotsky (1879–1940)