Fundamentals of Time and Relativity

Spacetime Algebra

The base vectors ${e_{μ}; μ = 0, 1, 2, 3}$ that span Minkowski space, satisfy the orthonormality relation $e_{μ} \cdot e_{ν} = g_{μ ν}$ ; the dot indicates the Minkowski inner product. In order to be able to construct a basis for geometrical objects such as the anti-symmetric electromagnetic field tensor and other tensors of this kind, one may extend the definition of the base vectors by endowing them with the product rule

9.9

e_{μ} \cdot e_{ν} : = \frac{1}{2} (e_{μ} e_{ν} + e_{ν} e_{μ}) = \frac{1}{2} {e_{μ} {,e}_{ν}} = g_{μ ν}

This means that the base vectors now satisfy the same algebraic rules as the Dirac matrices. However, the new objects are not matrices, but are to be regarded as four separate vectors with a clear geometric meaning in Minkowski space.

In addition to (9.9), an anti-symmetric wedge product can be defined that is the generalization of the Euclidean vector cross product to relativistic 4-vector space:

9.10

e_{μ} \land e_{ν} : = \frac{1}{2} (e_{μ} e_{ν} - e_{ν} e_{μ})

The multiplicative properties of the ${e_{μ}}$ can then be summarized in the product rule

9.11

e_{μ} e_{ν} = \frac{1}{2} (e_{μ} e_{ν} + e_{ν} e_{μ}) + \frac{1}{2} (e_{μ} e_{ν} - e_{ν} e_{μ}) = e_{μ} \cdot e_{ν} + e_{μ} \land e_{ν}

By forming all distinct products of the ${e_{μ}}$ one obtains a complete basis for this spacetime algebra (STA) consisting of the $2^{4} = 16$ linearly independent elements:

9.12

I, e_{μ}, e_{μ} \land e_{ν}, e_{μ} i, i

The identity arises as $I = {(e_{0})}^{2}$ and the spacetime pseudo-scalar $i$ is defined by the unit four-volume

9.13

i : = e_{0} e_{1} e_{2} e_{3} = e_{0} \land e_{1} \land e_{2} \land e_{3}

It follows that $i^{2} = - 1$ . Since Minkowski space is of even dimension, $i$ anti-commutes with all odd-grade elements and commutes with all even-grade elements.

It is convenient to introduce a reciprocal basis of vectors defined by the algebraic inverse:

9.14

e^{μ} : = {(e_{μ})}^{- 1}, e^{μ} e_{ν} = e^{μ} \cdot e_{ν} = δ_{ν}^{μ}

This is an equally valid basis of vectors that satisfies the same metric relation $e^{μ} \cdot e^{ν} = g^{μ ν} = g_{μ ν}$ . It follows that $e^{0} = e_{0}$ and $e^{i} = - e_{i}$ for $i = 1, 2, 3$ . The components $v^{μ} = e^{μ} \cdot v$ in the reciprocal basis are the contravariant components of the vector $v$ , while $v_{μ} = e_{μ} \cdot v$ are the covariant components.

The vector product (9.11) combines the non-associative dot and wedge products into a single associative product, called the geometric product.
To emphasize the connection with the Dirac algebra, one often uses the notation ${γ_{μ}; μ = 0, 1, 2, 3}$ for basis vectors satisfying the product rule (9.11).