Minkowski metric

by Stéphane Haussler

The Minkowski metric describes a four-dimensional flat spacetime, in contrast to the curved spacetime of general relativity. Concretely, the doubly covariant metric tensor is calculated using the dot product of all basis vectors \(∂_μ \cdot ∂_ν = g_{μν}\). It is described by a matrix where temporal and spatial components are distinguished by their signs. For flat spacetime the metric tensor \(g\) is noted \(η\). In that case, the only non-zero components of the matrix representation are the diagonal elements. The components are either \(+1\) for time, or \(-1\) for space, thereby using metric signature \((+, -, -, -)\). The metric tensor for four-dimensional Minkowski space is often represented by the same matrix for both contravariant and covariant forms:

\[\begin{split}η = \left[ \begin{alignedat}{6} + & 1 \quad& & 0 \quad & & 0 \quad & & 0 \\ & 0 \quad& - & 1 \quad & & 0 \quad & & 0 \\ & 0 \quad& & 0 \quad & -& 1 \quad & & 0 \\ & 0 \quad& & 0 \quad & & 0 \quad & - & 1 \\ \end{alignedat} \right]\end{split}\]

The drawback of this notation is that the metric tensor is given either in doubly covariant form \(η_{μν}\), or doubly contravariant form \(η^{μν}\), both represented by the same matrix following the row-major convention. This row/column representation does not necessarily adhere to standard matrix multiplication rules but often does because the non-diagonal elements are zero! To strictly follow matrix multiplication rules, we need an object that takes in a contravariant vector \(v^{μ}\) and outputs a contravariant vector \(v^{μ}\). Concretely, we would need a mixed covariant-contravariant \(η^{μ}{}_{ν}\) tensor. Using musical notation and the free matrix representation, we can resolve this by explicitly expressing the metric tensor with its basis, making the zero elements unnecessary:

\[\begin{split}η^{♯♯} = \left[ \begin{alignedat}{3} + ∂_t ⊗ ∂_t & & & \\ & - ∂_x ⊗ ∂_x & & \\ & & - ∂_y ⊗ ∂_y & \\ & & & - ∂_z ⊗ ∂_z \\ \end{alignedat} \right]\end{split}\]

We can then reduce the representation to the much more compact form:

\[\begin{split}η^{♯♯} = \left[ \begin{alignedat}{3} + & \, ∂_t & \, ⊗ & \, ∂_t \\ - & \, ∂_x & \, ⊗ & \, ∂_x \\ - & \, ∂_y & \, ⊗ & \, ∂_y \\ - & \, ∂_z & \, ⊗ & \, ∂_z \\ \end{alignedat} \right]\end{split}\]

Equivalently and with the same procedure, we express the doubly covariant metric tensor with:

\[\begin{split}η^{♭♭} = \left[ \begin{alignedat}{3} + & \, dt & \, ⊗ & \, dt \\ - & \, dx & \, ⊗ & \, dx \\ - & \, dy & \, ⊗ & \, dy \\ - & \, dz & \, ⊗ & \, dz \\ \end{alignedat} \right]\end{split}\]

Applying the metric to a vectors and covectors

We can flatten a basis vector with the flat operator \(♭\):

\[(∂_μ)^♭ = η_{μν} dx^ν\]

Or sharpen a basis covector with the flat operator \(♯\):

\[(dx^μ)^♯ = η^{μν} ∂_ν\]

Applying the metric to exterior product of vectors and covectors

The musicality of exterior products can also be sharpened or flattened. The steps are:

• Distribute the musical operators.

• Apply the musical operators.

• Use linearity to reorder numerical components to the front of the expressions.

\[(∂_μ ∧ ∂_ν)^{♭♯} = (∂_μ)^♭ ∧ (∂_ν)^♯ = (η_{γμ} dx^γ) ∧ ∂_ν = η_{γμ} dx^γ ∧ ∂_ν\]

\[(∂_μ ∧ ∂_ν)^{♯♭} = (∂_μ)^♯ ∧ (∂_μ)^♭ = ∂_μ ∧ (η_{γν} dx^γ)\ = η_{γν} ∂_μ ∧ dx^γ\]

\[(∂_μ ∧ ∂_ν)^{♭♭} = (∂_μ)^♭ ∧ (∂_ν)^♭ = (η_{δμ} dx^δ) ∧ (η_{γν} dx^γ) = η_{δμ} η_{γν} dx^δ ∧ dx^γ\]

Applying the metric to tensor products of vectors and covectors

Following the same steps as for the exterior product, the musicality of the tensor product can also be sharpened or flattend:

\[(∂_μ ⊗ ∂_ν)^{♭♯} = (∂_μ)^♭ ⊗ (∂_ν)^♯ = (η_{γμ} dx^γ) ⊗ ∂_ν = η_{γμ} dx^γ ⊗ ∂_ν\]

\[(∂_μ ⊗ ∂_ν)^{♯♭} = (∂_μ)^♯ ⊗ (∂_ν)^♭ = ∂_μ ⊗ (η_{γν} dx^γ) = η_{γν} ∂_μ ⊗ dx^γ\]

\[(∂_μ ⊗ ∂_ν)^{♭♭} = (∂_μ)^♭ ⊗ (∂_ν)^♭ = (η_{δμ} dx^δ) ⊗ (η_{γν} dx^γ) = η_{δμ} η_{γν} dx^δ ⊗ dx^γ\]