The _Minkowski Metric_ is a [[metric tensor]] that defines the four-component inner product $\mathbf{\eta}(\mathbf{x},\mathbf{y})=\eta_{\alpha\beta}\,x^\alpha y^\beta$ where $x^\alpha$ and $y^\beta$ are components of a pair of [[4-vector]]s, $\mathbf{x}$ and $\mathbf{y}$ and $\eta_{\alpha\beta}$ are components of a $4\times 4$ matrix given as $\mathbf{\eta}=\begin{pmatrix}-1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$ or $\mathbf{\eta}=\begin{pmatrix}1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix}$ where we find that the sign choice is a matter of convention chosen by different authors and we will make no attempt at maintaining a consistent convention. # distance ## Integral measure # Minkowski Space The Minkowski metric defines an associated [Inner product space](Inner%20products.md) called a [Minkowski Space](Minkowski%20Space). # Lorentz Group The group of linear transformations that preserves the [[Minkowski Space]] [Inner products](Inner%20products.md) is the [Lorentz Group](Lorentz%20Group.md). #Mechanics/SpecialRelativity