# Definitions The *Lorentz inner product* is usually defined as a [[Quadratic forms|quadratic form]] on $\mathbb{R}^{n+1}$ of signature $(n,1)$ whose matrix representation is given by $J= \text{diag}(-1,1,1\dots,1)$, i.e. the quadratic form defined by $\langle \cdot , \cdot\rangle_{L}:\mathbb{R}^{n+1}\times \mathbb{R}^{n+1} \to \mathbb{R}$ by $\begin{align} \langle \mathbf{x}, \mathbf{y} \rangle_{L} &= -x_{1}y_{1} +x_{2}y_{2} + x_{3}y_{3}+\dots +x_{n+1}y_{n+1} \\ &= \mathbf{x}^{T} J \mathbf{y}^T. \end{align}$ Although it is called an inner product it is not positive definite (positive definiteness is equivalent to signature of $(n+1,0)$) while it satisfies the other axioms. The *Lorentz norm* ... TBC # Applications See [[Hyperbolic space]]. See [[Lorentz transformations]].