Minkowski space

Remarks

Example: $\R^4$ with $\eta(x, y) = -x^0 y^0 + x^1 y^1 + x^2 y^2 + x^3 y^3$ is Minkowski space, written $\R^{1,3}$. Check: $\eta$ is symmetric and bilinear, its matrix $\operatorname{diag}(-1, 1, 1, 1)$ is invertible, so $\eta$ is nondegenerate, and it has one negative and three positive eigenvalues. Non-example: $\R^4$ with $-x^0 y^0 - x^1 y^1 + x^2 y^2 + x^3 y^3$ fails clause 3, since its signature is $(2, 2)$. Many physics texts use the opposite sign convention $\operatorname{diag}(1, -1, -1, -1)$. Often Minkowski space means the affine space of events whose translation space is $(V, \eta)$; it is then a flat Lorentzian manifold. A vector $x$ is timelike if $\eta(x, x) < 0$, null (or lightlike) if $\eta(x, x) = 0$ and $x \ne 0$, and spacelike if $\eta(x, x) > 0$. In physics $x^0 = ct$, where $t$ is time and $c$ is the speed of light.