Minkowski Space

A 4-D space with the Minkowski Metric. Alternatively, it can be considered to have a Euclidean Metric, but with its Vectors defined by

$$\left[{\matrix{x_0\cr x_1\cr x_2\cr x_3\cr}}\right] = \left[{\matrix{ict\cr x\cr y\cr z\cr}}\right],$$ (1)

where $c$ is the speed of light. The Metric is Diagonal with

$$g_{\alpha\alpha}={1\over g_{\alpha\alpha}},$$ (2)

so

$$\eta^{\beta\delta}=\eta_{\beta\delta}.$$ (3)

Let $\Lambda$ be the Tensor for a Lorentz Transformation. Then

$$\eta^{\beta\delta}\Lambda^{\gamma}{}_{\delta} = \Lambda^{\beta\gamma}$$ (4)

$$\eta_{\alpha\gamma}\Lambda^{\beta\gamma} = \Lambda_{\alpha}{}^\beta$$ (5)

$$\Lambda_\alpha{}^\beta = \eta_{\alpha\gamma}\Lambda^{\beta\g... ...eta_{\alpha\gamma}\eta^{\beta\delta} \Lambda^\gamma{}_\delta.$$ (6)

The Necessary and Sufficient conditions for a metric $g_{\mu\nu}$ to be equivalent to the Minkowski metric $\eta_{\alpha\beta}$ are that the Riemann Tensor vanishes everywhere ( $R^\lambda{}_{\mu\nu\kappa}=0$) and that at some point $g^{\mu\nu}$ has three Positive and one Negative Eigenvalues.

See also Lorentz Transformation, Minkowski Metric

References

Thompson, A. C. Minkowski Geometry. New York: Cambridge University Press, 1996.