A 4-D space with the Minkowski Metric. Alternatively, it can be considered to have a Euclidean Metric, but
with its Vectors defined by
![\begin{displaymath}
\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],
\end{displaymath}](m_1282.gif) |
(1) |
where
is the speed of light.
The Metric is Diagonal with
![\begin{displaymath}
g_{\alpha\alpha}={1\over g_{\alpha\alpha}},
\end{displaymath}](m_1283.gif) |
(2) |
so
![\begin{displaymath}
\eta^{\beta\delta}=\eta_{\beta\delta}.
\end{displaymath}](m_1284.gif) |
(3) |
Let
be the Tensor for a Lorentz Transformation. Then
![\begin{displaymath}
\eta^{\beta\delta}\Lambda^{\gamma}{}_{\delta} = \Lambda^{\beta\gamma}
\end{displaymath}](m_1286.gif) |
(4) |
![\begin{displaymath}
\eta_{\alpha\gamma}\Lambda^{\beta\gamma} = \Lambda_{\alpha}{}^\beta
\end{displaymath}](m_1287.gif) |
(5) |
![\begin{displaymath}
\Lambda_\alpha{}^\beta = \eta_{\alpha\gamma}\Lambda^{\beta\g...
...eta_{\alpha\gamma}\eta^{\beta\delta}
\Lambda^\gamma{}_\delta.
\end{displaymath}](m_1288.gif) |
(6) |
The Necessary and Sufficient conditions for a metric
to be equivalent to the Minkowski metric
are that the Riemann Tensor vanishes everywhere (
) and that at
some point
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.
© 1996-9 Eric W. Weisstein
1999-05-26