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Minkowski Metric

In Cartesian Coordinates,

\begin{displaymath}
ds^2 = dx^2+dy^2+dz^2
\end{displaymath} (1)


\begin{displaymath}
d\tau^2 = -c^2\,dt^2+dx^2+dy^2+dz^2,
\end{displaymath} (2)

and
\begin{displaymath}
g_{\alpha\beta}\equiv \eta_{\alpha\beta} =\left[{\matrix{
-...
...0 & 1 & 0 & 0\cr
0 & 0 & 1 & 0\cr
0 & 0 & 0 & 1\cr}}\right].
\end{displaymath} (3)

In Spherical Coordinates,
\begin{displaymath}
ds^2 = dr^2+r^2\,d\theta^2+r^2\sin^2\theta\,d\phi^2
\end{displaymath} (4)


\begin{displaymath}
d\tau^2 = -c^2\,dt^2+dr^2+r^2\,d\theta^2+r^2\sin^2\theta\,d\phi^2,
\end{displaymath} (5)

and
\begin{displaymath}
g = \left[{\matrix{
-1 & 0 & 0 & 0\cr
0 & 1 & 0 & 0\cr
0 & 0 & r^2 & 0\cr
0 & 0 & 0 & r^2\sin^2\theta}}\right].
\end{displaymath} (6)

See also Lorentz Transformation, Minkowski Space




© 1996-9 Eric W. Weisstein
1999-05-26