Minkowski-space/Simultaneity/Remark

Two events ${}A$ and ${}B$ in a two-dimensional Minkowski-space. For the first observer (with space axis denoted by ${}x$ and time axis denoted by ${}ct$), the two events are simultaneous, but not for the second observer (with axis ${}x'$ and ${}ct'$.)

For a four-velocity ${}v$ of an observers ${}B$ with the decomposition

${}V=V_{v}\oplus \mathbb {R} v\,,$

the set of points (events) of the form ${}sv+V_{v}$ with a fixed ${}s\in \mathbb {R}$ is called the space at the time ${}s$. The points from this set are called simultaneous for the observer ${}B$. For another observer ${}C$ with the four-velocity ${}w$, these points are not simultaneous. The concept of simultaneity of the observer ${}A$ depends on his decomposition of the Welt ${}V$ into space component and time component. If, for example, the four-velocity of the second observer is given, with respect to a Minkowski-basis of the first observer, by ${}{\begin{pmatrix}{\frac {3}{4}}\\0\\0\\{\frac {5}{4}}\end{pmatrix}}$, then

${\frac {15}{4}}{\begin{pmatrix}{\frac {1}{3}}\\0\\0\\{\frac {1}{5}}\end{pmatrix}},{\begin{pmatrix}0\\1\\0\\0\end{pmatrix}},{\begin{pmatrix}0\\0\\1\\0\end{pmatrix}}$

is an orthonormal basis of the space component of the second observer. For the first observer, the events ${}{\begin{pmatrix}0\\0\\0\\0\end{pmatrix}}$ and ${}{\begin{pmatrix}1\\0\\0\\0\end{pmatrix}}$ are simultaneous, but they are not simultaneous for the second observer, because the first vector has the same description, and the second vector equals

${}{\begin{pmatrix}1\\0\\0\\0\end{pmatrix}}={\frac {75}{16}}{\begin{pmatrix}{\frac {1}{3}}\\0\\0\\{\frac {1}{5}}\end{pmatrix}}-{\frac {3}{4}}{\begin{pmatrix}{\frac {3}{4}}\\0\\0\\{\frac {5}{4}}\end{pmatrix}}\,.$

The time component of the second event with respect to the second observer vector is ${}-{\frac {3}{4}}$.