Can I use the relativity formula $(\Delta s)^2 = (c \Delta t)^2 - (\Delta r)^2$ even when observers did not start at the same origin at $t=t'=0$? I know this is constant, but is it constant even if the other origins of observers were not the same at $t=t'=0$?
$\begingroup$
$\endgroup$
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
Space-Time Interval: Suppose an observer measures two events as being separated in time by $\Delta t$ and a spatial distance $\Delta x$. Then the spacetime interval $(\Delta s)^2$ between the two events that are separated by a distance $\Delta x$ in space and by $\Delta ct =c\Delta t$ in the $ct$-coordinate is: $$(\Delta s)^2=(\Delta ct)^2-(\Delta x)^2$$
You can see not the absolute time but the time interval between two intervals is involved in the formula and So it doesn't matter whether or not $t=t'=0$ But both observers must measure the time interval between the same events.
$\begingroup$
$\endgroup$
Choosing t0 == t'0 is a completely free choice, it says nothing about the physics (all that matters is that you can choose). Not doing it just makes the mathematics messier.
answered Dec 19, 2020 at 17:00