When drawing a spacetime diagram,
it is conventional to choose units so that light pulses will appear as 450
lines on the diagram. One way to do this is to plot $ct$ along the vertical
axis, rather than just $t$. Then given the same units for $x$ and $ct$, the
event of light pulse represents the line of slope $+1 $or $-1$. In this procedure
thus, for an object travelling at constant velocity $v$ the line would be a
line passing through the origin with slope $c/v$.
In the figure above, the thin blue line represents the events of light pulse. The brown line($ct’$) represents the event for the objects moving with velocity $v$, this line is the trajectory of a moving observer. The other brown line($x’$) represents the events simultaneous with the event at $(0,0)$ for the moving observer $t’=0$; the slope of this line is $v/c$. These two axes represent the coordinate system for the object moving with velocity $v$.
When finding the $x'$ and $t'$ coordinates we have to be careful, since these axes do not appear to be perpendicular on the diagram. To find $x'$ coordinates of an event, we draw a line of constant $x'$ through the event and see where it intersects the $x'$ axis. Similarly, to find the $t'$ coordinate, we draw a line of constant $t'$ through the event and see where it intersects the $t'$ axis.
In the figure above, the thin blue line represents the events of light pulse. The brown line($ct’$) represents the event for the objects moving with velocity $v$, this line is the trajectory of a moving observer. The other brown line($x’$) represents the events simultaneous with the event at $(0,0)$ for the moving observer $t’=0$; the slope of this line is $v/c$. These two axes represent the coordinate system for the object moving with velocity $v$.
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