With this applet you can see what happens to a reference frame when it moves with speed [math]v[/math] relative to another. [math]B_1[/math] is an [b]event[/b] in the rest frame, identified by the black axis, whose coordinates in this frame are [math](D_1,C_1)[/math]. The same event in the moving frame, whose axis and grid are shown in red, have coordinates [math](F_1,E_1)[/math]. All lines parallel to the space axis represent simultaneous events in the corresponding frame. All lines parallel to the time axis represent events happening at the same space coordinates, but at different time. When a frame is Lorentz boosted, its axis are [i]pushed[/i] toward the bisector of the first quadrant. As a consequence, simultaneous events in a frame are no more simultaneous in the other.
Change the speed [math]v[/math] of the [b][color=#c51414]red[/color][/b] reference frame with respect to the [color=#444][b]black[/b][/color] one, acting on the slider. The speed is given in natural units (i.e. [math]c=1[/math]). You can also change the position of the event in the plane dragging it.