1.4 Minkowski spacetime from the origin.Although observers O and O',using frames S and S',would not generally agree about the coordinates of events,they would agree about which events were on the lightcone,which were inside the lightcone and which were outside.This agreement between observers makes lightcones very useful in discussions about which events might cause,or be caused by,other events. Going back to an ordinary two-dimensional spacetime diagram of the kind shown in Figure 1.18,it is straightforward to read off the coordinates of an event in frame S or in frame S'.The event 1 in the diagram clearly has coordinates (ct1,x1)in frame S.In frame S',it has a different set of coordinates.These can be determined by drawing construction lines parallel to the lines representing the primed axes.Where a construction line parallel to one primed axis intersects the other primed axis,the coordinate can be found.By doing this on both axes,both coordinates are found.In the case of Figure 1.18,the dashed construction lines show that,as observed in frame S',event 1 occurs at the same time as event 2,and at the same position as event 3. ct increasing V ct'-axis event 2 lightcone of event 0 event 1, increasing V event 3 x'-axis Figure 1.18 A spacetime diagram for frame S with four events,0,1,2 and 3.Event coordinates in S'can be found event 0 at(0,0) by drawing construction lines parallel to the appropriate axes. Another lesson that can be drawn from Figure 1.18 concerns the order of events. Starting from the bottom of the ct-axis and working upwards,it is clear that in frame S,the four events occur in the order 0,2,3 and 1.But it is equally clear from the dashed construction lines that in frame S',event 3 happens at the same time as event 0(they are simultaneous in S),and both happen at an earlier time than event 2 and event 1,which are also simultaneous in S'.This illustrates the relativity of simultaneity,but more importantly it also shows that the order of events 2 and 3 will be different for observers O and O'. At first sight it is quite shocking to learn that the relative motion of two observers can reverse the order in which they observe events to happen.This has the potential to overthrow our normal notion of causality,the principle that all observers must agree that any effect is preceded by its cause.It is easy to imagine observing the pressing of a plunger and then observing the explosion that it causes.It would be very shocking if some other observer,simply by moving 331.4 Minkowski spacetime from the origin. Although observers O and O % , using frames S and S % , would not generally agree about the coordinates of events, they would agree about which events were on the lightcone, which were inside the lightcone and which were outside. This agreement between observers makes lightcones very useful in discussions about which events might cause, or be caused by, other events. Going back to an ordinary two-dimensional spacetime diagram of the kind shown in Figure 1.18, it is straightforward to read off the coordinates of an event in frame S or in frame S % . The event 1 in the diagram clearly has coordinates (ct1, x1) in frame S. In frame S% , it has a different set of coordinates. These can be determined by drawing construction lines parallel to the lines representing the primed axes. Where a construction line parallel to one primed axis intersects the other primed axis, the coordinate can be found. By doing this on both axes, both coordinates are found. In the case of Figure 1.18, the dashed construction lines show that, as observed in frame S % , event 1 occurs at the same time as event 2, and at the same position as event 3. x ct x1 ct1 increasing V increasing V ct " -axis x " -axis event 1 event 2 event 3 event 0 at (0, 0) lightcone of event 0 Figure 1.18 A spacetime diagram for frame S with four events, 0, 1, 2 and 3. Event coordinates in S % can be found by drawing construction lines parallel to the appropriate axes. Another lesson that can be drawn from Figure 1.18 concerns the order of events. Starting from the bottom of the ct-axis and working upwards, it is clear that in frame S, the four events occur in the order 0, 2, 3 and 1. But it is equally clear from the dashed construction lines that in frame S % , event 3 happens at the same time as event 0 (they are simultaneous in S % ), and both happen at an earlier time than event 2 and event 1, which are also simultaneous in S % . This illustrates the relativity of simultaneity, but more importantly it also shows that the order of events 2 and 3 will be different for observers O and O % . At first sight it is quite shocking to learn that the relative motion of two observers can reverse the order in which they observe events to happen. This has the potential to overthrow our normal notion of causality, the principle that all observers must agree that any effect is preceded by its cause. It is easy to imagine observing the pressing of a plunger and then observing the explosion that it causes. It would be very shocking if some other observer, simply by moving 33