Simultaneity

The classical way of thinking about simultaneity is so ingrained in our everyday habits that we have a great deal of difficulty adjusting to what special relativity has to say about this subject. Indeed, understanding how relativity changes this concept is the single most difficult part of the theory -- once you understand this, you are well on your way to mastering relativity!

Figure: Sketch of coordinate axes for a moving reference frame, $x' ,~ ct'$. The meanings of the events A-E are discussed in the text. The lines tilted at $\pm 45^{\circ }$ are the world lines of light passing through the origin.

Before tackling simultaneity, let us first think about collocation. Two events (such as A and E in figure 4.5) are collocated if they have the same $x$ value. However, collocation is a concept that depends on reference frame. For instance, George is driving from Boston to Washington. Just as he passes New York he sneezes (event A in figure 4.5). As he drives by Baltimore, he sneezes again (event D). In the reference frame of the earth, these two sneezes are not collocated, since they are separated by many kilometers. However, in the reference frame of George's car, they occur in the same place -- assuming that George hasn't left the driver's seat!

Notice that any two events separated by a timelike interval are collocated in some reference frame. The speed of the reference frame is given by equation (4.2), where the slope is simply the slope of the world line connecting the two events.

Classically, if two events are simultaneous, we consider them to be simultaneous in all reference frames. For instance, if two clocks, one in New York and one in Los Angeles, strike the hour at the same time in the earth reference frame, then classically these events also appear to be simultaneous to instruments in the space shuttle as it flies over the United States. However, if the space shuttle is moving from west to east, i. e., from Los Angeles toward New York, careful measurements will show that the clock in New York strikes the hour before the clock in Los Angeles!

Just as collocation depends on one's reference frame, this result shows that simultaneity also depends on reference frame. Figure 4.5 shows how this works. In figure 4.5 events A and B are simultaneous in the rest or unprimed reference frame. However, in the primed reference frame, events A and C are simultaneous, and event B occurs at an earlier time. If A and B correspond to the clocks striking in Los Angeles and New York respectively, then it is clear that B must occur at an earlier time in the primed frame if indeed A and C are simultaneous in that frame.

The tilted line passing through events A and C in figure 4.5 is called the line of simultaneity for the primed reference frame. Its slope is related to the speed, $U$, of the reference frame by

$$slope = U/c ~~~ \mbox{(line of simultaneity)} .$$ (5.4)

Notice that this is the inverse of the slope of the world line attached to the primed reference frame. There is thus a symmetry between the world line and the line of simultaneity of a moving reference frame -- as the reference frame moves faster to the right, these two lines close like the blades of a pair of scissors on the $45^{\circ }$ line.

In Galilean relativity it is fairly obvious what we mean by two events being simultaneous -- it all boils down to coordinating portable clocks which are sitting next to each other, and then moving them to the desired locations. Two events separated in space are simultaneous if they occur at the same time on clocks located near each event, assuming that the clocks have been coordinated in the above manner.

In Einsteinian relativity this doesn't work, because the very act of moving the clocks changes the rate at which the clocks run. Thus, it is more difficult to determine whether two distant events are simultaneous.

Figure 4.6: World lines of two observers (O1, O2) and a pulsed light source (S) equidistant between them. In the left frame the observers and the source are all stationary. In the right frame they are all moving to the right at half the speed of light. The dashed lines show pulses of light emitted simultaneously to the left and the right.

An alternate way of experimentally determining simultaneity is shown in figure 4.6. Since we know from observation that light travels at the same speed in all reference frames, the pulses of light emitted by the light sources in figure 4.6 will reach the two equidistant observers simultaneously in both cases. The line passing through these two events, A and B, defines a line of simultaneity for both stationary and moving observers. For the stationary observers this line is horizontal, as in Galilean relativity. For the moving observers the light has to travel farther in the rest frame to reach the observer receding from the light source, and it therefore takes longer in this frame. Thus, event B in the right panel of figure 4.6 occurs later than event A in the stationary reference frame and the line of simultaneity is tilted. We see that the postulate that light moves at the same speed in all reference frames leads inevitably to the dependence of simultaneity on reference frame.