The Spacetime Interval PDF

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The Spacetime Interval We have seen earlier in this section that time intervals and lengths (= space intervals), quantities that were absolutes, or invariants, for relatively moving observers using the classical Galilean coordinate transformation, are not invariants in special relativity. The Lorentz transformation and the relativity of simultaneity lead observers in inertial frames to conclude that lengths moving relative to them are contracted and time intervals are stretched, both by the factor g. The question naturally arises: Is there any quantity involving the space and time coordinates that is invariant under a Lorentz transformation? The answer to that question is yes, and as it happens, we have already dealt with a special case of that invariant quantity when we first obtained the correct form of the Lorentz transformation. It is called the spacetime interval, or usually just the interval, Ds, and is given by Experiments with muons moving near the speed of light are performed at many accelerator laboratories throughout the world despite their short mean life. Time dilation results in much longer mean lives relative to the laboratory, providing plenty of time to do experiments. 1.30 or, specializing it to the one-space-dimensional systems that we have been discussing, 1.31 It may help to think of Equations 1-30 and 1-31 like this: The interval is the only measurable quantity describing pairs of events in spacetime for which observers in all inertial frames will obtain the same numerical value. The negative sign in Equations 1-30 and 1-31 implies that may be positive, negative, or zero depending on the relative sizes of the time and space separations. With the sign of , nature is telling us about the causal relation between the two events. Notice that whichever of the three possibilities characterizes a pair for one observer, it does so for all observers since is invariant. The interval is called time like if the time separation is the larger and spacelike if the space separation predominates. If the two terms are equal, so that , then it is called lightlike. Timelike Interval Consider a material particle or object, such as, the elephant in Figure 1-27, that moves relative to S. Since no material particle has ever been measured traveling faster than light, particles always travel less than 1 m of distance in 1 m of light travel time. We saw that to be the case in Example 1-8, where the

time interval between launch and birth of the baby elephant was 31.7 months on the S clock, during which time the mother elephant had moved a distance of 23.8c # months. Equation 1-31 then yields and the interval in S is months. The time interval term being the larger, is a timelike interval and we say that material particles have timelike worldlines. Such worldlines lie within the shaded area of the spacetime diagram in Figure 1-21. Note that in the elephant’s frame S9 the separation in space between the launch and birth is zero and Dt is 21.0 months. Thus, Ds = 21.0c # months in S9 too. That is what we mean by the interval being invariant: observers in both S and S9 measure the same number for the separation of the two events in spacetime. The proper time interval t between two events can be determined from Equations 1-31 using space and time measurements made in any inertial frame since we can write that equation as Since Dt = t when Dx = 0—that is, for the time interval recorded on a clock in a system moving such that the clock is located at each event as it occurs—in that case Notice that this yields the correct proper time t = 21.0 months in the elephant example. Spacelike Interval When two events are separated in space by an interval whose square is greater than the value of (cDt)2, then Ds is called spacelike. In that case it is convenient for us to write Equation 1-31 in the form

so that, as with timelike intervals, (Ds)2 is not negative.16 Events that are spacelike occur sufficiently far apart in space and close together in time that no inertial frame could move fast enough to carry a clock from one event to the other. For example, suppose two observers in Earth frame S, one in San Francisco and one in London, agree to each generate a light flash at the same instant, so that cDt = 0 m in S and Dx = 1.08  107 m. For any other inertial frame (cDt)2  0, and we see from Equation 1-33 that (Dx)2 must be greater than (1.08  107)2 in order that Ds be invariant. In other words, 1.08  107 m is as close in space as the two events can be in any system; consequently, it will not be possible to find a system moving fast enough to move a clock from one event to the other. A speed greater than c, in this case infinitely greater, would be needed. Notice that the value of Ds = Lp, the proper length. Just as with the proper time interval t, measurements of space and time intervals in any inertial system can be used to determine Lp. Lightlike (or Null)

Interval The relation between two events is lightlike if Ds in Equation 1-31 equals zero. In that case cDt = Dx 1-34 and a light pulse that leaves the first event as it occurs will just reach the second as it occurs. The existence of the lightlike interval in relativity has no counterpart in the world of our everyday experience, where the geometry of space is Euclidean. In order for the distance between two points in space to be zero, the separation of the points in each of he three space dimensions must be zero. However, in spacetime the interval between two events may be zero, even though the intervals in space and time may individually be quite large. Notice, too, that pairs of events separated by lightlike intervals have both the proper time interval and proper length equal to zero since Ds = 0. Things that move at the speed of light17 have lightlike worldlines. As we saw earlier (see Figure 1-22), the worldline of light bisects the angles between the ct and x axes in a spacetime diagram. Timelike intervals lie in the shaded areas of Figure 1-32 and share the common characteristic that their relative order in time is the same for observers in all inertial systems. Events A and B in Figure 1-32 are such a pair. Observers in both S and S9 agree that A occurs before B, although they of course measure different values for the space and time separations. Causal events, that is, events that depend on or affect one another in some fashion, such as your birth and that of your mother, have timelike intervals. On the other hand, the temporal order of events with spacelike intervals, such as A and C in Figure 1-32, depends on the relative motion of the systems. As you can see in the diagram, A occurs before C in S, but C occurs first in S9. Thus, the relative order of pairs of events is absolute in the shaded areas but elsewhere may be in either order....