1.3 Consequences of the Lorentz transformations Table 1.2 Events and intervals for length contraction. Event S(laboratory) S'(rest frame) 2 (t,x2) (t6,x2) 1 (t,x1) (t,x1) Intervals (0,x2-x1) (t%-t,2-x1) ≡（△t,△x) =(△t,△x) Relation to known intervals (0,L) (?,Lp) On this occasion,the one unknown interval is At',so the interval transformation rule that relates the three known intervals is Equation 1.33.Substituting the known intervals into that equation gives Lp =(V)(L-0).So the lengths measured in S and S'are related by L=LP/Y(V). (1.41) Figure 1.10 George Since y(V)>1,this result tells us that the rod is observed to be shorter in the Fitzgerald(1851-1901)was an laboratory frame than in its own rest frame.In short,moving rods contract.This is Irish physicist interested in an example of the effect known as length contraction.The effect is not limited to electromagnetism.He was rods.Any moving body will be observed to contract along its direction of motion, influential in understanding that length contracts. though it is particularly important in this case to remember that this does not mean that it will necessarily be seen to contract.There is a substantial body of literature relating to the visual appearance of rapidly moving bodies,which generally involves factors apart from the observed length of the body. Length contraction is sometimes known as Lorentz-Fitzgerald contraction after the physicists (Figure 1.4 and Figure 1.10)who first suggested such a phenomenon,though their interpretation was rather different from that of Einstein. ct个 Exercise 1.5 There is an alternative way of defining length in frame S based ct2 event 2 on two events,1 and 2,that happen at different times in that frame.Suppose that event 1 occurs at x =0 as the front end of the rod passes that point,and event 2 also occurs at =0 but at the later time when the rear end passes.Thus event 1 is V at (t1,0)and event 2 is at(t2,0).Since the rod moves with uniform speed V in cti frame S,we can define the length of the rod,as measured in S,by the relation event 1 L=V(t2-t1).Use this alternative definition of length in frame S to establish that the length of a moving rod is less than its proper length.(The events are represented in Figure 1.11.) Figure I.II An alternative set of events that can be used to 1.3.3 The relativity of simultaneity determine the length of a uniformly moving rod. It was noted in the discussion of length contraction that two events that occur at the same time in one frame do not necessarily occur at the same time in another frame.Indeed,looking again at Figure 1.9 and Table 1.2 but now calling on the interval transformation rule of Equation 1.32,it is clear that if the events 1 and 2 are observed to occur at the same time in frame S(so At =0)but are separated by a distance L along the r-axis,then in frame S'they will be separated by the time △t=-y(V)VL/c2. 271.3 Consequences of the Lorentz transformations Table 1.2 Events and intervals for length contraction. Event S (laboratory) S % (rest frame) 2 (t, x2) (t % 2 , x% 2 ) 1 (t, x1) (t % 1 , x% 1 ) Intervals (0, x2 − x1) (t % 2 − t % 1 , x% 2 − x % 1 ) ≡ (Δt, Δx) ≡ (Δt % , Δx % ) Relation to known intervals (0, L) (?, LP) On this occasion, the one unknown interval is Δt % , so the interval transformation rule that relates the three known intervals is Equation 1.33. Substituting the known intervals into that equation gives LP = γ(V )(L − 0). So the lengths measured in S and S % are related by L = LP/γ(V ). (1.41) Since γ(V ) > 1, this result tells us that the rod is observed to be shorter in the laboratory frame than in its own rest frame. In short, moving rods contract. This is an example of the effect known as length contraction. The effect is not limited to rods. Any moving body will be observed to contract along its direction of motion, though it is particularly important in this case to remember that this does not mean that it will necessarily be seen to contract. There is a substantial body of literature relating to the visual appearance of rapidly moving bodies, which generally involves factors apart from the observed length of the body. Figure 1.10 George Fitzgerald (1851–1901) was an Irish physicist interested in electromagnetism. He was influential in understanding that length contracts. 0 ct S x event 1 event 2 V V ct1 ct2 Figure 1.11 An alternative set of events that can be used to determine the length of a uniformly moving rod. Length contraction is sometimes known as Lorentz–Fitzgerald contraction after the physicists (Figure 1.4 and Figure 1.10) who first suggested such a phenomenon, though their interpretation was rather different from that of Einstein. Exercise 1.5 There is an alternative way of defining length in frame S based on two events, 1 and 2, that happen at different times in that frame. Suppose that event 1 occurs at x = 0 as the front end of the rod passes that point, and event 2 also occurs at x = 0 but at the later time when the rear end passes. Thus event 1 is at (t1, 0) and event 2 is at (t2, 0). Since the rod moves with uniform speed V in frame S, we can define the length of the rod, as measured in S, by the relation L = V (t2 − t1). Use this alternative definition of length in frame S to establish that the length of a moving rod is less than its proper length. (The events are represented in Figure 1.11.) ■ 1.3.3 The relativity of simultaneity It was noted in the discussion of length contraction that two events that occur at the same time in one frame do not necessarily occur at the same time in another frame. Indeed, looking again at Figure 1.9 and Table 1.2 but now calling on the interval transformation rule of Equation 1.32, it is clear that if the events 1 and 2 are observed to occur at the same time in frame S (so Δt = 0) but are separated by a distance L along the x-axis, then in frame S % they will be separated by the time Δt % = −γ(V )V L/c2 . 27