1.3 Consequences of the Lorentz transformations Rather than deal with ticking clocks,our discussion here will refer to short-lived sub-nuclear particles of the sort routinely studied at CERN and other particle ct' S physics laboratories.For the purpose of the discussion,a short-lived particle is event 2 considered to be a point-like object that is created at some event,labelled 1,and subsequently decays at some other event,labelled 2.The time interval between c△T these two events,as measured in any particular inertial frame,is the lifetime of the particle in that frame.This interval is analogous to the time between cti event 1 successive ticks of a clock. We shall consider the lifetime of a particular particle as observed by two different inertial observers O and O'.Observer O uses a frame S that is fixed in the laboratory,in which the particle travels with constant speed V in the positive x-direction.We shall call this the laboratory frame.Observer O'uses a frame S' 6 that moves with the particle.Such a frame is called the rest frame of the particle △x ct2 --→◆event2 since the particle is always at rest in that frame.(You can think of the observer O' as riding on the particle if you wish.) c△T According to observer O',the birth and decay of the(stationary)particle happen at the same place,so if event 1 occurs at (t,)then event 2 occurs at (2,) 。event 11 and the lifetime of the particle will be At'=tt.In special relativity,the time between two events measured in a frame in which the events happen at the same position is called the proper time between the events and is usually denoted by 工2 the symbol Ar.So,in this case,we can say that in frame S'the intervals of time and space that separate the two events are△t=△r=t-tand△r'=0. Figure 1.7 Events and intervals for establishing the According to observer O in the laboratory frame S,event 1 occurs at(t1,1)and relation between the lifetime of event 2 at(t2,2),and the lifetime of the particle is At =t2-t1,which we shall a particle in its rest frame (S') call AT.Thus in frame S the intervals of time and space that separate the two and in a laboratory frame (S). events are△t=△T=t2-tand△x=x2-x1. Note that we show the These events and intervals are represented in Figure 1.7,and everything we know coordinate on the vertical axis as about them is listed in Table 1.1.Such a table is helpful in establishing which of ‘ct'rather than‘t'to ensure that the interval transformations will be useful. both axes have the dimension of length.To convert time intervals Table I.I A tabular approach to time dilation.The coordinates of the events such as△rand△T to this are listed and the intervals between them worked out,taking account of any coordinate,simply multiply known values.The last row is used to show which of the intervals relates to a them by the constant c. named quantity (such as the lifetimes AT and AT)or has a known value(such as A.r'=0).Any interval that is neither known nor related to a named quantity is shown as a question mark. Event S(laboratory) S'(rest frame) 2 (t2,x2) (,x) 1 (t1,x1) (t,x) Intervals (t2-t1,x2-r1) (丝-，0) ≡（△t,△x) =(△t,△x Relation to known intervals (△T,?) (△r,0) Four of the interval transformation rules that were introduced in the previous section involve three intervals.But only Equation 1.36 involves the three known intervals.Substituting the known intervals into that equation gives 251.3 Consequences of the Lorentz transformations Rather than deal with ticking clocks, our discussion here will refer to short-lived sub-nuclear particles of the sort routinely studied at CERN and other particle physics laboratories. For the purpose of the discussion, a short-lived particle is considered to be a point-like object that is created at some event, labelled 1, and subsequently decays at some other event, labelled 2. The time interval between these two events, as measured in any particular inertial frame, is the lifetime of the particle in that frame. This interval is analogous to the time between successive ticks of a clock. We shall consider the lifetime of a particular particle as observed by two different inertial observers O and O % . Observer O uses a frame S that is fixed in the laboratory, in which the particle travels with constant speed V in the positive x-direction. We shall call this the laboratory frame. Observer O% uses a frame S % that moves with the particle. Such a frame is called the rest frame of the particle since the particle is always at rest in that frame. (You can think of the observer O% as riding on the particle if you wish.) According to observer O% , the birth and decay of the (stationary) particle happen at the same place, so if event 1 occurs at (t % 1 , x% ), then event 2 occurs at (t % 2 , x% ), and the lifetime of the particle will be Δt % = t % 2 − t % 1 . In special relativity, the time between two events measured in a frame in which the events happen at the same position is called the proper time between the events and is usually denoted by the symbol Δτ . So, in this case, we can say that in frame S % the intervals of time and space that separate the two events are Δt % = Δτ = t % 2 − t % 1 and Δx % = 0. According to observer O in the laboratory frame S, event 1 occurs at (t1, x1) and event 2 at (t2, x2), and the lifetime of the particle is Δt = t2 − t1, which we shall call ΔT. Thus in frame S the intervals of time and space that separate the two events are Δt = ΔT = t2 − t1 and Δx = x2 − x1. These events and intervals are represented in Figure 1.7, and everything we know about them is listed in Table 1.1. Such a table is helpful in establishing which of the interval transformations will be useful. ct S S " x " x " ct" x Δx c ΔT c Δτ event 1 event 2 x1 x2 ct1 event 1 ct2 event 2 ct1 ct2 " " Figure 1.7 Events and intervals for establishing the relation between the lifetime of a particle in its rest frame (S% ) and in a laboratory frame (S). Note that we show the coordinate on the vertical axis as ‘ct’ rather than ‘t’ to ensure that both axes have the dimension of length. To convert time intervals such as Δτ and ΔT to this coordinate, simply multiply them by the constant c. Table 1.1 A tabular approach to time dilation. The coordinates of the events are listed and the intervals between them worked out, taking account of any known values. The last row is used to show which of the intervals relates to a named quantity (such as the lifetimes ΔT and Δτ ) or has a known value (such as Δx % = 0). Any interval that is neither known nor related to a named quantity is shown as a question mark. Event S (laboratory) S % (rest frame) 2 (t2, x2) (t % 2 , x% ) 1 (t1, x1) (t % 1 , x% ) Intervals (t2 − t1, x2 − x1) (t % 2 − t % 1 , 0) ≡ (Δt, Δx) ≡ (Δt % , Δx % ) Relation to known intervals (ΔT, ?) (Δτ, 0) Four of the interval transformation rules that were introduced in the previous section involve three intervals. But only Equation 1.36 involves the three known intervals. Substituting the known intervals into that equation gives 25