Chapter I Special relativity and spacetime Two events that occur at the same time in some frame are said to be simultaneous in that frame.The above result shows that the condition of being simultaneous is a relative one not an absolute one:two events that are simultaneous in one frame are not necessarily simultaneous in every other frame.This consequence of the Lorentz transformations is referred to as the relativity of simultaneity. 1.3.4 The Doppler effect A physical phenomenon that was well known long before the advent of special relativity is the Doppler effect.This accounts for the difference between the emitted and received frequencies (or wavelengths)of radiation arising from the relative motion of the emitter and the receiver.You will have heard an example of the Doppler effect if you have listened to the siren of a passing ambulance:the frequency of the siren is higher when the ambulance is approaching (i.e.travelling towards the receiver)than when it is receding (i.e.travelling away from the receiver). Astronomers routinely use the Doppler effect to determine the speed of approach or recession of distant stars.They do this by measuring the received wavelengths of narrow lines in the star's spectrum,and comparing their results with the proper wavelengths of those lines that are well known from laboratory measurements and represent the wavelengths that would have been emitted in the star's rest frame. Despite the long history of the Doppler effect,one of the consequences of special relativity was the recognition that the formula that had traditionally been used to describe it was wrong.We shall now obtain the correct formula. Consider a lamp at rest at the origin of an inertial frame S emitting electromagnetic waves of proper frequency fem as measured in S.Now suppose that the lamp is observed from another inertial frame S'that is in standard configuration with S,moving away at constant speed V(see Figure 1.12).A detector fixed at the origin of S'will show that the radiation from the receding lamp is received with frequency frec as measured in S'.Our aim is to find the relationship between frec and fem. lamp detector The emitted waves have regularly positioned nodes(points of zero disturbance) that are separated by a proper wavelength Xem =fem/c as measured in S. Figure 1.12 The Doppler In that frame the time interval between the emission of one node and the effect arises from the relative next,At,represents the proper period of the wave,Tem,so we can write motion of the emitter and △t=Tem=l/fem receiver of radiation. Due to the phenomenon of time dilation,an observer in frame S'will find that the time separating the emission of successive nodes is At'=y(V)At.However, this is not the time that separates the arrival of those nodes at the detector because the detector is moving away from the emitter at a constant rate.In fact,during the interval At'the detector will increase its distance from the emitter by VAt'as measured in S',and this will cause the reception of the two nodes to be separated by a total time At'+VAt'/c as measured in S'.This represents the received period of the wave and is therefore the reciprocal of the received frequency,so we can write 六=a+g=ma(+2) 28Chapter 1 Special relativity and spacetime Two events that occur at the same time in some frame are said to be simultaneous in that frame. The above result shows that the condition of being simultaneous is a relative one not an absolute one; two events that are simultaneous in one frame are not necessarily simultaneous in every other frame. This consequence of the Lorentz transformations is referred to as the relativity of simultaneity. 1.3.4 The Doppler effect A physical phenomenon that was well known long before the advent of special relativity is the Doppler effect. This accounts for the difference between the emitted and received frequencies (or wavelengths) of radiation arising from the relative motion of the emitter and the receiver. You will have heard an example of the Doppler effect if you have listened to the siren of a passing ambulance: the frequency of the siren is higher when the ambulance is approaching (i.e. travelling towards the receiver) than when it is receding (i.e. travelling away from the receiver). Astronomers routinely use the Doppler effect to determine the speed of approach or recession of distant stars. They do this by measuring the received wavelengths of narrow lines in the star’s spectrum, and comparing their results with the proper wavelengths of those lines that are well known from laboratory measurements and represent the wavelengths that would have been emitted in the star’s rest frame. Despite the long history of the Doppler effect, one of the consequences of special relativity was the recognition that the formula that had traditionally been used to describe it was wrong. We shall now obtain the correct formula. Consider a lamp at rest at the origin of an inertial frame S emitting electromagnetic waves of proper frequency fem as measured in S. Now suppose that the lamp is observed from another inertial frame S % that is in standard configuration with S, moving away at constant speed V (see Figure 1.12). A detector fixed at the origin of S% will show that the radiation from the receding lamp is received with frequency frec as measured in S % . Our aim is to find the relationship between frec and fem. V y " x " y x lamp detector Figure 1.12 The Doppler effect arises from the relative motion of the emitter and receiver of radiation. The emitted waves have regularly positioned nodes (points of zero disturbance) that are separated by a proper wavelength λem = fem/c as measured in S. In that frame the time interval between the emission of one node and the next, Δt, represents the proper period of the wave, Tem, so we can write Δt = Tem = 1/fem. Due to the phenomenon of time dilation, an observer in frame S % will find that the time separating the emission of successive nodes is Δt % = γ(V ) Δt. However, this is not the time that separates the arrival of those nodes at the detector because the detector is moving away from the emitter at a constant rate. In fact, during the interval Δt % the detector will increase its distance from the emitter by V Δt % as measured in S % , and this will cause the reception of the two nodes to be separated by a total time Δt % + V Δt %/c as measured in S % . This represents the received period of the wave and is therefore the reciprocal of the received frequency, so we can write 1 frec = Δt % + V Δt % c = γ(V ) Δt ? 1 + V c @ . 28