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Taking the Lorentz transformation one step further By studying the pole-and-barn assignment last week, you now know that x and t are relative concepts in Special Relativity. Thereby keeping the speed of light c constant in vacuum in both inertial frames. S′ S y′ V y 0 ct 1 β ct = γ(V ) x0 x β 1 x x′ β = V /c s γ(V ) = Lambert van Eijck 1 1 − β2 TN2612 - intro special relativity - lecture 3 1 What is ux for an object observed to have ux 0 in S 0 ? S′ S u′ y′ V y x x′ Figure: An object is observed as moving with ux 0 based on (x 0 , t 0 ). How is the movement perceived in S? Lambert van Eijck TN2612 - intro special relativity - lecture 3 2 What is ux for an object observed to have ux 0 in S 0 ? Galilei transformation (GT): Lorentz transformation (LT): x = x 0 + Vt 0 x = γ(V )(x 0 + β ct 0 ) y = y0 y = y0 z = z0 z = z0 t = t0 ct = γ(V )(ct 0 + β x 0 ) ∆(x 0 + Vt 0 ) ∆(x) = ∆(t) ∆t 0 0 ∆x + V ∆t = = (ux 0 + V ) ∆t 0 ux = Dx/Dt=ux VDt/Dt=V ux =???? uy =???? uz =???? uy = uy 0 uz = uz 0 Lambert van Eijck TN2612 - intro special relativity - lecture 3 3 What is ux for an object observed to have ux 0 in S 0 ? Lorentz transformation (LT): check yourself: x = γ(V )(x 0 + β ct 0 ) ∆x = γ(V )(∆x 0 + β c∆t 0 ) y = y0 ∆y = ∆y 0 z = z0 ∆z = ∆z 0 ct = γ(V )(ct 0 + β x 0 ) ∆ct = γ(V )(c∆t 0 + β ∆x 0 ) beta=V/c -> beta*c = V 0 ∆x ∆x ∆x γ(V )(∆x 0 + βc∆t 0 ) ∆t 0 + V ux = = c= c = 0 c 0 0 ∆t ∆ct γ(V )(c∆t + β∆x ) c + β ∆x ∆t 0 ux 0 + V ux = 0 1 + Vcu2x Lambert van Eijck TN2612 - intro special relativity - lecture 3 4 What is uy , uz for an object observed to have uy 0 , uz 0 in S 0 ? 0 So we obtain for the velocity components: 0 ∆x = γ(V )(∆x + β c∆t ) ∆y = ∆y 0 ∆z = ∆z 0 ∆ct = γ(V )(c∆t 0 + β ∆x 0 ) ∆y ∆y = c uy = ∆t ∆ct ∆y 0 c = γ(V )(c∆t 0 + β∆x 0 ) uy 0 γ(V )(1 + V ux 0 ) c2 uz = uz 0 γ(V )(1 + V ux 0 ) c2 different from: ux = dividing top and bottom by Dt = uy = ∆y 0 ∆t 0 0 γ(V )(c + β ∆x ∆t 0 ) Lambert van Eijck ux 0 + V 0 1 + Vcu2x c TN2612 - intro special relativity - lecture 3 5 Particle collisions at relativistic speeds Put yourself on the left particle, so the lab-frame (x, y ) travels with β = −3/4 (to the left). The speed of the right particle is then: ux = ux 0 +V 0 1+ V u2x c ux S 3 4c − 43 c ux = − 34 c+− 34 c −3 1+( −3 ) 4 4 96 = − 100 c y x Lambert van Eijck TN2612 - intro special relativity - lecture 3 6 addition of velocities in the x direction at the sane ct, light (dark yellow) has travelled two x, while the object at u=0.5c has travelled 1. so it goes at half the speed of c Figure: frame S 0 moves with β = 0.5 wrt S, so it travels half the distance light would do, at any time ct. But also in the frame of S 0 an object at speed β 0 = u 0 /c = 0.5 travels half the distance of what light would do. How to recognize that in this plot? Lambert van Eijck TN2612 - intro special relativity - lecture 3 7 addition of velocities in the x direction Figure: frame S 0 moves with β = 0.5 wrt S and in the frame of S 0 an object traveling with β 0 = u 0 /c = 0.5 travels half the distance of what light would do (cf. red arrows). But for S the velocity of the object is clearly not (0.5 + 0.5)c. Lambert van Eijck TN2612 - intro special relativity - lecture 3 8 Example: particle collisions at relativistic speeds Put yourself on the left particle, so the lab-frame (x, y ) travels with β = −3/4 (to the left). The speed of the right particle is then: ux = ux 0 +V 0 1+ V u2x c ux S 3 4c − 43 c ux = − 34 c+− 34 c −3 1+( −3 ) 4 4 96 = − 100 c y x Lambert van Eijck TN2612 - intro special relativity - lecture 3 9 Doppler effect: a wave observed by a moving object L v u The thin grey circles depict the maxima of the wave emitted from the source. They leave the source in a period T after each other. How is the wave and its frequency perceived by the moving object using a non-relativistic (Galileian) transformation? I 1st maximum leaves source: t1 I 2nd maximum leaves source: t2 = t1 + T with f = 1/T Lambert van Eijck TN2612 - intro special relativity - lecture 3 10 Doppler effect: a wave observed by a moving object Galilei with observer moving: t/T = number of periods blue line: moving observer Galilei with source moving (ambulance): 14 12 solid red line: source moving dotted red line: wave from source 10 8 t/T t2' 9 t1' 8 6 7 4 0 t/T t2 t1 6 2 0 1 u= speed of the wave v = speed of observer 2 x/L 3 4 ∆t 0 = t2 0 − t1 0 Dt' (blue) is the uT 0 perceived period from the observer moving at u ∆t = u−v v ⇒ f 0 = (1 − )f u Lambert van Eijck 5 5 4 3 2 1 0 −6 −4 −2 0 x/L 2 4 6 will yield: f 0 = f /(1 − vu ) TN2612 - intro special relativity - lecture 3 11 Relativistic Doppler effect of EM radiation We will assume the ’ambulance’ case again: a light source is static at some position r 0 in S 0 and we want to study the Doppler effect in S, while S 0 is moving at speed u. ur= radial component u = speed of frame ut= tangent component In S 0 the source has a proper frequency f0 and proper period T0 . Lambert van Eijck TN2612 - intro special relativity - lecture 3 12 Relativistic Doppler effect of EM radiation I The wave sent out by the light source has its first crest at t0 0 and the second crest at t1 0 = t0 0 + 1/f0 . I If the time t0 0 corresponds to some time t0 in S, it will take an extra r /c to arrive at the origin of S: t1 = t0 + r /c, with r as measured in S. I In S, the second crest comes t2 = t0 + source. γ(u) f0 γ(u) f0 later and + r2 /c, with r2 the position of the moved r and r2 are components of the radiation, so like the radius of a sphere originating in the source of the radiation Lambert van Eijck TN2612 - intro special relativity - lecture 3 13 Relativistic Doppler effect of EM radiation 1 instead of 2 If |r | >> u∆t then the length of r2 can be approximated as r2 ≈ r + ur ∆t. γ(u) + r2 /c f0 γ(u) = t0 + + (r + ur ∆t)/c f0 γ(u) γ(u) = t0 + + (r + ur )/c f0 f0 t2 = t0 + Lambert van Eijck TN2612 - intro special relativity - lecture 3 14 Relativistic Doppler effect of EM radiation T = t2 − t1 = and f = γ(u) (1 + ur /c) f0 f0 γ(u)(1 + f=1/T ur c ) I Only the radial component ur appears in the equations I If ur = 0, there is still a Doppler effect. There is no classical analogue of this effect. Lambert van Eijck TN2612 - intro special relativity - lecture 3 15 Examples of the Doppler effect Lambert van Eijck TN2612 - intro special relativity - lecture 3 16 Examples of the Doppler effect A police officer gives a person a ticket for driving through a red traffic light. The driver says that the police man is mistaken. According to the driver the light was green. Question: What was the velocity of the car, approaching the traffic light? Given: the wavelengths of red and green light are 690 and 530 nm, respectively. Lambert van Eijck TN2612 - intro special relativity - lecture 3 17