General Physics I. Lecture 20: Lorentz Transformation. Prof. WAN, Xin ( 万歆 )

Transcription

1 General Physics I Lecture 20: Lorentz Transformation Prof. WAN, Xin ( 万歆 ) xinwan@zju.edu.cn

2 Outline Lorentz transformation The inariant interal Minkowski diagram; Geometrical meaning of Lorentz transformation

3 Galilean Transformation (track) (train) If two eents happen at the same place O in the train frame, then in the track frame the distance OO between them is equal to the time between them, multiplied by the speed 0 of the train along the tracks. Familiar? (Relatie elocity 0 of frames) OO' = 0 t t = t: Asymmetry in space and time.

4 Galilean Transformation If two eents happen at the same place O in the train frame, then in the track frame the distance OO between them is equal to the time between them, multiplied by the speed 0 of the train along the tracks. Familiar? (Relatie elocity 0 of frames) OO' = 0 t Now, try to exchange time and space.

5 Beyond Galilean Transformation If two eents happen at the same place O in the train frame, then in the track frame the distance OO between them is equal to the time between them, multiplied by the speed 0 of the train along the tracks. Familiar? (Relatie elocity 0 of frames) OO' = 0 t If two eents happen at the same time t = 0 in the train frame, then in the track frame the time T F between them is equal to the distance D F between them, multiplied by the speed 0 /c 2 of the train along the tracks. Familiar? Peculiar? T F = D F 0 /c 2 If we choose proper units such that c = 1, space and time are interchangeable.

6 Lorentz Transformation Expect to find a linear map t = at ' + b y y ' x = c t ' + d where a, b, c, and d depend on. z Frank x z ' Mary How to determine a, b, c, and d? The Galilean transformation: t = t ' x = + t '

7 Lorentz Transformation Expect to find a linear map t = at ' + b y y ' x = c t ' + d where a, b, c, and d depend on. z Frank x z ' Mary 1. Relatie speed : For the origin in Frank's frame, x = 0 = t ' c/ d = x = d( + t ')

8 The Tale of Two Frames Track Frame (Frank's) 0 -T M L F t = 0 t ' = T m Train Frame (Mary's) 0 0 L M 0 T F = L m D F D M T M 0 L F T F = D F /c 2 By the principle of relatiity, T F T F T F ( T M ) = L M /c 2 D B

9 Lorentz Transformation Expect to find a linear map t = at ' + b y y ' x = c t ' + d where a, b, c, and d depend on. z Frank x z ' Mary 2. Rear clock ahead t = 0 t ' = /c 2 b/a = /c 2 t = a(t ' + /c 2 )

10 Length Contraction Track Frame (Frank's) 0 -T M L F 0 0 Train Frame (Mary's) t ' = L M 0 T F D F D M T M 0 L F = D m L M = D M = s D F T F T F where s = 1 2 /c 2 D F x = D F

11 Lorentz Transformation Expect to find a linear map t = at ' + b y y ' x = c t ' + d where a, b, c, and d depend on. z Frank x z ' Mary 3. Length contraction t ' = 0 = sx d = 1/ s

12 Time Dilation Stick Frame Clock Frame 0 = L 0 sl T' T' T L T t = T sl t ' = T ' In the stick frame (at rest), T = L/ for clock to arrie at the right end. In the clock frame, T ' = s( L/) (time dilation)

13 Lorentz Transformation Expect to find a linear map t = at ' + b y y ' x = c t ' + d where a, b, c, and d depend on. z Frank x z ' Mary 4. Time dilation = 0 t ' = st a = 1/ s

14 Lorentz Transformation Expect to find a linear map t = at ' + b y y ' x = c t ' + d where a, b, c, and d depend on. z Frank x z ' Mary Relatie speed : Rear clock ahead: Length contraction: Time dilation: c/ d = b/a = /c 2 d = 1/ s a = 1/ s t ' + /c2 t = s x = t ' + s

15 Lorentz Transformation t = t ' + /c2 1 2 /c 2 y y ' x = + t ' 1 2 /c 2 x Sole for t, x : t x /c2 t ' = 1 2 /c 2 z When Frank c z ' Mary = x t 1 2 /c 2 t ' = t = x t inerse transformation: - recoer the Galilean transformation.

16 Lorentz Transformation The complete transformation 1) t ' = γ(t x/c 2 ) y y ' 2) = γ(x t) 3) 4) y ' = y z ' = z where γ = /c 2 z Frank x z ' Mary We went all the way from the postulates to the fundamental effects, and finally to the Lorentz transformation. One may also start from the postulates to derie the Lorentz transformation first, then to the relatiistic effects. See, e.g., Classical Dynamics of Particles and Systems by S. T. Thornton and J. B. Marion.

17 Example: Length Contraction Consider a rod of length l lying along the x-axis of an inertial frame K. An obserer in system K' moing with uniform speed along the x-axis measures the length of the rod in the obserer's own coordinate system by determining at a gien instant of time t' the difference in the coordinates of the ends of the rod, x'(2) x'(1). t ' = 0 What is the rod in our train-track problem? (1) = 0 (2) = D m

18 Example: Length Contraction According to Lorentz transformation (Eq. 2) (2) (1) = [ x(2) x(1)] [t (2) t (1)] 1 2 /c 2 Can you translate the equation to the language of our train-track problem? t (2) = T F x(1) = 0 where x(2) x(1) = l. Because t'(2) = t'(1), Eq. 1 leads to t (2) t (1) = [ x(2) x(1)]/c 2 x(2) = D F Translate to:

19 Example: Length Contraction The length l' as measured in the K' system is therefore l ' = (2) (1) = [x(2) x(1)] 1 2 /c 2 = sl where s = 1 2 /c 2 Therefore, to the obserer in K', objects in K appear contracted. Similarly, to a stationary obserer in K, objects in K' also appear contracted. Put together, to an obserer in motion relatie to an object, the dimensions of objects are contracted by a factor s.

20 The Inariant Interal Under Galilean transformation the distance between two eents is inariant, and Newton's laws are inariant. (Δ s) 2 = (Δ x) 2 + (Δ y ) 2 + (Δ z) 2 Under Lorentz transformation, we postulate the interal between eents is inariant, and we will consider dynamics in the coming lecture. (Δ s) 2 = c 2 (Δ t) 2 (Δ x) 2 (Δ y) 2 (Δ z) 2 to be proed on the next slide

21 Proof of the Inariant Interal For simplicity we drop D and ignore y and z directions. According to the (inerse) Lorentz transformation c 2 t 2 x 2 = c2 (t ' + /c 2 ) /c 2 ( + t ')2 1 2 /c 2 = t '2 (c 2 2 ) 2 (1 2 /c 2 ) 1 2 /c 2 = c 2 t ' 2 2 s 2

22 Timelike, Spacelike & Lightlike Timelike separation: s 2 > 0 It is possible to find a frame in which the two eents happen at the same place. s/c is called the proper time. It is possible for a particle to trael from one eent to the other. Spacelike separation: s 2 < 0 It is possible to find a frame in which the two eents happen at the same time. s is called the proper distance, or proper length. Lightlike separation: s 2 = 0 In eery frame a photon emitted at one of the eents will arrie at the other.

23 Ex: Time Dilation Let frame K' moe at speed with respect to frame K. Consider two eents at the origin of K', separated by time t'. The separation between the eents is in K: (x, t) = (t, t) and in K': (x', t') = (0, t').

24 Ex: Time Dilation Again, in K: (x, t) = (t, t) and in K': (x', t') = (0, t'). The inariant interal implies c 2 t ' 2 0 = c 2 t 2 2 t 2 t = Note that x' = 0. t ' 1 2 /c 2

25 Lorentz Transformation The complete transformation 1) t = γ(t ' + /c 2 ) y y ' 2) x = γ( + t ') 3) 4) y = y ' z = z ' z Frank x z ' Mary Or, 1) 2) 3) 4) ct = γ[(ct ') + β ] x = γ[β(ct ') + ] y = y ' z = z ' β = /c γ = = /c β 2

26 Space-Time Squeezing in (ct ', ) = (1,0) (ct, x) = (γ, γ β) (ct ', ) = (0,1) (ct, x) = (γ β, γ) Unit lengths for the ct' and x' axes change to ( ct x ) = ( γ γ β γ β γ ) ( ct ' ) = γ ( 1 β β 1 ) ( ct ' ) γ 1 + β 2 x X X Frank = 1 + β2 1 β 2 ct ct ' X X Mary ct '

27 Minkowski Diagram x What does now mean? What does here mean? q ct ' Well, they depend on the choice of frame. q ct tan θ = β

28 Minkowski Diagram x now What does now mean? What does here mean? here ct ' In the rest frame K, now means t = const, and here means x = const. ct

29 Minkowski Diagram x now here What does now mean? What does here mean? ct ' ct In the moing frame K', now means t' = const, and here means x' = const.

30 Length Contraction, Again right end same time in K C B left end x O q A same time in K' tan θ = β ct ' ct Suppose the length of the stick is 1 in the K' frame. This means the length of OA is OA = 1 + β2 1 β 2 The length measured by Frank in the K frame is the length of OB. OC = OAcos θ BC = AC tanθ AC = OAsinθ OB =?

31 Length Contraction, Again same time in K x same time in K' Therefore, the length of the stick measured in the K frame is OB = OC BC right end C B q A ct ' = 1 β 2 which is the standard length contraction result. Note that you can figure out left end O tan θ = β ct cosθ = β 2

32 Length Contraction, Again same time in K x same time in K' Suppose the length of the stick is 1 in the K frame, i.e. Therefore, OA = 1 right end left end A O q E tan θ = β ct ' ct OE = OA/cosθ = 1 + β 2 The length measured by Mary in the K' frame is in units of the x' axis, or OE 1 + β 2 / 1 β 2 = 1 β 2

33 Rotation in Space ( x y ) = ( cosθ sin θ sinθ cosθ ) ( y ' ) y y ' Show that if we define Frank x Rose tanh ϕ = β = /c The Lorentz transformations in the matrix form become ( ct x ) = ( cosh ϕ sinh ϕ sinh ϕ cosh ϕ ) ( ct ' )

34 Exercise Consider a spaceship leaing the Earth at a coasting speed of 0.8c, where c is the speed of light in acuum. The spaceship will trael to another planet 4 light years away and return immediately with the same speed after it arries. Three years, in the Earth frame, after the spaceship was launched, a light signal is sent from the Earth to the traeling spaceship. Suppose a clock on the spaceship is set to zero when it departs. When the spaceship receies the signal, what time is it on the clock (in units of years)? Neglect the time it takes the spaceship to turn around.