2.3 The Lorentz Transformation Eq.

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1 Announcement Course webpage Textbook PHYS-2402 Lecture 3 HW1 (due 9/13) Chapter 2 20, 26, 36, 41, 45, 50, 51, 55, 58 Sep. 6, The Lorentz Transformation Eq. We can use γ to write our transformations. Chapter 2 Special Relativity 1. Basic Ideas 2. Consequences of Einstein s Postulates 3. The Lorentz Transformation Equations 4. The Twin Paradox 5. The Doppler Effects 6. Velocity Transformation 7. Momentum & Energy 8. General Relativity & a 1 st Look at Cosmology 9. The Light Barrier 10. The 4 th Dimension Frame S Frame S 4

2 2.3 The Lorentz Transformation Eq. What if v << c? We get the classical results! 5 Lorentz Transformation of Distances and Time Intervals This time using Lorentz Transformations IMPORTANT: space and time distances mix together

3 Lorentz Transformation of Distances and Time Intervals 0 = 0 0 Lorentz Transformation of Distances and Time Intervals -2 m = 0 = -2 m -2 m 0 0

4 Proper Time (Δt 0 ) The time measured in the frame where all events occur in the same location in that frame: x 1 = x 2 The time difference from our Lorentz Transformation would then be Proper Length The length measured in a frame where the object being measured is at rest, so that it doesn t matter WHEN we measure the end points. But in any MOVING frame, S, the ends must be established SIMULTANEOUSLY to obtain a meaningful length: t 1 = t 2 The length difference from our Lorentz Transformation would then be In more general form, In more general form, L 0 is measured in the frame where the object at rest Consequence 3: Time Dilation t 0 : the time difference in the frame in which the events occur at the same location 13 Consequence 3: Length Contraction 14 SUMMARY Einstein s Postulates of Relativity: Twin Paradox Michelson- Morley Experiment NO AETHER! Consequences of Einstein s Postulates: 1. Relative Simultaneity 2. Time Dilation 3. Length Contraction EXPERIMENT The Theory of Relativity As an object approaches the speed of light, time slows down. (Moving clocks are slow) & (Moving rulers are short)

5 A trip from Earth to Planet Hollywood Twin Paradox One twin stays at home. One twin travels on a spaceship at very high speeds. Relativity says traveling twin will age more slowly. But one can say the twin on Earth is traveling w.r.t. the twin in the spaceship and should be the younger. This is the paradox. Who is really younger. Answer: Traveling twin because of accelerations for the traveling twin non inertial frame.. 17 Homer stays on Earth. Loner travels 10 light-years at 80% of the speed of light. Beta (β = v/c) is the velocity of the object compared to the speed of light. (β=0.8) Gamma (γ) is the effect of traveling at speeds close to c Effect of Time on Spaceship Velocity (v) = 0.8c therefore β =v/c= 0.8 γ= β² (β² = 0.8²= 0.64) 1 - β² = = 0.36 and the of 0.36 = 0.6!! As Viewed From Earth γ = = 1²/³ = 5/3 Without Relativity.. x = vt or t = x/v x = 10 light-years traveled (10yr*c) v = velocity = (0.8c) t = time. t = 10yr*c/0.8c = 12.5 years each way. There and back makes the trip 12.5 x 2 or 25 years!!

6 As Viewed From Earth With Relativity Physical Results of Trip Homer sees Loner s clock is running slow by??? γ = 5/3!! Therefore Loner s clock reads 25 years γ n Homer on Earth ages 25 years!! n Loner, traveling at 80% the speed of light ages 15 years!! 25 5/3 = 15 years!! t 0 : the time difference in the frame in which the events occur at the same location The traveling twin is younger! As Viewed From Spaceship Loner sees distance of planets contracted by γ= 5/3 In Loner s frame, distance is 10 light years 5/3 10 5/3 = 6 light years. Therefore t = x/v= 6/0.8 = 7.5 years each way. There and back is 7.5 x 2 = 15 year trip for Loner!! Anna travels away and back at v = 0.8 c (! = 5/3) Bob stays home

7 The Key: Round Trip! PARADOX: Seemingly absurd or self-contradictory thought, often true statement (Oxford Dictionary) The Key: Round Trip! ACCELERATION Change of inertial frames

8 Minkowski diagram of the twin paradox. There is a difference between the trajectories of the two twins: the trajectory of the ship is equally divided between two different inertial frames, while the Earth-based twin stays in the same inertial frame Velocity Transformation We may now relate velocity in different frames. We know that the classical transformation u = u-v is wrong. The correct one is a straightforward application of the Lorentztransformation eq. S (x,u,t) S (x,u,t )! u u u,u = velocity of an object moving relative to a frame; quantities (position, velocity, time) have different value in different frame * We reserve the symbol v exclusively for the relative speed between the two frames!!! u v + u Classical transformation is wrong!!

9 Lorentz Transformations NOTE that coordinates orthogonal to the direction of motion stay the same What about y and z coordinates? (x - direction of motion) NO (Lorentz) Length Contraction in directions other than along the direction of relative motion Lorentz Transformations Lorentz Transformations Relativistic velocity Transformations Relativistic velocity Transformations u x, u y u z BUT ALL Components of the Velocity Vector Transform! BUT ALL Components of the Velocity Vector Transform! WHY?

10 Because: The time transforms independently of the direction of motion, coordinates do not, and velocity combines both u (velocity of an object in frame S ) is the differential displacement in that frame divided by the differential time interval in that frame dt / dt u

11 Parallel to the Direction of Relative motion Orthogonal to the Direction of Relative motion ux = c Classical Limit? (both u and v << c) O.K. ux = c uy = 0 uz = 0

12 The Doppler Effect The Doppler Effect So far we learned about the transformation of: space coordinates and time velocity Now, let s study the relativistic transformation of frequency (how does the light appear in a moving reference frame?) So far we learned about the transformation of: space coordinates and time velocity Now, let s study the relativistic transformation of frequency (how does the light appear in a moving reference frame?) Doppler Effect Special Case:!= 0 The source of light moves away from the observer (shift to lower frequencies) Red Shift

13 1 1 f = = 1 β = f 1 β T T 0 Special Case:!= 90 deg. The source of light moves perpendicular with respect to the observer Transvere Red Shift The origin of the transverse Doppler effect is time dilation, this is a pure relativistic effect, no counterpart in classical mechanics. No classical analog