2.4 The Lorentz Transformation
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1 Announcement Course webpage Textbook PHYS-2402 Lecture 4 Jan. 27, 2015 Lecture Notes, HW Assignments, Physics Colloquium, etc The Lorentz Transformation We can use γ to write our transformations. Chapter 2 Special Relativity Frame S 1. Basic Ideas 6. Velocity Transformation 2. Consequences of Einstein s Postulates 7. Momentum & Energy 3. The Lorentz Transformation Equations 8. General Relativity & a 1st Look at Cosmology 4. The Twin Paradox 9. The Light Barrier 5. The Doppler Effects 10. The 4th Dimension Frame S 4
2 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) One twin stays at home. Twin Paradox 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.. A trip from Earth to Planet Hollywood Homer stays on Earth. Loner travels 10 light-years at 80% of the speed of light. (speed of light = c) Beta (β) 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 light speed (c) has on time (t) or distance (x) 7
3 Effect of Time on Spaceship Planet Hollywood Loner Homer Velocity (v) = 0.8c therefore β =v/c= 0.8 γ = β² (β² = 0.8²= 0.64) Earth 1 - β² = = 0.36 and the of 0.36 = 0.6!! γ = = 1²/³ = 5/3 As Viewed From Earth! 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!! As Viewed From Earth! With Relativity! Homer sees Loner s clock is running slow by! γ = 5/3!!! Therefore Loner s clock reads 25 years γ"! 25 5/3 = 15 years!!
4 Physical Results of Trip " Homer on Earth ages 25 years!! " Loner, traveling at 80% the speed of light ages 15 years!! 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!! The traveling twin is younger! Anna travels away and back at v = 0.8 c (! = 5/3) Bob stays home PARADOX: Seemingly absurd or self-contradictory thought, often true statement (Oxford Dictionary)
5 The Key: Round Trip! The Key: Round Trip! ACCELERATION Change of inertial frames
6 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 Lorentz transformation 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!! What about y and z coordinates? (x - direction of motion)
7 Lorentz Transformations Lorentz Transformations Relativistic velocity Transformations NOTE that coordinates orthogonal to the direction of motion stay the same NO (Lorentz) Length Contraction in directions other than along the direction of relative motion u x, u y u z BUT ALL Components of the Velocity Vector Transform! Lorentz Transformations Relativistic velocity Transformations Because: The time transforms independently of the direction of motion, coordinates do not, and velocity combines both BUT ALL Components of the Velocity Vector Transform! WHY?
8 dt / dt u u (velocity of an object in frame S ) is the differential displacement in that frame divided by the differential time interval in that frame
9 Parallel to the Direction of Relative motion Orthogonal to the Direction of Relative motion Classical Limit? (both u and v << c) O.K. ux = c u and/or v ~ c? ux = c uy = 0 uz = 0 Please try to figure out yourself
10 Another way of expressing the Lorentz Transformation Equations 4-vectors x x + y y + z z! The LENGTH is not INVARIANT, i.e. it is not conserved under Lorentz transformations xx + yy + zz x x + y y + z z x x + y y + z z (ct )(ct ) Is there anything that IS INVARIANT under Lorentz Transformations?! 4-dimensional LENGTH xx + yy + zz = xx + yy + zz (ct)(ct) Another way of expressing the Lorentz Transformation Equations 4-vectors
11 Another Example: Next Lecture We will see this soon
2.3 The Lorentz Transformation Eq.
Announcement Course webpage http://highenergy.phys.ttu.edu/~slee/2402/ Textbook PHYS-2402 Lecture 3 HW1 (due 9/13) Chapter 2 20, 26, 36, 41, 45, 50, 51, 55, 58 Sep. 6, 2016 2.3 The Lorentz Transformation
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