Announcements. Muon Lifetime. Lecture 4 Chapter. 2 Special Relativity. SUMMARY Einstein s Postulates of Relativity: EXPERIMENT

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1 Announcements HW1: Ch.2-20, 26, 36, 41, 46, 50, 51, 55, 58, 63, 65 Lab start-up meeting with TA tomorrow (1/26) at 2:00pm at room 301 Lab manual is posted on the course web *** Course Web Page *** Lecture Notes, HW Assignments, Schedule for thephysics Colloquium, etc.. Outline:! Basic Ideas Lecture 4 Chapter. 2 Special Relativity! Consequences of Einstein s Postulates! The Lorenz Transformation Equations! The Twin Paradox! The Doppler Effect (later)! Velocity Transformation! Momentum & Energy! General Relativity & a 1 st Look at Cosmology SUMMARY Einstein s Postulates of Relativity: Muon Lifetime Michelson- Morley Experiment NO AETHER! Consequences of Einstein s Postulates: 1. Relative Simultaneity 2. Time Dilation 3. Length Contraction EXPERIMENT Muon (µ) is an elementary particle similar to electron, but heavier (will learn more in Chapter 11)

2 Muon Lifetime Muon Lifetime Muons are created abundantly in elementary particle showers in the atmosphere, initiated by energetic cosmic rays (photons, particles and nuclei). Atmosphere Cosmic Ray Muons originate from decays of particles called pions (!) that are the primary products in these showers Neutrinos Earth Particle shower CONCLUSION: According to Classical Physics,muons should not reach the ground! BUT THEY DO

3 The distance between the creation and detection points There are many-many experiments that prove the consequences of the two postulates of relativity Twin Paradox! 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. The Theory of Relativity As an object approaches the speed of light, time slows down. (Moving clocks are slow) & (Moving rulers are short)! This is the paradox. Who is really younger.! Answer: Traveling twin because of accelerations for the traveling twin non inertial frame.. 12

4 A trip from Earth to Planet Hollywood!!Homer stays on Earth. Planet Hollywood Loner Homer!!Loner travels 10 light-years at 80% of the speed of light. (speed of light = c)!!beta (!) is the velocity of the object compared to the speed of light. (!=0.8c)!!Gamma (") is the effect of traveling at speeds close to light speed (c) has on time (t) or distance (x) Earth Effect of Time on Spaceship! Velocity (v) = 0.8c therefore!! =v/c= 0.8 "= 1 1 -!# (!# = 0.8#= 0.64) 1 -!# = = 0.36 and the $ of 0.36 = 0.6!! " = = 1#/% = 5/3 As Viewed From Earth!! Without Relativity..!! x = vt or t = x/v!! x = mlight years traveled!! v = velocity.!! t = time.!! T = 10/0.8c = 12.5 years each way.!! There and back makes the trip 12.5 x 2 or 25 years!!

5 As Viewed From Earth! With Relativity Physical Results of Trip! Homer sees Loner s clock is running slow by! " = 5/3!!! Therefore Loner s clock reads 25 years "!! Homer on Earth ages 25 years!!!! Loner, traveling at 80% the speed of light ages 15 years!!! 25 5/3 = 15 years!! The traveling twin is younger! As Viewed From Spaceship! Loner sees distance of planets contracted by "= 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!! 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 v + u u = velocity of an object moving relative to a frame u = velocity of an object in frame S Classical transformation is wrong!!

6 Galilean Transformation y K y K v z O z O x = x vt is wrong!! x x x = x vt y = y z = z t = t 21 What about y and z coordinates? (x - direction of motion) 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 WHY? Velocity Vector Transform!

7 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

8 Parallel to the Direction of Relative motion Orthogonal to the Direction of Relative motion Another way of expressing the Lorentz Transformation Equations 4-vectors x x + y y + z z! xx + yy + zz Classical Limit? (both u and v << c) O.K. The LENGTH is not INVARIANT, i.e. it is not conserved under Lorentz transformations

9 Another way of expressing the Lorentz Transformation Equations 4-vectors Another Example: Next Lecture We will see this soon Relativistic Dynamics Newton s 2 nd Law: Relativistic Momentum the momentum of a particle, m is invariant (does not depend on the velocity) expressed in terms of 3-vectors, invariant under G.Tr. (but not L.Tr.!) Relativistic form of the 2 nd Law (introduced by Einstein): Outline:! Relativistic Momentum! Relativistic Kinetic Energy! Total Energy! Momentum and Energy in Relativistic Mechanics! Thursday: General Theory of Relativity! Next Week Quantum Physics where definition of the momentum in relativistic mechanics Example: Calculate the momentum of an electron moving with a speed of 0.98c. By ignoring relativistic effects, one would get

10 Relativistic Kinetic Energy