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A lecture series on relativity theory and quantum mechanics

The Relativistic Quantum World. A lecture series on relativity theory and quantum mechanics. Marcel Merk. University of Maastricht, Sept 16 – Oct 7, 2013. The Relativistic Quantum World. Sept 16: Lecture 1: The Principle of Relativity and the Speed of Light

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A lecture series on relativity theory and quantum mechanics

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  1. The Relativistic Quantum World A lecture series on relativity theory and quantum mechanics Marcel Merk University of Maastricht, Sept 16 – Oct 7, 2013

  2. The Relativistic Quantum World Sept 16: Lecture 1: The Principle of Relativity and the Speed of Light Lecture 2: Time Dilation and Lorentz Contraction Sep 23: Lecture 3: The Lorentz Transformation Relativity Lecture 4: The Early Quantum Theory Sep 30: Lecture 5: The Double Slit Experiment Lecture 6: Quantum Reality Quantum Mechanics Oct 7: Lecture 7: The Standard Model Lecture 8: The Large Hadron Collider Standard Model Lecture notes, written for this course, are available: www.nikhef.nl/~i93/Teaching/ Literature used: see lecture notes. Prerequisite for the course: High school level mathematics.

  3. The Story Sofar Postulates of Special Relativity Two observers in so-called Inertial frames, i.e. they move with a constant relative speed to each other: The laws of physics for each observer are the same The speed of light in vacuum for each observer is the same “Absolute velocity” is meaningless

  4. The Story Sofar Time dilation: Lorentz contraction: g Speed v/c

  5. Real life example • Muon particles have a half-lifetime of 1.56 μs. • In the atmosphere they are created at 10 km height with a speed: v = 0.98 c • They can reach the surface because: • As seen from an observer on earth • they live a factor 5 longer • As seen from the muon particle the • distance is a factor 5 shorter

  6. Lecture 3 The Lorentz Transformation • “Imagination is more important than knowledge.” • Albert Einstein

  7. Coordinate Systems “Do not worry about your difficulties in mathematics. I can assure you mine are still greater.” -Albert Einstein A reference system or coordinate system is used to determine the time and position of an event. Reference system S is linked to observer Alice at position (x,y,z) = (0,0,0) An event is fully specified by giving its coordinates and time: (t, x, y, z) Reference system S’ is linked to observer Bob who is moving with velocity v with respect to Alice. The event has: (t’, x’, y’, z’) How are the coordinates of an event, say a lightning stroke in a tree, expressed in coordinates for Alice and for Bob? (t, x, y, z) (t’, x’, y’, z’)

  8. Space-time diagram Bob drives from Maastricht to Amsterdam with 100 km/h Alice drives from Maastricht to Amsterdam with 200 km/h t “Events”: 12h00 Departure Alice & Bob 1 h Arrival Bob Arrival Alice 11h00 “World lines”: 1 h Bob’s world-line Alice’s world-line (x,t) 10h00 x 200 km Amsterdam Maastricht Events with space-time coordinates: (x,t) More general: it is a 4-dimensional space: (x,y,z,t)

  9. Coordinate transformation How does Alice’s trip look like in the coordinates of the reference system of Bob? t t’ 12h00 12h00 1 h 1 h (x2,t2) (x2’,t2’) 11h00 11h00 1 h 1 h (x1’,t1’) (x1,t1) 10h00 x 10h00 x’ 200 km 100 km Alice as seen from Maastricht S = fixed reference system in Maastricht Alice as seen from Bob S’ = fixed reference to Bob

  10. Coordinate transformation How does Alice’s trip look like in the coordinates of the reference system of Bob? t t’ 12h00 12h00 1 h 1 h (x2,t2) (x2’,t2’) 11h00 11h00 1 h 1 h (x1’,t1’) (x1,t1) 10h00 x 10h00 x’ Classical (Galilei transformation): Relativistic (Lorentz Transformation): 200 km 100 km with:

  11. Lorentz Transformations Hendrik Anton Lorentz (1853 – 1928) Dutch Physicist in Leiden (Nobelprize 1902 with Pieter Zeeman) To explain the Michelson-Morley experiment he assumed that bodies contracted due to intermolecular forces as they were moving through aether. (He believed in the existence if the aether) Einstein derived it from the relativity principle and also saw that time has to be modified.

  12. Let’s go crazy and derive it…

  13. Classical Limit Lorentz Transformation: With the new variables: fraction of light-speed Relativistic factor It becomes: For every day life experience: velocities much less than lightspeed In every day life we do not see the difference between the classical and the relativity theory

  14. Relativity Revisited

  15. Paradoxes Case 1 : Sherlock & Watson A murder scene being investigated. Alice enters a room and from the doorstep shoots Bob, who dies. (Thought experiment!) Sherlock stands at the doorstep (next to Alice) and observes the events. Alice shoots at t = tA from position x = xA Bob dies at t = tBat position x = xB Watsonpasses by on a fast train and sees the same scene. He sees: Alice shoots at t’ = tA’ from position x’ = xA’ Bob dies at t’ = tB’ at position x’ = xB’

  16. Causality Sherlock: Alice shoots at Bob from xA at time tA Bob dies on position xB and time tB What does Watson see at v=0.6c? ct ct’ (xB,ctB) ctB (xA’,ctA’) (xB’,ctB’) (xA,ctA) 0 x x’ To make the calculation easy let’s take xA = 0 and tA = 0 : xB 0 This is what Watson sees: 0 ctB’ If distance xB > ctB / 0.6 then ctB‘ < 0 : Bob dies before Alice shoots the gun! xB’ 0

  17. What is wrong? The situation was not possible to begin with! Light-speed ct Time-like Space-like (xB,ctB) (xA,ctA) 0 x 0 Nothing can travel faster then the speed of light, also not the bullet of gun. The requirement: xB > ctB / 0.6 implies a speed v = xB/tB > c / 0.6 = 1.67 c ! Causality is not affected by the relativity theory!

  18. Paradox 2: A ladder in a barn? Alice runs towards a barn with v = 0.8 c (g =1.66). She carries a 5 m long ladder. Bob stands next to 4 m deep barn. Will the ladder fit inside? 4 meters 5 meters Alice: no way! She sees a L/g = 2.4 m deep shed Bob: sure, no problem! He sees a L/g = 3 m long ladder

  19. Paradox 2: A ladder in a barn? Alice’s experiment: Bob’s experiment: Bob and Alice don’t agree on simultaneity of events. Alice claims the back door is opened before the front door is closed. Bob claims both doors are closed at the same time. *But will it stay inside?! To stay inside the ladder must stop (negative accelleration) Compressibility lightspeed The ladder has been fully inside The ladder has not been fully inside

  20. Paradox 3: The Twin Paradox Identical twins: Jim andJules Jules Jim “Jules+2y” “Jim+10y” Jules travels to a star with v = 99.5% of c (g = 10) and returns to Jim on earth after a trip of one year. Jim has aged 10 years, Jules only 1 year. Jim understands this. Due to Jules’ high speed time went slower by a factor of 10 and therefore Jim has aged more than Jules. But Jules argues: the only thing that matters is our relative speed! As seen from his spaceship time goes slower on earth! He claims Jim should be younger! Who is right? Answer: special relativity holds for constant relative velocities. When Jules turns around he slows down, turns and accelerates back. At that point time on earth progresses fast for him, so that Jim is right in the end.

  21. Paradox 3: The Twin Paradox • 1971: A real experiment! • Joseph Hafele & Richard Keating tested it with 3 atomic cesium clocks. • One clock in a plane westward around the earth (against earth rotation) • One clock in a plane eastward around the earth (with earth rotation) • One clock stayed behind in the lab. The clock that went eastward was 300 nsec behind, in agreement with relativity.

  22. Addition of Velocities . w= v= With which speed do the ball and the outfielder approach each other? Intuitive law (daily experience): 30 m/s+10 m/s=40 m/s Galileo Galilei More formal: Observer S (the Batter) observes the ball with relative velocity: w Observer S’ (the running Outfielder) observes the ball with relative velocity: w’ If the velocity of S’ with respect to S is: v This is the Galileian law for adding velocities. What is the relativistic correct formula? w’=w+v

  23. Derive the laws for adding speed . Lorentz Transformation: Galilei Transformation:

  24. Relativistic Transformation law . . w= v= Relativistic: 0.8c 0.6c Bob in a rocket passes a star with 0.8 c Alice in a rocket passes a star with 0.6 c In opposite directions. What is their relative speed?

  25. How about Alice seeing light from Bob? How fast does the light go for Alice?

  26. E=mc2 Consider a box with length L and mass M floating in deep space. A photon is emitted from the left wall and a bit later absorbed in the right wall. Center of Mass (CoM) of box + photon must stay unchanged. Action = - Reaction: photon momentum is balanced with box momentum Time it takes the foton Distance that the box has moved C.O.M. does not move: box compensated by photon C.O.M. Substitute the above equations Equivalence of mass and energy!

  27. Conclusions Relativity • Simple principle: • Laws of physics of inertial frames are the same. • Speed of light is the same for all observers. • Consequences: • Space and time are seen differently for different observers. • Bob’s time is a mixture of Alice’s time and space and vice versa. • Bob’s space is a mixture of Alice’s time and space and vice versa. • Time dilation and Lorentz contraction • Energy and mass are equivalent General Relativity: inertial mass = gravitational mass

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