Special Relativity. Mário Pimenta Udine, September 2009

Transcription

1 Special Relativity Mário Pimenta Udine, September 29

2 any observer doing experiments cannot determine whether he is at rest or moving at a steady speed Galileo Galilei

3 Salviatius : Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though doubtless when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still. Galileo Galilei Dialogue Concerning the Two Chief World Systems, translated by Stillman Drake, University of California Press, 1953, pp (Second Day).

4 Isaac Newton There exists an absolute space, in which Newton's laws are true. An inertial frame is a reference frame in relative uniform motion to absolute space. All inertial frames share a universal time. The fundamental laws of physics are the same in all inertial frames

5 Not easy to get an inertial frame a!!! no fictional forces aa

6 Frames in rotation W r Coriolis Force Centrifugal force Euler force

7 Coriolis force corrents Low pressions (North)

8 Foucalt pendulum Pantheon, Paris

9 Really difficult Solar system Milky way Great attractor

10 A possible candidate??? CMB dipole COBE, ΔT/T ~ 1-3 CMB rest frame COBE, ΔT/T~1-5 After dipole subtraction Penzias and Wilson, 1965

11 The true inertial frame The free-fall elevator! General relativity

12 Galileo Transformations x x V t y y z z t t v x v x V t v y v y v z v z a x a x a y a z a z a z

13 Instantaneous forces F G m m 1 r 2 2 no time scale! Crab SNR (154) any change in any object in theuniversewillbefeltat once in all points of the Universe

14 The speed of light Galileo Roemer 167 Fizeau ~ 185

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16 Michelson -Morley Michelson 1887 try to detect the predicted fringe shift due to the Earth movement (3 km/s) in the aether. Resolution 8 km/s.

17 The laws of physics are the same for all observers in uniform motion relative to one another The speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the source of the light. Albert Einstein

18 Simultaneity S S ' A B Two events simultaneous for one observer may not be simultaneous for another observer

19 Time dilatation S (bus) S (Lab) v Δt/2 2 Δ t Δ t /!!! ( 1 v 2 / c

20 Hard to test it

21 Muons at the Earth surface Muons are produced on the upper layers of the atmosphere but they do arrive at the Earth surface!!!

22 Length contraction Δ t 2 l / c Δ t Δt 1 + Δt 2 Δ t (l + v Δt 1 ) / c + (l - v Δt 2 ) / c 2 Δ l Δ l!!! ( 1 v 2 / c

23 Fotografia tirada em 1912 durante o Grande Prémio de França por Jacques-Henri Lartigues

24 End first lecture

25 Space-time

26 Try to save the aether Maxwell equations are not invariant under the Galileo transformations! Michelson and Morley experiment failed to detector the aether wind! Francis FitzGerald Joseph Larmor FitzGerald introduced the hypothesis of the length contraction. Larmor published the complete Lorentz transformation two years before Lorentz, and predicted time dilatation. Lorentz published the Lorentz transformations in 1895, 1899 and 194 Hendrick Lorentz

27 / 1 ) / / ( / 1 ) / ( c v c x v t t z z y y c v t v x x The Lorentz transformations ) ( ) ( x z z y y x x β τ γ τ β τ γ β v / c τ c t γ 1 / 2 1 β

28 The γ factor

29 A clock in S Praga astronomical clock A x A τ A { { B ct x B B τ A { { A x A τ ct ct x B B γ τ β γ B time dilatation! ) ( ) ( x x x + + β τ γ τ β τ γ

30 A ruler in S ) ( ) ( x x x + + β τ γ τ β τ γ B x A τ A { { B B x B L τ A { { A x A τ L L x B B β γ τ γ B B A B T V L x x + ) ( L τ B + β ( L ) L L β γ β γ / γ L L Length contraction!

31 Space-time diagrams S S ct B ct B A A L B L A B A

32 Space-time interval light cone s 2 Δ 2 τ -(Δ 2 x+ Δ 2 y+ Δ 2 z) is a Lorentz invariant! s 2 > time-like interval s 2 < space-like interval s 2 light interval if s 2 > : (s 2 ) proper time (τ ) If s 2 < o : (-s 2 ) rest length (L ) future past present

33 Velocity transformations ) ( ) ( x z z y y x x + + β τ γ τ β τ γ dτ dx c dt dx v x / / τ d dx c dt dx v x / / c V / β 2 / 1 ) ( ) ( c V v V v dx d d dx c v x x x β τ γ τ β γ ) / (1 ) ( 2 c V v v dx d dz c v x z z + + γ β τ γ ) / (1 ) ( 2 c V v v dx d dy c v x y y + + γ β τ γ!!! c v c v x x V!!! v v V +

34 Four vectors, 1 1 γ -γβ -γβ γ τ x y z τ x y z ν μ ν μ X X Λ ν μ μν X X s g g μν

35 The twin paradox One of the twin leaves on a space journey during which he travels close to the speed of light, while his sister remains on Earth. On his return the space traveller will find that his sister has aged more than himself! The paradox arises because it can be argued that the sister is moving near the speed of light relative to her brother and so the brother should be getting older instead.

36 The Lorentz transformations ) ( ) ( x z z y y x x + + β τ γ τ β τ γ β v / c τ c t γ 1 / 2 1 β

37 Simultaneity planes the simultaneity planes in two frames are not the same!

38 The travel of the brother When the traveller twin inverts his movement changes his simultaneity plane!

39 End second lecture

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41 2 Emc It followed from the special theory of relativity that mass and energy are both but different manifestations of the same thing -- a somewhat unfamiliar conception for the average mind. Furthermore, the equation E is equal to m c-squared, in which energy is put equal to mass, multiplied by the square of the velocity of light, showed that very small amounts of mass may be converted into a very large amount of energy and vice versa. The mass and energy were in fact equivalent, according to the formula mentioned before. This was demonstrated by Cockcroft and Walton in 1932, experimentally.

42 momentum p r must be conserved in all the references frames!!! v p γ v m r m v v / c CM frame B rest frame Classical mechanics Relativistic mechanics m - particle rest mass v of a body under a constant force

43 Energy Kinetic energy Total energy E γ m c Rest energy 2 E m c Rest mass is just a form of energy! 2 Low β E k c E k 2 γ m c m c m v ( ( ) v 2c )

44 Fission and fusion Fission the splitting of an heavy atom Fusion the fusing of light atoms into heavy atom Binding energy mass Kinetic Energy

45 Particle Physics Collides high energy particles and observes what comes out! Kinetic Energy mass An example: the DELPHI- the first WW event

46 Energy-momentum Lorentz transformation p E / c p r is a four vector! E / c p x p y p z γ -γβ -γβ γ 1 1 E / c p x p y p z Useful formulae p 2 (E/c) 2 - p r 2 m c 2 E p r 2 2 c 2 +m 2 c 4 β p r γ E /( E / c) /( m c 2 ) two body (p 1 +p 2 ) (p 3 +p 4 ) E CM 2 (p 1 +p 2 ) if m << E (p 1 p 3 ) 2 ~ - 4E 1 E 3 sin 2 (θ/2) 4

47 Mandelstam Variables u (p 1 p 4 ) 2 s + t + u m m 2 2 +m 3 2 +m 4 2

48 Particle Physics examples(i) λ mass

49 Particle Physics examples(ii) Fix target vs collider

50 Particle Physics examples(iii) π decay

51 m The light is emitted by the lamp and is absorbed in the wall. Does the wagon moves? a) Yes - the light carries Energy and momentum! b) NO there is no external Force!

52 The photon - γ m γ E γ p γ c E γ h ν h c/ λ Compton effect Energy- momentum conservation hc/λ i + m e c 2 hc/λ f + ((m e c 2 ) 2 +(p e c) 2 ) h/λ i h/λ f cos(θ) + p e cos(φ) h/λ f sin(θ) + p e sin(φ) λ f λ i h/(m e c) (1 cos(θ))

53 Doppler effect observer at rest, source in movement t source T signal T S x 1 x x observer rest frame

54 Doppler effect observer at rest, source in movement T t signal observer rest frame source T S x 1 x x x x1 T TS v VSTS v V TS v v Classical Mechanics v signal velocity V S source velocity T S T S v f f v V Relativistic Mechanics v T S γ T S f f γ ( v V ) if v c f S T S c + V c V S S S f S

55 Doppler effect source at rest, observer in movement t source T signal T S x 1 x x observer rest frame

56 Doppler effect source at rest, observer in movement T t signal source T S x 1 x x observer rest frame T VTS vr V TS vr vr Classical Mechanics T v S R T T S V S + x x v v R 1 Relativistic Mechanics T v S R γ T S V + v Vv 1+ 2 c if v v R signal velocity (obs) v signal velocity (medi) V observer velocity c T S v + V f f v v + V f f γ v f c + V c V S f S

57 The GPS System

58 Cosmological redshift The galaxies are not moving,isthespace itself that is in expansion!

59 End third lecture

60 The speed of light has been an elusive thing through out history. In 167, Ole Roemer stumbled upon a measurement that was accurate to 1% in the negative direction of the now accepted value. Roemer was an astronomer studying Jupiter, specifically its moon Io. He noticed that as the year went by, the time at which Jupiter eclipsed the moon seemed to grow as the two planets moved apart. He reasoned that this time delay was due to the extra time it took light to travel extra distance. This experiment was reproduced using a smaller change in time, 3 days, and greater precision in the measurements of distance between the planets, obtained from pre-existing sources. This will hopefully yield a more accurate value for c. An additional value for c was obtained through a Fitzeau wheel. Fitzeau, in the mid 18 s, used a rotating toothed wheel and a light, concentrated through lenses, measured the speed of light to about 1% of the accepted value in the positive direction. In a Fitzeau wheel, the light is directed through the teeth, 36 in this case, of a quickly rotating wheel, 1, RPM, reflected off a plain mirror back to the wheel, at which point it passes through a different tooth. The time it takes for the wheel to rotate from one tooth to the next is equal to the round trip time of the light. The round trip distance and the time were then used to calculate an additional value for c. The experiment is ongoing; results are pending. The predicted result of the Jupiter calculation is close to 2.9*1^8, and the predicted result of the Fitzeau wheel calculation 3.1*1^8.