Paradoxes in Special Relativity Paradoxes in Special Relativity. Dr. Naylor

Transcription

1 October 2006 Paradoxes in Special Relativity Paradoxes in Special Relativity Dr. Naylor 1

2 102 years after Einstein s A. Einstein, Special theory of relativity 2

3 Paradoxes? Twin Paradox Time dilation Barn-pole Paradox Lorentz Contraction Relativity of simultaneity 3

4 Subtleties? Different observers disagree on NOW Observer O uses two clocks to measure s single clock, and vice versa ) disagreement on NOW There is only relative simultaneity Spacetime diagrams help a great deal! 4

5 Asymmetry ) relative simultaneity Fig: 5

6 Twin paradox Terence stays on Earth while Stella makes a 14yr round trip into space; 7yr outward journey. Terence Stella V=24/25=0.96 γ = 1 v 2 = 1 (0.96) 2 = 0.28 Assuming that Stella is moving, then Terence sees Stella s proper time Δτ as t = τ = 14 1 v = 50yrs 6

7 However, can t Stella argue that the Earth was traveling with respect to her ship? Terence? Stella V=24/25=0.96 Conventional answer: SR does not say that all frames of references are equivalent, only inertial frames! Stella must accelerate to v=0.96 then change direction and then slow down to v=0 back at Earth. 7

8 Spacetime diagrams However, SR allows for infinite accelerations and we can assume that Stella instantaneously changes direction (No GR required)! For Stella, as she changes her frame she sees time jump from A to C v ¼ 0.5 As v increases the jump becomes larger because lines of simultaneity get steeper! Fig: 8

9 C 25yrs Terence A P t D Stella returns B Stella leaves Terence s line of simultaneity At faster speeds this jump gets larger! Note that Stella only covers a very small part of the spacetime of Terence: Terence = Δ PBD Stella = Δ PBA + Δ BCD x For Stella, Terence s time is PA = 2yrs, PC=48yrs, CD=2yrs For Terence, Stella s time is PB=7yrs=BD 9

10 Digression: Relativistic Doppler effect Next week will will give a more rigorous derivation!!! Light source (f e ) Next wave meets at time delay t e = λ e c λ e v c = 1 (1 v/c)f e λ e = c f e v Stella However, due to time dilation Stella will measure the time between waves as t o = t e 1 v 2 /c 2 = 1 v2 /c 2 (1 v/c)f e = 1 f o Thus, Stella observes frequency f o = 1 v/c 1 + v/c f e ( 1 v c ) f e v c 0 Non-relativistic limit 10

11 Imagine Stella and Terence send laser light pulses to each other every second ) f e =1 Stella sees more blueshifted light Replace v by v for blue-shifts 1 + v/c 1 v/c red-shift 1 v/c 1 + v/c Terence see more red-shifted light Thus, Terence ages more! Still confused? Terence to Stella Stella to Terence Fig: 11

12 Down to bad coordinates? Why is Stella is surprised that Terence has aged? Bad spacetime coordinates! Consider an example in 2D Euclidean space y Analogy taken from Schutz s book D C Imagine measuring the line AD in x-y frame, but at point B you rotate the axes by an angle θ to frame θ A B x Clearly then you would begin at point C and measure CD Total will be AB+CD AD For Stella to realize this fact she must keep smb on the outward journey for (see page 8): AD/0.28 = 48/0.28 ¼ 171yrs! 12

13 The problem with Lorentz contraction Length contracted pole/ladder Length contracted garage/barn Ref: This leads to P.T.O

14 Lorentz contraction paradoxes? Various kinds have been devised We shall look at barn-pole (or ladder-garage) type paradoxes l S =20m Terence Stella v=0.8c Barn b T =15m Key point is that length and time are linked so length contraction leads to time dilation and hence relative simultaneity 14 14

15 Barn-Pole: double door variation Problem is only with concept of NOW, there is only relative simultaneity As we can see Stella and Terence disagree on the times when both doors are actually open and shut! Ref: Barn (Terence s) frame Pole (Stella s) frame 15

16 Double door spacetime diagram Blue and red bands show the barn & pole spacetime, respectively. Front of the pole hits back of barn at event A. D is the point where the end of the pole enters the barn AB is simultaneous in barn frame so this will be what the barn sees as the pole length at the time of event A and thus, the pole fits in the barn The above diagram is in the rest frame of the barn, with x and t being the barn frame. The pole frame is for a person sitting on the front of the pole (axes x and t ). Ref: However, from the point of view of the pole, AC is the pole length and thus, the back of the pole is outside the barn. 16

17 Barn-pole: single door variation Consider a 20m pole which an Olympic athlete (Stella) runs with at speed v=0.8c into a barn of length 15m? l S =20m Terence Stella v=0.8c Barn b T =15m Pole fitting into length contracted barn. Ref: Finite transmission speed (v=c) of the shock wave prevents the pole from behaving rigidly and thus, Stella and Terence disagree on the time the door shuts; however, both agree that the door does shut! 17

18 Single door spacetime diagram In barn frame rod stops simultaneously all along its length. Barn frame sees the ladder as AB, but the pole frame sees the pole as AC. When the back of the pole enters the garage at point D, it has not yet felt the effects of the impact. Spacetime diagram when one of the doors remains shut: Ref: According to someone at rest with respect to the back of the pole, the front of the ladder will be at point E and will see the ladder as DE. The length in the pole frame is not the same as CA which is the rest length of the pole before impact. (See previous slide.) 18

19 Questions on the barn-pole 1. What length does Terence measure for the pole? [Hint, Lorentz length contraction formula:] 2. After Stella enters the barn how long does Terence measure before the pole hits the wall? 3. Is the interval in Terence s frame timelike or spacelike? 4. According to Stella the barn length is 1. b S =? 19

20 Answers 1. What length does Terence measure for the pole? Using the Lorentz contraction formula ) l T = 1 v 2 l S = 20 1 (0.8) 2 = = 12m 2. After Stella enters the barn how long does Terence measure before the pole hits the wall? In c=1 units, (15-12)/0.6=3.75m But 1m = 1/( ) s ) s 3. The interval to Terence is spacelike : Δs B 2 =-Δt 2 + Δx 2 =-(3.75) 2 +(15) 2 =211m 2 > 0 4. According to Stella the barn length is b S = = 9m 20

21 References and final comment References and final comment John Baez s web page for many useful discussions on physics go to SR and twin paradox Wikipedia has many nice diagrams Both of these web cites discuss a myriad of paradoxes in SR including the Barn-pole paradox, e.g., For criticism of Rindler s Man in grate paradox see Even 101 years later, SR still causes much debate and sometimes controversy However, this is only due to our Newtonian view of the universe! 21