Twin Paradox and Velocity Addition in Relativity

1. Setup: 1 open four browser screens. • for class: http://www.physics.byu.edu/Courses/CourseList.aspx • For twin http://physics.syr.edu/courses/modules/LIGHTCONE/LightClock/#twins • MU 43:14 (velocities) 11.22 minute • MU 43:19 Twin • MU 44: 9’15” to 14’ 30” billiards • Looking at appearances: http://www.anu.edu.au/Physics/Savage/TEE/site/

2. Twin paradox, Relative Velocity, and Momentum Class 3: Everything should be made as simple as possible but not simpler. A. Einstein. Which of these are invariant quantities? time, distance, velocity, c, frequency, charge, mass, momentum, force?

3. Iclickers FAQ • You can use before registering. (We rerun the recording often the first two weeks of class.) • You can register at class website or directly at http://gardner.byu.edu/222f/quizid.html • Registering at the Iclicker company site does NOT do the job for class. • Today unless noted otherwise questions are participation points (pp).                Register your transmitter ID

4. Did you complete at least 70% of Chapter 1: 5-7?Not pp Einstein once gave a talk on some new idea of his at a meeting of the German Physical Society. After he had spoken and the chairman had respectfully invited questions a young man got up at the back of the hall and said something like ‘What Professor Einstein has told us is not so stupid. • Yes • No But the second equation does not strictly follow from the first.It needs an assumption, which is not proved, and moreover it is not invariant the way it should be. . .’ Everyone in the hall had turned round and was staring at the bold young man. Except Einstein; he was facing the blackboard and thinking. After a minute he turned round and said ‘What the young man in the back has said is perfectly correct; you can forget everything I have told you today!’

5. Plan: Reading was 1.5-1.7 • Review: EXAMPLE 1.11 (page 44 of book) or Velocities and space-time diagrams 43:14 starts at min 11.22' • Twin paradox MU or as viewed in light cone space time diagram. http://physics.syr.edu/courses/modules/LIGHTCONE/LightClock/#twins • E & M effects: Story & Problem • “Relativistic momentum changes” and discussion of mass as invariant.

6. Relative Velocity

7. Shall we do velocity examples? • Yes, do one or more • Yes, let it be one from HW. • No, but let’s discuss formulae etc. • I want to see MU 43 11.22’ on this. • No, let’s go on. Some of the issues to discuss are how do we get the signs right? How do we read the notation? How looks in ST?

8. 4. (4 pts) The nucleus of a particular atom, initially at rest in the laboratory system, is unstable and disintegrates into two particles. Particle one moves to the left at a speed of 0.80c, and particle two moves to the right at a speed of 0.98c. (a) What is the velocity of particle one with respect to an observer at rest with particle two? (b) An observer at rest with respect to the laboratory system finds that both particles are unstable. Particle one decays after 6.6 μs. Particle two decays after 6.0 μs. What are the lifetimes of the two particles from a reference frame in which particle two is at rest?

9. Relativistic Addition of Velocities Galilean relative velocities cannot be applied to objects moving near the speed of light. But 1st step is Galilean. vab = vad + vdb • Einstein’s modification is • The denominator is a correction based on length contraction and time dilation • Vab is velocity of a in ref. frame of b, etc.

10. 1. In classical physics, what is the velocity of one of the objects in the reference frame of the other object? (pp)0.5 c→ ←0.99 c • 0.49c • 1.49c

11. Why can’t it be 1.49 c really? • What is relativistic velocity? • Vab= [0.5c+0.99c]/ [1+(.5)(.99)/c2]= 0.9966c • Another example?

12. 2. In classical physics, what is the velocity of one of the objects in the reference frame of the other object? (pp)a0.99 c→ b0.98 c→ • 0.01c • 1.97c

13. What is velocity of a in b, Vab?a0.99 c→ b0.98 c→ • We are “d” • Vad = 0.99, • Vbd =0.98c; • But Vdb= -Vbd = -0.98c • Vab=[0.99c -0.98c]/[1+(0.99)(-0.98)/c2] = 0.34c

14. 3. See ex. 1.11 in text Spacecraft Alpha α is moving with v=0.90c with respect to the earth. If spacecraft Beta β passes α at relative speed 0.50c in the same direction, what is speed of β with respect to earth? I am going to have them both going in the +x direction. _____________ β→ _____ α→ vab = (vad + vdb)/(1+vadvdb /c2); here a is β and b is earth. d is α. Now go in and put in names for symbols with goal of getting the signs right. vad is speed of β with respect to α= 0.50c. vdb is speed of α with respect to earth = 0.90c. βab = (0.5+0.9)/(1+0.5*0.9) =1.4/(1.45) = 0.97; I use βab ≡ vab/c Many people use the expression “Laboratory Frame” where “earth” is used in this problem. (1.11 in book)

15. How shall we review twin paradox? Facts: • Both see that the others clock (biology) is running slow on way out. • The fact that Henry turns around makes his journey asymmetric. Quiz: A. diagrams, B. See Mechanical universe 43:19; C. No MU. I saw it but want to see more web items. D. Several of above E. go on.

16. Fig 39-10, p.1257

17. Fig 39-12, p.1259

18. Here is an electronic rendering of “twin-paradox”. It is like example 1.4 in text (page 19) • http://physics.syr.edu/courses/modules/LIGHTCONE/LightClock/#twins

19. Shall we review the Barn-Pole Paradox?(Simultaneity) A. Yes B. No The observer (Albert) in the “barn stationary frame” sees an instant when the shortened pole is all the way in the barn. The runner (Henri) with the pole sees the shortened barn headed at him. There is no instant when the pole is all the way in the barn. Why no splinters? He claims that there is a period of time when the leading edge of the pole is out the back door and the trailing edge still is not in the front door of the barn. WHO IS RIGHT?

20. Review of barn paradox • Albert sees that there is a way to have both doors closed for an instant. • Henri sees that the barn is length contracted, not his pole, and thus there is no way to have both doors closed without the pole crashing into one or other.

21. What Albert photographs (using an imagined kind of instantaneous light). Barn doors are the green.Pole ends are marked with cyan. Fig 39-13a, p.1261

22. They switched colors. Naughty What runner “sees”. His pole ends are marked with the green. (Vertical in time.) Barn doors are in cyan. Fig 39-13b, p.1261

23. Resolution? Events are at different times for Henri than Albert. • The observer (Albert) in the “barn frame” sees an instant when the shortened pole is all the way in the barn. • The runner (Henri) with the pole sees the shortened barn headed at him. There is no instant when the pole is all the way in the barn. Why no splinters? He claims that there is a period of time when the leading edge of the pole is out the back door and the trailing edge still is not in the front door of the barn. Both are RIGHT: Simultaneity depends on reference frame

24. Section 1.6 (Why was E&M Einstein’s way to get to relativity?) Did you get the email about viewing MU? MU 43: min. 1’ 43”- 5’40” and 24’ 15” Please do this Quiz: I viewed MU 43: 1’ 43”- 5’40” A. Yes. B. No, but I will. Quiz counts

25. E&M: A natural way for relativity Consider the case of cars headed north to SLC at Each has a charge of 50 nC. 1 car passes me standing on the side of the rode each second. What kind of field could be measured? • An Electric B. A Magnetic C. Both D. Neither Now suppose you are in one of the cars. What kind of field could be measured riding in car? Same choices. More ideas at http://www.cavehill.uwi.edu/fpas/cmp/online/p20b/P20B%20Notes.htm pp. 65-67.

26. Section 1.7: Force & momentum • F=ma? = mdv/dt? • The trouble is that v depends on frame and time does too! • Key idea: F=ma when v=0; but is there another way to think about force? • Rewrite F= dp/dt. (This is really what Newton had in the first place.) • p=γmv

27. Watch Out for “Relativistic Mass” • Some older treatments of relativity maintained the conservation of momentum principle at high speeds by using a model in which the mass of the particle increases with speed.  You might still encounter this notion of “relativistic mass” in your outside reading, especially in older books.  Be aware that this notion is no longer widely accepted and mass is considered as invariant, independent of speed.  The mass of an object in all frames is considered to be the mass as measured by an observer at rest with respect to the object.

28. Please do this Quiz: I viewed the MU online on relativistic billiards (44: 9’15” to 14’ 30”) • Yes. • No, but I will. • No, and I want us to view it now instead of rest of material.

29. Relativistic Linear Momentum To account for conservation of momentum in all inertial frames, the definition must be modified to satisfy these conditions • The linear momentum of an isolated particle must be conserved in all collisions • The relativistic value calculated for the linear momentum p of a particle must approach the classical value mu as u approaches zero • u is the velocity of the particle, m is its mass

30. Relativistic Form of Newton’s Laws • The relativistic force acting on a particle whose linear momentum is p is defined as F = dp/dt. • This preserves classical mechanics in the limit of low velocities. • It is consistent with conservation of linear momentum for an isolated system both relativistically and classically. • Looking at acceleration it is seen to be impossible to accelerate a particle from rest to a speed v³c. more below.

31. If we apply a constant force on an object, the acceleration of the object, as it approaches the speed of light, will _______. (pp) • decrease • increase • remain constant

32. Constant Force (begin at rest) F=p/t ► p= Ft ► γmv =Ftv<<c ► v increasesv→c ► γ increases t= γmv/Fas v→c, ► γ →∞, ► t→∞v never reaches c.

33. Speed of Light, Notes • The “speed of light” is the speed limit of the universe. (Or is it the other way around?) • It is the maximum speed possible for energy and information transfer • Any object with mass must move at a lower speed

34. At some point look at appearances: http://www.anu.edu.au/Physics/Savage/TEE/site/

35. Next time: Relativistic Energy • The definition of kinetic energy requires modification in relativistic mechanics • E = mc2 – mc2 • This matches the classical kinetic energy equation when u << c • The term mc2 is called the rest energy of the object and is independent of its speed • The term mc2 is the total energy, E, of the object and depends on its speed and its rest energy

36. Relativistic Energy – Consequences • A particle has energy by virtue of its mass alone • A stationary particle with zero kinetic energy has an energy proportional to its inertial mass • This is shown by E = K + mc2 • A small mass corresponds to an enormous amount of energy

37. Energy and Relativistic Momentum It is useful to have an expression relating total energy, E, to the relativistic momentum, p • E2 = p2c2+ (mc2)2 When the particle is at rest, p = 0 and E = mc2 Massless particles (m = 0) have E = pc • The mass m of a particle is independent of its motion and so is the same value in all reference frames • m is often called the invariant mass

38. Lack of simultaneity is best seen with long distances. • Imagine signal being sent off from a relativistic spacecraft part of way between stars. (LOOK at figure on overhead.) • If ½ way in earth-star reference frame received at same time. • but c is c, so from spacecraft received by star first since it is coming at you.

39. Simultaneous Events • MU 42:20