Special Relativity

1. Special Relativity Jim Wheeler PHYX 4550: Advanced Mechanics

2. Postulates of Special Relativity Spacetime is homogeneous and isotropic All inertial frames are equivalent The paths and speed of light are universal

3. Postulate 1: Homogeneity and Isotropy We assume that, in empty space, any two points are equivalent and any two directions are equivalent. In particular, this means that there exists a class of inertial frames which move with constant relative velocities. Further, the spatial origins of these frames may be represented by straight lines in a spacetime diagram.

4. Postulate 2: Inertial Frames The equivalence of inertial frames implies that there is no absolute motion or absolute rest. Only relative velocities can be of physical relevance. One consequence is that there is no such thing as “vertical” or “horizontal” in a spacetime diagram.

5. Postulate 3: The speed of light In agreement with Maxwell’s theory of electromagnetism, and in conflict with Newton’s laws of mechanics, we assume the speed of light in vacuum relative to any observer in an inertial frame takes the same value, c = 2.998 x 108 m/s = 1 light second / second This is particularly striking, given the equivalence of inertial frames.

6. A picture of spacetime

7. How we think about it Time flows sort of upward Space goes side to side

8. What we all agree on Light (moving right) Postulate 3: Constancy of the speed of light

9. What we all agree on more light (moving left) Postulate 3: Constancy of the speed of light

10. Postulate 3: Every observer sees light move the same distance in a given amount of time We can choose our units so that light moves at 45 degrees. (e.g., light seconds & seconds) Lotsa light

11. Suppose some observers move in straight linesPostulate 1: Spacetime is homogeneous and isotropic Observers move slower than light. An observer moving along in spacetime World line of an observer

12. Postulate 2: Equivalence of inertial frames Observers move along in spacetime

13. An observer moving along in spacetime How might an observer label points in spacetime?

14. Some observers have watches An observer marking time They can mark progress along their world line. A watch gives a perfectly good way to label points along an observer’s world line.

15. Some observers have watches An observer marking time Postulate 1: Homogeneity and isotropy. We assume “good” clocks give uniform spacing.

16. How might an observer label other points in spacetime? Observer A

17. How might an observer label other points in spacetime? Observer Al They can send out light signals.

18. How might an observer label other points in spacetime? Observer Ali There needs to be dust or something, so some light comes back. They can send out light signals.

19. Labeling points in spacetime Observer Alic +2 s Suppose they send out a signal two seconds before noon… …and the signal reflects and returns at two seconds after noon. Noon = 0 s dust -2 s

20. The time of a remote event. Observer Alice +2 s The observer assumes the reflection occurred at noon. (0, x) 0 s -2 s

21. The distance of a remote event Observe Alice +2 s The light took 2 seconds to go out, and two seconds to come back. The dust must be 2 light seconds away at t = 0. (Postulate 3: Constancy of the speed of light) (0, 2) 0 s -2 s

22. Spacetime coordinates O serve Alice +2 s (0 s, 2 ls) 0 s -2 s

23. Spacetime coordinates serve Alice +2 s The observer can send out other signals, at various times, in various directions. (0, 2) 0 s -2 s

24. Spacetime coordinates serv Alice +2 s (0, 2) 0 s Sometimes there will be dust in just the right places. -2 s

25. Spacetime coordinates ser Alice +2 s (0, 3) (0, 2) (0, 1) (0, 0) (0, -1) (0, -2) We assign coordinates to each point where a reflection occurs. -2 s

26. Spacetime coordinates se Alice In this way we find all points that are labeled by t = 0, but different values of x. This is what we mean by the x-axis. +2 s (0, 3) (0, 2) (0, 1) (0, 0) (0, -1) (0, -2) -2 s

27. Spacetime coordinates s Alice In this way we find all points that are labeled by t = 0, but different values of x. This is what we mean by the x-axis. +2 s (0, 3) (0, 2) (0, 1) (0, 0) (0, -1) (0, -2) -2 s

28. Spacetime coordinates Alice In this way we find all points that are labeled by t = 0, but different values of x. This is what we mean by the x-axis. +2 s (0, 3) (0, 2) (0, 1) (0, 0) (0, -1) (0, -2) -2 s

29. Spacetime coordinates Alice In this way we find all points that are labeled by t = 0, but different values of x. This is what we mean by the x-axis. +2 s (0, 3) (0, 2) (0, 1) (0, 0) (0, -1) (0, -2) -2 s

30. Spacetime coordinates Alice In this way we find all points that are labeled by t = 0, but different values of x. This is what we mean by the x-axis. +2 s (0, 3) (0, 2) (0, 1) (0, 0) (0, -1) (0, -2) -2 s

31. Spacetime coordinates Alice In this way we find all points that are labeled by t = 0, but different values of x. This is what we mean by the x-axis. (2, 0) (0, 3) (0, 2) (0, 1) (0, 0) (0, -1) (0, -2) (-2, 0)

32. Spacetime coordinates Alice In this way we find all points that are labeled by t = 0, but different values of x. This is what we mean by the x-axis. (2, 0) (1, 0) (0, 3) (0, 2) (0, 1) (0, 0) (0, -1) (0, -2) (-1, 0) (-2, 0)

33. Spacetime coordinates Alice (3, 0) In this way we find all points that are labeled by t = 0, but different values of x. This is what we mean by the x-axis. (2, 0) (1, 0) (0, 3) (0, 2) (0, 1) (0, 0) (0, -1) (0, -2) (-1, 0) (-2, 0)

34. Spacetime coordinates Alice (4, 0) (3, 0) In this way we find all points that are labeled by t = 0, but different values of x. This is what we mean by the x-axis. (2, 0) (1, 0) (0, 3) (0, 2) (0, 1) (0, 0) (0, -1) (0, -2) (-1, 0) (-2, 0)

35. Spacetime coordinates Alice (5, 0) (4, 0) (3, 0) In this way we find all points that are labeled by t = 0, but different values of x. This is what we mean by the x-axis. (2, 0) (1, 0) (0, 3) (0, 2) (0, 1) (0, 0) (0, -1) (0, -2) (-1, 0) (-2, 0)

36. Spacetime coordinates Alice (5, 0) (4, 0) (3, 0) In this way we find all points that are labeled by t = 0, but different values of x. This is what we mean by the x-axis. (2, 0) (1, 0) (0, 4) (0, 3) (0, 2) (0, 1) (0, 0) (0, -1) (0, -2) (0, -3) (-1, 0) (-2, 0)

37. Spacetime coordinates Alice (5, 0) (4, 0) (3, 0) In this way we find all points that are labeled by t = 0, but different values of x. This is what we mean by the x-axis. (2, 0) (1, 0) (0, 4) (0, 3) (0, 2) (0, 1) (0, 0) (0, -1) (0, -2) (0, -3) (0, -4) (-1, 0) (-2, 0)

38. Spacetime coordinates (0, 4) (1, 4) (0, 3) (1, 3) (1, 2) (0, 2) (1, 1) (0, 1) (1, -1) (0, -1) (1, -2) (0, -2) (0, -3) (1, -3) (0, -4) (1, -4) Alice (5, 0) (4, 0) (3, 0) We assign coordinates to other points in the same way. (2, 0) (1, 0) (0, 0) (-1, 0) (-2, 0)

39. Spacetime coordinates (0, 4) (0, 3) (0, 2) (0, 1) (0, -1) (0, -2) (0, -3) (0, -4) Alice (5, 0) (4, 0) (3, 0) (2, 0) (1, 0) (0, 0) (-1, 0) We now have coordinate labels for each point in spacetime. (-2, 0)

40. Spacetime coordinates (0, 4) (0, 3) (0, 2) (0, 1) (0, -1) (0, -2) (0, -3) (0, -4) Alice (5, 0) (4, 0) (3, 0) (2, 0) (1, 0) (0, 0) (-1, 0) Notice that the paths of light move one light second per second (-2, 0)

41. Alice’s coordinate axes t Alice Alice x

42. Alice and Bill’s coordinate axes t Alice Alice Bill t’ Since the coordinate system is constructed using only the postulates, similar coordinates can be constructed for any inertial frame x x’

43. Spacetimeterminology

44. Events Alice Generic points in spacetime are calledevents. An event is characterized by both time and place

45. The light cone Alice

46. The light cone Alice The t and x axes make equal angles with light paths. j j

47. The light cone x t Alice If we add another spatial dimension the light cone really looks like a cone. y

48. The light cone x t Alice Think of the light cone as the surface of an expanding sphere of light. y

49. The light cone x t Alice Think of the light cone as the surface of an expanding sphere of light. y

50. The light cone x t Alice Think of the light cone as the surface of an expanding sphere of light. y