Special Theory of Relativity

1. Special Theory of Relativity Mr. Finn April, 2008

2. What is “time”? • Newton: • “Absolute, true, and mathematical time, of itself, and from its own nature flows equably without regard to anything external” • Einstein: • time is simply the rate at which a clock ticks and each clock keeps its own time • time is relative, not absolute - dependent on motion, status of observer • “The only reason for time is so that everything doesn't happen at once.”

3. What is “space”? • Newton: • “Absolute space, in its own nature, without regard to anything external, remains always similar and immovable.” • Einstein • space is like a fabric that bends, warps • length is relative, depending on motion/status of observer

4. Outline • What is light? • Postulates • Consequences • Paradoxes

5. #1. Riding a light wave I wonder what you would see if you could ride on a beam of light?

6.  = dot product = “divergence”  = cross product = “curl” Maxwell’s Equations Electric fields diverge from charges Electric fields curl around changing magnetic fields Magnetic fields do notdiverge from anything Magnetic fields curl around moving charges and changing electric fields

7. No charges present and solution is not E=B=0 Interesting Solution… Solution consists of changing electric and magnetic fields curling around each other while propagating through space at speed of:

8. If you were to “ride a light wave”… Wonder what I will see? Oops! … then the light wave would cease to exist!

9. AKA: inertial frames of reference (FoR) #2. Postulates of Special Relativity • Relativity Postulate: “…the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.” • Speed of Light Postulate: “… light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.” A. Einstein, “Zur Elektrodynamik bewegter Körper,” Annalen der Physik. 17:891, 1905.

10. #3. Some Consequences • Relativity of Simultaneity • Time order of events depends on your point of view! • Time Dilation • Moving clocks tick slower than stationary ones • Length Contraction • Meter sticks get shorter the faster they move

11. Both ships receive order simultaneously #3.1 Relativity of Simultaneity Radio signal travels at the speed of light in all directions. Eat Lunch NOW! FoR = Flag Ship

12. Relativity of Simultaneity Radio signal still travels at the speed of light in all directions. Eat Lunch NOW! FoR = Observer

13. Relativity of Simultaneity

14. Relativity of Simultaneity

15. Trail ship receives signal first!! Relativity of Simultaneity

16. Lead ship receives signal second!! Relativity of Simultaneity

17. How can simultaneous events not happen at the same time? • Both observers see light moving at same speed • Flag ship: signal travels same distance at same speed  simultaneous • Space Station: signal travels different distances at same rate  not simultaneous • Classically – light would not move at same speed • Space Station: light travels faster to lead ship, slower to rear ship so signal still simultaneous! • Two events are not causally connected • order of events is irrelevant! More importantly - they CANNOT be causally connected.

18. Proper time = time interval in IFoR where events are at same location Time interval between same events in moving IFoR Relative velocity between two IFoR #3.2 Time Dilation

19. Mathematics of Time L observer at rest relative to clock

20. Moving clock t = time for this clock seen by observer L x observer - clock moving at v relative to him

21. Moving clock Math Speed of light is the same in both FoR: - moving clock - stationary clock Pythagorean Theorem D = c t x = v t

22. Time slows? Really?! • Do all clocks slow at same rate? • Or is time dilation true for only “light clocks?” • What about “biological” clocks - your metabolism? • Does time slow or only clocks? • Do you perceive time slowing as you move? • Can you measure a change in rate of time? • How do you see other people’s clocks who are moving relative to you? • How do they see your clocks? • Who is right? • How can you resolve the argument?

23. How fast to see effect? • speed of sound = 360 m/s •   1.000 000 000 000 72 • shuttle in orbit  7700 m/s •   1.000 000 000 329 • 10% of speed of light = 3  107 m/s •   1.00504 • 50% of speed of light = 1.5  108 m/s •   1.155

24. Length as seen in moving IFoR Proper or rest length of an object #3.3 Length Contraction

25. Both ships receive order simultaneously and nothing changes New order: Speed Up! Burn engine for 1 s IFoR = Flag Ship

26. From another point of view IFoR = Observer

27. Repeat this order! Distance between ships decreases since they are not accelerating simultaneously! IFoR = Observer

28. Distances get shorter! Space ships get shorter! All distances get shorter – even those between atoms. IFoR = Observer

29. Two Points of View IFoR = Flag Ship IFoR = Observer

30. No Sideways Contraction Before firing Gun’s perspective Bullet’s perspective

31. #3.4 Invariant Interval • Time between two events is relative • Time dilates or gets longer with speed • Distance between two events is relative • Distances contract or get shorter with speed • Is anything absolute? YES!! Note: space (x,y,z) and time (t) must be combined.

32. Spacetime • “Distance” between two events • Measure in meters… • … or in seconds • c = unit conversion between space and time • 3.0  108 meters of space = 1 second of time Explains why c is such a fundamental constant. It links space to time.

33. Spacetime Diagram Limits for tangent to world line  =ct (m) world line for particle world line for light ray (45°) x (m)

34. y y x x #3.5 Relative Velocities v u = velocity from S u = velocity from S

35. E = mc2 is energy objects have without moving #3.6 E=mc2 (Finally!) • What is energy? • First encountered with W = F x = K using F=ma • K = 1/2 m v2 • Do the same calculation with relativistic quantities and get:

36. #4. Paradoxes • Twin Paradox • Which twin is younger if all motion is relative? Why can’t the twin on the spaceship say he remained at rest? • Can you fit a 20-m pole in a 5-m barn?

37. Twin Paradox • Twins on Earth • Bob stays on Earth • Alice leaves in spaceship and travels at close to the speed of light • turns around and returns home • Time for trip different for twins • Time slows for Alice (time dilation) and is younger than twin Bob when she returns • If all motion is relative, Bob also moves relative to Alice - why can’t he age slower?? • Is the situation symmetrical?