Unit 8B : Special Relativity

1. Unit 8B: Special Relativity Motion through space is related to motion in time.

2. Space-Time Newton and others thought of space as an infinite expanse in which all things exist. Einstein theorized both space and time exist onlywithin the universe. There is no time or space “outside.” Einstein reasoned that space and time are two parts of one whole called space-time. special theory of relativity: time is affectedbymotion at constantvelocity; mass and energy are related

3. Space-Time You are moving through time at 24 hr/day. This is only half the story. Instead, think “moving through space-time.” • In motion, you travelthroughspace, but mostly still throughtime. • At rest, you travel only through time.

4. Space-Time If you could travel the speed of light… c= 3 x 108 m/s • Light travels only through space and is timeless. • A photon moving at light speed, takes no time to travel!

5. Space-Time If in motion at the maximum rate through space:c (time would standstill) If at rest, you are traveling at the maximum rate through time: 24 hr/day

6. Space-Time Whenever we move through space, we alter our rate of moving into the future (time). time dilation: stretching of time caused by motion in space Stationary Clock Moving Clock 0.87c (87% speed of light)

7. The First Postulate of Special Relativity Einstein reasoned all motion is relative and there is nooneabsoluteframe of reference. A spaceship cannot measure its speed alone, but only relative to other objects. Which is at rest? Which is moving?

8. The First Postulate of Special Relativity There is no physical experiment we can perform to determine our state of uniform motion. Pool on a ship: No need to compensate for the ship’s speed.

9. The First Postulate of Special Relativity • Many experiments can detect acceleratedmotion, but according to Einstein, none can detect the state of uniform motion. • 1st Postulate of S.R: • laws of physics are the same whether movingor at rest • all motion is relative, there is nooneabsoluteframe of reference

10. The Second Postulate of Special Relativity Einstein asked: “What would a light beam look like if you traveled along beside it?” The beam would be at rest to such an observer, but… Einstein was convinced this was impossible. If an observer could travel near the speed of light, he would still measure the light as moving away at 3 x 108 m/s, because…

11. The Second Postulate of Special Relativity 2nd Postulate of S.R.: speed of light is constantregardless of the speed of the flashlight or observer.

12. The Second Postulate of Special Relativity A flash of light is emitted from the station at 300,000 km/s (c) (3 x 108 m/s) (186,000 mi/s) The speed of a light flash emitted by either the spaceship or the space station is measured as c by observers on the ship or the space station. Everyone who measures the speed of light will get the same value, c.

13. The Second Postulate of Special Relativity • The fixed speed of light is what unifies space and time. • For any motion through space, there is a corresponding passage of time. • The ratio of space to time for light is the same for all who measure it.

14. Quick Quiz! • What Einstein discovered about space and time is that they… • are separate entities. • are parts of one whole. • follow an inverse-square law. • are special to space travelers.

15. Quick Quiz. • Einstein stated that the laws of physics are… • different depending on the situation. • common sense applied to microscopic and macroscopic things. • the same in all frames of reference. • the same in all uniformly moving frames of reference.

16. Quick Quiz. • Einstein’s 2nd postulate tells us that the speed of light… • depends on one’s frame of reference. • is a constant in all frames of reference. • changes depending on the time of day. • slows in a transparent medium.

17. Time Dilation Einstein proposed that time can be stretched depending on the motion between the observer and the events being observed. The stretching of time is time dilation.

18. Time Dilation • Consider a “light clock”: • empty tube with a mirror at each end • flash of light bounces between parallel mirrors • flash bounces indefinitely • “ticks off” equal intervals of time

19. Time Dilation • A moving ship contains a light clock. • An observer moving with the spaceship observes the light flash moving vertically. • An observer who is passed by the moving ship observes the flash moving along a diagonal path.

20. Time Dilation • Recall the 2nd Postulate of S.R. : • “speed of light measured by any observer as c.” • sincethe speed of light can notincrease, we must measure more time between bounces! more distance more time less distance less time = c = same c

21. Time Dilation • From outside, one tick of the clock takes longerthan inside the spaceship. • The spaceship’s clock has sloweddown. • BUT, for observers insidethe spaceship, it has not slowed.

22. Time Dilation Einstein showed the relation between the time t0 in the observer’s frame of reference and the relative time t measured in another frame of reference is: v= relative velocitybetween the observer and observed c= speed of light time at rest of observer time relative to observer

23. Time Dilation You fly at Mach 10 (3300 m/s) away from an observer on the ground, then return at Mach 10 after 5 min. What is the observed time difference (time dilation)? (5 min) 3300 m/s 2 3x108 m/s 1 – 0.0000000001 = 1 5 min (or 0 min dilated)

24. Time Dilation You fly at c/2 away from an observer on the ground, then return at c/2 after 5 min. What is the observed time difference (time dilation)? (5 min) 1.5x108 m/s 2 3x108 m/s 1 – (0.502) = 0.75 6.7 min (or 1.7 min dilated)

25. Time Dilation More distance covered (greater speed), then more time stretch.

26. Time Dilation • Timein the movingframe of reference slows as viewedfromourframe of reference. • The heartbeats of the spaceship occupants will have a slower rhythm. • All eventson the movingship will be observed by us as slower. • We say that time is stretched—it is dilated.

27. Time Dilation How do people on the ship view their own time? There’s no way to tell uniform motion from rest. They have no clue that we view their time as dilated from our frames of reference.

28. Time Dilation 1 The Twin Trip 2 3 4

29. Time Dilation If a twin travels at 50% cfor 1 yr, 1.15 yrs will have elapsed on Earth. If the twin travels at 87% c for a yr, 2 yrs will have elapsed on Earth. At 99.5% c, 10 Earth yrs would pass in 1 yr. At this speed, the traveling twin would age only 1 yr while the stay-at-home twin ages 10 yrs.

30. Time Dilation A clock reads 12 noon. An observer in space who later receives the light says, “Oh, it’s 12 noon on Earth now.” You in the trolley on earth and the distant observer see 12 noon at different times. If the trolley travels as fast as light, then it will keep up with the information that says “12 noon.” Traveling at the speed of light, then, tells the time is always 12 noon at the village square. Time at the village square is frozen!

31. Time Dilation If the trolley is not moving, you see the clock move into the future at the rate of 60 seconds per minute. If you move at the speed of light, you see seconds on the clock taking infinite time. These are two extremes. What’s in between? When you return, will the effects of going and coming cancel out? No! Time will be stretched. Your watch and the clock will disagree. This is time dilation.

32. Time Dilation think! Does time dilation mean that time really passes more slowly in moving systems or that it only seems to pass more slowly? Explain.

33. Time Dilation think! Does time dilation mean that time really passes more slowly in moving systems or that it only seems to pass more slowly? Explain. Answer: It is not merely an illusion resulting from motion. Time really does pass more slowly in a moving system (according to outside observers).

34. Time Dilation think! If you were moving in a spaceship at a high speed relative to Earth, would you notice a difference in your pulse rate?

35. Time Dilation think! If you were moving in a spaceship at a high speed relative to Earth, would you notice a difference in your pulse rate? Answer: No. There is a relativistic effect between you and Earth. You’d find their pulse rate slower than normal (and they would find your pulse rate slower than normal). Relativity effects are always attributed to “the other guy.”

36. Quick Quiz! • When we speak of time dilation, we mean that time… • compresses with speed. • stretches with speed. • is a constant at all speeds. • is related to space.

37. Quick Quiz. • If you travel at speeds close to the speed of light, then, compared with your friends who “stay at home,” you are… • older. • younger. • no younger nor no older. • longer.

38. Length Contraction For moving objects, space as well as time undergoes changes. time dilation: stretching of time observed outside the motion length contraction: observed shortening of objects moving at higher speeds (like nearspeed of light) The amount of contraction is related to the amount of time dilation. For normal speeds, the amount of contraction is much too small tobe measured.

39. Length Contraction At 87% c, a meter stick appears to be 0.5 m At 99.5% c, it appears to you to be 0.1 m As speed approaches the speed of light, the measured length of an object contracts closer and closer to zero.

40. Length Contraction From inside the spaceship, its length is 1 m. Observers from the ship see our meter sticks contracted. The effects of relativity are always attributed to “the other guy.”

41. Length Contraction The contraction of speeding objects is the contraction of spaceitself. Space contractsin only one direction, the direction of motion. Lengths along the direction perpendicular to this motion are the same.

42. Length Contraction Relativistic length contraction can be stated mathematically: v= velocity of object relative to observer c= speed of light Length relative to observer Length at rest What if v = 0 ? (at rest relative to observer)

43. Length Contraction At 87% cfor v, we find… L = 0.5L0 At 99.5% cfor v, we find… L =0.1L0

44. Length Contraction If an object could reach the speed c, its length would contract to zero. c is the universal speed limit that no object can reach. L = L0 L = 0.5L0 L = 0 L = 0.1L0

45. One of the most celebrated outcomes of special relativity is the discovery that mass and energy are one and the same thing— as described by E = mc2.

46. Mass and Momentum in Relativity Momentum equals mass times velocity: p = mv To Newton, infinite momentum would mean infinite speed. Einstein showed that a new definition of momentum is required: v= velocity of an object c= speed of light

47. Mass and Momentum in Relativity What happens as vapproaches c? The denominator approaches zero, and… the momentum approaches infinity! mv 0 1 – 1 Nothing that has mass can be pushed to the speed of light. (cis the universal speed limit) Newton’s definition of momentum is valid at low speeds (b/c 1 – (v/c)2 = 1). 1 – 0 = 1

48. Mass and Momentum in Relativity When a particle is pushed close to the speed of light, it acts as ifits mass were increasing, because its momentum increases more than its speed increases. rest mass: (m) is a constant property of an object no matter what speed it has rest energy: (E) according to E = mc2, a piece of matter has an “energy of being” Mass is a kind of potential energy.

49. The Correspondence Principle If a new theory is to be valid, it must account for the verified results of the old theory. correspondence principle: new theory and old theory must overlap and agreewhere the results of the old theory have been fully verified. Newton Einstein p = mv

50. The Correspondence Principle Relativity equations reduce to Newtonian values for small everyday speeds (v <<< c) …for very small v…