Understanding Special Relativity: Time Dilation, Length Contraction, and Relativistic Velocity

1. Relativity By: Jennifer Doran

2. What was Known in 1900 • Newton’s laws of motion • Maxwell’s laws of electromagnetism

3. Newton’s Laws Predicted that the speed of light should depend on the motion of the observer and the light source Maxwell’s Laws Predicted that light in a vacuum should travel at a constant speed regardless of the motion of the observer or source Contradiction Between Laws

4. Two Postulates of Special Relativity • The laws of physics are the same for all non-accelerating observers • The speed of light in a vacuum is constant for all observers, regardless of motion

5. Inconsistencies with Classical Mechanics • Newton’s Laws state that v+u=V • Einstein’s postulate says that the speed of light is independent of the motion of all observers and sources.

6. Classical Lorentz Transformation x = x' + u t' ; y = y' ; z = z' ; t = t' OR x' = x – u t ; y = y' ; z = z' ; t = t' Then v = v' + u And a = a' BUT this is only good for u<<v. To make the transforms relativistic, assume: x = Γ (x' + u t') x' = Γ (x – u t)

7. Finding Γ A light pulse starts at S at t = 0; S' at t' = 0 So in S, x = ct, and in S‘, x' = c t', by Einstein’s second postulate Then ct = Γ (c t' + u t') = Γ (c + u) t' And ct' = Γ (c - u) t Substitute for t', then ct = [Γ(c + u)Γ(c - u)t]/c Then Γ2 = c2 / (c2-u2) = 1/(1- u2/c2) Γ = 1/√(1- u2/c2)

8. Time Dilation • Δt = Γ•Δtp • Δtp is the proper time, the time between events which happen at the same place. • Since Γ is always greater than 1, all clocks run more slowly according to an observer in relative motion.

9. Time Dilation Example • An astronaut in a spaceship traveling away from the earth at u = 0.6c decides to take a nap. He tells NASA that he will call them back in 1 hour. How long does his nap last as measured on earth?

10. Time Dilation Example • Δtp= 1 hour • 1 – (u/v)2 = 1 – (0.6)2 = 0.64 • Therefore, Γ = √(1/0.64) = 1.25 • Δt = 1.25 hours

11. Length Contraction • L = (1/Γ)•Lp • Lp is the length of the object in the reference frame in which the object is at rest. • An object in motion at relativistic speeds will always appear shorter to a stationary observer than when the object is at rest.

12. Length Contraction

13. Length Contraction Example In the reference frame of a muon traveling at u = 0.999978c, what is the apparent thickness of the atmosphere? (To an observer on earth, the height of the atmosphere is 100 km.)

14. Length Contraction Example • L = (1/Γ)•Lp • Lp = 100 km • L = 100 km•√(1-(0.999978)2) • L = 0.66 km = 660 m

15. Relativistic Velocity Addition • When objects are moving at relativistic speeds, classical mechanics cannot be used. • v = (v' + u) / (1 + v'u/c2) • v = velocity addition • v' = velocity of object moving in the reference frame of u. • u = motion of object

16. Relativistic Velocity Addition Example • A spaceship moving away from Earth at a speed of 0.80c fires a missile parallel to its direction of motion. The missile moves at a speed of 0.60c relative to the ship. What is the speed of the missile as measured by an observer on Earth?

17. Relativistic Velocity Addition Example • Cannot use classical mechanics. • v is the velocity we are looking for. • u = 0.80c = the velocity of the spaceship • v' = 0.60c = the velocity of the missile in the reference frame of the spaceship • v = (v' + u) / (1 + v'u/c2) • v = (0.6c + 0.8c) / (1 + (0.6c)(0.8c)/c2) • v = 1.40c/1.48 = 0.95c

18. Relativistic Velocity Addition Example • A star cruiser is moving away from the planet Mars with a speed of 0.58c and fires a rocket back towards the planet at a speed of 0.69c as seen from the star cruiser. What is the speed of the rocket as measured by an observer on the planet?

19. Relativistic Velocity Addition Example • v is the velocity we are looking for. • u = 0.58c = the velocity of the spaceship • v' = -0.69c = the velocity of the rocket in the reference frame of the star cruiser • v = (v' + u) / (1 + v'u/c2) • v = (0.58c - 0.69c) / (1 + (0.58c)(-0.69c)/c2) • v = -0.11c/0.5998 = -0.18c • Compared to –0.11c

20. Twin Paradox • A clock in motion runs slower than one at rest, including biological clocks. • Apparent paradox with one twin staying on earth and other going on a relativistic journey. • Solution

22. Twin Paradox Example • Twin sisters Betty and Ann decide to test the relativity theory. Ann stays on earth. On January 1, 2000, Betty goes off to a nearby dwarf star that is 8 light years from Earth. Betty travels there and back at 0.8c.

23. Twin Paradox Example • What is Betty’s travel time to the star according to Ann? • t = d/v = 8 light years / 0.8c = 10 years • So Betty’s total travel time according to Ann is 20 years.

24. Twin Paradox Example • As soon as Betty reaches the star, she sends her sister an e-mail message saying, “Wish you were here!” via radio. When, from Ann’s perspective, does Ann receive the e-mail from Betty? • 2010 + 8 yrs = 2018

25. Twin Paradox Example • When Ann receives Betty’s e-mail message, what is the date on it? • Length contraction Γ = 1/√(1- (0.8c)2/c2) • So Γ = 0.6 • 8 light years • 0.6 = 4.8 light years • t = d/v = 4.8 light years/0.8c = 6 years • So the e-mail will have the date of January 2006.

26. Twin Paradox Example • How much younger is Betty than Ann when Betty returns from the star? • For Betty, her return date is 2012, but for Ann, she returns in 2020. • Ann is now 8 years older than Betty.

27. Twin Paradox Example

28. Newtonian Gravity • Quote from Newton • Two masses in Newton’s theory — inertial mass and gravitational mass • Newton saw no reason why these masses should be equal • BUT they are!!!

29. Einstein on Gravity “I was sitting in a chair in the patent office at Bern when all of a sudden a thought occurred to me: if a person falls freely he will not feel his own weight. This simple thought made a deep impression on me. It impelled me toward a theory of gravitation.” -Albert Einstein

30. Principle of Equivalence • There is no local experiment that can be done to distinguish between the effects of a uniform gravitational field in a non-accelerating inertial frame and the effects of a uniformly accelerating reference frame.

31. Principle of Equivalence • Free space view is the same as the free fall view

32. General Relativity • The presence of matter causes spacetime to warp or curve. • Gµv = 8G/c4•Tµv

33. Schwarzschild’s Solution • Found the first exact solution of one of Einstein’s equations • The metric (distance relation) is: • ds2 = -c2(1-2MG/c2r)dt2 + dr2/(1-2MG/c2r) + r2(dq2+sin2qdf2). • Describes space-time geometry around a spherical object of mass M • Basis of tests of general relativity

34. Testing the General Theory of Relativity • Deflection of Starlight • First tested during total solar eclipse in 1919

35. Slowing Down of Clocks by Gravity • Clock closer to the mass measures a shorter elapsed proper time than a clock that is further out. • Rc=2MG/c2, where Rc is the Schwarzschild radius.