Chapter 28: Relativity Brent Royuk Phys-111 Concordia University Classical Mechanics Translational Rotational s = r x = vt = t vt = r v = vo + at = o + t at = r x = v ot + 1 2 at 2 θ = ω ot + 1 2 αt 2 v 2 = v o2 +2 ax ω 2 = ω o2 + 2αθ F τ = r F = rf sin 2 m I = m i r i F NET = F = m a τ NET = I α W = Fd cosθ W = K TOT = 1 2 mv 2 + 1 2 Iω 2 K = 1 2 mv 2 K rot = 1 2 Iω 2 P av = ΔE Δt p = m v F t = m v P = Fv L = I ω t = I P = τω Classical vs. Modern The more important fundamental laws and facts of physical science have all been discovered, and these are now so firmly established that the possibility of their ever being supplanted in consequence of new discoveries is exceedingly remote... Our future discoveries must be looked for in the sixth place of decimals. - Albert. A. Michelson, speech at the dedication of Ryerson Physics Lab, U. of Chicago 1894 3 1
Classical vs. Modern When I began my physical studies [in Munich in 1874] and sought advice from my venerable teacher Philipp von Jolly... he portrayed to me physics as a highly developed, almost fully matured science... Possibly in one or another nook there would perhaps be a dust particle or a small bubble to be examined and classified, but the system as a whole stood there fairly secured, and theoretical physics approached visibly that degree of perfection which, for example, geometry has had already for centuries. - from a 1924 lecture by Max Planck (Sci. Am, Feb 1996 p.10) 4 Classical vs. Modern There is nothing new to be discovered in physics now. All that remains is more and more precise measurement -Lord Kelvin, 1900 5 The Correspondence Principle Text 22178 and your message to 37607 or PollEv.com 6 2
Introduction What s relative about relativity? 7 Relativity Billy-Bob s Pickup Truck Galilean Relativity Inertial Rest Frames A place where Newton s Laws work. A lab on a pickup truck? A boxcar? The space shuttle? Earth s surface? What is light s reference frame? c = 3 x 10 8 m/s The search for the luminiferous ether. 8 Relativity 9 3
Relativity 10 Relativity 11 Michelson-Morley Experiment 12 4
Einstein s Two Postulates 1. The Principle of Relativity: The laws of physics are the same in all inertial frames of reference. Galilean relativity There s no way you can tell you re moving. 2. The Invariance of c: The speed of light is the same in all IRFs, independent of relative motion. This is the surprising one. Justification? Maxwell and experiment. Herman Bondi: The irrelevance of motion and the uniqueness of light. Special vs. General Any objections? 13 The Relativity of Simultaneity Gedanken 14 Time Dilation The Light Clock Derivation, p. 1001-02 Notation: β = v/c Result: Δt =γ Δt o γ = 1 1 v 2 c 2 ( ) 1 2 = 1 β 2 15 5
Time Dilation Calculating Gamma 1 γ = 1 v 2 c 2 = ( 1 β ) 2 1 2 16 Time Dilation Why dilation? Is this a change in measured time or actual time? So which observer is right? 17 Time Dilation Example: Given β = 0.8, Sammy the Spaceman s watch tics off 10 s. What do our watches say during that interval? Proper time = the moving clock Events and spacetime Four Possible Permutations for Time dilation: Sammy s says 10, what do ours say? Ours say 10, what does Sammy s say? Sammy sees ours say 10, what does his say? During ten of Sammy s, how much time does he see pass on ours? Remember the first postulate? 18 6
Time Dilation Example 28.1: Suppose a cosmic ray colliding with a nucleus in the Earth s upper atmosphere produces a muon that has a velocity v = 0.95 c. The muon then travels at constant velocity and lives 1.52 µs as measured in the muon s frame of reference. (You can imagine this as the muon s internal clock.) How long does the muon live as measured by an Earth-bound observer? 19 Time Dilation Example: Hafele and Keating, 1972: atomic clocks on commercial airplanes Time dilation was verified within 5-10% Observed time differences: east: -40 ns, west: +275 ns. (compared to ground) When atomic clocks are transported, they get out of sync. GPS satellites have to be synchronized with time dilation effects accounted for NOVA Episode #2612: On a plane trip to London, a clock gained 40 ns. Joe Hafele: Suppose you were to live for 100 years and you would spend your entire life on one of these aircraft, flying around the world, you could expect to be younger than people who did not do that by about one ten-thousandth of a second. 20 Cultural Manifestations 21 7
Cultural Manifestations 22 Cultural Manifestations 23 Cultural Manifestations 24 8
Length Contraction Measured lengths shrink by the gamma factor for different reference frames. This occurs only along the direction of relative motion. This is not just something that happens to fast meter sticks. Length is really relative to the observer. Equation: L = L o /γ Example: At what speed would a flying meter stick appear to be half a meter long? 25 Length Contraction http://www.glenbrook.k12.il.us/gbssci/phys/mmedia/specrel/lc.html 26 Visual Distortion 27 9
Length Contraction Visual Appearance: Antony Searle, http://www.anu.edu.au/physics/searle/ 28 Length Contraction Visual Appearance: Antony Searle, http://www.anu.edu.au/physics/searle/ 29 Length Contraction Visual Appearance: Antony Searle, http://www.anu.edu.au/physics/searle/ 30 10
Velocity Addition Classically, u = v + u. Relativistically, we get: u = v + u' 1 + vu' c 2 33 Relativistic Time Travel How long would it take to go 100 ly at v = 0.9 c? The Twin Paradox Bobby and Ricky are twins. When they are 20 years old, Bobby leaves on a space flight to a star that is 20 light years away, traveling very close to the speed of light. At this extreme speed, the gamma factor for time dilation is equal to 20. What happens? 34 The Pole in a Barn Paradox 36 11