Introduction to Relativity & Time Dilation The Principle of - PDF document

Introduction to Relativity & Time Dilation • The Principle of Newtonian Relativity • Galilean Transformations • The Michelson-Morley Experiment • Einstein’s Postulates of Relativity • Relativity of Simultaneity • Time Dilation • Homework 1

The Principle of Newtonian Relativity • The laws of mechanics must be the same in all inertial frames of reference. • An inertial frame is one in which Newton’s 1 st law is valid. • Any frame moving with constant velocity with re- spect to an inertial frame must also be an inertial frame. • This does not say that the measured values of physical quantities are the same for all inertial observers. • It says that the laws of mechanics, that relate these measurements to each other, are the same. 2

Two Inertial Reference Frames • The observer in the truck sees the ball move in a ver- tical path when thrown upward. • The stationary observer sees the path of the ball to be a parabola. • Their measurements differ, but the measurements sat- isfy the same laws. 3

Galilean Transformations • Consider an event that occurs at point P and is ob- served by two observers in different inertial reference frames S and S ′ , where S ′ is moving with a velocity v relative to S as shown below • The coordinates for the event as observed from the two reference frames are related by the equations known as the Galilean transformation of coordinates x ′ = x − vt y ′ = y z ′ = z t ′ = t 4

Galilean Addition of Velocities • Suppose a particle moves a distance dx in a time in- terval dt as measured by an observer in S • The corresponding distance dx ′ measured by an ob- server in S ′ is dx ′ = dx − vdt • Since dt = dt ′ , we have dx ′ dt ′ = dx dt − v or u ′ x = u x − v 5

Michelson-Morley Experiment • In the 19 th century, physicists believed light, like me- chanical waves, required a medium to propagate through and they proposed the existence of such a medium called the ether • The ether would define an absolute reference frame in which the speed of light is c • The Michelson-Morley experiment was designed to show the presence of the ether 6

Michelson-Morley Experiment (cont’d) • The ether theory claims that there should be a time difference for light traveling to mirrors M 1 and M 2 • No time difference was observed! 7

Einstein’s Postulates • The Relativity Postulate: The laws of physics are the same for observers in all inertial reference frames. – Galileo and Newton assumed this for mechanics. – Einstein extended the idea to include all the laws of physics. • The Speed of Light Postulate: The speed of light in a vacuum has the same value c in all directions and in all inertial reference frames. 8

�✁ ✂✄ Tests of the Speed of Light Postulate • Accelerated electron experiment: Bill Bertozzi (MIT) showed this in 1964 by independently measuring the speed and kinetic energy of accelerated electrons 6 Kinetic energy (MeV) Ultimate speed ✞✆✞✟ 4 2 ☎✆☎✝ 0 1 2 3 8 Speed (10 m/s) • π 0 → γγ decay experiment (CERN 1964) v = c γ v = 0.99975c π 0 v = c γ 9

Relativity of Simultaneity • Two lightning bolts strike the ends of a moving box- car. • The events appear to be simultaneous to the observer at O , who is standing on the ground midway between A and B . • The events do not appear to be simultaneous to the observer O ′ riding on the boxcar, who claims the front end of the car is struck before the rear. • A time measurement depends on the reference frame in which the measurement is made. 10

Time Dilation 1 • The observer at O ′ measures the time interval be- tween the two events to be ∆ t p = 2 d c • The two events occur at the same location in O ′ s ref- erence frame, and she needs only one clock at that location to measure the time interval, so we call this time interval the proper time. • The observer at O uses two synchronized clocks, one at each event, and measures the time interval to be � 2 + d 2 �� 1 ∆ t = 2 L c = 2 2 v ∆ t c 11

Time Dilation (cont’d) � 2 + �� 1 � 2 � 1 ∆ t = 2 2 v ∆ t 2 c ∆ t p c 1 4 c 2 ∆ t 2 = 1 4 v 2 ∆ t 2 + 1 4 c 2 ∆ t 2 p ∆ t 2 = c 2 ∆ t 2 c 2 − v 2 � � p c ∆ t p ∆ t = √ c 2 − v 2 ∆ t p ∆ t = � � 2 � v 1 − c • It is convenient to define the speed parameter as β = v c 1 and the Lorentz factor as γ = √ 1 − β 2 • Then the time dilation expression can be written as ∆ t = γ ∆ t p • Since we must have v < c , γ > 1 , and ∆ t > ∆ t p 12

Time Dilation (cont’d) • All clocks will run more slowly according to an ob- server in relative motion (this includes biological clocks). • Time dilation has been tested and confirmed on both the microscopic (lifetimes of subatomic particles) and macroscopic (flying high precision clocks in airplanes) levels. 13

Example The elementary particle known as the positive kaon (K + ) has, on average, a lifetime of 0.1237 µ s when stationary- that is, when the lifetime is measured in the rest frame of the kaon. If a positive kaon has a speed of 0.990c in the laboratory, how far can it travel in the lab during its lifetime? 14

Example Solution The elementary particle known as the positive kaon (K + ) has, on average, a lifetime of 0.1237 µ s when stationary- that is, when the lifetime is measured in the rest frame of the kaon. If a positive kaon has a speed of 0.990c in the laboratory, how far can it travel in the lab during its lifetime? ∆ t p ∆ t = � � 2 � v 1 − c ∆ t = 0 . 1237 × 10 − 6 s � 2 = 8 . 769 × 10 − 7 s � � 0 . 990 c 1 − c 3 . 00 × 10 8 m/s 8 . 769 × 10 − 7 s � � � � d = v ∆ t = (0 . 990) = 260 m 15

Homework Set 16 - Due Wed. Oct. 20 • Read Sections 9.1-9.4 • Answer Questions 9.2 & 9.4 • Do Problems 9.1, 9.2, 9.6, 9.9 & 9.13 16