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 inertialframes of reference.

• An inertial frame is one in which Newton’s 1st law isvalid.

• 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 physicalquantities are the same for all inertial observers.

• It says that the laws of mechanics, that relate thesemeasurements 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 bea 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 referenceframes S and S ′, where S ′ is moving with a velocityv relative to S as shown below

• The coordinates for the event as observed from thetwo reference frames are related by the equations knownas 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

oru′

x = ux − v

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Michelson-Morley Experiment

• In the 19th century, physicists believed light, like me-chanical waves, required a medium to propagate throughand they proposed the existence of such a mediumcalled the ether

• The ether would define an absolute reference framein which the speed of light is c

• The Michelson-Morley experiment was designed toshow the presence of the ether

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Michelson-Morley Experiment (cont’d)

• The ether theory claims that there should be a timedifference for light traveling to mirrors M1 and M2

• No time difference was observed!

7

Einstein’s Postulates

• The Relativity Postulate: The laws of physics are thesame 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 avacuum has the same value c in all directions and inall inertial reference frames.

8

Tests of the Speed of Light Postulate

• Accelerated electron experiment: Bill Bertozzi (MIT)showed this in 1964 by independently measuring thespeed and kinetic energy of accelerated electrons

Speed (10 m/s)8

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1 2 3

2

4

6

0

Kin

etic

ene

rgy

(MeV

)

Ulti

mat

e sp

eed

• π0 → γγ decay experiment (CERN 1964)

π0v = 0.99975c

v = cγ

v = cγ

9

Relativity of Simultaneity

• Two lightning bolts strike the ends of a moving box-car.

• The events appear to be simultaneous to the observerat O, who is standing on the ground midway betweenA and B.

• The events do not appear to be simultaneous to theobserver O′ riding on the boxcar, who claims the frontend of the car is struck before the rear.

• A time measurement depends on the reference framein which the measurement is made.

10

Time Dilation 1

• The observer at O′ measures the time interval be-tween the two events to be

∆tp =2d

c

• The two events occur at the same location in O′s ref-erence frame, and she needs only one clock at thatlocation to measure the time interval, so we call thistime interval the proper time.

• The observer at O uses two synchronized clocks, oneat each event, and measures the time interval to be

∆t =2L

c=

2√

(

1

2v∆t

)2+ d2

c

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Time Dilation (cont’d)

∆t =2

√

(

1

2v∆t

)2+

(

1

2c∆tp

)2

c1

4c2∆t2 =

1

4v2∆t2 +

1

4c2∆t2p

(

c2 − v2)

∆t2 = c2∆t2p

∆t =c∆tp√c2 − v2

∆t =∆tp

√

1 −(

vc

)2

• It is convenient to define the speed parameter as β = vc

and the Lorentz factor as γ = 1√1−β2

• Then the time dilation expression can be written as

∆t = γ∆tp

• Since we must have v < c, γ > 1, and ∆t > ∆tp

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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 boththe microscopic (lifetimes of subatomic particles) andmacroscopic (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 µs when stationary-that is, when the lifetime is measured in the rest frameof the kaon. If a positive kaon has a speed of 0.990c inthe laboratory, how far can it travel in the lab during itslifetime?

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Example Solution

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

∆t =∆tp

√

1 −(

vc

)2

∆t =0.1237 × 10−6s

√

1 −(

0.990cc

)2= 8.769 × 10−7s

d = v∆t = (0.990)(

3.00 × 108m/s) (

8.769 × 10−7s)

= 260 m

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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

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