Mon. Jan. 5 – Physics Lecture #16Relativity – Basic Postulates

0) Overview of Quarter – Special Relativity, E&M, Unification

1) Basic Postulates

2) Relative velocities

3) Light Clocks

4) Comparing time between events – time dilation

5) Comparing length of object – length contraction

Office hours• Rachel: Wed noon – 1:00, (Lab 2 3rd Floor)• Krishna: Thu 2:00 – 3:00, (Lab 2 3rd Floor)

Tutor hours TBA

Learning Goals

Be able to state the basic, fundamental principles of relativity, and be able to explain how the various aspects of relativity all follow logically from these basic principles.

Use the velocity transformations to relate the velocities of objects or of reference frames.

Be able to relate time intervals in two different reference frames using the proper time relation if one of the observers is at both events.

Be able to relate length and distance measurements in two different reference frames using the length contraction relation if one of the observers is at rest with respect to the distance/length being measured.

1) Basic Postulates

History of measurements of speed of lightWikipedia “speed of light”

Fowler, UVA http://galileoandeinstein.physics.virginia.edu/Overview and Lecture Index Lecture 18 The Speed of Light

The Principle of Relativity:

the laws of physics are the same for observers in different inertial reference frames.

The invariance of the speed of light:

The speed of light in a vacuum c is measured to be 3.0 x 108 m/s by any observer in any inertial reference frame. (really 299792458 m/s exactly)

2) Relative velocities (Review Mazur Ch. 6 as needed)

You’re on a (very) long train moving at constant velocity of 5 m/s with respect to the ground. You throw a ball at 10 m/s with respect to the train.

What does a ground-based observer measure as the speed of the ball?

You’re on a (very) long train moving at constant velocity of 0.5c (1.5 x 108 m/s) to the right with respect to the ground. You shine a pulse of light to the left. The speed of light is 1.0c in your reference frame. What does a ground-based observer measure as the speed of the light pulse?

1. 0.5c (to the left)

2. 1.0c (to the left)

3. 1.5c (to the left)

4. 0.5c (to the right)

5. 1.0c (to the right)

6. 1.5c (to the right)

Little Engines that Could

A Red Train travels to the right at 80 mph, as measured by observers standing next to the train tracks. A Black Train travels to the left at 60 mph, away from the Red Train, as measured by those same observers.

According to observers on the Red Train, how fast is the Black Train moving?

Little Engines that Could, Really Fast

A Red Train travels to the right at 0.8c, as measured by observers standing next to the train tracks. A Black Train travels to the left at 0.6c, away from the Red Train, as measured by those same observers.

According to observers on the Red Train, how fast is the Black Train moving?

t = 0 s

Train rest frame

xball =

yball =

3) Light Clocks

L

mirror

Event 1: pulse from emitter

Pulse bounces off mirror

Event 2: pulse back to emitter

emitter

Let’s put this apparatus on a train.

Let’s have the train moving to the right at constant speed

L

Observers moving with light clock

Observers not moving with light clock

L

Distance traveled: Ld 2

tcd Ld 2tcd

Distance traveled:

Dt time between Event 1 and Event 2

Dt’ time between Event 1 and Event 2

4) Comparing time between events – time dilation

How did Einstein interpret the fact that d’ > d in this case? Note that d is the distance the light travels in the rest frame of the train (my perspective) between the two events, and d’ is the distance the light travels in the frame where the train is moving (your perspective) between the two events.

1. This isn’t the case. d’ must be the same as d.

2. If d’ > d, then c’ must be > c (i.e., the light pulse is moving faster from your perspective than from my perspective).

3. If d’ > d, then c’ must be < c (i.e., the light pulse is moving slower from your perspective than from my perspective).

4. If d’ > d, then Dt’ must be > Dt.

5. If d’ > d, then Dt’ must be < Dt.

A B

2ABtc

2

ABtc

2Dtc

2ABtv

2

ABtv

Mission to Alpha Centauri

NASA sends out an interstellar mission to the nearby star Alpha Centauri, which is 4 light years away as measured by observers on the Earth. The crew travels at a speed of 0.8c, relative to the Earth and Alpha Centauri. a) How long will it take for the crew to leave the Earth and arrive at Alpha Centauri, according to people on the Earth (and Alpha Centauri – we’ll assume that Earth and Alpha Centauri are effectively at rest with respect to each other)?

How does the duration of the trip measured by the crew Dtcrew compare to the duration Dtearth as measured by people back on earth?

1. Dtcrew = Dtearth

2. Dtcrew < Dtearth

3. Dtcrew > Dtearth

Mission to Alpha Centauri

NASA sends out an interstellar mission to the nearby star Alpha Centauri, which is 4 light years away as measured by observers on the Earth. The crew travels at a speed of 0.8c, relative to the Earth and Alpha Centauri.

Mission to Alpha Centauri

NASA sends out an interstellar mission to the nearby star Alpha Centauri, which is 4 light years away as measured by observers on the Earth. The crew travels at a speed of 0.8c, relative to the Earth and Alpha Centauri. b) How long will it take for the crew to leave the Earth and arrive at Alpha Centauri, according to the crew?

5) Comparing length – length contraction

A crew of astronauts is traveling to Alpha Centauri, which is 4 lt-yrs away as measured by the people on the Earth. They are traveling at a speed of 0.8c. We just found that the trip takes 5 years according to observers on the Earth, and takes 3 years according to the astronauts.

How is it possible that the ship could get to Alpha Centauri in less than 4 years, according to the astronauts?

1. From the crew’s perspective, their relative speed is 4/3c.

2. It only seems to the crew as though the trip took 3 years; it actually takes them 5 years.

3. From the crew’s perspective, the distance from the Earth to Alpha Centauri is less than 3 light years.

Rest Frame of Earth and Alpha Centauri

Rest Frame of Rocket Ship

Commuter Train

According to people sitting on a train, the train is 125 m long. The train is zipping along at 0.6c, according to workmen working on a nearby bridge (the bridge is measured to be 100 m long by the workmen.) a) How long is the train, according to the workmen?

b) How long is the bridge, according to the people on the train?