Physics 55Monday, November 28, 2005

1. Introduction to Special Relativity2. Simultaneity is not absolute3. Time dilation, space contraction, relativistic momentum4. Introduction to General Relativity5. Einstein’s equivalence principle

Events

1. TUNL tour Tuesday, Nov. 29, 5:15-6 pm.

2. Observation sessions this Thursday and Friday nights from 10pm-midnight.

3. Last quiz this Friday, Dec 2.

4. Course evaluations on Monday, Dec 5.

5. All extra credit projects due this Monday, Dec. 5.

6. Last class is Friday, Dec. 9.

7. Review sessions during exam week.

8. “I want answers!” Discussion Board on Blackboard.

Deep Puzzles Associated With LightLight is an electromagnetic wave that travels at 300,000,000 meters in a second (denoted by letter c after the Latin word celeritas).

But what is “waving” as light moves through space?

What is the speed of light measured with respect to?

Many Waves Travel at Speeds Independent of Source

But: speed with which waves strike an object like the rowboat here depends on the velocity of the rowboat. If you are moving toward the source of the waves with speed v and the waves have speed vw, you see the waves arriving with speed v+vw. If you move away from the source, you see waves with speed v-vw. (For these reasons, a supersonic jet can not hear the sound of its engines!) But not so for light!

Special Relativity (1905): Postulates

Complements rather than supplants Newton’s ideas, only important when speeds become close to speed of light.

Postulates:1. Laws of nature are the same in all inertial frames of

reference. (It is impossible to determine your speed just by making measurements in a lab.)

Note: an inertial frame of reference is a laboratory for which Newton’s first law of motion is observed to hold, i.e., an object for which the total force is zero moves in a constant direction with constant speed.

2. The speed of light is same for all observers.

Special Relativity (1905): Deductions

Deductions:• Time and space are not absolute.• Simultaneity is not absolute.• Time slows down the faster an object moves!• Lengths become shorter the faster an object moves!• Speeds add in a funny way: .6c + .6c is .88c, not 1.2c.• Mass increases the faster an object moves!• E=mc2

2

( v) becomes v

1 (v / )

d m d mF F

dt dt c

Relative Motion at Low Relative Speeds

Relative Motion For Light

Measured speed of light is same for you and Jackie!

Question: how does someone measure the speed of light?

The Speed of Light c Is the Maximum Possible Speed and Is the Same for All Observers

Note: Light does change if the source is moving, e.g., the wavelength and frequency f of light change (this is the Doppler shift), but the observed speed of light is independent of motion.

Experimental Confirmation thatSpeed of Light Must Be Absolute

For visual binary star system with one star moving toward and away from Earth , we would see two arcs of light instead of two points of light if speed of light depended on motion of source, contrary to observation.

Albireo

Implication of Speed of Light as Maximum Speed:Limits to Computing, Need for Parallel Computing

In one nanosecond (one billionth of a second), light travels about 0.3 m (1 foot) so all parts of computer must be within 1 foot radius to avoid waiting for information to arrive. So maximum speed of light forces computers to become tiny in order to become fast.

But the it is hard to remove heat from many circuit elements so close together, one reason why processors become hotter and hotter over time (you can cook an egg with 3 GHz chip it is so hot).

Modern integrated circuit with 200,000,000 elements

Making computers (here a teraflop IBM computer) massively parallel is one of the few ways to solve the speed and heat issues associated with the speed of light.

Implication of Maximum Speed is c: Travel to the Stars is Hard

Fastest current man-made object: Voyager 1, launched in 1970s, now moving at 3.6 AU/year (~40,000 mph) and nearest star is 250,000 AU away (4.4 light-years) so time to get to Alpha Centauri would be about 70,000 years. Voyager is presently about 100 AU from the Sun.

Simultaneity Is Not Absolute

Insight: Observer at point O has no knowledge of lightning strikes at points A and B until light (or sound or some effect) propagates through space to point O. The blue arcs indicate the fronts of light waves spreading out from points A and B.

Simultaneity is Not Absolute: IIEinstein’s Example Based on Two Trains

Observer O says lightning struck simultaneously at points A2 and B2, but observer O1 disagrees, says that lightning struck at B1 then struck at A1. Here you can see that the waves from B1 have already passed by O1, but the waves from A1 have yet to catch up to O1 because O1 is moving away from A1.

Time Dilation Thought Experiment

You see Jackie’s light clock as taking longer to tick specifically because c is independent of v: light has to follow a longer path but the speed of light is always the same.

Einstein’s Time Dilation Formula

2

moving frame rest frame

v1t t

c

v=0.900c gives 2.3 fold dilationv=0.990c gives 7.1 fold dilationv=0.999c gives 22 fold dilation

V=0.9999995 gives 1000 fold dilate

Implication of Time Dilation: Travel to Stars Makes You Younger Compared to Someone On Earth

With rocket moving at 0.999c, round-trip time to Vega about 25 light-years away would be 50 years for someone on Earth. But person on rocket sees distance to Vega contracted to 25(1/22) ~ 1 ly so time to go to Vega and back would be about 2 years.

If the twins were 30 years old at launch time, one would be 30 + 25 + 25 = 80 years old, the other 30 + 1 + 1 = 32 years old when reunited!

Recommended Novel

Implication of Time Dilation: Muon Showers Require Underground Experiments

Average duration of muon is two microseconds (s). At speed of light and if time dilation didn’t occur, muon would cover average distance of 600 m before disintegrating into an electron and two neutrinos.

But average speed of created muons is 0.999c so muons live about 20 times longer, which means they travel distance of about 20 x 600 = 12 km, and now can reach surface of Earth.

Data: 570 muons/hour observed on mountain top of 6000 ft (1800 m) would see about 35/hour at sea level which is 6 ms away. But instead about 400 muons/hour observed at sea level in accord with relativity.

Relativistic Length Contraction

2

rest c

v1'

LL

v/c=0.8 gives 0.6 contractionv/c=0.995 gives ~0.1 contraction

PRS Question: How Does the Angle Change?

A person in a rocket is lying on a bed that is tilted up from the horizontal by an angle when the rocket is at rest. If the rocket were now to zoom by you at close to the speed of light, you would measure the angle of the bed to be 1. larger than 2. smaller than 3. unchanged

Relativity Paradox: Does the Train Fit Inside the Tunnel or Not?

V=0.8c

Sqrt[1-(v/c)2] = 0.6

Neat Relativistic Optical Effectshttp://www.anu.edu.au/Physics/Searle/

Implications of Relativity: Design of 21st Century Research Devices like Loch

Large Hadron Collider (LHC)27 km (17 miles) long ring on Swiss-French border.

Relativistic Addition of Velocities

Jackie sees Bob coming to her with speed 0.988c, not 1.7c!

How fast is Bob approaching Jackie if Bob approaches you at 0.8c, Jackie approaches you at 0.9c?

AB BCAC

AB BC2

2

v vv

v v1

.9 .8 =

.9 .81

.988

cc c

c cc

c

Relativistic Addition Formula

Explore some properties: what are some questions we can ask of this formula to see if it makes sense?

2BCAB

BCABAC vv

1

vvv

c

PRS Question

When you are traveling in a rocket ship at speed v=0.5c past the Earth, you measure Sue traveling in a rocket ship past you in the same direction at speed v=0.7c, and Sue measures Bob traveling past her in a rocket in the same direction at speed v=0.9c.

What speeds v1 and v2 should you use in the addition formula to determine what you measure for Bob’s speed?

1. v1=0.5c, v2=1.6c 2. v1=0.889c, v2=0.9c 3. v1=0.7c, v2=0.9c 4. v1=0.2c, v2=0.4c 5. Don’t know what to do…

You Sue Bob

Variations of the Question

When you are traveling in a rocket ship at speed v=0.5c past the Earth, you measure Sue traveling in a rocket ship past you in the same direction at speed v=0.7c, and Sue measures Bob traveling past her in a rocket in the same direction at speed v=0.9c.

What speeds v1 and v2 should you use in the addition formula to determine Earth’s speed as measured by Sue?

What is Bob’s speed with respect to Earth?

You Sue Bob