Modern Topics in Physics:Special Relativity

2012 Fall Semester

Wade Naylor

Images from “The Cosmic Perspective,” Pearson copyright 2010

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What are the major ideas of special relativity?

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• Special Relativity (1905)– Usual notions of space and time must be

revised for speeds approaching light speed (c).

– E = mc2

• General Relativity (1915) [next time]– Expands the ideas of special theory to

include a surprising new view of gravity

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Einstein’s Theories of Relativity

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• No material object can travel faster than light.• If you observe something moving near light speed:

– Its time slows down.– Its length contracts in direction of

motion.– Its mass increases.

• Whether or not two events are simultaneous depends on your perspective.

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Key Ideas of Special Relativity

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What is “relative” about relativity?

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• Motion is not absolute—we must measure speed of one object relative to another.

• Example: A plane moving at 1670 km/hr from east to west would appear from space to be standing still.

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Relativity of Motion

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1. The laws of nature are the same for everyone.

2. The speed of light is the same for everyone.

All of relativity follows from these two ideas!

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Absolutes of Relativity

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• As children, we revised our ideas of “up” and “down” when we learned that Earth is round.

• Relativity forces us to revise how we think of “space” and “time.”

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Making Sense of Relativity

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How did Einstein think about motion?

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• Motion can be defined with respect to a particular frame of reference.

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Reference Frames

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• Einstein claimed that light should move at exactly c in all reference frames (now experimentally verified). 11

Absoluteness of Light Speed

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What’s surprising about the absoluteness of the speed of light?

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• Einstein explored the consequences of the absoluteness of light speed using “thought experiments.”

• The consequences will be easiest for us to visualize with thought experiments involving spaceships in freely floating reference frames (no gravity or acceleration).

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Thought Experiments

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Relativity of Motion at Low Speeds

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Relativity of Motion at Low Speeds

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Relativity of Motion at High Speeds

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c + 0.9c = c !?!

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Light Speed is Absolute

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Why can’t we reach the speed of light?

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• Suppose you tried to catch up to your own headlight beams.

• You’d always see them moving away at speed c.

• Anyone else would also see the light moving ahead of you.

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Trying to Catch Up to Light

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• Is there a loophole? • What if you’re somehow moving away from a

distant planet faster than the speed of light? • In that case, you have no way of detecting that the

planet is there.• Although there are some phenomena that move

faster than light, no information can be communicated faster than the speed of light.

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Special Topic: What if Light Can’t Catch You?

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Insert TCP 6e Figure S2.11 unannotated

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How does relativity affect our view of time and space?

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• Thinking about the motion of a ball on a train will prepare us for the next thought experiment.

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Path of Ball in a Stationary Train

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• Someone outside the train would see the ball travel a longer path in one up-down cycle.

• The faster the train is moving, the longer that path would be.

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Path of Ball in a Moving Train

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• We can perform a thought experiment with a light beam replacing the ball.

• The light beam, moving at c, travels a longer path in a moving object.

• Time must be passing more slowly there.

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Time Dilation

t02= t2 � v2

c2

c2t02+ v2t2 = c2t2

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The Time Dilation Formula*

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• Time will appear to pass more slowly in a moving object by an amount depending on its speed.

• Time almost halts for objects nearing the speed of light.

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The Time Dilation Formula*

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• In your reference frame, red and green lights on other spaceship appears to flash simultaneously.

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Simultaneous Events?

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• But someone on the other spaceship sees the green light flash first—simultaneity is relative!

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Simultaneous Events?

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• Similar thought experiments tell us that an object’s length becomes shorter in its direction of motion.

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Length Contraction

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• A force applied to a rapidly moving object produces less acceleration than if the object were motionless.

• This effect can be attributed to a mass increase in the moving object.

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Mass Increase

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Velocity of first ship in your frame = v1

Velocity of second ship in frame of 1st = v2

Velocity of second ship in your frame:

v1+ v2

1+v1

c× v2

c

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Velocity Addition*

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Time dilation : tʹ′ = t 1−v 2

c 2

Length contraction : lʹ′ = l 1−v 2

c 2

Mass increase : mʹ′ =m

1−v 2

2c32

Formulas of Special Relativity*

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kinetic energyMass-energy of object at rest

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Deriving E = mc2 *

€

m =m0

1− v 2

c 2

#

$ %

&

' (

≈ m0 1+12v 2

c 2

#

$ %

&

' ( for small v

Total energy = mc 2 ≈ m0c2 +

12m0v

2

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Do the effects predicted by relativity really occur?

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• First evidence for absoluteness of speed of light came from the Michelson-Morley experiment performed in 1887.

• Time dilation happens routinely to subatomic particles that approach the speed of light in accelerators.

• Time dilation has also been verified through precision measurements in airplanes moving at much slower speeds.

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Tests of Relativity

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• Prediction that E = mc2 is verified daily in nuclear reactors and in the core of the Sun.

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Tests of Relativity

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• If speed of light were not absolute, binary stars would not look like two distinct points of light.

• You can verify relativity by simply looking through a telescope at a binary star system.

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Test Relativity for Yourself

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• If speed of light were not absolute, you would see the car coming toward you reach the collision point before the car it struck.

• There is no paradox if light speed is same for everyone.

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A Paradox of Non-relativistic Thinking

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How can we make sense of relativity?

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• According to you, time slows down in a moving spaceship.

• According to someone on that spaceship, your time slows down.

• Who is right?• You both are, because time is not absolute

but depends on your perspective.

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Making Sense of Relativity

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• As children, we learned that “up” and “down” are relative.

• Relativity tells us that “time” and “space” are relative.

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Toward a New Common Sense

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How does relativity offer us a ticket to the stars?

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• The distance to Vega is about 25 light-years.

• But if you could travel to Vega at 0.999c, the round trip would seem to take only 2 years!

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A Journey to Vega

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• At that speed, the distance to Vega contracts to only 1 light-year in your reference frame.

• Going even faster would make the trip seem even shorter!

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A Journey to Vega

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• However, your twin on Earth would have aged 50 years while you aged only 2 years.

• Time and space are relative!

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A Journey to Vega

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• How can we make sense of relativity?– We need abandon our old notions of

space and time as absolute and adopt new a new common sense in which time and space depend on your perspective.

• How does special relativity offer us a ticket to the stars?– For someone moving near light speed,

distances appear to become shorter because of length contraction.

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Conclusion...

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Homework: A paradox?

• Terence stays on Earth and Stella goes to space say for 25 + 25 years at about 0.99c.• Relative to Terence who waits 50 years,

Stella is moving away from him and her clock slows down (time dilation) and appears to take 1+1 years!

• However, relative to Stella, Terence moves away at 0.99c and while she ages 25+25 years he ages only 1+1 years!

• Who is right? Think about it...

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