http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 1

Einstein’s theory of special relativity

•  Homework 1s due at 1:00pm on Friday in the wood cabinet just inside the physics help room (G2B90)

•  Last year’s Nobel Prize winner David Wineland (CU and NIST) is giving the colloquium at 4pm in G1B20. As last week there are cookies at 3:45pm.

Announcements:

Today we will derive time dilation (and maybe length contraction)

Albert Einstein (1879—1955):

http://www.colorado.edu/physics/phys2170/

For Colloquium:Schrödinger's Cat A cat is placed into a steel chamber, along with a device containing a vial of a radioactive substance. If a single decay occurs during the test period, a relay mechanism will break the vial and kill the cat. The observer cannot know whether or not an atom of the substance has decayed, and consequently, cannot know whether the vial has been broken, the released, and the cat killed. Since we cannot know, according to quantum mechanics, the cat is both dead and alive, in what is called a superposition of states. Only when we break open the box and learn the condition of the cat is the superposition is lost, and the cat becomes one or the other (dead or alive). This is called quantum indeterminacy or the observer's paradox: the observation or measurement itself affects an outcome. (There is no single outcome unless it is observed.)

Physics 2170 – Fall 2013 2

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What is the plot?

11/04/10 The Large Hadron Collider

3

Time (days)

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 4

Time Dilation Today: •  Clocks run at different

rates in different reference frames

•  By how much? •  Proper time •  We are going to ditch

trains and move on to faster moving objects. Examples are often space ships where we can have large effects.

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 5

A light flash starts a tick and when the light reflects off the mirror and back down to a detector it fires off another light flash. So a light clock “ticks” each time light makes a complete circuit: up and down.

For Lucy, how long does each tick take?

ΔtL = 2d/c

This is the time between two events: Event 1: The light flashes Event 2: The reflected light reaches the detector

Use “Light Clock” on a spaceship

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 6

Ricky is on Earth watching Lucy’s spaceship fly by at a speed v.

Lucy’s spaceship flies by Earth at speed v

Ricky has observers set up at the location of the two events. Using their synchronized clocks he finds that a time of ΔtR passes between the light flash and when the reflected light reaches the detector.

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 7

According to Ricky, how far does the light beam travel between the two events (light flash and reflected light hitting the detector)?

Clicker question 1 Set frequency to AD

A. B. C. D. E. None of A-D

Pythagorean theorem (X2 for two triangles)

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 8

According to Ricky, how far does the light beam travel between the two events (light flash and reflected light hitting the detector)?

Clicker question 1 Set frequency to AD

A. B. C. D. E. None of A-D

Pythagorean theorem (X2 for two triangles)

Note, this length is also equal to c·ΔtR

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 9

Algebra to give us the time dilation formula

Squaring both sides gives

Collecting terms gives

Dividing both sides by gives

Dividing by and taking the square root gives

Note the time measured by Lucy is

Therefore, the relation between the time observed by Ricky (on Earth) and Lucy (in the spaceship) is:

Ricky sees light travel a distance of

The distance traveled by light can also be written:

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 10

The relationship between the time between two events in Ricky’s frame (on Earth) and the time between the same two events in Lucy’s frame (on spaceship) is:

Time Dilation

Define and so then

Clicker question 2

Q. According to Ricky, the time between the two events (light flash and reflected light reaching detector) is

A. greater than what is observed by Lucy B. less than what is observed by Lucy C. equal to what is observed by Lucy

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 11

The relationship between the time between two events in Ricky’s frame (on Earth) and the time between the same two events in Lucy’s frame (on spaceship) is:

Time Dilation

Define and so then

and therefore

Clicker question 2

Q. According to Ricky, the time between the two events (light flash and reflected light reaching detector) is

A. greater than what is observed by Lucy B. less than what is observed by Lucy C. equal to what is observed by Lucy

So . They are equal only when v = 0.

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 12

Compare the two frames

There is something special about the two events in Lucy’s frame!

Both events occur at the same location in Lucy’s frame.

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 13

Any observer moving with respect to this clock sees it run slowly (i.e., time intervals are longer). This is time dilation:

If two events occur at the same location, then the time between them can be measured by a single observer with a single clock (the “Lucy time” in our example). The time measured between these types of events is called the proper time,

Proper time and time dilation

Example: any given clock never moves with respect to itself. It keeps proper time in its own frame.

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 14

Q. Carol and Vicki are identical twins. While Vicki stays on Earth, Carol departs for the star Sirius, 8 light-years away, traveling at a speed v = 0.8c (Note γ = 5/3). According to observers in Vicki’s frame, how long does the trip take?

A. 6 years B. 8 years C. 10 years D. 16.67 years

v

Vicki Carol

Set frequency to AD Clicker question 3

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 15

Q. Carol and Vicki are identical twins. While Vicki stays on Earth, Carol departs for the star Sirius, 8 light-years away, traveling at a speed v = 0.8c (Note γ = 5/3). According to observers in Vicki’s frame, how long does the trip take?

A. 6 years B. 8 years C. 10 years D. 16.67 years

v

Vicki Carol

Set frequency to AD Clicker question 3

The distance is 8 light-years and the speed is 0.8 c so we get

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 16

Q. Vicki stays on Earth and Carol departs for the star Sirius, 8 light-years away, traveling at a speed v = 0.8 c (γ = 5/3). According to Carol, how long does the trip take?

A. 6 years B. 8 years C. 10 years D. 16.67 years

v

Vicki Carol

Set frequency to AD Clicker question 4

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 17

Solving for the proper time gives

Q. Vicki stays on Earth and Carol departs for the star Sirius, 8 light-years away, traveling at a speed v = 0.8 c (γ = 5/3). According to Carol, how long does the trip take?

A. 6 years B. 8 years C. 10 years D. 16.67 years

v

Vicki Carol

Set frequency to AD Clicker question 4

Carol’s clock is the only one present at both events (leaving Earth and arriving at Sirius). Therefore, she is measuring the proper time! This tells us which time is which in the time dilation formula.

Follow the proper time!

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 18

We found that only 6 years pass in Carol’s reference frame as she travels 8 light-years to Sirius at 0.8 c. We say that every inertial frame is valid so won’t she observe the space ship going faster than the speed of light (8 light years in 6 years!)? This discrepancy is resolved by length contraction – moving objects are shorter (in the direction of motion)

Length contraction

v

Vicki Carol

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 19

How come we never notice the effect of time dilation?

Time Dilation

How much time will your watch lose on a non-stop flight from Los Angeles to Sydney (11 hours at 300 m/s)?

Not usually noticeable but in fact measured in 1971 by carrying 4 atomic clocks onboard a jet airplane around the world + synchronized to a reference clock. Found 275 +/- 7 ns discrepancy with reference clock.

If our watch measures 11 hours, the time recorded on the ground will be longer: = 11 hours + 20 ns

Time dilation is

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Example – Muon Lifetime •  Muon has a mean lifetime of 2.2µs when at rest. •  What is the mean lifetime if it is traveling at speed 0.6c = 1.8 x 108 m/s ?

Most muons passing through us have a much higher speed and thus live much longer.

Physics 2170 – Fall 2013 20

€

Δt =Δt0

1− v2

c 2

= 2.8 ×10−6 sAnswer is

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 21

We found that only 6 years pass in Carol’s reference frame as she travels 8 light-years to Sirius at 0.8 c. We say that every inertial frame is valid so won’t she observe the space ship going faster than the speed of light (8 light years in 6 years!)? This discrepancy is resolved by length contraction – moving objects are shorter (in the direction of motion)

Length contraction

v=0.8c

Vicki Carol

8 light-years

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 22

Carol sees a star approaching at 0.80c which gives

and measures the length between Earth and star as

So in Carol’s frame the ship covers 4.8 light years in 6 years which works out to a velocity of 4.8/6 = 0.8c, same as the velocity seen on earth by Vicki.

Length contraction

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 23

Length of an object

... -3 -2 -1 0 1 2 3 ...

This stick is 3m long. I measure both ends at the same time in my frame of reference.

Or not. It doesn’t matter, because the stick isn’t going anywhere.

But as we know, “at the same time” is relative – it depends on how you’re moving.

This length, measured in the stick’s rest frame, is its proper length.

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 24

Length of an object

... -3 -2 -1 0 1 2 3 ...

Another observer comes whizzing by at speed v. This observer measures the length of the stick, and keeps track of the time.

S

0

v S′

Event 1 – Origin of S′ passes left end of stick.

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 25

Length of an object

... -3 -2 -1 0 1 2 3 ...

S

0

v

S′

Event 1 – Origin of S′ passes left end of stick. Event 2 – Origin of S′ passes right end of stick.

How many observers are required to measure the time between these two events in reference frame S and S′?

http://www.colorado.edu/physics/phys2170/ Physics 2170 – Fall 2013 26

Length of an object

... -3 -2 -1 0 1 2 3 ...

S

0

v

S′

Event 1 – Origin of S′ passes left end of stick. Event 2 – Origin of S′ passes right end of stick.

How many observers are required to measure the time between these two events in reference frame S and S′?

For S: 2 observers at 0 and 3. For S′: just one at the origin