Announcements• Homework: Supplemental Problems

• 2nd Project is due at the final exam which is Tuesday May 5 at 1:30 – 3:30pm. A list of potential projects is posted.

• Final Exam will be another hour exam covering the material we covered last week and this week. It will be the second thing during the final exam period (project presentation first)

The Need for Relativity

0 0

1c

First published in a four part paper by James Clerk Maxwell between 1861 and 1862. Maxwell showed that light was an electromagnetic wave whose propagation speed was equal to the inverse square root of the product of the two fundamental constants in his equations.

The Need for Relativity II

Performed in 1887 by Albert Michelson and Edward Morley at Case Western Reserve University to detect the motion of Earth through the luminiferous aether

Given that light was an electromagnetic wave, a medium for the wave was postulated: the luminiferous aether.

In 1905 Albert Einstein figured out the problem

He published a paper titled “On the Electrodynamics of Moving Bodies” which laid out the Special Theory of Relativity. In it he showed how things behave differently if their relative speeds are close to the speed of light.

Newton assumed motions transformed according to

equations Galileo developed

Galileo assumed t = t’. Turns out that isn’t correct. Since x’ also

depends on t, it is incorrect, too.

The LorentzTransformation

2

2 2

2 21 1

vxtx vt cx y y z z t

v vc c

2

2 2

2 21 1

vxtx vt cx y y z z t

v vc c

0 0x x at t t

In relativity, we are describing an “event”

, , , , , ,x y z t and x y z t

An event is described by eight quantities

ExampleA rocket is moving in the +x direction at 0.7c to an observer on the ground. At t = t’ = 0 an observer in the rocket has the same position as an observer on the ground (x = x’ = 0). At t = 1.0x10-7 s the observer in the rocket sees a flash goes off 3.0m from that him. Describe the event in each reference frame.

Solution 1Easiest to choose the rocket as the unprimed reference frame. Then x = 3.0m, y = 0m and z = 0m. The light had to travel 3.0m to reach the observer at 7.0x10-7s so the event occurred earlier by

Thus

So

88

3.01.0 10

3.0 10 ms

ms

7 8 81.0 10 1.0 10 9.0 10t s s s

8, , , 3.0 ,0 ,0 ,9.0 10x y z t m m m s

Solution 2Since we used the rocket as the unprimed reference frame, the velocity of the primed reference frame is -0.7c. Since the relative motion is in the x and x’ direction only, y’ and z’ are both zero.

8 8

2 2

22

3.0 0.7 3.0 10 9.0 1030.7

0.71 1

msm sx vt

x mv cc c

8

828

27

2 2

22

0.7 3.0 10 3.09.0 10

3.0 101.36 10

0.71 1

ms

ms

msvx

tct sv cc c

Final Solution

8

7

, , , 3.0 ,0 ,0 ,9.0 10

, , , 30.7 ,0 ,0 ,1.36 10

x y z t m m m s

x y z t m m m s

Consequences of Special Relativity

2

20 1 vL Lc

Length contraction: two observers do not see the length of a moving object as the same. If L0 is the length of the object an observer that is stationary with respect to it measures then the moving observer will see its length as

This length contraction is only in the direction of the relative motion

Consequences of Special Relativity 2

Time dilation: to observers moving with respect to each other do not see time moving at the same rate. If D t0 is the time interval as measured by an observer stationary with respect to the clock, a moving observer will measure the time interval as

0

2

21

tt

vc

DD

Watch videos on Simultaneous Events in Relativity and Time Dilation in Relativity

ExampleJohn sits on a asteroid in space and observes his friend Jane fly by him in a rocket moving at 0.8c. He sees Jane holding a meter stick and a clock but they don’t seem to match his meter stick and clock. How long does he see Jane’s meter stick as and at what rate does he see her clock moving?

SolutionSince we have the velocity of the rocket with respect to John, use him as the unprimed reference frame. For the calculation use L0 = 1.0m and D t0 = 1.0s

22

2 20

0

2 2

22

0.8

0.81 1.0 1 1.0 1 .64 0.6

1.01.67

0.81 1

v c

cvL L m m mc c

t st s

v cc c

DD

Velocity TransformationsGiven the velocity u of a particle with respect to observer O in the unprimed system.

2 2

2 2

2 2 2

1 1

1 1 1

y zx

x y zx x x

v vu uu v c cu u uvu vu vu

c c c

With the reverse transforms from primed to unprimed

2 2

2 2

2 2 2

1 1

1 1 1

y zx

x y zx x x

v vu uu v c cu u uvu vu vu

c c c

ExampleJohn sits on a asteroid in space and observes his enemy Jane fly by him in a rocket moving at 0.8c. After Jane flies by he fires his super cannon that hurls a projectile at Jane at 0.9c (muzzle speed he measures). How fast does Jane see the projectile moving toward her?

SolutionLet John be the unprimed reference frame. The v = 0.8c and ux = 0.9c. Since there is no uy or uz, we only need to transform the x velocity.

2 2

0.9 0.80.357

0.9 0.81 1

xx

x

u v c cu c

vu c cc c

So, Jane sees the projectile approaching her at a speed on 0.357c, not 0.1c as you would expect from classical physics