Chapter 37 – Introduction to Special Relativity

I.Introduction

A.In 1905 Einstein published a paper, “On the Electrodynamics of Moving Bodies,” proposing what is now called the “Special Theory of Relativity.” The theory is based on two postulates:

1.The laws of physics are the same in all inertial frames of reference.

2.The speed of light in a vacuum is the same for all observers – c = 3 x 108 m/s.

B.Postulates sound reasonable, but sometimes contradict common sense.

1.Laws of physics postulate: perform experiments

  • pour coffee while stationary and while in a 757 traveling at 600 mi/hr
  • throw a ball upward while stationary and while moving
  • play pool while stationary and while moving

2.For constant light speed postulate: consider a moving source and/or a moving observer

  • moving source and stationary observer
  • stationary source and moving observer
  • Evidence:

- rotating sun -

- starlight observed by earth orbiting around the sun at ~ 30 km/s

C.Speed of light being constant seems contradictory because for all other wave phenomena, a medium is needed to support the oscillations of the wave, and the waves travel at a constant velocity relative to the medium. In the case of light, the early scientists called this medium that supported the light wave oscillations “ether.” Michelson and Morley designed and performed experiments to detect the effect of this “ether,” but came up with a “null” result.

Here’s a simplified version of the Michelson-Morley experiment:

As the earth orbits around the sun, the earth is traveling through “ether.” From the point of view of a person on the earth, the ether is whizzing past the earth (like the air whizzing past a moving car). If the earth is traveling at a velocity v, then the ether wind is moving past the earth is moving at a velocity v. Assume two orthogonal laser beams start at the same point, travel the same distance L to plane mirrors, then reflect back to their starting points. If “ether” is present then speed of light is constant relative to the ether, and the time it takes the light to travel along the two paths will be different.

Horizontal beam velocities:

to right:c – v

to left:c + v

Vertical beam velocities:

II.Kinematical Consequences of Michelson and Morley’s “null” result – Time Dilation.

A.Take O as a stationary frame of reference and O’ as a moving frame of reference attached to a spaceship moving with a velocity, u, in the horizontal direction. (The clocks in each frame of reference are synchronized.)

Examine what an observer in each frame of reference sees when the flash of light travels to the mirror and back.

B.What does the person in the spaceship observe? How much time does it take the light to travel to the mirror and back? Call this time to, the proper time – the elapsed as measured by an observer who sees the event occurring at one point in space.

C.What does a person outside the spaceship observe as the spaceship is passing by? How much time does it take the light to travel to the mirror and back? Call this time t.


D.The equation for time dilation:

,

which is also written as , where and .

Note that u < c, and so 0 < , and t > to.

E.Examples.

1.You are standing on the side of a road. Your friend passes you in a car traveling at 60 mi/hr (~27 m/s), and tells you that it took her one minute to travel from point A to point B. You tell her that you measured a different time as she traveled from A to B. How different are the two times?

Remember that the proper time is the time measured by the person who sees the event as taking place at one point in space. Who is that?

Be aware of the approximation: , when x < 1.

2.Answer question 1 if the speed of the car is 0.1c.

3.Answer question 1 if the speed of the car is 0.9c.

4.Suppose your friend is traveling past you at 0.9c, and she notices that one minute elapsed on your watch. According to her, how much time elapsed on her watch?

III.Kinematical Consequences of Michelson and Morley’s “null” result – Length Contraction.

A.Let lo be the length of an object that is at rest in the O frame of reference. This length, lo, corresponds to the distance the spaceship travels past O in the time it takes the light beam to travel to the mirror and back. In this case lo is called the proper length – the length measured by an observer who is at rest relative to the object.

B.From the point of view of an observer in the spaceship (O’), the object appears to be moving to the left at a velocity u. When the light flashes, the left end of the object (A) is at the position of the flash. In the time it takes the light to travel to the mirror and return, the object moves until the right end (B) of the beam is at the position of the flash.

C.Combine the two observations.

D.Examples

1.The length of a car is measured to be 20 ft when at rest. How long is the car to an observer who sees the car traveling past at a speed of

a.60 mi/hr (~27 m/s)?

b.0.9c?

2.The distance from Santa Monica College to the beach is 2.0 km. A rocket car built by some physics students drives from SMC to the beach traveling at 0.8c. According to the students in the car, how far did they travel?

3.A physics student measures a square-shaped object to be one meter by one meter, and therefore has an area of one square meter. The object is then placed aboard a spaceship and travels past him at 0.8c. What is the area of the object?

4.Evidence of length contraction and time dilation: Particles, called muons, are created when cosmic rays collide with atoms in the upper atmosphere. After the collision they are found to be traveling with an average speed of 0.998c. Some of these muons are found to reach the surface of the earth 9000 m below where they were produced. The average lifetime of a muon is 2 x 10-6 sec. At a speed of 0.998c and a lifetime of 2 x 10-6 sec, the muon should travel only ~ 600 m (0.998c x 2 x 10-6). So, how is it possible for the muons to reach the surface of the earth?

a.observations from the ground:

b.observations traveling with the muon:

5.The twin paradox: An astronaut is scheduled to travel from the earth to a distant star system that is 6 light years away. Her twin is a rock star who remains on earth. The astronaut enters a spaceship and travels from earth to the distant star system and back. The people on earth measure the speed of the spaceship to be 0.6c in both directions. How old she upon returning to earth? How old is the rock star?

a.according to the people on earth:

b.according to the astronaut on the spaceship:

c.Why the discrepancy?

IV.Other Consequences of Special Relativity

A.Simultaneity: Events that are simultaneous to one person are not simultaneous to another person who is moving with respect to the first.

1.Imagine one car of a train traveling past an observer on the ground. A second person is inside the train. Lightning hits the ends of the train at the same time according to the person on the ground. What does the person inside the train observe?

a.ground observer:

b.observer in the train

2.A meter stick travels past you at 0.9c. You hold your hands 0.5 m apart; bring them down very quickly as the meter stick passes you. Will your hands hit the meter stick?

a.your point of view:

b.meterstick’s point of view:

c.Who’s right? Why?

V.Lorentz Transformations and the Velocity Transformations

A.Galilean Transformations with motion in the x and x’directions (clocks synchronized, t = 0 and t’ = 0 when origins coincide):

B.Lorentz Transformations: O’ frame moving with a velocity u in the x direction relative to the O frame.

y’ = yy = y’

z’ = zz = z’

These equations reduce to the Galilean transformations when uc.

C.Velocity Transformations:

1.In order to use these equations:

-identify the event (what it is you want to determine), identify O (the stationary frame of reference), identify O’ (the moving frame of reference)

-u (the velocity of O’ as measured by O)

-vx , vy and vz (the x, y and z components of the velocity of the event as measured by O)

-v’x , v’y and v’z (the x, y and z components of the velocity of the event as measured by O’)

2.Example: A spaceship is traveling past the earth at a speed of 0.6c. It sends out a beam of light in the same direction it is traveling. How fast is the beam of light traveling as measured by a person on earth?

3.Example: Two spaceships, A and B, are traveling toward each other. As measured by a person on the earth, spaceship A is traveling at 0.8c and spaceship B is traveling at 0.9c. How fast is spaceship B traveling as measured by a person on spaceship A?

VI.Momentum Consequences

When examining momentum and taking relativity into account, we find that the momentum of a particle of mass m and traveling with a velocity v must be written as

.

Notice that if the velocity of the particle doubles, the momentum does not just double, but increases by more rapidly. This implies that the mass is changing as a result of the velocity change – the mass of a particle depends on the velocity. The value of mass m in the above equation is referred to as the “rest mass” of the particle – the mass when the particle is at rest relative to the person making the measurement.

VII.Newton’s Second Law Consequences

A.We’ve been writing Newton’s Second Law as: .

B.More Properly Newton’s Second Law should be written as:

, where .

Taking the derivative, we get

,

and

.

Note that as v  c, the acceleration a  0. Therefore c is the maximum speed.

VIII.Energy Consequences

The work-energy theorem indicates that the work done accelerating a particle equals the change in kinetic energy of the particle. When relativity is taken into account, the work-energy theorem predicts that the kinetic energy of a particle starting from rest and traveling at a velocity v is different from our classical value.

Work-energy theorem: , when starting from rest.

, where motion is along the x-axis.

, but .

Upon integrating

or rearranged

Rearranging further, we get

total energy.

Or, in an alternative form: given by the expression

.

The mc2 term is called the “rest” energy of the particle and represents the energy that is available if all the mass m is converted into energy.

Examples.

1.How much energy is available if the entire mass of a 1 kg object could be converted into energy?

E = mc2 = (1 kg)(3 x 108 m/s)2 = 9 x 10 16 Joules

2.When a proton and a neutron are combined (fused) to form a deuteron, mass disappears. Where did it go? How much energy results from this fusion?

Mass of proton=1.007825 u1 u = 1.66 x 10-27 kg

Mass of neutron=1.008665 u = 931.5 MeV

Mass of deuteron=2.014102 u

3.When a neutron is absorbed by Uranium-235 nucleus , the nucleus undergoes fission, one group of fission byproducts of many possible groups of fission byproducts are Krypton, Barium, and three neutrons . When the mass of the neutron and the Uranium nucleus together is compared to the mass of the byproducts, some mass is missing. Where did it go?

A typical fission reaction produces about 3 x 10-11 Joules of energy. The explosion of a TNT molecule releases about 5 x 10-18 Joules of energy.

4.a.Find the total energy of an electron that is accelerated through a potential difference of 2500 volts.

b.How fast will the electron be traveling?

IX.Magnetism

A.Consider a wire carrying a current to the right. Then the positive charges +q in the wire are traveling to the right. Also say that there is a charge +Q traveling parallel to the wire to the right.

B.From the point of view of the charge +Q:

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