Chapter 35:Special Theory of Relativity

Outline:IntroductionTime DilationAddition of VelocitiesLength ContractionMomentum and Energy

“Classical” or “Newtonian” physics does a good job of describing thebehavior of particles at low speeds. But as speeds start to approach thatof light we must use The Special Theory of Relativity proposed byEinstein in 1905.

Classical physics represents the “low speed limit” of relativistic physics.

The Postulates of RelativityThe Special Theory of Relativity is based on twopostulates:• The laws of physics are the same in all inertialreference frames.• The speed of light in a vacuum is ALWAYSmeasured to be

c = 299,792,458 m/sindependent of the motion of the observer or themotion of the source of light.Note: Relativity Theory is counter-intuitive!

Reference FramesThe frame of reference for a person or object

is the coordinate system which moves withthe person or with the object.

Think of the frame of reference as a set of xyzaxes which are attached to the object.

An inertial reference frame is one in whichNewton’s Laws are valid. All inertialreference frames move with constantvelocity relative to one another.

Addition of Velocities: Common Sense

How do we normally deal with velocities in differentreference frames?

vx/y is the velocity of object x compared to yIn general: v1/3 = v1/2 + v2/3

Example:A train is moving in the x direction at a speed of 10 m/s.A woman is walking toward the back of the train at 3m/s. What is the speed of the woman relative to theground outside the train?

The Speed of Light

But what if the spaceship fires a beam of light in the samedirection it is moving? The observer measures the speed ofthe light to be c, not c+1000 m/s! All observers measure thesame speed of light.

We need Relativity Theory to explain this!

In the picture, a spaceshipfires a missile, which has arelative velocity of 3000 m/srelative to an observer onEarth.

In Relativity Theory, the distance betweentwo points and the time interval betweentwo events depend on the frame of referencein which they are measured!

There is no absolute measure of time.There is no absolute measure of length.

Time is Relative

Two events which appear to be simultaneous in one reference frameare in general not simultaneous in a second frame moving withrespect to the first.In this picture, the stationary observer, Justin, observes the lightningstrike simultaneously at A and B. Juliane thinks the light hits firstat B and then at A. Why?

Time interval measurements depend on the frame inwhich they are measured.

Time Dilation

Two observers, each in their own referenceframe moving with a relative velocity willnot agree on how fast time passes. Eachwill think the other’s clock is wrong.

Moving clocks run slow.

This effect is known as time dilation.

The proper time Δt0 is the time interval between twoevents as measured by an observer who sees theevents occur at the same place. Let the “proper”frame move with velocity u with respect to anotherframe. Δt = γ Δt0

where

!

" =1

1# u2 /c 2

Proper Time

Which is larger, Δt or Δt0?

Example 1An astronaut at rest on Earth has a heartbeat rate of 70beats/min. When the astronaut is traveling in a spaceship at0.90c, what will this rate be as measured by (a) an observer alsoin the ship and (b) an observer at rest on the Earth.

Example 2

A millionairess was told in 1992 that she had exactly 15 years tolive. However, if she travels away from the Earth at 0.8 c and thenreturns at the same speed, the last New Year's day the doctors expecther to celebrate is:

A) 2001B) 2003C) 2007D) 2010E) 2017

Length ContractionThe distance between two points depends on the frame

of reference in which it is measured.The proper length l0 is the length of the object measured

in the frame of reference in which the object is at rest.If the object is moving in a reference frame, its length l

will be measured to be less than its proper length:

This is known as relativistic length contraction.

!

l =l0

"= l

01#

u2

c2

Example 3A friend in a spaceship travels past you at a high speed. He tells youthat his ship is 20 m long and that the identical ship you are sitting inis 19 m long. According to your observations, (a) how long is yourship, (b) how long is his ship, and (c) what is the speed of yourfriend's ship?

Example 4The proper length of one spaceship is three times that of another.The two spaceships are traveling in the same direction and, whileboth are passing overhead, an Earth observer measures the twospaceships to have the same length. If the slower spaceship ismoving with a speed of 0.35c, determine the speed of the fasterspaceship.

Example 5A supertrain of proper length 100 m travels at a speed of 0.95cas it passes through a tunnel having proper length 50 m. Asseen by a trackside observer, is the train ever completely withinthe tunnel? If so, by how much?

Relativistic Addition of VelocitiesIn relativity theory we must modify the way we addvelocities: The correct relativistic addition of velocities is:

where vx/y is the velocity of x with respect to y.

In the textbook, they use:!

va / b

=va / d

+ vd / b

1+va / dvd / b

c2

!

V =v1

+ v2

1+ v1v2/c

2

What if va/d and vd/b are both the speed of light?

v1 = velocity of object relative to S´ frame (vo/S´)

V = velocity of object relative to S frame (vo/S)

v2 = velocity of S´ frame relative to S frame (vS´/S)

Example 6Spaceship R is moving to the right at a speed of 0.70c withrespect to the Earth. A second spaceship, L, moves to the leftat the same speed with respect to the Earth. What is the speedof L with respect to R?

Example 7A spaceship is traveling at 0.95c with respect to the Earth. Insidethe spaceship, a man shines a flashlight in the direction thespaceship is moving. What is the velocity of the light from theflashlight with respect to Earth?

Relativistic Momentum

In order for momentum to always be conserved inrelativistic interactions, we must modify ourclassical expression for momentum to instead read:

m is the rest mass of the object.

!

! p =

m! v

1" v2/c

2= # m

! v

Example 8Calculate the momentum of a proton moving with aspeed of 0.5c.

Relativistic EnergyThe total energy E of an object moving with velocity v is:

Mass is another form of energy!The energy of an object at rest is known as the rest energy

Erest = mc2

The total energy is comprised of the rest energy and the kineticenergy:

E = mc2 + K

What is the relativistic formula for kinetic energy?

!

E = "mc 2 =mc

2

1# v2 /c2

Momentum and EnergyThe relationship between total energy E andmomentum p is given by

E2 = (pc)2 + (mc2)2

For a massless particle such as a photon,

E = pc

Example 9

An electron moves with a speed of 0.80c. Calculate its (a) restenergy, (b) total energy, and (c) kinetic energy.

Example 10A proton in a high-energy accelerator is given a kinetic energyof 50.0 GeV. Determine (a) the momentum and (b) the speedof the proton