The Special Theory of Relativity

An Introduction to One the Greatest Discoveries

The Relativity Principle

Galileo Galilei 1564 - 1642

Problem: If the earth were moving wouldn’t we feel it? – No

The Copernican Model

The Ptolemaic Model

The Relativity Principle

A coordinate system moving at a constant velocity is called an inertial reference frame.

v

The Galilean Relativity Principle: All physical laws are the same in all inertial reference frames.

Galileo Galilei 1564 - 1642

we can’t tell if we’re moving!

Electromagnetism

James Clerk Maxwell 1831 - 1879

A wave solution traveling at the speed of light

c = 3.00 x 108 m/s

Maxwell: Light is an EM wave!

Problem: The equations don’t tell what light is traveling with respect to

Einstein’s Approach to Physics

Albert Einstein 1879 - 1955

1. Gedanken (Thought) Experiments

E.g., if we could travel next to a light wave, what would we see?

2. “The Einstein Principle”:

If two phenomena are indistinguishable by experiments then they are the same thing.

Einstein’s Approach to Physics2. “The Einstein Principle”:

If two phenomena are indistinguishable by experiments then they are the same thing.

A magnet moving A coil moving towards a magnet

Both produce the same current

Implies that they are the same phenomenon

towards a coil

Albert Einstein 1879 - 1955

current current

Einstein’s Approach to PhysicsAll physical laws (like electromagnetic equations)

depend only on the relative motion of objects.

A magnet moving A coil moving towards a magnet

Implies that we can only measure relative motions, i.e., motions of objects relative to other objects.

By the “Einstein Principle” this means all that matters are relative motions!

towards a coil

current currentEx) samecurrent

Einstein’s Approach to Physics1. Gedanken (Thought) Experiments

E.g., if we could travel next to a light wave, what would we see?

c

c

We would see an EM wave frozen in space next to us

Problem: EM equations don’t predict stationary waves

Albert Einstein 1879 - 1955

ElectromagnetismAnother Problem: Every experiment measured the speed of light to be c regardless of motion

The observer on the ground should measure the speed of this wave as c + 15 m/s

Conundrum: Both observers actually measure the speed of this wave as c!

Special Relativity Postulates

1.The Relativity Postulate: The laws of physics are the same in every inertial reference frame.

2.The Speed of Light Postulate: The speed of light in vacuum, measured in any inertial reference frame, always has the same value of c.

Einstein: Start with 2 assumptions & deduce all else

This is a literal interpretation of the EM equations

Special Relativity PostulatesLooking through Einstein’s eyes:

Both observers (by the postulates) should measure the speed of this wave as c

Consequences:

Time behaves very differently than expected

Space behaves very differently than expected

Time DilationOne consequence: Time Changes

Equipment needed: a light clock and a fast space ship.

Time Dilation

In Bob’s reference frame the time between A & B is Δt0

Sallyon earth

Bob

Beginning Event B

Ending Event A

D

Δt0

Time Dilation

In Bob’s reference frame the time between A & B is Δt0

Sallyon earth

Bob

Beginning Event B

Ending Event A

tt

ligh of speed the

eledlight trav distance the0

D

Δt0

c

D2

Bob

Time DilationIn Sally’s reference frame the time between A & B is Δt

Bob

A BSallyon earth

Δt

Bob

Time DilationIn Sally’s reference frame the time between A & B is Δt

A BSallyon earth

22 2 22 2 2

2

v ts D L D

Length of path for the light ray:

c

st

2and

Δt

Time Dilation

22 2 22 2 2

2

v ts D L D

Length of path for the light ray:

c

st

2and

and solve for Δt:

22 /1

/2

cv

cDt

cDt /20

Time measured by Bob

22

0

/1 cv

tt

Time Dilation

22

0

/1 cv

tt

Δt0 = the time between the two events measured by Bob

Δt = the time between the two events measured by Sally

v = the speed of one observer relative to the other

Time Dilation = Moving clocks slow down!

If Δt0 = 1s, v = .9999 c then: s 7.709999.1

s 12

t

Time Dilation

Bob’s watch always displays his proper time

Sally’s watch always displays her proper time

How do we define time?

The flow of time each observer experiences is measured by their watch – we call this the proper time

If they are moving relative to each other they will not agree

Time DilationA Real Life Example: Lifetime of muons

Muon’s rest lifetime = 2.2x10-6 seconds

Many muons in the upper atmosphere (or in the laboratory) travel at high speeds.

If v = 0.9999 c. What will be its average lifetime as seen by an observer at rest?

s 105.19999.1

s 102.2

/1

4

2

6

22

0

cv

tt

Length Contraction

Bob’s reference frame:The distance measured by the spacecraft is shorter

Sally’s reference frame:

Sally

Bob

The relative speed v is the same for both observers:

22

0

/1 cv

tt

220 /1 cvLL

t

Lv

0

0t

L

t 0t

Length Contraction

Sally

Bob

220 /1 cvLL

t 0t

L0 = the length measured by Sally

L = the length measured by Bob

Length Contraction =

If L0 = 4.2x1022 km, v = .9999 c then km 100.6 20LTo a moving observer all lengths are shorter!

SummaryEinstein, used Gedanken experiments and the

“Einstein Principle” to formulate the postulates of special relativity:

1. All physical laws are the same in all inertial reference frames

2. The constancy of the speed of light

The consequences were that

1. Moving clocks slow down

2. To a moving observer all lengths are shorter.

Special Relativity & Beyond The special theory of relativity dramatically changed

our notions of space and time.

Because of this, mechanics (like notions of energy, momentum, etc.) change drastically, e.g., E=mc2.

Special relativity only covers inertial (non-accelerated) motion. To include acceleration properly we must incorporate gravity. This theory is known as the general theory of relativity which is Einstein’s greatest contribution to physics.

Real Life Application of RelativityIn Global Positioning Satellite (GPS) general

relativistic corrections are needed to accurately predict the satellite’s clock which ticks slower in orbit. Without it you GPS would be off by at least 10 kilometers. With the corrections you can predict positions within 5-10 meters

http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html