Special Relativity

SPH4U

Review of Scientific “Theories”

Recall discussion from the first day of class A scientific “theory” is a proposed

explanation/description for observed facts It is possible for a theory to be a good

approximation or have some usefulness even if it is not fully correct

One of the best examples is “Newtonian” Physics vs. Relativity & Quantum Mechanics

Newtonian Physics

Physics principles as explained by Newton and others Newton’s 3 Laws and

Law of Gravity Maxwell’s Equations of

Electromagnetism Equations for motion,

momentum, kinetic energy, etc. discussed earlier in this class

Underlying foundations of space and time as absolute

Relativity and Quantum Mechanics

New physics as described by Einstein and others, most of the work done in the early 1900sTime dilation, length contractionUncertainty principleBohr Theory of the Atom

Different fundamental assumptions about the Universe

The Special Theory of Relativity

Aimed to answer some burning questions:Could Maxwell’s equations for electricity

and magnetism be reconciled with the laws of mechanics?

Where was the aether?

The Conflict

Newtonian physics seems to describe the world as we are used to it

However, several experiments as well as some hypothetical arguments signaled some problems

Relativity and Quantum Mechanics improve upon Newtonian physics

Newtonian Physics

Newtonian physics accurately describes the Universe when…Speeds are not too largeGravity is not too strongYou are at a macroscopic level, i.e. not

dealing with individual molecules/atoms

Newtonian Physics, cont.

Under the conditions of the previous slide, there is no reason to use anything other than Newtonian physicsEquations give the same results to high

accuracyExample: Trajectories of satellites and space

probes use Newtonian physics

Relativity

Relativity is a set of physics concepts and laws deduced by primarily by Albert Einstein

Special Relativity Published by Einstein in 1905 “Special” case with no forces/acceleration

General Relativity Published by Einstein in 1915 Extension of previous theory to include forces

What ISN’T Relativity?

Relativity does not simply mean “everything is relative”

On the contrary, relativity says certain things are relative, and other things are absolute

Relativity also tells us by how much those certain things are relative and in what way

Experimental and Theoretical Need for Relativity

Michelson-Morley ExperimentSpeed of light is the same regardless of the

Earth’s motion through the aether (“absolute space”)

Maxwell’s Equations of ElectromagnetismPredict very unusual things, like magnetic

fields with “loose ends”, when speeds are extremely large

Michelson-Morley Experiment

For a long time, scientists believed in an “aether”—absolute space

In the Michelson-Morley experiment, the speed of light was measured “with” and “across” the “flow of the aether” as the Earth moved through it

Michelson-Morley ExperimentFlashContrary to expectations, however, the

speed of light was the same both “with” and “across”!

Theoretical Foundations of Relativity

To explain all of these things, Einstein came up with new laws of physics based on two assumptions The laws of physics are the same in all inertial (non-

accelerating) frames The speed of light is the same as measured by all

observers in all inertial frames Einstein took these principles “on faith” The principles and their implications have

passed subsequent experimental testing

Relatively Speaking

What do Einstein’s two assumptions imply? All motion is relative Relativity of simultaneity Relativistic velocity addition Time dilation Length contraction Relativistic mass increase E = mc2

Who is moving?

All Motion is Relative, cont.

You and your friend Jackie like to travel in bizarre spherical spaceships

Who is moving? Who is stationary?

All Motion is Relative, cont.

In spite of our everyday intuition, the only velocities that can be measured are relative velocities

Examples:Relative to the surface of the EarthRelative to the SunRelative to a distant galaxy

Galilean Relativity

1,000,000 ms-1 1,000,000 ms-1

■ How fast is Spaceship A approaching Spaceship B?

■ Both Spaceships see the other approaching at 2,000,000 ms-1.

■ This is Galilean or Classical Relativity.

Einstein’s Special Relativity

1,000,000 ms-1

0 ms-1

300,000,000 ms-1

Both spacemen measure the speed of the approaching ray of light. How fast do they measure the speed of light to be?

Nothing Can Go Faster Than The Speed of Light

Addition of Velocities In normal circumstances,

if you are moving and throw an object, an outside observer will see the object at a different velocity Straight-forward velocity

addition But all observers

measure the speed of light to be the same

Velocity Additions Do Not Apply to Light Even if you are

moving away from your friend at a very high velocity, you will both see a light beam moving at c.

21

v uu

vuc

velocity of object 1 relative to you

= velcity of object with respect to object 1

v

u

Nice to know formula

Relativistic Velocity Additions A formula for adding

velocities exists, but it is not required for the course.

The formula works such that you can never get velocities greater than c

For small velocities, is approximately the same as just adding the velocities

2

22

0.75 0.75 1.50.96

01 .75 0.75 1 .751

c cc c

v uu

vuc c

2

2

0.8 0.90.8 0.9

1

1.7

1 .8 .7

1

0.988

c cc c

c

v uu

vu

c

c

21

velocity of moving frame velocity of object in moving framevelocity of object in rest frame

velocity of moving frame velocity of object inmoving frame

v uu

v u

c

Relativistic Velocity Additions

Relativity of Simultaneity

Two lights an equal distance from M go off

A passing train carries M’ M’ sees the light from B first M see the light flashes at the

same time M’ is moving in the direction

of B This relativity is determined

by the speed of light and the relative motion of the objects/observers

Relativity of Simultaneity

Events which are simultaneous in one frame may not be in another!

Each observer is correct in their own frame of reference

The Lorentz Factor

Calculating length contraction, time dilation, and other quantities requires calculating the Lorentz factor

= v/cIf v = 99% of c, then = 0.99

is always < 1 1

2 2

2

1 1

11vc

The Lorentz Factor, cont.

Some examples:v = 0.1% of c = 1.0000005v = 1% of c = 1.00005v = 10% of c = 1.005v = 50% of c = 1.155v = 90% of c = 2.294v = 99% of c = 7.089v = 99.9% of c = 22.37

Time Dilation

2 sD c t

2s

Dt

c

This time is known as Proper Time. Because the clock is rest the frame of the occurring event. The Proper Time interval between two events is always the time interval measured by an observer for whom the two events take place at the same position.

0 is also written as st t Note:

d vtD

A clock using light pulses to keep time. Every time the pulse returns, a unit of time has passed

Distance = Velocity x Time

V

Time Dilation

2 22

2 2M Mc t v t

D

2 2

2

2 2M Mc t v t

D

2

2 2 2

2Mt c v D

2 2

2 22Mt D

c v

2

2

22

2

4

1M

Dt

vc

c

2s

Dt

c Now since

2

2

2

1M

Dt

vc

c

2

21

sM

tt

vc

D

½ vDt

½ cDt

L=vt

We are now watching the clock move horizontally with velocity v. We will examine one cycle, more specifically one-half of one cycle. During a cycle of the light photon the clock will have moved horizontally a distance L, and if we calculate the distance travelled by the light in this one cycle (a upside down V), the distance would be c times the time we measured for the cycle, that is ct. So for one-half cycle the distance travelled by the light is ½ ct.

This says a moving clock run slow. If ts =1 then you watching it move would notice it taking more than 1s (tm >1) on your clock, so ts runs slow.

Time Dilation Example

2

21

sM

tt

vc

You and a friend are having a eating contest. Your friend is on a train traveling at speed v=0.9 c. By her watch, she finishes her food in 5 seconds. Determine the time you measure, if you are standing still at the train station.

2

2

5

(0.9 )1

cc

5

1 .81

11.5 seconds

Since eating is happening on the train, that is the “proper” time, ts=5.

Time Dilation Example 2Now it is your turn to eat. According to your watch you finish your food in 5 seconds. How long does your friend think it took you to finish the food?

Now eating is happening at the station, so that is the “proper” time, again ts=5.

Both people think they won!

2

21

sM

tt

vc

2

2

5

(0.9 )1

cc

5

1 .81

11.5 seconds

Your friend would consider you to be moving. Remember the proper time is where the event and clock are together

Space TravelAlpha Centauri is 4.3 light-years from earth. (It takes light 4.3 years to travel from earth to Alpha Centauri). How long would people on earth think it takes for a spaceship traveling v=0.95c to reach A.C.?

M

dt

v 4.3 light-years

0.95 c 4.5 years

How long do people on the ship think it takes?People on ship have ‘proper’ time since they see

earth leave, and Alpha Centauri arrive. Dts

2

21

sM

tt

vc

2

21s M

vt t

c

24.5 1 .95

Dts = 1.4 years

Space Travel

Another approach that solves any special relativity problem by treating space and time as spacetime.

The only requirement is that both separated units are recorded in the same units. (i.e.: light seconds, light minutes, light years, …)

Space TravelAlpha Centauri is 4.3 light-years from earth. (It takes light 4.3 years to travel from earth to Alpha Centauri).

How long would people on earth think it takes for a spaceship traveling v=0.95c to reach A.C.?

M

dt

v 4.3 light-years

0.95 c 4.526 years

How long do people on the ship think it takes?

2 2 2 2Rocket time interval Rocket event seperation Earth time interval Earth space seperation

2 2 2 2

2 2

Rocket time interval 0

Rocket t

4.526 years 4.3

1.99ime interval

Rocket time in

8

=1.4 yter el rva a

years

years

An amazing technique is to place time and space in the same units then use the following relativistic formula:

Time Dilation Review

Time flows more slowly in a moving frame as observed by an outside observer

But remember motion is relative

If you and I are moving past each other I see your clock moving

more slowly But you also see mine

moving more slowly…!!!

Length Contraction

Objects moving relative to an outside observer appear contracted in the direction of their motion as measured by the observer

Length Contraction, cont.

If you and I move past each other in some sweet sports cars I measure your sports

car as being shorter You measure my sports

car as being shorter Only applies to the

direction of motion We see our sports cars

as still being the same height

Length Contraction

v=0.1 c

v=0.8 c

v=0.95 c

Length Contraction Example

People on ship and on earth agree on relative velocity v = 0.95 c. But they disagree on the time (4.5 vs 1.4 years). What about the distance between the planets?

Earth/Alpha: d0 = v t

= .95 (3x108 m/s) (4.5 years)= 4x1016m (4.3 light years)

Ship: d = v t

= .95 (3x108 m/s) (1.4 years)= 1.25x1016m (1.3 light years)

2

21M s

vL L

c

Length in moving frame

Length in object’s rest frame

Twin Paradox

Twins decide that one will travel to Alpha Centauri and back at 0.95c, while the other stays on earth. Compare their ages when they meet on earth.

Earth twin thinks it takes 2 x 4.5 = 9 years

Traveling twin thinks it takes 2 x 1.4 = 2.8 years Traveling twin will be younger!

Note: Traveling twin is NOT in inertial frame!

Question

You’re eating a burger at the interstellar café in outer space - your spaceship is parked outside. A speeder zooms by in an identical ship at half the speed of light. From your perspective, their ship looks:(1) longer than your ship

(2) shorter than your ship

(3) exactly the same as your ship

2

21M s

vL L

c

Always <1

Ls > LM

In the speeder’s reference frame

In your reference frame

Comparison:Time Dilation vs. Length Contraction

Dto = time in same reference frame as event i.e. if event is clock ticking, then to is in the reference

frame of the clock (even if the clock is in a moving spaceship).

Lo = length in same reference frame as object length of the object when you don’t think it’s moving.

2

21m s

vL L

c

2

0 21v

t tc

L0 > L Length seems shorter from “outside”

t > toTime seems longer

from “outside”

2

0 21v

L Lc

2

21s m

vt t

c

Relativistic Mass Increase

Einstein made two other surprising discoveries…Mass must increase with speed, as viewed by

an outside observerDue to conservation of momentum

There is “leftover” energy even when the object is at rest

Due to conservation of energyE = mc2

Relativistic Mass

2

21

sM

mm

vc

0

2

21

mm

vc

Rest massRest mass

Actually written

E = mc2

E = mc2 = m0c2

This E is the total energy of an objectWhen the object is at rest…

v = 0 = 1E = m0c2 (“rest mass energy”)

The reason that energy can be released through fusion/fission

Total EnergyRelativistic kinetic energy is the extra energy an object with mass has as a result of its motion:

total rest KE E E

We can solve this for the Kinetic energy of an object:

220

02

21

K total restE E E

m cm c

vc

Relativistic Momentum

Relativistic Momentum

2

21

mvp

vc

Note: for v<<c p=mv

Note: for v=c p=infinity

Relativistic Energy2

2

21

mcE

vc

Note: for v=0 E = mc2

Objects with mass can’t go faster than c!

Note: for v<<c E = mc2 + ½ mv2

Note: for v=c E = infinity (if m<> 0)

QuestionCalculate the rest energy of an electron (m=9.1x10-31 kg) in joules.

20 0E m c

2

31 80

14

9.1 10 3.0 1

8 2 1

0

. 0

mE kg

s

J

Calculate the electron’s Kinetic energy if it is moving at 0.98c.2

2002

21K

m cE m c

vc

1414

2

2

13

8.2 108.

3.3 1

2 100.98

1

0

K

JE J

c

c

J

Simultaneous?

At Rest

Moving

YES

NO

Simultaneous depends on frame!

A flash of light is emitted from the exact center of a box. Does the light reach all the sides at the same time?

Simultaneous?

Many times, questions are concerned with the determination of the spatial interval and/or the time interval between two events. In this case a useful technique is to subtract from each other the appropriate Lorentz contraction describing each event.

2

2

2

' ' ' '

1

b a b a

b a

vt t x x

ct tvc

Three Other Effects

3 strange effects of special relativityLorentz TransformationsRelativistic Doppler EffectHeadlight Effect

Lorentz Transformations

Lorentz Transformations

■ Light from the top of the bar has further to travel.

■ It therefore takes longer to reach the eye.

■ So, the bar appears bent.

■ Weird!

Doppler Effect

The pitch of the siren:Rises as the ambulance approachesFalls once the ambulance has passed.

The same applies to light!Approaching objects appear blue (Blue-shift)Receding objects appear red (Red-shift)

Headlight effect

Beam becomes focused. Same amount of light concentrated in a

smaller area Torch appears brighter!

V

Warp

Program used to visualise the three effects

Fun stuff

Eiffel Tower Stonehenge

Summary0

2

2

2

0 2

0

2

2

0

2

2

20 0

220

02

2

1

1

1

1

1total K

tt

vc

vL L

cm

mvc

m vp

vc

E m c

m cE m c E

vc

2

2

2

' ' ' '

1

b a b a

b a

vt t x x

ct tvc

2

2

21

b a b a

b a

vt t x x

ct tvc

21

v uu

vuc

UnderstandingAn observer has a pendulum that has a period of 3.00 seconds. His friend who happens to own a spaceship (with cool engines), zooms by the stationary pendulum. If the speedometer of the spaceship says 0.95c, what will the friend measure are the period of the pendulum?

Since I am with the pendulum, my measured time is the Proper Time.

0

2

21

tt

vc

2

2

3

0.951

s

c

c

9.6s

This makes sense because a moving clock would run slower from my perspective. So the pendulum would have a period of 9.6s.

Understanding Vega is 25 light-years

away Travel to Vega at 0.999c

The length would appear contracted to you

About 1 light-year Make the trip in ~1 light-

year (each way) as measured by you

Earth would measure 25 years each way

You would spend 2 years (your time) travelling and arrive 50 years in the future Earth time.

UnderstandingYou throw a photon (3x108 m/s). How fast

do I think it goes when I am:Standing still

Running 1.5x108 m/s towards

Running 1.5x108 m/s away

Strange but True!

3x108 m/s

3x108 m/s

3x108 m/s

Understanding

A 1.0 m long object with a rest mass of 1.0 kg is moving at 0.90c. Find its relative length and mass

2

21M s

vL L

c Use length contraction formula:

2

2

0.901.0 1

1.0 0.4346

0.44

M

cL m

c

m

m

Mass increase formula:0

2

21

mm

vc

2

2

1.02.3

0.901

kgkg

C

c

UnderstandingFor a 1.0 kg mass moving at 0.90c. Find the rest energy and kinetic energy of the object

For rest energy, Use energy formula: 20 0E m c

2

80

16

1.0 3.0 10

9.0 10

mE kg

s

J

For Kinetic energy, Use relativistic energy formula:

20

2

21

m cE

vc

1617

2

2

9.0 102.0 10

0.901

JJ

c

c

20

17 16

17

2.0 10 9.0 10

1.1 10

KE E m c

J J

J

Now:2

0 KE m c E

Therefore

UnderstandingA person’s pulse rate is 65 beats per minute.a) If the person is on a spaceship moving at 0.10c, what is the pulse rate as

measured by a person on Earth?b) What would the pulse rate be if the ship were moving at 0.999c?

a) Use time dilation:2

21

sM

tt

vc

2

2

0.015min0.015min 65

min0.101

beats

c

c

a) Use time dilation:2

21

sM

tt

vc

2

2

0.015min0.336min 3.0

min0.9991

beats

c

c

1 10.015

65t

f

We need time for a heart beat

UnderstandingA muon at rest has an average lifespan of 2.20 x 10 -6 sa) What will an observer on Earth measure as its lifespan if it travels at 0.990c?b) What distance would we observe it travel before disintegrating?c) What distance would it travel if relativistic effects were not taken into account?

a) Time dilation: 0

2

21M

tt

vc

65

2

2

2.20 101.56 10

0.9901

ss

c

c

b) Distance formula Md vt 8 50.990 3.0 10 1.56 10 4630m

s ms

c) Distance formula sd vt 8 60.990 3.0 10 2.2 10 653m

s ms

2

20 2

1 4630 1 .990 653v

L L m mc

To Show consistency

This is the distance the muon measures it travels before disintegrating

Understanding

You measure the length of an object as 100m when it passes you at 0.90c. What is its length when at rest?

2

0 21v

L Lc

Use length contraction formula:

0 2

2

2

2

0

100

0.901

100

0.4346

23

1

0

mL

c

c

m

LL

v

m

c

http://onestick.com/relativity

UnderstandingAs a rocket ship sweeps past the Earth with speed v, it sends out a light pulse ahead of it. How fast does the light pulse move according to the people sitting on the Earth?

2 21 1

moving frame object inmoving frameobject in rest frame

moving frame object inmoving frame

v uv uu

vu v uc c

21

1

object in rest frame

v cu

vcc

v cvc

v cv cc

c

Understanding

A train 0.5 km long (as measured by an observer on the train, therefore this is the proper length) is travelling at 100 km/h. Two lightening bolts strike the ends of the train simultaneously as determined by an observer on the ground. What is the time separation as measured by the observer on the train?

Units:1000 1

100 27.783600

km m h m

h km s s

We are given that tb-ta=0 and what we want to determine is tb’-ta’

2

2

2

' ' ' '

1

b a b a

b a

vt t x x

ct tvc

UnderstandingA train 0.5 km long (as measured by an observer on the train, therefore this is the proper length) is travelling at 100 km/h. Two lightening bolts strike the ends of the train simultaneously as determined by an observer on the ground. What is the time separation as measured by the observer on the train?

28

2

28

27.78500

3.0 100

27

'

.781

3.0

'

10

b a

ms mms

ms

t t

ms

28

27.78500

3.0

' '

10b a

ms mms

t t

131.54' ' 10b at t s

The negative sign reminds us that even a occurred after event b