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Drouin Secondary College VCE Physics Unit 3: Relativity

VCE Physics

Unit 3

Topic 3

Special Relativity

The World at the Speed of Light.

Einstein’s Contribution.


Unit Outline

To achieve the outcome the student should demonstrate the knowledge and skills to:

describe Maxwell’s prediction that the speed of light depends only on the electrical and magnetic

properties of the medium it is passing through and not on the speed of the source or the speed of the

medium;

contrast Maxwell’s prediction with the principles of Galilean relativity (no absolute frame of reference, all

velocity measurements are relative to the frame of reference);

interpret the results of the Michelson Morley experiment in terms of the postulates of Einstein’s special

theory of relativity;

- the laws of physics are the same in all inertial frames of reference

- the speed of light has a constant value for all observers;

compare Einstein’s postulates and the postulates of the Newtonian model;

use simple thought experiments to show that

- the elapse of time occurs at different rates depending on the motion of the observer relative to

the event;

- spatial measurements are different when measured in different frames of reference;

explain the concepts of proper time and proper length as quantities that are measured in the frame of

reference in which the objects are at rest;

explain movement at speeds approaching the speed of light in terms of the postulates of Einstein’s

special theory of relativity;

model mathematically time dilation, length contraction and mass increase with respectively the equations

t = toγ, L = Lo/γ, m = moγ where γ = 1/(1-v2/c2)1/2

explain the relation between the relativistic mass of a body and the energy equivalent according to

Einstein’s equation E = mc2

explain the equivalence of work done to increased mass energy according to Einstein’s equation E =

mc2

compare special relativistic and non relativistic values for time, length and mass for a range of situations.

_____________________________________________

1.0 Galilean Relativity

One of the earliest of the great minds to ponder motion, both on Earth and in the heavens, was

Galileo Galilei.

He developed the principle of Galilean Relativity.

This is best shown with a simple example:


Imagine an observer in a house by the sea shore and another in the windowless hull of a ship.

Neither will be able to determine that the ship is moving at constant velocity by comparing the

results of experiments done inside the house or on the ship.

In order to determine motion these observers must look at each other.

Generalizing these observations Galileo postulated his relativity hypothesis:

any two observers in inertial frames of reference with respect to one another will obtain the

same results for all mechanical experiments.

There is no ______________ inertial frame of reference: all velocity measurements are relative

to the frame of reference.

FRAMES OF REFERENCE

Frames of reference can be of 2 types:

1. Inertial Frames. These are systems (or groups of objects) which are either at _________

or moving with constant ______________________.

2. Non Inertial Frames. These are systems which are ___________________________

1.1 Galilean Motion

In Galileo’s world, the idea of relative motion is clearly understood.

This can be shown with a simple example.

A train carriage is travelling to the right at a constant velocity of 25.0 ms-1.

A boy standing in the carriage throws a ball to the right at a constant velocity of 5.0 ms-1.

The boy in the carriage sees the ball travel away from him at ___________ ms-1

But, an observer standing beside the track, sees the ball moving to the right at ________ ms -1.

So what is the ball’s “correct” speed ?

5.0 ms-1 or 30.0 ms-1?

VTRAIN = 25.0 ms-1

VBALL = 5.0 ms-1

Stationary observer


BOTH answers are _____________________.

There is no single “correct” answer. The speed of an object depends on where the observer is

when the speed was measured.

Remember, according to Galileo, there is no absolute inertial frame of reference: all velocity

measurements are relative to the frame of reference.

1.2 Isaac Newton The next great mind to influence mankind’s understanding of the

operation of the universe was Isaac Newton (1642 – 1727), when

he developed his 3 laws, first mentioned in his 1687 book

Philosophiae naturalis principia mathematica (or just Principia).

Law 1 (The Law of Inertia)

A body will remain at rest, or in a state of ____________ motion,

unless acted upon by a net external ___________.

Law 2 The acceleration of a body is directly proportional to

net force applied and inversely proportional to its ___________.

Mathematically, a = F/m more commonly written as F = ma

Law 3 (Action Reaction Law)

For every action there is an equal and opposite ________________.

These Laws explained Galilean relativity and using Newton's laws, physicists in the 18th and

19th century were able to predict the motions of the planets, moons, comets, cannon balls, etc.

In classical Newtonian mechanics, time was universal and absolute.

1.3 The Clouds Gather For more than two centuries after its inception (in about the 1680’s), the Newtonian view of the

world ruled supreme, to the point that scientists developed an almost blind faith in this theory.

And for good reason: there were very few problems which could not be accounted for using this

approach.

Nonetheless, by the end of the 19th century, new experimental evidence, difficult to explain

using the Newtonian theory, began to accumulate, and the novel theories required to explain

this data would soon replace Newtonian physics.

Isaac Newton, aged 26


1.4 19 th Century Clouds

In 1884 Lord Kelvin (of temperature scale fame) in a lecture delivered in Baltimore, Maryland,

mentioned the presence of “Nineteenth Century Clouds'' over the physics of the time, referring

to certain problems that had resisted explanation using the Newtonian approach.

Among the problems of the time were:

a) __________ had been recognized as a wave, but the properties (and the very

existence!) of the medium that conveys light appeared inconsistent.

b) The equations describing _________________ and magnetism were inconsistent

with Newton's description of space and time.

c) The orbit of Mercury, which could be predicted very accurately using Newton's

equations, presented a small but disturbingly unexplained discrepancy between

the observations and the calculations.

d) Materials at very _________ temperatures do not behave according to the

predictions of Newtonian physics.

e) Newtonian physics predicted that an oven at a stable constant temperature has

infinite energy.

1.5 The Revolution

The first quarter of the 20th century witnessed the creation of the revolutionary theories which

explained these phenomena.


They also completely changed the way we understand Nature.

The first two problems concerning the nature of light and electricity and magnetism required the

introduction of the Special Theory of Relativity.

The third item concerning Mercury’s orbit required the introduction of the General Theory of

Relativity.

The last two items low temperature materials and infinite energy ovens can be understood only

through the introduction of a completely new mechanics: _______________ mechanics.

The new theories that superseded Newton's had the virtue of explaining everything Newtonian

mechanics did (with even greater accuracy) while extending our understanding to an even wider

range of phenomena.

1.6 Maxwell’s Contribution Throughout 1700’s and 1800’s, many individual laws about electricity and magnetism had been

discovered, such as Coulomb’s law of

electrostatic force.

James Clerk Maxwell (1831-1879) had, by 1855, unified some laws and finally by 1873 had

found that all of these laws could be summarised by _______ partial differential equations.

A triumph of unification!

(Which of course is the holy grail of Physics)

One interesting consequence of Maxwell’s unification is that you can calculate the velocity of

electromagnetic waves based on properties of capacitors and inductors.

c = Speed of EM Waves

μ0 = Permeability of free space

ε0 = Susceptibility of free space

In Maxwell’s own words:

This velocity is so nearly that of light, that it seems we have strong reasons to conclude that

light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance

in the form of waves propagated through the electromagnetic field according to electromagnetic

laws.


Chapter 2

2.0 19 th Century Physics

Around this time, Physicists were trying to find a way to measure the ABSOLUTE VELOCITY of

an object relative to some fixed point which was COMPLETELY AT REST.

But what, in our universe, is completely at rest ?

Certainly not the __________, which as well as spinning on its axis at 500 ms-1 (1800 kmh-1),

travels around the __________ at 30 kms-1 (108,000 kmh-1).

The sun, of course, is in orbit around the centre of our __________________ at 250 kms -1

(900,000 kmh-1).

And our galaxy is in some kind of orbit amongst the other galaxies (velocity unknown).

SO MUCH FOR USING THE EARTH AS A STATIONARY LABORATORY.

2.1 The Ether By the 1880’s scientists knew that waves transferred energy from one place to another and their

movement depended upon them travelling through a ______________ (water waves in water,

sound waves in air and other materials).

This led them to believe that ALL waves required a medium for travel, and so to development of

the concept of the Luminiferous _______________, (or aether) which was the name given to

the medium through which light supposedly travelled from the sun to earth.

The ether was a hypothetical medium in which it was believed that electromagnetic waves

(visible light, infrared radiation, ultraviolet radiation, radio waves, X-rays, would propagate.

2.2 The Speed of Light

In 1887, Albert A. Michelson and Edward W. Morley working at the Case School in Cleveland,

Ohio, tried to measure the _________________ of the ether, (or more precisely the speed of

the Earth through the ether).

They expected to find the speed of light (symbol, c) _____________ depending on its direction

with respect to the “ether wind”. This result would accord with Galilean relativity.


Michelson explained his experiment to his children this way: two swimmers race; one struggles

upstream and back while the other swims the same distance across and back. The second

swimmer will always win, if there is any current in the river.

The result of the Michelson-Morley experiment was that the speed of the Earth through the

ether (or the speed of the ether wind) was _______________________.

Therefore, they also showed that there is no need for any ether at all, and it appeared that the

speed of light (in a vacuum) was independent of the velocity of the observer!

2.3 Michelson Morley in Detail The experiment was set up using a “monochromatic” (single colour) light source split into two

beams.

Using an interferometer floating on a pool of _____________, they tried to determine the

existence of an ether wind by observing interference ________________ between the two light

beams. One beam travelling with the "ether wind" as the earth orbited the sun, and the other at

90º to the ether wind.

The interference fringes produced by the two reflected beams were observed in the

_________________. It was found that these fringes did ______ shift when the table was

rotated. That is, the time required to travel one leg of the interferometer never varied with the

time required to travel its normal counterpart. They _____________ got a changing interference

pattern.

ether


The travel times for the two beams were compared in a very sensitive manner.

If the travel times were different the two beams, when combined, would have produced an

“interference pattern”.

This is the same as the pattern produced when a monochromatic beam of light is allowed to

pass through two narrow slits.

NO ______________ IN THE PATTERN COULD EVER BE DETECTED WHEN THE EQUIPMENT TURNED THROUGH 900

Michelson and Morley repeated their experiment many times up until 1929, but always with the

same results and conclusions.

Michelson won the Nobel Prize in Physics in 1907. Probably the only prize ever awarded for a

failed experiment.

The result proved to be an extremely perplexing and frustrating to the physicists of the day who

firmly believed in the ether theory.

The result proved, beyond doubt, that the speed of light is _______________, no matter how

fast an observer was travelling when measuring it.

In other words, it led to the death of the ether concept and, more importantly, the death of

Galilean Relativity

It took nearly 20 years to develop the theory to match this experimental result.

Chapter 3 3.1 Einstein’s Insight

It was Einstein who finally found an answer to the seemingly unbelievable result – that the

speed of light in inertial frames of reference is always the ______________.

The answer was to change the understanding of the term simultaneity.

Incident Light

Double Slit

Interference Pattern


Two physical events that occur simultaneously in one inertial frame are only simultaneous in

any other inertial frame if they occur at the same time and at the same place. This means:

TIME IS RELATIVE!

The figures to the left, seen from two different inertial frames,

help clarify the concept of simultaneity:

Fig 1:

In the inertial frame of the wagon, the lamps are switched on

simultaneously and the two light impulses reach the girl at the

same time

Fig 2:

In the inertial frame of the observer outside the wagon, it

seems that the _________ lamp is switched on first, although

for the girl in the wagon the lamps are switched on

simultaneously.

3.2 Introducing Relativity

Einstein developed the theory of Special Relativity in 1905 and the more

comprehensive and far more complex theory of General Relativity about

10 years later.

Special Relativity deals with large velocity differences between frames of

reference (Inertial Frames).

General Relativity deals with large acceleration differences between

frames of reference (Non inertial Frames)

At low speeds, Newton’s laws are adequate to explain motion.

But the relativity theories need to be applied to objects travelling at or near c, the speed of light.

Speeds of objects Inertial Frames Non inertial Frames

Very much less than c Newton’s Laws Newton’s Laws

Plus Fake Forces

Close to c Special Relativity General Relativity

Fig 1

Fig 2


3.3 Special Relativity • The theory of Special Relativity was developed by Einstein in 1905 when, as a 26 year

old, he was working as a clerk in the Swiss Government Patents Office.

• Basically the theory states:

1. The laws of physics are _______________ for all observers, provided they are moving at

constant velocity with respect to one another, i.e., they are all in inertial frames of reference.

2. The SPEED OF LIGHT is _____________________. This is true no matter how fast the

observer is travelling relative to the source of light.

This theory was completely at odds with the classical physics of Aristotle, Galileo and Newton.

After studying the results of the Michelson - Morley experiments, Einstein proposed the

following:

THE SPEED OF LIGHT IS ALWAYS THE SAME, REGARDLESS OF WHO MEASURES IT

AND HOW FAST THEY ARE GOING RELATIVE TO THE LIGHT SOURCE.

From this simple statement a number of startling consequences arise:

Chapter 4 4.0 Time Dilation The first of these consequences is known as “Time ______________”

It requires that, depending on the motion of an observer, time must pass at _____________

rates.

Two observers, one stationary, the other moving near the speed of light, observe the same

event.

In order for each to get the same ___________ for the event, each must see it occur during

different _____________ intervals.

The faster the observer travels the slower the rate at which his time appears to pass to

stationary observer.

In order to demonstrate this change in the rate at which time passes, let us produce a simple

clock.

A photon of light (travelling at c) is bouncing backwards and

forwards between two parallel mirrors.

One back and forth motion of the photon represents one ______ of

the clock.

Photon Clock


An observer, stationary in space with respect to the Sun, sees the Earth (with its attached

“clock”), go zooming past on its orbit around the sun.

In the time, we (standing on Earth), see the photon bounce back and forth once, the space

observer sees the Earth move a little way along its orbit path.

Hence, if the photon is to strike the mirrors, the space observer requires it to travel on a

_______________ path, as shown.

Clock as seen by Earth bound observer Clock as seen by Space observer

Since the photon MUST travel at the Speed of Light, c, the only logical outcome for the space

observer is to conclude that the photon on Earth takes a ____________ TIME to cover the

APPARENTLY LONGER DISTANCE it needs to travel.

The space observer thus concludes the Earth clock runs slow compared to his clock.

This same argument holds true for the earth bound observer, who would see the space

observer’s clock running slow.

Earth Earth Earth

Photon

Short Distance Speed = c

Earth Earth Earth

Long Distance

Speed = c

Remember, Speed = distance time

If this has to stay the same

and this gets biggerthis must also get

bigger

Thus, MOVING CLOCKS RUN SLOW.


The mathematical representation of Time Dilation is shown in the formula:

t = γto

γ is called the “_____________ Factor”

where:

t = moving observer’s time as measured by the stationary observer.

to = time measured by stationary observer’s clock. (“proper time”)

v = speed of moving observer.

c = Speed of Light.

• The formula has a number of consequences:

• If v << c, the term v2/c2 approaches zero and the square root term approaches 1.

• Thus t = to and no change in time (the rate at which time passes) is observed.

• As v approaches c (say v = 0.9c), the stationary observer sees the moving observer’s

clock tick over only 0.4 sec for every 1 second on his own clock.

• If v = c, the term v2/c2 = 1 and the square root term becomes zero. Dividing a number by

zero equals infinity.

• Thus, when v = c the time interval becomes ______________. In other words, time stops

passing.

4.1 The Twins Paradox

Twins, Adam and Eve, are thinking how they will age if one of them goes on a space journey,

travelling at say 0.866c.

Will Eve be younger, older, or remain the same age as her brother if she does a round trip of

some years duration ?

Assume that Adam and Eve’s clocks are synchronized before Eve leaves.

where 1

1 -c2 v2γ =

to

1 -c2 v2t =So


At 0.866c, Adam will “see” Eve’s time pass at exactly _________ the rate his time passes. So

when Eve returns, she will have aged by 1 year for every 2 years Adam has aged.

Thus, Eve is younger than Adam.

However, can you turn the discussion around and say that Eve has been at rest in her space-

ship while Adam has been on a "space journey" with planet Earth?

In that case, Adam must be younger than Eve at the reunion!

Adam is at rest all the time on Earth, i.e., he is in the same inertial frame all the time, but Eve is

not - she will have felt forces when her space-ship accelerates and retards, and Adam will not

feel such forces. So the argument is not an interchangeable one. The travelling twin is the

younger upon their reunion.

P.S. Eve's space-ship has to consume fuel, which means that it costs to keep yourself young!

4.2 Length Contraction The 2nd consequence of light having a constant speed is Length __________________.

An observer sees two set squares, one stationary in his inertial frame,

the other in an inertial frame moving near the speed of light.

How does the speed difference affect the apparent size of the set square ?

Remember the stationary observer sees the moving “frames” clock running slow. To get the

same value for c in each frame, he must measure the length of the set square (in the direction

of travel) to be shorter than his own stationary ruler.

Remember: Speed = distance

time

v = 0 v = 0.8c

y

x


IMPORTANT NOTE:

The length contraction only occurs in the direction of travel (x direction) and measurements at

right angles to that direction are unaffected ! (no contraction in the y direction)

The mathematical representation of Length Contraction is shown in the formula.

L = Lo/γ

Where:

L = Length of moving object as measured by stationary observer.

Lo = Length of stationary object measured by stationary observer. (“Proper Length”)

v = speed of moving object.

c = Speed of Light.

The formula has a number of consequences:

1. If the v << c, the square root term approaches 1 and the length is unaffected, i.e. L = Lo

2. As v approaches c, v2/c2 approaches 1 and the square root term approaches 0. Thus, the

length approaches 0 i.e. L = 0.

So, a photon of light travelling at c from the Sun to the Earth makes the journey in ____ time

and travels ______ distance !!!!!!

The moving observer’s view

of the length contracted world

The stationary observer’s view of

the length contracted Superman

4.3 Mass Dilation

The third effect of the invariance of the speed of light is mass ____________________.

As the speed of an object _________________ so too does its ____________ !!!!!!

Under Newton, mass is an ________________ quantity for each object and it is conserved,

never changing for each object. This invariance of mass is the basis of Newton’s 2nd Law (F =

ma), and our own every day experience seems to verify that mass is absolute.

Lo( 1 -c2 v2

So, L = )


Under Einstein mass is whatever we measure it to be.

We must use an operational definition for mass.

He showed that the mass of an object depends on how fast the object is moving relative to a

stationary observer.

4.4 Mass – Newton v Einstein

Newtonian physics gives good results at

speeds less than ______ of the speed of light.

Einstein’s relativity deals with faster speeds.

The mass of an object does ______ change

with speed, it changes only if we cut off or add

a piece to the object .

As an object moves faster its mass

_____________. (As measured by a stationary

observer).

F = ma means that to accelerate a mass

requires a force, by supplying sufficient force

you can make an object go as fast as you like.

Mass approaches _______________ as

speed approaches c. To reach c would require

infinite force.

Kinetic Energy = ½mv2, since mass does not

change an increase in KE means an increase

in speed.

Since mass changes with speed, a change in

K.E. must involve both a change in speed and

a change in mass.

At speeds close to c most of the change

occurs to the mass.

4.5 Mass - How Fast, How Heavy ? The mass of an object at rest is called its rest mass (m0)

At low velocities the increase in mass is small.

An object travelling at 20% of the speed of light (60,000 kms-1) has an apparent mass only ____

% greater than its rest mass (m0).


Mathematically:

m = γm0

where:

m = Apparent Mass of the object

m0 = Rest Mass of the object

v = speed of object.

c = Speed of Light.

1. When v << c, the square root term approaches 1, and m = m0

2. As v approaches c, the square root term approaches 0, and m approaches ____________.

There is insufficient energy in the universe to accelerate even the smallest ______________ up

to the speed of light !!!!!!!!!!

0 20 40 60 80 100

m0

2m0

4m0

6m0

% of Speed of Light

Speed of object as seen by a stationary observer

m0

1 -c2 v2So, m =


4.6 Energy & Mass

Increasing the speed of a ________ requires energy.

The fact that feeding energy into a body increases its mass suggests that the rest mass m0 of a

body, multiplied by c2, can be considered as a quantity of energy.

The truth of this is best seen in interactions between elementary particles. For example, if a

positron and an electron ______________ at low speed (so there is very little kinetic energy)

they both disappear in a flash of electromagnetic radiation.

This EM radiation can be detected and its energy measured.

It turns out to be 2m0c2 where m0 is the mass of the electron (and the positron).

So each particle must have possessed so called “rest energy” of m0c2

Einstein recognised the fundamental importance of the interchangeability of ___________ and

_____________ which is summarised in his famous equation:

E = mc2

where m is the Apparent Mass.

4.7 Rest Energy

If an object is at rest it possesses “rest mass energy” or more simply “rest energy”

Einstein’s equation is then written as:

E = m0c2

Where E = Energy (joules)

m0 = Rest Mass (kg)

c = 3 x 108 ms-1

How much energy does 1 kg of mass, at rest, represent ?

E = m0c2

= (1)(3 x 108)2

= 9 x 1016 Joules

This represents the average annual output of a medium sized Power Station

A Hiroshima sized atomic bomb releases about 1014 Joules, (100,000 billion joules).

How much mass has been converted ?


E = m0c2

Thus m0 = (1014)/(3 x 108)2

= 1.1 x 10-3 kg

= 1.1 g

As can be seen a tiny mass converts to a huge amount of energy

4.8 Moving Mass

As a mass begins to move it possesses BOTH rest mass energy AND energy of motion (Kinetic

Energy).

Expressing Einstein’s equation as:

E = mc2

Includes both rest mass and kinetic energy

The Kinetic Energy of a fast moving particle can be calculated from:

K.E. = mc2 – m0c2

The relativistic energy of a particle can also be expressed in terms of its momentum (p) in the

expression:

E = mc2 = (p2c2 + m02c4)½

This is essentially defining the kinetic energy of an object as the excess of the object’s energy

over its rest mass energy.

For low velocities this expression approaches the non relativistic kinetic energy expression.

For v/c << 1,

KE = mc2 – m0v2 ≈ ½ m0v2

As an object’s speed increases more and more of the energy goes into increasing _________

and less and less into increasing ____________.

Chapter 5 5.0 The Speed of Light. A Limit ?


These equations together are called “The Lorentz Transforms”.

Each Lorentz Transform has a limiting factor.

If v > c, then:

• t becomes negative, and time runs _____________________ !!!!!! (the

bullet hits you BEFORE it is fired from the gun).

• L becomes negative, and an object has a length less than zero!!!!!,

• m becomes negative and objects have a mass less than zero!!!!!

• Thus, c (the speed of light) is the limiting factor.

• Speeds greater than c are not possible.

5.1 Relativistic Speed Addition

Imagine that you are standing between two space-ships moving away from you.

One space-ship moves to the left with a speed of 0.75 c

(relative to you) and the other one moves to the right also

with a speed of 0.75 c (again relative to you). At what

speed will each space-ship see the other moving away?

0.75 c + 0.75 c = 1.5 c? No, their relative speed will be

0.96 c (according to the relativistic addition of velocities),

and it cannot, of course, be faster than the speed of light c.

In classical Newtonian mechanics, two different velocities

and are added together by the formula

v” = v’ + v

where v” is the sum of the two velocities.

However, in special relativity, the velocities are added together as

This formula is called the relativistic addition of velocities.

Note that if v’ = c and/or v = c, then v” = c, and for small velocities v, v’ << c, then the classical

formula is regained.

5.2 Special Relativity Experimental Proofs

v =0.75c v’=0.75c

v” = 0.96c

v” = v’ + v 1 + v.v’

c2


Experimental proof of the for each of the areas of Time Dilation, Length Contraction and Mass

Dilation are available on Earth. These are shown below.

5.3 Relativistic Doppler Effect The Doppler Effect:

Motion towards or away from a source will cause a change in the observed ______________, f

(or wavelength, λ) as compared to the emitted frequency.

All wave phenomena (e.g., water, sound, and light) behave in this way.

Suppose a source emits light of frequency f (or wavelength λ, remembering that c = fλ). Then,

an observer moving with a speed v away from the source, will observe the frequency:

This formula is called the relativistic Doppler formula. Note that f < f0 for all 0 < v < c, i.e., the

frequency which the observer sees, is smaller than the "original" frequency in the inertial frame

of the source.

Observers moving away from the source will see a _______________ in the frequency of the

light, since light with lower frequencies are "more red" and light with higher frequencies are

"more blue." While observers moving towards the source will see a corresponding

________________.

If you are driving towards a red traffic light (λ0 = 650 nm) at a speed of approximately v = 0.17 c,

the traffic light will actually appear to be green (λ = 550 nm)! (0.17 c is approximately 5.0 x 107

ms-1.)

f = f0 c – v c + v


5.5 Special Relativity Conclusion • I leave the last word to Einstein himself who, when asked

to describe Special Relativity in laymen's terms, said:

• “Put your hand on a hot stove for a minute, and it seems

like an hour.

• Sit with a pretty girl for an hour and it seems like a minute.

• That’s relativity”

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