Conceptual Introduction to Special Relativity

ConceptualConceptual
Introduction toIntroduction to
Special RelativitySpecial Relativity
www.fisicaconceptual.net

Part 1:Part 1:
Introduction, postulatesIntroduction, postulates
and first consequencesand first consequences

What can I expect
from this presentation?
What it is not: the typical
explanation of relativity
in 2 minutes
When we stay
at a superficial level,
we cannot deeply understand,
neither enjoy, nor value,
the meaning and implications
of relativity.

Can I understand relativity?
It is possible with a bit of effort, as long as
you allow your previous ideas to change.
Do I need to know Mathematics?
Mathematical demonstrations in
Special Relativity are not complex. This
presentation almost does not use them.
Do I need to know Physics?
Physics knowledge is almost not required
to understand the basics of the theory.
In this presentation we will only use the
formula v=e/t

Then
Where is the problem?
Why do people not understand
relativity?
The difficulty is that this theory
seems to go against our experience.
Moreover, it is very hard to adapt to this
new way of interpreting reality.
Many reasoning mistakes are made
since we are still thinking in terms of
absolute space and time.

Disclaimers
It is necessary to forget
all previous experience:
Almost everything we think about
space and time is false...
If we focus on following our experience
we will feel confused, and misled.
Nevertheless, relativity is correct!
It has been proved in many scenarios, with an
unimaginable precision, and all experimental results
verify the assumed postulates and accord with
theoretical predictions.

We should be guided by thought.
We will start from only two
postulates. We will see that they are
very easy and reasonable.
We must have a clear idea about what those postulates are.
And from them we will deduce what they imply.
the conclusions that we achieve will be inevitable,
as strange as they may seem.
Therefore, if these postulates are correct,

Free yourself from
experience:
It is not applicable to what
we will do.
Be open minded.
Use logic to deal with the
intellectual problems we are
going to face.
A new reality awaits us...
To sum up:

2) Test the theory’s coherence
(despite the paradoxes that will appear)
Teaching about the relativist concept of mass and energy
is not an objective of this presentation
(because it requires some knowledge in physics).
But still, the foundations of Relativity will be taught.
Objectives:
1) Comprehend the relativity
of space and time
- Where they come from
- What they mean and imply

One of the
most beautiful
(and significant)
experiments in Physics
The Michelson-
Morley experiment
Wikipedia
… despite the fact that
it was a “failure”

Historical context:
as sound needs air to spread itself, it was thought that
light needed something too (the aether).
If it is still,
from the front and the back
it gets away at 340m/s
At what speed does sound get away from a plane?
If the plane is moving at 240 m/s,
sound will get away from
the front at 100 m/s and
from the back at 540 m/s

Similarly, Michelson and Morley
wanted to measure the speed
the Earth moves in relation to the aether.
If the Earth is not moving in relation to the aether,
it does not matter where we point,
light will get away from the Earth at the same speed.
If the Earth moves in relation to the aether,
the speed will be different depending on where
the light beam is going.
This would allow us to measure the Earth’s speed.

The experiment was reviewed and done again,
improving it and changing the designs time after time,
and the results continued to show that
the speed of light remained same.
The light speed constant was key to the
development of relativity.
The result made no sense: it was as if
the Earth stood still in relation to the aether,
but obviously it cannot be still...
But, no variation in the speed of light
was detected...

Postulate of the light speed constant:
The speed of light is always the same,
no matter the speed of the emitter or the observer.
Relativity postulate:
All inertial reference frames (r.f.) are equivalent.
There is no preferred frame.
Notation:
We call inertial reference frame (r.f.) all frames with
constant speed.
Inertial reference frame examples:
- a car in a highway at a constant speed of 100 km/h
- a spaceship at a constant speed of 50000 km/h

Both postulates have been checked over and over again.
In the Einstein era there were already many proofs which
showed that both were valid (for example the result of the
Michelson-Morley experiment is in both postulates).

This postulate was already, partly, accepted by all
physicists. It was known as Galileo’s Relativity Principle.
What happened was that, until Einstein this principle was
only used in the context of mechanics.
Einstein generalized this principle. Since then,
it is also used in the context of electromagnetism
(therefore, in the context of light too).
With this postulate, the result of the Michelson-Morley
experiment can be pretty well understood:
the Earth being idle and moving are equivalent.
Light must have the same speed in both cases.

With this postulate, the result of the Michelson-Morley
experiment can be pretty well understood:
even if the Earth is moving in relation to the Sun,
light emitted from it will always travel at the same speed.
The speed of light = 300 000 km/s
Notation: Speed of light = c

There is no point in trying to measure the Earth’s speed in
relation to the aether, nor to any absolute space. That
absolute reference frame does not exist (nor does the aether).
The Michelson-Morley experiment was pursuing something
impossible.
The phrase: “stand still” lacks any sense, unless you indicate
in relation to what it is still. Standing still in relation to space
also makes no sense, because there is no absolute space.
Immediate consequence:
There is neither an absolute reference frame nor
absolute space

The classical Law of Speed Addition is false
0,1 c
As the laser beam moves away
from the spaceship at c, does it
move away from earth at 1.1c?
No. It also moves away at c
Immediate consequence:

From the centre of the
spaceship two light beams
are fired in opposite directions
Consequence:
Demonstration:
For the observer inside
the ship both beams hit
both walls at once.
Simultaneity is relative

v Seen from the Earth,
both laser beams travel
at the same speed:
the speed of light.
But, seen from the Earth,
the spaceship is moving
towards the right at a speed v.
Consequence: Simultaneity is relative

v From Earth, the left beam
will hit the wall sooner.
Thus, simultaneous events
inside the spaceship
(both beams hitting the walls),
aren’t simultaneous
for a observer on the Earth.
Consequence: Simultaneity is relative

In the second part of
this presentation we
will continue to explore
the consequences of
the assumed
postulates
We will reach a
significant result:
Time Dilation

This presentation is published under the license:
Creative Commons Reconocimiento-NoComercial-
CompartirIgual 3.0 Unported License
(http://creativecommons.org/licenses/by-nc-sa/3.0/)
For use beyond this license you can request permission here:
http://www.fisicaconceptual.net
Author: Juan Antonio Martínez-Castroverde Pérez
Licentiate in Physics.
Secundaria and Bachillerato teacher
English translation: Carlos Rodríguez Pérez
María José Lorenzana Sánchez

Part 2:Part 2:
Time DilationTime Dilation

Lets continue
exploring the
consequences of
the assumed
postulates.
We will reach a significant result:
Time Dilation

Before we continue, it is convenient to
distinguish between what actually happens in
the universe and what we can see; these are
not always simultaneous.
The reason why sometimes they do not coincide is that light,
even though it travels very fast, it takes time to propagate.
When something happens very close to us, we see what
really happens, practically, instantly.
But when the incident occurs very far from us, it will be a
reality even though light has not reached us yet and, for that
reason, we have not seen it.

For example: if the Sun were to go out, it
would take us around 8 and a half minutes
to stop receiving light.
The Sun would have truly gone out, but
we would still live, oblivious, as if nothing
happened in the time that light takes to
reach us.
There are two ways of analysing the surprising
conclusions that Relativity offers:
Considering what the different observers see or,
as we will do in this presentation, considering what is
really happening, even if the observer cannot see it yet.
We will use expressions such as “if we could see”, in
order to reference that reality, which we will not always
be able to confirm instantly.

Thought experiment:
r.f. spaceship
On the left, what can be seen
inside the spaceship when
a red laser beam is shot.
From when the laser is shot,
reflected in the mirror and bounced back,
one second elapsed
according to the astronaut's watch.
If we could “magically” see what happens
inside a spaceship that travels at high speed,
What would we see?

v=
e
t
→ t=
e
v
Using:
t=
e
c
We conclude
that, for light:
Using the Light speed invariance postulate, we can see
there is a direct relationship between the distance travelled
by light and the time it takes to do so:
The longer the distance travelled
by light, the more time has elapsed

v
Seen from the earth, the spaceship travels at high speed
and the laser beam follows the trajectory shown below.
From the earth, it
can be seen that
the second hand of
the spaceship’s
clock advances one
mark (1s) from
when the laser
beam is shot until it
bounces back.
But: does a clock on the
earth advance 1 s from when
the laser beam is shot
until it bounces back?
r.f. Earth

For both observers the speed of light is the same,
but the distance travelled by light is different:
v
r.f. Earth
r.f. Spaceship

Since: t=
e
c
The perception of the passage of time
will be different for each observer...
v
r.f. Earth
r.f. Spaceship

A clock on the earth advances more than
1 second since the laser beam is shot until
it bounces back
v
t=
e
c
r.f. Earth
r.f. Spaceship

Conclusion:
Δt '=Δt
√1−
v2
c
2
An observer on the earth that could see what
happens inside the spaceship, would see that
time in the spaceship passes slower, like in
“slow motion”: time dilates
If the observer on the earth looks at a clock in the
spaceship and sees that it advances Δt', while the
clock on the earth advances Δt, the relationship
between them is:
(this relationship can easily
be deducted using The
Pythagorean Theorem)

Δt '=Δt
√1−
v2
c
2
More than the equation itself,
what needs to be highlighted
now are two extreme cases:
v=0 → Δt ' =Δt
√1−
02
c2
→ Δt ' =Δt
v=c → Δt '=Δt
√1−
c2
c
2
→ Δt '=0
As a spaceship approaches the speed of light, we see that
time in it “freezes” increasingly.

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
0
10
20
30
40
50
60
Spaceship's velocity (in c)
For example: If a spaceship travels at 0,866 c, when the
clock on the earth advances 1 hour, we would see the
clock on the spaceship advance 30 minutes.
For each hour on earth,
How many minutes pass in the spaceship?

t=
e
c
Time Dilation is a logical need of the
light speed invariance postulate
v
r.f. spaceship
r.f. Earth
It is not an optical illusion!

But...
In the spaceship
time elapses as usual!
Only the observer on the
Earth watches the dilation of
time inside the spaceship.
(Equivalence postulate
of all inertial reference frames)

v
Paradox
(Proper Time / Improper Time)
From the spaceship
it is the Earth that moves
When the astronaut looks at
the Earth he sees that time
goes slower there.
r.f.
spaceship

When my watch advances 2s,
the time in the spaceship
advances 1s

advances 1s
the time on the Earth
advances 1s

advances 1s
advances 1s
¿absurd?

NO it is not absurd!!
¿absurd?

Bare in mind that:
✔ Paradox = seemingly contradictory statement
✔ Time Dilation has been proven experimentally
For example, particle physicist prove it daily, when they measure particles
that travel at speeds close to c, they would disintegrate before being able
to make those measurements if Time Dilation did not exist.
NO it is not absurd!!
¿absurd?

How is it not absurd?
Think about it!
Give yourself a chance!

The explanation is...
Time is
relative!

What does that mean?
The explanation is...
Time is
relative!

Lets consider a simple example:
80 km/h100 km/h
Is it absurd to say that the car travels
at a 100 km/h speed in relation to the asphalt
and a 20 km/h speed in relation to the bus?

It is not absurd,
since speed is relative

It is not absurd,
It is absurd to say that
the car travels at a speed of
100 km/h and 20 km/h
in relation to the asphalt
simultaneously.
100 km/h
20 km/h
However...

100 km/h
20 km/h
It is not absurd,
However...
It is absurd to say that
the car travels at a speed of
100 km/h and 20 km/h
in relation to the asphalt
simultaneously.

Similarly, it is absurd to say that both statements about
the passing of time are right “simultaneously”

advances 1s
advances 1s
Similarly, it is absurd to say that both statements about
the passing of time are right “simultaneously”

Time is relative means
that the passing of time
depends on each observer.

As a consequence,
Every statement about how time passes
only has validity in that r.f.

No statement has universal validity
because there is no absolute time.
As a consequence,
Every statement about how time passes
only has validity in that r.f.

But…if the spaceship stops and the passenger
descends to the Earth, then both observers will
share the same r.f. and their past experiences are
incompatible!!!
Paradox
(Twin Paradox)

The most famous version of this paradox is the Twin
Paradox...
Be patient. The time to address this paradox has not
come yet. We will come back to it later.
Paradox
(Twin Paradox)
But…if the spaceship stops and the passenger
descends to the Earth, then both observers will
share the same r.f. and their past experiences are
incompatible!!!

In the 3rd
part of the
presentation we will
continue to explore the
consequences of the
assumed postulates.
We will reach another
significant result:
Length
Contraction
v

This presentation is published under the license:
Creative Commons Reconocimiento-NoComercial-
CompartirIgual 3.0 Unported License
(http://creativecommons.org/licenses/by-nc-sa/3.0/)
For use beyond this license you can request permission here:
http://www.fisicaconceptual.net
Author: Juan Antonio Martínez-Castroverde Pérez
Licentiate in Physics.
Secundaria and Bachillerato teacher
English translation: María José Lorenzana Sánchez

Part 3:Part 3:
Length ContractionLength Contraction

Length
Contraction
0,866 c 0,990 c

Lets continue
exploring the
consequences of
the assumed
postulates
We will now reach another significant result:
Length Contraction

Consequence: Length contraction
Lets assume a trip to a planet, 10 light-years away,
made by a spaceship going at a speed of 0,866c

Seen from the Earth:
- The trip lasts 11,55 years
- The spaceship moves away
at a speed of 0,866 c
- At the end of the trip the
spaceship’s clock indicates
5,77 years (half of the time)

Seen from the
spaceship:
- The speed has
been 0,866c
- The trip lasts 5,77
years

Seen from the
spaceship:
- The speed has
been 0,866c
- The trip lasts 5,77
years
However, inside the spaceship, the 5.77 years have
passed normally. Its time flow is as valid as the time
flow on the Earth.

How is it possible that travelling at the same speed
observed from the Earth the trip is shorter from the
spaceship?

v=
e
t
=
e
2t '
v '=
e'
t '
Notation:
Being v, e y t the speed of the spaceship, the distance
travelled and the time it takes seen from the r.f. Earth
and v' e' y t' the same but seen from the r.f. spaceship
e'
t '
=
e
2t '
→
v '=v
e '=
e
2
How is it possible that travelling at the same speed
observed from the Earth the trip is shorter from the
spaceship?
Time Dilation

This result does not only affect the
distance, but space itself:
Seen from the spaceship,
The space has contracted in the
direction of the speed.
Therefore, not only the distance
contracts, the planets that the spaceship
travels by also contract, adopting the
shape of a lentil.
The answer is clear:
On the spaceship the trip is shorter because the
distance between the two planets has been
reduced by half.
0,866 c

However,
for an observer in any of the planets, it is the
spaceship that moves and it will be the spaceship
that contracts in the direction of its speed.
Both the objects we see go
by and the space they are in
(relative space) contract in
the direction of the speed in
which we see them move.
Conclusion:
Seen from the planets,
the Earth remains unchanged

0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1
0
10
20
30
40
50
60
70
80
90
100
Spaceship's velocity (in c)
Example:
If a spaceship goes by at 0,866 c, its length is reduced by half
Seen from the Earth,
How big is a 100 m long spaceship?

Disclaimer: length contraction is not an
“optical illusion” due to speed. It is real...
it is a logical consequence of the assumed
postulates.
Lets design a thought experiment to
illustrate what length contraction is real
means.
Moreover, we will come across a shocking paradox
with an even more shocking solution.
Lets put it into perspective:

advances 1s
advances 1s
How did we solve this paradox?

advances 1s
advances 1s
How did we solve this paradox?
Each statement is only valid in its r.f.
Each
statement
is “isolated”
from the
other

We propose now a thought experiment
that will “connect” those two different r.f.
How will we solve the paradox now?

If we had two spaceships, exactly the same but
different colours:
Here we see
the two
spaceships
when idle,
before starting
to travel.
We propose now a thought experiment
that will “connect” those two different r.f.
How will we solve the paradox now?

In order to prove that length contraction is real, they
come up with the following experiment:
The spaceships pass by each-other at a relative speed
of 0,866 c
r.f. R
v
Seen from r.f. R, the blue spaceship is the one moving
at 0,866 c and, consequently, the blue spaceship
contracts to half its length.

When they cross paths, as the marks of the
spaceships coincide, the red spaceship shoots two
deadly laser beams.
r.f.R
Supposedly there is no problem, since the blue
spaceship has contracted by half:
it can be confirmed that length contraction is true.
v

...however, from the blue spaceship, (r.f. B)
the one that moves and contracts
Is the Red Spaceship!!!
v
Would you accept the red spaceship shooting
the laser beams when the marks coincide?
Paradox
(of the ladder and the barn)
r.f. B

Do not worry, if seen from the red spaceship you do
not die, you cannot die: it would not make sense to die
in one reference frame and not the other...
r.f. B
v
But we come across a seemingly impossible situation.
How can we explain all this?

The explanation comes from another shocking
outcome of Relativity previously obtained:
v
Simultaneous events in a reference frame (shooting
the two laser beam when the marks coincide) are not
simultaneous in another r.f.
Simultaneity is relative
r.f. B

From the blue spaceship,
the chain of events is as follows:
v
The red spaceship shoots the first laser beam
without reaching the blue spaceship.
r.f. B

Both marks coincide, but there is no shot.
v
r.f. B

r.f. B
v
The red spaceship shoots
the second laser beam
without reaching the blue spaceship.

r.f. R
The spaceships experiment confirms that
length contraction is real
(the blue spaceship is not destroyed)
and a seemingly absurd situation has been overcome.
v

Is it time for a break?
You may be surprised or a bit confused...
We have seen that time and space, which we always
considered absolute… in reality they are not.
Even more surprising, simultaneity, which we also
thought was absolute, is not either...

A problem arises:
If we were wrong
in our way of
understanding concepts
as simple as space, time
and simultaneity,
we reach a point in which
we doubt everything.
Aside from the
assumed postulates,
What is left as true?

But have patience:
we will leave
most of these
considerations
for the appendix.
Take it easy:
Things that are given
as true in this
presentation and that
maybe you are now
questioning.
There are things
we can be sure about.

In the 4th
part of the
presentation we will
continue to explore the
consequences of the
assumed postulates.
We will solve one of the
most famous paradoxes of
Special Relativity:
The Twin Paradox

Part 4:Part 4:
Twin ParadoxTwin Paradox

Twin Paradox
Mark (L) and Scott (R) are the only twins to have traveled to space. (NASA)
http://www.theverge.com/2016/3/1/11138102/scott-kelly-year-in-space-health-effects-return-to-earth

Lets continue
exploring the
consequences of
the assumed
postulates
We will see that the consequences of the
theory, far from creating contradictions,
they enable us to solve one of the most
famous paradoxes of Special Relativity.

One twin goes on a space trip while the other one stays on
the Earth.
Seen from the Earth, the travelling twin ages much slower.
But seen from the spaceship the Earth is the one moving,
therefore the twin on the Earth is the one ageing much
slower.
Twin Paradox
What happens when they meet?

First of all we should point out that the event is
not symmetrical. The travelling twin experiments
acceleration (for example, to be able to come back).
Due to that acceleration a “re-adjustment” of what can
be seen from the spaceship is made, so that both
observers agree that the twin on the Earth ages more.
But...
What is really happening? How do we “fix” the
different views on the passage of time?

Then, for maximum clarity, we propose a thought
experiment where we are able to illustrate what
really happens in the most simple and visual way
possible.
Disclaimer: The resolutions of this paradox usually
offered in works about Special Relativity are of a
certain degree of complexity.
In order to do this; an specific case, ideal for this simple
visual representation, has been chosen: the relative
speed between the two observers is 0,866c. Therefore,
the space of the other r.f. is contracted by half and the
time in the other r.f. advances half as fast.

B
TB
A
TA
v
r.f. B
Lets suppose that, from r.f. B, when mark A
coincides with mark B, the clock in A shows TA
and the clock in B shows TB.
Before we continue, lets make a notation, to
present the result we are going to use:
The blue and red rectangles are two objects
that, when idle, are the same length.

Important result: “Law of Connection”
B
TB
A
TA
vr.f. B
When mark A coincides with mark B,
the clock in A and the clock in B show a value
that does not depend on the r.f. from where
it is looked at. (demonstration in the appendix)
Now we can go back to the Twin paradox...

We present the leads of our rationale:
8,66 light-years
A B
The Earth (A) and another inhabited planet (B):
A' B'
A spaceship (A') and another one going in the opposite direction (A'')
A'' B''

A B
The rectangles represent the space of each r.f.
8,66 light-years
A B
A' B'
A'' B''
Space is relative! From each r.f. you can see a space.
What do the rectangles represent?

A B
8,66 light-years
A B
A' B'
A'' B''
The blue rectangle represents the space, as seen
from r.f. A (or from r.f. B).
r.f. A is the same r.f as r.f. B since A does not move in
relation to B (they are going at the same speed)

A B
8,66 light-years
A B
A' B'
A'' B''
The Red rectangle represents the space, as seen
from r.f. A'. For example, the imaginary point B' is always
8,66 light years from A', seen from A'.

A B
8,66 light-years
A B
A' B'
A'' B''
The Blue rectangle is like a structure connecting
planets A and B extending towards the right.
If it makes understanding easier, we can see the
rectangles as a real material structure:

A B
8,66 light-year
A B
A' B'
A'' B''
The Red rectangle is like an extension of the red spaceship,
thus it will move with it.
The Green rectangle is like structure that will move with the
green spaceship.

A B
8,66 light - years
A B
A' B'
A'' B''
Yes! These imaginary structures (the blue, the red and the
green) will each have a huge length:
double 8,66 light-years.
But this is not worrying, since this is a thought experiment.

A B
8,66 light-years
A B
A' B'
A'' B''
R.f. A' is the same r.f. as r.f. B' since A' does not move in
relation to B' (they are going at the same speed).
Equally r.f. A'' is the same r.f as r.f. B'' since they are going
at the same speed.

A B
Seen from any r.f.,
the clocks of that same r.f. are synchronised:
A B
A' B'
r.f. A
0 00
0 0 0
r.f. A'
What “can be seen” from
What “can be seen” from

A B
The trip seen from the Twin on Earth r.f. (A)
A B
A' B'
0,866 c
The distances in r.f. A' contract by half.
0
0 0 0
We suppose that when A' is in front of A
their respective clocks show 0.
8,66 light - years
r.f. A

A B
The trip for the twin on Earth (A) lasts 10 years:
A B
A' B'
0,866 c
If he could see the travelling twin’s clock (A') he would see
that time advances half as fast.
5
10 10 10
When A' gets to B his clock shows 5 years.
8,66 light - years
t=
e
v
Both the observer in A'
and the one in B agree
with what both of their
clocks show.

A B
The green spaceship comes into scene, with its associated
space and the shown speed.
A B
A' B'
0,866 c
5
10 10 10
8,66 light - years
A'' B''
5
0,866 cWe suppose that
the twin instantly
jumps from A' to
A'' with his clock.
Now in A'' shows
5 years.

Consider: Is it valid to suppose that the twin jumps from
A' to A'' instantly?
0,866 c
5
A'' B''
5
0,866 cWe can object
that, due to
relative speed,
that is impossible,
A' B'
And if that happened
in a very little amount
of time the
acceleration needed
would be huge.
Can the implications that the acceleration
may have be ignored?

To avoid this issue, lets modify our thought experiment:
0,866 c
5
A'' B''
5
0,866 c
We will now
suppose that,
instead of the
twin jumping to
the other
spaceship, in the
A' B'
Illustrated situation
he sends a light signal
with the time of his clock
to the green spaceship
(which is in A'')
Since light does not need acceleration, and A' is very close
to A'', light reaches nearly instantly. This way it is
reasonable to suppose that in the illustrated situation the
clocks indicate the values shown.

A B
The clock in A'' shows,
consequently, the
travel time from when
A' left from A
A B
A' B'
0,866 c
5
10 10 10
8,66 light - years
A'' B''
5
0,866 c

A B
Since the “return” is symmetrical:
A B
A'
0,866 c
10
20 20 20
8,66 light - years
A'' B''
10
0,866 c

A B
Therefore, the “travelling clock” shows
10 years, while the clock on Earth
shows that 20 years have gone by.
A B
A'
0,866 c
10
20 20 20
8,66 light - years
A'' B''
10
0,866 c Due to The Law of Connection, both
the twin in A, and the “twin” in A''
agree with those values.
The “travelling twin” is younger.

A B
Up until now there is nothing new.
We knew that when A looks at A' (or A'') he sees
how time there advances twice as slow.
A B
A'
0,866 c
10
20 20 20
8,66 light - years
A'' B''
10
0,866 c The issue was that from A' ’s point of
view (or A'') the one moving is A,
it is the time in A that advances
twice as slow.

A B
How is it possible then that the clock in A shows
20 years and the clock in A'' shows 10 years and
both observers agree?
A B
A'
0,866 c
10
20 20 20
8,66 light - years
A'' B''
10
0,866 c

0
A
To understand what happens we would have to look
from multiple r.f.. Lets start considering
what can be seen from r.f. A' :
A B
0
0 ? ?
0,866 c
A' B'
8,66 light - years
As it had been established, and due to the Law of
connection, when A is in front of A' both their clocks
show 0.
All the clocks in
r.f. A' are
synchronized:
r.f. A'

0
A
However, as we previously mentioned, what is
simultaneous in r.f. A, for example when all its clocks
show 0, is not seen from r.f. A'
A B
0
0 7,5 15
0,866 c
A' B'
8,66 light - years
Seen from r.f. A', for the speed and distance values
chosen, the clocks in r.f. A show what the illustration
reveals (see appendix).
Now, it is r.f. A that
contracts by half.
The relative speed
is the same.
4,33
light - years

5
A
Applying t=e/v, seen from r.f. A', the spaceship’s trip to
planet B has lasted 5 years. That value is shown in the
clocks of r.f. A'
A B
5
2,5 10 17,5
0,866 c
A' B'
8,66 light - years
Seen from r.f. A', each clock in r.f. A has advanced
at half the speed.
Each of these
clocks has added
2,5 years,as the
illustration shows:
4,33
Light - years

5
A
Seen from r.f. A', and as it should happen (Law of
Connection), when A' coincides with B each clock
shows the same that was seen from r.f. A
(5 for A' and 10 for B)
A B
5
2,5 10 17,5
0,866 c
A' B'
So far, the twin on Earth is younger than the
travelling twin!!!

5
A
Now r.f A''comes into play, which moves at 0,866 c
towards the left in relation to r.f. A.
A B
5
2,5 10 17,5
0,866 c
A' B'
A'' B''
Now we see the lengths
of r.f. A'' reduced to no
more than 15%0,98974 c
Applying the
relativist law of
composition
of speeds,
we obtain,
in relation to r.f. A',
the shown speed..

5
A
In this moment the travelling twin jumps to r.f. A'',
bringing with him his clock. The clock in A'' now shows 5.
A B
5
2,5 10 17,5
0,866 c
A' B'
A'' B''
5
0,98974 c

5
A
We will now draw the situation as it would be seen
From r.f. A'' :
A B
5
2,51017,5
0,866 c
A' B'
A'' B''
5
0,98974 c
r.f. A''
The clock in B shows the
same as before, but now
something significant
happened:
the other clocks
in r.f. A have
changed!!!

5
A
In other words: when B, A' y A'' coincide, from r.f A'' the
twin on Earth is 15 years older than from r.f. A'
A B
5
2,51017,5
0,866 c
A' B'
A'' B''
5
0,98974 c
Simply due
to changing the r.f.
of the travelling twin
the twin on Earth
ages 15 years!!!

5
A
This result can seem shocking. In the appendix we will
justify why the clocks that (from a r.f.) we can “see coming”
A B
5
2,51017,5
0,866 c
A' B'
A'' B''
5
0,98974 c
show more than
the ones we have
“beside us” and
the ones “moving
away from us”
show less.

10
A
From now on, what follows is pretty simple:
A'' comes back in 5 years and, from the point of view
of r.f. A'',
A B
10
520 12,5
0,866 c
A'' B''
The clocks in r.f. A
advance twice as
slow:
When the travelling
twin gets to the Earth,
he is not surprised
that his twin brother
is older.

The key difference between what each twin lives should
be clear now: the twin on Earth never changes r.f. while
the travelling twin changes r.f. in order to come back.
This is what allows us
to solve the paradox.
Once the paradox has been
solved, the internal
consistency of the theory
is proven once again...

¡Congratulations!
This introduction to
Special Relativity ends here.
But for those who want more,
they can continue delving into
Special Relativity in the appendix…

Part 5:Part 5:
AppendixAppendix

Appendix
Do you have any questions?
You might find the answer in what
follows...
For those who want to continue
delving into Special Relativity

We would have probably never questioned this, but we
cannot rely on anything any more..
The answer is yes: both observers agree that their
relative speed is v.
We had already used this result in previous arguments.
Question:
If from a spaceship A, we see another spaceship
B approaching us at a speed v, an observer in B,
does he agree that our spaceship is approaching
his at that same speed v?

Since both r.f. are the same for all purposes we
come across a symmetrical event.
Therefore, there is no reason why one observer
would measure a different speed than the other one.
Thereby, the only logical possibility is that they both
measure the same (v).
Demonstration:
The demonstration is related to the
First Postulate:
There is no preferred r.f..

Answer: Yes. The Principle of Causation is so basic
it is not questioned. In fact, next, we will use this
principle to make deductions.
Moreover, the Special Relativity Theory is tested to
make sure its conclusions never contradict the
Principle of Causation.
Question:
Does the Principle of Causation remain?
meaning: Are all effects preceded by the
causes that induce them?

Question:
Lets suppose that, from r.f. B, we see that when
the mark A coincides with the mark B the clock
in A shows TA and the clock in B shows TB.
B
TB
A
TA
v
From r.f. A, when the mark B coincides with the mark A
¿Would we see that the clock in A shows TA
and the clock in B shows TB?
r.f. B

Demonstration:
It should be like that due to the Principle of
Causation.
If it were not true, we could find situations
where this principle is contradicted.
Answer: Yes. When the mark A coincides with
the mark B the clock in A and the clock in B have the
same value which does not depend on the r.f. from
where it is looked at.
Lets imagine, for example, the following
thought experiment:

The point A and point B are now two holes and
the clocks that show TA and TB are now two grey
circles with both holes that spin, indicating with
their turn the time elapsed.
B
TB
A
TA
vr.f. B
TB

Lets suppose that from r.f. B the 4 holes coincide,
allowing light to go through the shared hole.
Lets suppose that the light going through the hole
activates a mechanism which causes an explosion.
BTB
A
TA
vr.f B
TB

That said, the explosion needs to occur from any r.f.,
and since there cannot be an effect (the explosion)
without a cause (the 4 hole coinciding), that cause
must be present in all r.f.
Therefore, in any r.f. when the hole A
coincides with the hole B
the time in their clocks will be TA y TB.
This result has been named
Law of Connection,
since it “connects” the values of
the clocks which are in different r.f..

Effectively, this can be achieved as follows: there are
a lot of observers in that r.f., each of them with a clock.
When they all see the first observer reset his chronometer,
they will all do the same. Since light takes some time to
reach them they will have to add that time to their clock.
Question:
Is time at all points of a r.f., seen by an
observer in that r.f., always the same?
Answer:
Yes, if we consider this means that we can
make all the clocks in that r.f. synchronised for an
observer in that r.f..

¡Important! Lets not forget that those clocks
had previously been synchronised.
What happens is so shocking, especially when the
travelling twin changes spaceship, that verifying that
the values shown are correct and could not be different
is convenient.
Question:
What do the clocks show in a different r.f.?
To solve the twin paradox,
apart from changing the r.f. of the travelling twin,
the values of the different clocks in the r.f. of the
planets has also been key.

When leaving the Earth, seen from r.f. A', the clocks of
both r.f. show what is quoted below (the chronometer
resets to 0 when A coincides with A'):
0
AA B
0
0 ? ?
0,866 c
A' B'
8,66 Light - years
4,33
Light - years
r.f. A'

5
AA B
5
2,5 10 ?
0,866 c
A' B'
8,66 Light - years
4,33
Light - years
When reaching planet B, seen from r.f A', the clocks of
both r.f. show what is quoted below. Showing 10 in B is
necessary due to the Law of Connection
(an observer from r.f. A sees those values for B y A' )
r.f. A'
In A shows 2,5 because, seen from r.f. A', the clock on A
advances half as fast (and goes from 0 to 2,5).

AA B
2,5 10 ?
0,866 c
As well as the clock in A goes from 0 to 2,5 (seen from
r.f. A') the clock in B has also advances half as fast as
the one in A' and its value increased by 2,5 years,
therefore when leaving the earth it should show 7,5
years...
5
AA B
0 7,5 ?
0,866 c
r.f. A'0
r.f. A'

AA B
2,5 10 17,5
0,866 c
The value of the clocks in C we can propose it through
symmetry (if from A to B advances 7,5 years, there is
no reason why it would vary differently from B a C)
5
AA B
0 7,5 15
0,866 c
r.f. A'0
C C
r.f. A'

AA B
2,5 10 17,5
0,866 c
We can see the clocks that “are coming” show
more years (7,5) than the clock we have beside us,
while the clocks that “have already passed by” always
show less years (7,5 less) than the one next to us.
5
AA B
0 7,5 15
0,866 c
r.f. A'0
C C
r.f. A'

AA B
2,5 10 17,5
0,866 c
This is why, when changing spaceships, the twin on Earth
must age “all at once” according to what the travelling twin
sees. There is no other possibility. In other words:
According to the Law of Connection, when the trip ends,
both twins will see that 20 years have gone by according to
the clock on the Earth and 10 years according to the
spaceship’s clock. To achieve this, it is necessary that the
green spaceship begins the return when “seeing” 17,5 in
A’s clock.
5
C AA B
5
2,51017,5
0,866 c C
r.f. A' r.f. A''

AA B
2,5 10 17,5
0,866 c
In conclusion:
The law of Connection (and, therefore, the Principle of
Causation) together with Time Dilation, imply that the
clocks’ values in “the other r.f.” vary similarly to what we
have previously mentioned.
In doing so, we have proven once again that simultaneity
is relative (the clocks, in r.f. A, are synchronised).
5
C AA B
5
2,51017,5
0,866 c C
r.f. A' r.f. A''

Now for real...
Next, the credits...

F.A.Q. http://humewoodcouncil.com/f-a-q/
Lighthouse image: http://elfarofomentolector.blogspot.com.es/
Paratrooper image:
https://recortespalomon.wordpress.com/2013/06/28/que-empiezen-las-susto-olimpiadas/
Credits:
http://www.agenciasinc.es/Entrevistas/Diez-preguntas-para-
entender-la-teoria-de-la-relatividad-general-de-Einstein licencia
Creative Commons BY 3.0

Target: https://www.timtul.com/
Arrow: https://pixabay.com/es/derecho-flecha-siguiente-297788/
Exclamation point (alert):
https://pixabay.com/es/alerta-bot%C3%B3n-se%C3%B1al-s%C3%ADmbolo-icono-309217/
Foundations:
http://elmaestrodecasas.blogspot.com.es/2011/06/construccion-de-cimientos.html
Gears:
https://pixabay.com/es/pi%C3%B1ones-engranajes-parrillas-gris-306402/

Brain and head:
http://icon-icons.com/es/icono/la-cabeza-el-cerebro-las-ciencias-la-ciencia/53022#96
Question mark:
https://pixabay.com/es/ayuda-info-pregunta-questionmark-147419/
Thinking: http://educainternet.es/pictures/3568
https://sciencesprings.wordpress.com/2014/04/18/fr
om-fermilab-proving-special-relativity-episode-3/
Dark Filament of the Sun
https://apod.nasa.gov/apod/ap100522.html
Créditos: NASA / Goddard / SDO AIA Team

HELP button
https://pixabay.com/es/ayuda-bot%C3%B3n-rojo-de-emergencia-153094/
Parachute that does not open:
https://glubdark.files.wordpress.com/2010/08/c2a1mi-paracaidas-no-se-
abreeeee.png
Blue Marble, Eastern Hemisphere
Credit: NASA
http://earthobservatory.nasa.gov/IOTD/view.php?id=84214
Puzzle:
https://pixabay.com/es/rompecabezas-piezas-pieza-concepto-308908/
Verification symbol:
https://pixabay.com/es/aprobado-bot%C3%B3n-de-verificaci%C3%B3n-151676/

Bench:
https://pixabay.com/es/banco-banco-del-parque-sentarse-1992997/
Applause:
https://pixabay.com/es/aplausos-palmas-manos-negro-bravo-297115/
Speleologist:
https://pixabay.com/es/otras-cosas-gurtov-astronauta-metro-978064/
Cave:
https://pixabay.com/es/cueva-taladro-monta%C3%B1a-145615/
The End is Near:
http://onlyconnectparke.blogspot.com.es/2015/05
/the-end-is-near-do-you-agree.html
Worker with questions:
http://www.freepik.es/vector-gratis/trabajador-con-dudas_834551.htm

Teacher pointing at the blackboard:
http://www.freepik.es/iconos-gratis/profesor-apuntando-pizarra_726348.htm
The End: https://openclipart.org/detail/246761/the-end

Conceptual Introduction to Special Relativity

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