Special relativity lecture notes
January 2023
1 lecture 1: A first look at special relativity
1.1 Galilean relativity
Let us begin by defining what is meant by a reference frame: a coordinate
system in which experiments can make position and time measurements on
an object. In special relativity we are restricted to inertial reference frames;
therefore, the velocity is constant with no acceleration. When we imagine a
model for a question it is important to remember that the coordinate axes
infinitely extend in all directions and the experimenter is modeled as staying
stationary in the reference frame. Moving onto events, these are defined as:
physical event that takes place at a definite point in space and at a definite
instant in time. Where and when an event occurs can be measured in different
reference frames at different times and positions. Moving onto the equations for
Galilean transformations, these equations apply if reference frame S ′ is moving
with a velocity v relative to the S reference frame, parallel to the x and x′ axes.
x = x′ + vt (1)
y = y′ (2)
z = z′ (3)
To convert these functions for calculating coordinates of an event in S given
the coordinates in the S’ frame. If we differentiate these equations with respect
to t we get expressions for the velocity in the S reference frame:
ux = u′x + v (4)
uy = u′y (5)
uz = u′z (6)
The Galilean principle of relativity: the laws of mechanics are the same in all
reference frames. This concludes this part of the lecture that involves Galilean
relativity.
1
, 1.2 Einstein’s special relativity
First, consider what light propagates through, some believed that there was an
ether that light traveled through. This was until Maxwell’s equations showed
that light was an electromagnetic wave. These equations showed that an electric
field can produce magnetic fields and that magnetic fields can produce electric
fields. These processes can couple together to produce waves. These waves
will travel through the vacuum at a constant speed of 3 · 108 . This solved the
problem of needing the ether to explain light. However, raised many questions
about the behavior of light in a range of reference frames. To round off this
lecture, Einstein’s principle of relativity: all laws of physics are the same in all
inertial reference frames.
2 Lecture 2: Continued Relativity
2.1 Relativity of time
This lecture has a heavy focus on the relativity of time. Most of this lecture
is going to be considered through thought experiments rather than through
explanation and complicated mathematics.
To begin let us consider the situation where the observer stands at the side of
the road and can see a tree, a lamppost, and a cyclist riding up the road. In
this case, event 1 is the bike passing the tree, this event happens at t = 0 and
x = xtree . Allow the distance between the tree and the lamppost to be ∆x.
Now consider the bike passing the lamppost, the time taken to travel from the
tree to the lamppost is ∆t. In this event, x = xlamp = xtree + ∆x. This leads
to the following equation for the measurement of the bike’s speed:
∆x
u= (7)
∆t
Now consider the same scenario from a new reference frame (the s′ reference
frame). The bike passes the tree at t′ = 0 and x′ = x′tree . Now moving
onto event 2 in this scenario, the bike passing the lamppost, t′ = ∆t′ and at
x′ = x′lamp = x′tree +∆x′ . Common sense tells us that at these speeds ∆t′ = ∆t.
This leads to the equation for the cyclist’s speed in the s′ reference frame as:
∆x′
u′ = (8)
∆t′
This shows that the value of u′ is not equal to the value of u. This shows that
the time taken for an event to occur depends on the reference frame.
2.2 Events and observations
The coordinates for an event are defined relative to the reference frame, not the
observer. Now let us visualize the scenario where we have a tree being struck
by lightning with an observer watching from a distance (D). The moment the
2
January 2023
1 lecture 1: A first look at special relativity
1.1 Galilean relativity
Let us begin by defining what is meant by a reference frame: a coordinate
system in which experiments can make position and time measurements on
an object. In special relativity we are restricted to inertial reference frames;
therefore, the velocity is constant with no acceleration. When we imagine a
model for a question it is important to remember that the coordinate axes
infinitely extend in all directions and the experimenter is modeled as staying
stationary in the reference frame. Moving onto events, these are defined as:
physical event that takes place at a definite point in space and at a definite
instant in time. Where and when an event occurs can be measured in different
reference frames at different times and positions. Moving onto the equations for
Galilean transformations, these equations apply if reference frame S ′ is moving
with a velocity v relative to the S reference frame, parallel to the x and x′ axes.
x = x′ + vt (1)
y = y′ (2)
z = z′ (3)
To convert these functions for calculating coordinates of an event in S given
the coordinates in the S’ frame. If we differentiate these equations with respect
to t we get expressions for the velocity in the S reference frame:
ux = u′x + v (4)
uy = u′y (5)
uz = u′z (6)
The Galilean principle of relativity: the laws of mechanics are the same in all
reference frames. This concludes this part of the lecture that involves Galilean
relativity.
1
, 1.2 Einstein’s special relativity
First, consider what light propagates through, some believed that there was an
ether that light traveled through. This was until Maxwell’s equations showed
that light was an electromagnetic wave. These equations showed that an electric
field can produce magnetic fields and that magnetic fields can produce electric
fields. These processes can couple together to produce waves. These waves
will travel through the vacuum at a constant speed of 3 · 108 . This solved the
problem of needing the ether to explain light. However, raised many questions
about the behavior of light in a range of reference frames. To round off this
lecture, Einstein’s principle of relativity: all laws of physics are the same in all
inertial reference frames.
2 Lecture 2: Continued Relativity
2.1 Relativity of time
This lecture has a heavy focus on the relativity of time. Most of this lecture
is going to be considered through thought experiments rather than through
explanation and complicated mathematics.
To begin let us consider the situation where the observer stands at the side of
the road and can see a tree, a lamppost, and a cyclist riding up the road. In
this case, event 1 is the bike passing the tree, this event happens at t = 0 and
x = xtree . Allow the distance between the tree and the lamppost to be ∆x.
Now consider the bike passing the lamppost, the time taken to travel from the
tree to the lamppost is ∆t. In this event, x = xlamp = xtree + ∆x. This leads
to the following equation for the measurement of the bike’s speed:
∆x
u= (7)
∆t
Now consider the same scenario from a new reference frame (the s′ reference
frame). The bike passes the tree at t′ = 0 and x′ = x′tree . Now moving
onto event 2 in this scenario, the bike passing the lamppost, t′ = ∆t′ and at
x′ = x′lamp = x′tree +∆x′ . Common sense tells us that at these speeds ∆t′ = ∆t.
This leads to the equation for the cyclist’s speed in the s′ reference frame as:
∆x′
u′ = (8)
∆t′
This shows that the value of u′ is not equal to the value of u. This shows that
the time taken for an event to occur depends on the reference frame.
2.2 Events and observations
The coordinates for an event are defined relative to the reference frame, not the
observer. Now let us visualize the scenario where we have a tree being struck
by lightning with an observer watching from a distance (D). The moment the
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