The theory of special relativity states two postulates:
the laws of physics are equal for every inertial observer,
the speed of light is constant for all inertial observers.
Common sense tells us that the sum of two objects' speeds is done by simply adding them together. However, this naïve assumption would break the second postulate of special relativity—resulting in the speed of light changing whenever an object moves. So, how do we derive the correct formula for adding speeds?
This is where linear transformations come into play.
In two-dimensional space, eigenvectors are two vectors that do not change directions in a linear transformation.
Eigenvectors will be crucial in deriving our formula, as our transformation matrix must obey two requirements:
the observer's speed shall become zero, relative to the observer,
the speed of light shall remain constant.
We can create a graph of lines, each representing a different speed. The horizontal axis represents time; Meanwhile, the vertical axis represents position. Therefore, the slopes of these lines represent the object's speed.
(The two thick black lines represent the speed of light. No, this is not at all related to light dispersion—I just wanted to make it look pretty.)
Before we begin, let's assign variables to some terms.
c will represent the velocities of light—traveling forward (1, 1) or backward (1, -1),
A will be our 2×2 transformation matrix (c is not to be confused with c),
λ will be the eigenvalues of A,
v will be a vector that represents the observer's velocity from a reference point O, with v being its speed,
and v' will be a vector representing the observer's velocity according to themself, with 0 being the speed and p representing the scale (a practically unimportant variable).
Our goal is to find a matrix transformation A such that:
The observer's speed becomes zero. In other words, A transforms (1, v) into (p, 0).
The speed of light stays constant. In other words, (1, 1) and (1, -1) are eigenvectors of A.
Let us begin.
Our second postulate gives the following equation:
We can substitute the values of I, A, and c into this equation,
giving us
λ₁ - a - b = 0 ...(1)
-c + λ₁ - d = 0 ...(2)
Additionally, we can substitute c with the negative velocity of light,
which gets us
λ₂ - a + b = 0 ...(3)
-c - λ₂ + d = 0 ...(4)
Subtracting (3) from (1) gives us
λ₁ - a - b - λ₂ + a - b = 0
λ₁ - λ₂ - 2b = 0
b = ½(λ₁ - λ₂)
Adding (2) and (4) gives us
-c + λ₁ - d - c - λ₂ + d = 0
-2c + λ₁ - λ₂ = 0
c = ½(λ₁ - λ₂)
Hence, b = c
Adding (1) and (3) gives us
λ₁ + λ₂ - 2a = 0
a = ½(λ₁ + λ₂)
Subtracting (4) from (2) gives us
λ₁ + λ₂ - 2d = 0
d = ½(λ₁ + λ₂)
Hence, a = d
Our first postulate gives the following equation:
With matrix multiplication, we find that
c + dv = 0
c = -dv
We can substitute everything we know into our matrix A, which gets us:
Let's define two new variables.
u: the speed of a moving object from a reference point O
u': the speed of the object according to our observer
(Δs and Δt are distance and time changes)
Now that we've got the values of Δs' and Δt', we can substitute them into u'.
Now you know how to add near-light-speed velocities together!
This is just the beginning. We haven't even touched time dilation, length contraction, etc. You could derive formulas for those from the matrix we've got, but who the hell wants to do that LMFAO