Deriving Einstein’s Special Relativity with High-School Math
What if I told you that using nothing more than high school algebra, you could derive one of the most profound theories in all of physics? In this article, we’re going to do exactly that: derive Einstein’s special relativity equations from scratch using mathematics you learned in school.
But first, we need to understand the fundamental compromise that makes this journey necessary.
The Speed of Light: An Unbreakable Universal Constant
From Maxwell’s equations (or from our previous mental models), we’ve established something extraordinary: light travels at exactly the same speed — 299,792,458 meters per second — regardless of how you move relative to the light source.
Think about this carefully:
- If you’re standing still relative to a flashlight, you measure the light speed as c
- If you run towards the flashlight at velocity v, you still measure the light speed as c (not c + v)
- If you run away from the flashlight at velocity v, you still measure the light speed as c (not c — v)
This is completely unlike classical motion! If someone throws a ball at 20 m/s from a train moving at 30 m/s, you’d expect to see the ball moving at 50 m/s (20 + 30). But light doesn’t behave this way. No matter how fast you move toward or away from a light source, you always measure the same speed c.
This isn’t negotiable. It’s written into the very fabric of reality itself and confirmed by countless experiments.
If you want to understand why the speed of is a constant please click here to refer to my previous post.
But here’s the problem: this creates an impossible situation with our everyday understanding of motion.
Imagine Alice standing on a platform and Bob on a train moving at 100 km/h. If Alice shines a flashlight forward:
- Classical thinking says: Bob should see the light moving at c — 100 km/h (slower)
- Reality says: Bob sees the light moving at exactly c (same as Alice)
Something has to give. Since the speed of light is non-negotiable, we must make a profound compromise.
The Great Compromise: No Universal Time or Length
Here’s the revolutionary insight: there is no such thing as universal time or universal length measurements.
What does this mean? It means that space and time themselves work differently for different observers:
- Time runs differently: Alice’s clock ticks at one rate, while Bob’s clock (as observed by Alice) ticks at a different rate. When Alice’s clock shows “5 seconds” have passed, Bob’s clock might show only “4.8 seconds” have passed for the same physical event
- Space stretches differently: Alice’s meter stick and Bob’s meter stick are different lengths when compared to each other. What Alice measures as “10 meters” using her ruler, Bob might measure as “9.5 meters” using his ruler for the same physical object
- Both are experiencing reality correctly: Alice isn’t wrong and Bob isn’t wrong. Each observer’s clocks and rulers work perfectly in their own frame (and they don’t feel anything odd)— it’s just that the units themselves are different relative to each other
Think of it this way: it’s not that Alice and Bob are measuring incorrectly. Rather, Bob’s seconds are literally different from Alice’s seconds, and Bob’s meters are literally different from Alice’s meters. Time itself flows at different rates for them, and space itself has different scales for them.
But if different observers experience time and space differently, there must be some relationship that connects them. The question is: what kind of mathematical relationship connects how Alice experiences spacetime to how Bob experiences spacetime?
Why the Relationship Must Be Linear
This is where mathematics guides us with iron logic. The relationship between different observers’ measurements of space and time cannot be quadratic, cubic, or any higher-order equation. Here’s why:
Reason 1: Additivity of Measurements If Alice measures two consecutive time intervals as t₁ and t₂, then Bob should measure them as some functions f(t₁) and f(t₂). The total time Alice measures is t₁ + t₂, and the total time Bob measures should be f(t₁ + t₂).
But by the principle of measurement additivity: f(t₁ + t₂) = f(t₁) + f(t₂)
This is called Cauchy’s functional equation, and for continuous functions, the only solutions are linear: f(t) = At for some constant A.
Reason 2: Reversibility If we can transform from Alice’s measurements to Bob’s, we must also be able to transform back. With quadratic or higher-order transformations, this reversibility becomes problematic — you can get multiple solutions or lose information.
Simple Example: If f(x) = x², then f(2) = f(-2) = 4. Given the output “4,” we can’t uniquely determine whether the input was 2 or -2. This irreversibility violates the principle that physical transformations should be invertible.
Reason 3: The Group Property Physical transformations must satisfy what mathematicians call the “group property.” If Alice transforms to Bob’s frame, and Bob transforms to Charlie’s frame, this should be equivalent to Alice directly transforming to Charlie’s frame. Linear transformations naturally satisfy this property, while nonlinear ones generally don’t.
Therefore, by mathematical necessity, the relationship between different observers’ measurements must be of the form:
x’ = A₁x + A₂t
t’ = B₁x + B₂t
where A₁, A₂, B₁, B₂ are constants to be determined (given a fixed speed).
Note: x’ = A₁x + A₂t + A₃ and t’ = B₁x + B₂t + B₃ with constants A₃ and B₃ is more generic but Cauchy’s functional equation makes us to choose A₃ and B₃ as zero.
Now our job is to figure out what these constants are, using only the constraint that light speed is constant for all observers.
Constraint: Relative Motion
Here’s where we need to think physically. Bob’s train is moving at velocity v relative to Alice. What does this mean mathematically?
Key insight: The origin of Bob’s coordinate system (x’ = 0) is moving at velocity v in Alice’s coordinate system.
Let’s work through a concrete example:
- At t = 0: Bob is at position x = 0 (both coordinate origins coincide)
- At t = 1 second: Bob is at position x = v (he’s moved distance v)
- At t = 2 seconds: Bob is at position x = 2v
- At any time t: Bob is at position x = vt
Now, Bob’s position is always x’ = 0 in his own frame (he’s always at his own origin). For example, if Bob considers x’ = 0 to be the left end of his train (which is moving to the right), that left end is always his starting point or reference — it’s always at x’ = 0 in his coordinate system. But for Alice watching from the platform, that same left end of the train is moving at velocity v. So we need:
When x’ = 0: A₁x + A₂t = 0, which gives us x = -A₂t/A₁
But we know physically that x = vt when x’ = 0.
Therefore: vt = -A₂t/A₁
This gives us: A₂ = -A₁v
Our transformation becomes:
x’ = A₁(x — vt)
t’ = B₁x + B₂t
The Magic Happens: Applying Light’s Constant Speed
Now comes the revolutionary part. We know that light travels at speed c in all reference frames. Let’s see what this constraint tells us about our unknown constants.
Case 1: Light Pulse Moving Right
Consider a light pulse moving in the positive x-direction:
- In Alice’s frame (S): x = ct
- In Bob’s frame (S’): x’ = ct’ (same light, same speed c)
Substituting into our transformation:
ct’ = A₁(ct — vt) = A₁t(c — v) … (1)
ct’ = c(B₁x + B₂t) = c(B₁ct + B₂t) = ct(B₁c + B₂) … (2)
From equations (1) and (2): A₁(c — v) = c(B₁c + B₂) … (Equation I)
Case 2: Light Pulse Moving Left
Now consider light moving in the negative x-direction:
- In Alice’s frame: x = -ct
- In Bob’s frame: x’ = -ct’
Following the same logic:
-ct’ = A₁(-ct — vt) = -A₁t(c + v)
-ct’ = -ct(B₁c — B₂)
This gives us: A₁(c + v) = c(B₁c — B₂) … (Equation II)
Solving the System
From Equations I and II:
A₁(c — v) = c(B₁c + B₂)
A₁(c + v) = c(B₁c — B₂)
Adding these equations:
2A₁c = 2cB₁c
Therefore: B₁ = A₁/c
Subtracting the first from the second:
2A₁v = -2cB₂
Therefore: B₂ = -A₁v/c
Our transformation is now:
x’ = A₁(x — vt)
t’ = (A₁/c)x + (-A₁v/c)t = (A₁/c)(x — vt) = A₁(t — vx/c²)
Finding the Final Constant A₁
We still need to determine A₁. Here’s where the principle of relativity gives us the final piece of the puzzle.
The Key Insight: If Alice sees Bob moving to the right at velocity v, then Bob sees Alice moving to the left at velocity v. Both observers are equally valid — there’s no “preferred” frame of reference.
This means the transformation from Bob’s frame back to Alice’s frame must have exactly the same mathematical form, except with v replaced by
-v (since Bob sees Alice moving in the opposite direction).
So if our transformation from Alice to Bob is:
x’ = A₁(x — vt)
t’ = A₁(t — vx/c²)
Then the transformation from Bob back to Alice must be:
x = A₁(x’ + vt’) [note: +v instead of -v]
t = A₁(t’ + vx’/c²) [note: +v instead of -v]
The inverse transformation should be:
x = A₁(x’ + vt’)
t = A₁(t’ + vx’/c²)
Substituting our expressions for x’ and t’:
x = A₁[A₁(x — vt) + vA₁(t — vx/c²)]
x = A₁²[x — vt + vt — v²x/c²]
x = A₁²x(1 — v²/c²)
For consistency, we need: A₁²(1 — v²/c²) = 1
Therefore: A₁ = 1/√(1 — v²/c²)
We call this the Lorentz factor: γ = 1/√(1 — v²/c²). 😀 😀
The Final Result: Einstein’s Lorentz Transformation
Our complete transformation is:
x’ = γ(x — vt)
t’ = γ(t — vx/c²)
where γ = 1/√(1 — v²/c²)
These are the famous Lorentz transformations — the mathematical heart of Einstein’s special relativity!
The Physical Consequences: Why Reality is Stranger Than Fiction
From these simple equations flow all the mind-bending consequences of relativity (some of which you have heard of):
Time Dilation
For a clock at rest in Alice’s frame, Δx = vΔt,
A clock in Bob’s frame: Δt’ = γ(Δt — v²Δt/c²) = γΔt(1 — v²/c²) = Δt/γ
Therefore: Δt = γΔt’
Moving clocks run slow! Bob’s clock ticks slower as observed by Alice.
Length Contraction
For a ruler at rest in Bob’s frame with proper length L₀: L₀ = Δx’ = γΔx
Therefore: L = L₀/γ
Moving objects contract in the direction of motion!
The Speed Limit of the Universe
Notice what happens as v approaches c: γ approaches infinity. And if you go faster than c, then γ becomes an imaginary number. Therefore, the speed of light is truly the cosmic speed limit.
The Profound Insight: Mathematics Reveals Truth
What we’ve discovered is extraordinary. Starting with nothing more than:
- The definition of relative motion
- The constancy of light speed
We were forced — by mathematics itself — into Einstein’s theory of relativity. We didn’t choose these equations; the structure of spacetime itself demanded them.
This is the power of mathematical physics: when we encode the right physical principles into mathematics, the math reveals truths about reality that we could never have guessed. The universe speaks in the language of mathematics, and when we learn to listen, it tells us the most beautiful stories.
Conclusion: From School Math to Cosmic Truth
We’ve just taken a journey from simple linear algebra to one of the most profound theories in all of science. Using nothing more than high school mathematics, we’ve derived equations that govern the behavior of space and time themselves.
The next time someone tells you that math is just abstract symbol manipulation with no connection to reality, remind them of this: the same algebra you learned in school contains within it the secrets of the cosmos. Mathematics isn’t just a tool for describing reality — it’s the language in which reality is written.
The constancy of light speed, that single fact we discussed in our previous post, has led us to a complete revolution in our understanding of space and time. And we discovered it all using math you already know.
That’s the true beauty of physics: the deepest truths about our universe are often hiding in plain sight, waiting to be revealed by elegant applications of mathematical reasoning to careful observation of the world around us.