Why Velocities Don't Simply Add
In Newtonian physics, velocities add: if you're on a train moving at 50 km/h and you walk forward at 5 km/h, you're moving at 55 km/h relative to the ground. Simple addition.
But at high speeds, this breaks down spectacularly. If a spaceship travels at 0.9c and fires a missile forward at 0.9c (in its own frame), the missile doesn't travel at 1.8c relative to Earth. It travels at about 0.995c. The speed of light is a hard limit that cannot be exceeded.
The Relativistic Formula
Einstein showed that velocities combine using:
w = (u + v) / (1 + uv/c²)
In natural units where c = 1, this simplifies to w = (u + v) / (1 + uv). This is called the Einstein velocity addition formula.
Why This Formula?
It comes directly from the Lorentz transformation. When you boost to a moving frame and then boost again, the combined transformation is another Lorentz boost, but the velocities don't add linearly — they combine through this formula.
The denominator (1 + uv) grows faster than the numerator (u + v) as velocities increase, ensuring the result never exceeds c.
Special Cases
Low Speeds (u, v ≪ c)
When both velocities are much less than c, uv ≈ 0, so the denominator ≈ 1. The formula reduces to w ≈ u + v, recovering Newtonian addition. This is why we don't notice relativistic effects in daily life.
One Velocity is Light Speed
If v = c (or u = c), then:
w = (u + c) / (1 + u·c/c²) = (u + c) / (1 + u/c) = c(u/c + 1) / (u/c + 1) = c
No matter what u is, the result is always c. Light travels at c in every reference frame, regardless of the motion of the source or observer. This is the second postulate of special relativity, built into the formula.
Both Velocities Near c
Even if u = v = 0.9c, the result is w = 1.8c / (1 + 0.81) = 1.8/1.81 ≈ 0.995c. Still less than c! As velocities approach c, the denominator grows rapidly, always keeping the result below the speed of light.
Rapidity: The Linear Alternative
There's a clever quantity called rapidity (φ) where velocities do add linearly:
φ = arctanh(β) = ½ ln[(1+β)/(1-β)]
If two velocities have rapidities φ₁ and φ₂, the combined rapidity is simply φ₁ + φ₂. Then convert back: β = tanh(φ).
Why does this work? Because Lorentz transformations form a group under composition, and rapidity is the additive parameter for this group. Mathematically, it's the "natural" coordinate for boosts, even though velocity is more intuitive.
Things to Try
Build Up to Light Speed: Start with u = 0.5c. Keep adding v = 0.5c (by using the result as the new u). Notice how quickly the gains diminish. After just a few steps, you're already close to c and further additions barely help.
Symmetric Cases: Try u = v = 0.5c, then 0.7c, then 0.9c. Watch how the classical prediction grows linearly (1.0c, 1.4c, 1.8c) but the relativistic result asymptotically approaches c (0.8c, 0.94c, 0.995c).
One Fast, One Slow: Set u = 0.99c and v = 0.01c. The result is only slightly higher than 0.99c. Even a tiny nudge when you're already near c hardly makes a difference. The closer you get to c, the harder it is to go faster.
Compare Rapidity: Look at the rapidity panel. Notice how φ grows unbounded as β → 1. You can keep adding rapidity forever (φ → ∞), but the velocity asymptotically approaches c. Rapidity is the "unlimited" quantity; velocity is bounded.
Light Speed Test: Use the "One at Light Speed" preset. No matter what the other velocity is, the result is always exactly c. This is a direct consequence of the constancy of light speed.
Physical Intuition
Why can't you exceed c by adding velocities? It's not that the formula is artificially capping things — it's that simultaneity changes. When you measure a velocity, you're dividing distance by time. But distance and time depend on your reference frame.
From your rest frame, an object moving at u has a certain velocity. But if you boost to a frame moving at v, your notion of "simultaneous" changes (relativity of simultaneity), distances contract, and times dilate. When you account for all these effects properly, the combined velocity comes out to the Einstein formula.
Real-World Applications
Particle Colliders
In the Large Hadron Collider, protons travel at 0.999999991c (γ ≈ 7460). When two such protons collide head-on, their relative velocity is not ~2c — it's approximately 0.999999999999c. The collision energy is determined by γ, not velocity, which is why the LHC can produce such high energies despite "only" accelerating to ~c.
Cosmic Rays
Ultra-high-energy cosmic rays can have γ > 10¹¹. Their velocity is extraordinarily close to c, but still less than c. Even though their rapidity is huge (φ ≈ 25), their velocity asymptotes to light speed.
Astrophysical Jets
Material ejected from black holes and quasars can move at 0.99c or faster. When we observe these jets, we must use relativistic velocity addition to calculate their motion relative to surrounding gas or other jet components. Classical addition would give wildly wrong results.
Why c is Special
The speed of light isn't special because light travels at it. Light travels at c because c is the fundamental speed limit of spacetime itself. Massless particles (photons, gluons, gravitons) travel at c; particles with mass travel slower. But c is built into the geometry of spacetime.
If we ever discover a massless particle that travels slower than c, or a massive particle that travels faster, that would break special relativity. In over a century of experiments, no violation has ever been found. The 2011 "faster-than-light neutrino" claim was traced to a loose fiber optic cable — once fixed, the neutrinos traveled at < c as expected.
Velocity addition is one of the most counterintuitive predictions of relativity, yet it's been verified countless times. Every particle accelerator experiment confirms it. Nature doesn't allow simple addition of velocities. The Einstein formula isn't an approximation or a correction — it's the correct law, and Newtonian addition is the approximation that only works at low speeds.