Special Relativity

Demand that the speed of light be the same for every observer, and the familiar Newtonian world — absolute time, simple velocity addition, rigid rulers — falls apart. What replaces it is a geometry of spacetime in which moving clocks run slow, moving meter sticks shrink, and mass is a kind of frozen energy.

1. Why special relativity exists

By the 1890s, physics had a problem it couldn't look away from. Maxwell's equations — the laws of electricity and magnetism, written down in the 1860s — predicted electromagnetic waves that travel at exactly one speed, about $3 \times 10^8$ meters per second. Every physicist assumed that speed was measured relative to some medium, the same way the speed of sound is relative to air. They named the medium the luminiferous ether and went looking for it.

In 1887, Albert Michelson and Edward Morley built a precision interferometer in the basement of a Cleveland chemistry building. The Earth rotates and orbits the Sun, so their lab had to be moving through the ether at tens of kilometers per second. An interferometer should see a fringe shift as the apparatus rotates. They found nothing. Not "a small shift." Not "something subtle." Nothing to the precision of their instrument. They rebuilt the experiment, refined it, and kept finding nothing.

Something was very wrong. Lorentz and Fitzgerald tried to patch the ether theory by proposing that objects physically contract in the direction of motion, by just the right factor, to hide the effect. Einstein, in 1905, went the other way. Instead of rescuing the ether, he threw it out. His paper "On the Electrodynamics of Moving Bodies" started from two assumptions and rederived everything.

Why should you, in 2026, care? Three reasons:

• GPS. The satellites in your phone's location service carry atomic clocks that run at a different rate than clocks on the ground, partly because of special relativity, partly because of general relativity. Without relativistic corrections, GPS would drift by about 10 km per day and would have been useless on day one.
• Particle physics. Anything more energetic than a flu shot is doing relativistic physics. The LHC accelerates protons to 99.9999991% of the speed of light. Their effective mass, collision energy, and lifetime are all relativistic.
• Electromagnetism itself. Magnetic forces are what you see when you Lorentz-transform an electric field. The magnet on your fridge is a relativistic effect you can touch.

This page assumes you remember basic algebra, a little trigonometry, and the idea that graphs can have axes. We will build the ideas from scratch and keep the math honest.

2. Vocabulary cheat sheet

These are the symbols you'll see repeatedly. Glance at them now.

One warning: a lot of introductory books use a "relativistic mass" $m_{\text{rel}} = \gamma m$ that grows with speed. Modern physics does not. Mass means rest mass, full stop. Energy is what grows.

3. The two postulates

Einstein's 1905 paper starts from two assumptions. They look innocent. They are not.

The first postulate is Galileo's old idea: physics on a smoothly moving ship is the same as physics in port. Newton had already assumed this. The second one is the bomb. If you are at rest and I drive past you at half the speed of light while shining a flashlight forward, common sense says the beam leaves me at $c$ and reaches you at $c + v/2$. Postulate 2 flatly denies this. You measure the beam at $c$, and I measure the beam at $c$, full stop.

Something has to give. What gives is the comfortable notion that "the time between two events" and "the distance between two events" are the same in every frame. They are not. Let's see how fast that unravels.

Consider a simple thought experiment: a "light clock." Inside a rocket moving past you with velocity $v$, a pulse of light bounces between two mirrors separated by a height $L$. In the rocket's frame, the round-trip time is

Now you watch the rocket fly past. In your frame, the pulse does not travel straight up and down. During the time the pulse is rising, the rocket has moved forward. The light takes a diagonal path. Since it must still travel at speed $c$ (that's postulate 2), the diagonal takes longer than the vertical trip would. So the clock on the rocket, as seen from outside, ticks slower than a clock next to you. That is time dilation. We'll derive the exact factor in the next section.

4. Time dilation and length contraction

Let's make the light-clock argument quantitative. In the rocket frame, the pulse covers $2L$ in time $\Delta \tau = 2L/c$. In your frame — call it the lab frame — the pulse travels a diagonal of length $D$ during a time $\Delta t$. During that time, the rocket has moved forward by $v\,\Delta t/2$ (for the upward leg of the trip). By the Pythagorean theorem,

Solve for $\Delta t$. Multiply out, collect terms, and you get

Length contraction is the geometric companion. A rod of length $L_0$ at rest has that length only in its own frame. Viewed from a frame where the rod is moving along its length at speed $v$, its length shrinks:

Worked example — muons from the sky

Muons are unstable particles with a mean lifetime of $\tau_0 \approx 2.2 \ \mu\text{s}$ when at rest. Cosmic rays hitting the upper atmosphere create muons at altitudes of about 15 km. Without relativity, a muon moving at nearly $c$ would travel at most $c \tau_0 \approx (3 \times 10^8)(2.2 \times 10^{-6}) \approx 660$ m before decaying. It should never reach the ground. But muon detectors at sea level see a flood of them. Why?

Time dilation. At a typical energy the muons have $\gamma \approx 20$. In the lab frame, their lifetime is stretched to $20 \times 2.2 \ \mu\text{s} = 44 \ \mu\text{s}$, and their reach to $c \gamma \tau_0 \approx 13$ km. A respectable fraction survives.

In the muon's own frame, the muon lives for its usual $2.2 \ \mu\text{s}$. But length contraction squeezes the 15 km atmosphere down to $15/\gamma \approx 0.75$ km, which the muon covers easily. Different explanations in the two frames, identical physical outcome: the muon reaches the detector. This is the Rossi–Hall experiment (Mount Washington, 1941) and it has been reproduced thousands of times since.

Worked example — GPS clocks

A GPS satellite orbits at about 3.87 km/s. That gives $\beta \approx 1.29 \times 10^{-5}$ and $\gamma - 1 \approx 8.3 \times 10^{-11}$. Over a day of $86{,}400$ seconds, the satellite clock lags the ground clock by roughly $7 \ \mu\text{s}$ from special relativity alone. (General relativity adds a larger, opposite correction we'll cover on the next page.) Uncorrected, that would corrupt your position fix by hundreds of meters a day.

5. Lorentz transformations and simultaneity

Time dilation and length contraction are consequences of a deeper rule — the Lorentz transformation, which tells you how to convert the coordinates of an event from one inertial frame to another. Let $S$ and $S'$ be two frames; $S'$ moves at velocity $v$ along the $x$-axis of $S$. An event has coordinates $(t, x, y, z)$ in $S$ and $(t', x', y', z')$ in $S'$. Then

The inverse transformation just flips the sign of $v$ — same formulas with $v \to -v$.

Relativity of simultaneity

Set $\Delta t = 0$ between two events separated by $\Delta x$ in frame $S$. In $S'$, the time between them is

Velocity addition

Applying the Lorentz transformation to both a position and a displacement gives the correct rule for adding velocities. If frame $S'$ moves at $v$ relative to $S$, and an object moves at $u'$ along $x$ in $S'$, then in $S$ the object moves at

6. Interactive: Lorentz boost

Drag the slider to change $\beta = v/c$ and watch the spacetime grid tilt. The cyan lines are lines of constant $t'$ (events the boosted observer calls "simultaneous"), and the violet lines are lines of constant $x'$ (events at fixed location in the boosted frame). Both tilt by the same angle, so the diagonal $x = ct$ (the light cone) is preserved. The fixed pink dot is an event $E$ at $(t, x) = (1.0, 0.0)$ in the lab frame — watch how its boosted coordinates $(t', x')$ change.

A few things the picture teaches you that words don't:

• The boosted time axis tilts toward the light line; so does the boosted space axis. Both tilt by the same angle from their unboosted versions.
• At $\beta = 0$ the grid is upright — no boost.
• As $\beta \to \pm 1$ the grid collapses onto the light line; the geometry is degenerate because nothing with mass can travel at $c$.
• The event $E$ sits still while the grid moves underneath it. That is the whole point: the event is fixed in the universe, but its labels depend on who is labeling.

7. Momentum, energy, and $E = mc^2$

Newton's definitions of momentum ($p = mv$) and kinetic energy ($\tfrac{1}{2} m v^2$) are not conserved in all inertial frames once you take Lorentz transformations seriously. The quantities that are conserved, and that reduce to the Newtonian ones for small speeds, are

Set $v = 0$ and you get the most famous equation in physics:

There is a useful identity relating energy, momentum, and mass, with no $\gamma$ in sight:

Worked example — LHC proton energy

A proton at the Large Hadron Collider has total energy $E = 6.5 \ \text{TeV}$. Its rest energy is $m c^2 \approx 0.938 \ \text{GeV} = 0.000938 \ \text{TeV}$. So $\gamma = E / (m c^2) \approx 6928$. Its speed comes from $\gamma = 1/\sqrt{1 - \beta^2}$, giving $\beta \approx 1 - 1/(2 \gamma^2) \approx 1 - 1.04 \times 10^{-8}$. That's about 2.7 meters per second slower than light. In 27 km, the beam finishes its lap in about $89.6 \ \mu\text{s}$ — but a beam of light would finish only $2.4 \ \text{fm}$ sooner. Relativity is what the accelerator engineers work in, every shift.

8. Four-vectors and invariants

The algebra of special relativity becomes much easier if you stop treating time and space as separate things and write them as a single four-component object. A four-vector is any quantity that transforms like the spacetime coordinates under a Lorentz boost. The two you'll meet first are the position four-vector and the energy-momentum four-vector:

The Minkowski inner product is how you take the "length" of a four-vector. It is not the Euclidean dot product; it has a minus sign:

Apply it to the position four-vector and you get the spacetime interval:

Similarly, $P \cdot P = (E/c)^2 - p^2 = (mc)^2$. That's the energy–momentum relation again, one line. The algebra loves four-vectors.

9. Paradoxes resolved

The twin paradox

Alice stays on Earth. Bob gets in a rocket, flies to Proxima Centauri at $\beta = 0.8$, and comes back. Each twin "sees" the other's clock running slow by $\gamma = 1.667$. Who is actually older when Bob returns?

Bob. Bob is demonstrably younger. There is no symmetry between the twins, because Bob accelerates — he turns his rocket around. Alice stays in a single inertial frame the whole time; Bob does not. The spacetime path Alice takes between "Bob leaves" and "Bob returns" is a straight line in spacetime. Bob's path is an out-and-back kink. In Minkowski geometry, straight-line paths maximize proper time (note: maximize, not minimize — the minus sign flips the intuition from Euclidean geometry). Bob's bent path covers less proper time than Alice's straight one. He has aged less.

Numerically: if the round trip takes 12.5 years in Alice's frame (Proxima is 4 light-years away, out and back at $0.8 c$ is $2 \times 4 / 0.8 = 10$ light-years of travel, plus a turnaround), Bob ages only $12.5 / \gamma = 7.5$ years. Alice is 5 years older when they hug. This is not a thought experiment with ambiguities. It has been verified with atomic clocks flown on airliners (Hafele–Keating, 1971) and atomic clocks flown on satellites every day.

The barn and the pole

A runner carries a 10-meter pole and runs at $\beta = 0.6$ ($\gamma = 1.25$) toward a barn whose front and back doors are 8 meters apart. In the barn's frame, the pole is length-contracted to $10/1.25 = 8$ m and just barely fits. We close both doors at the same instant, and for a moment the pole is fully inside. In the runner's frame, the barn is length-contracted to $8/1.25 = 6.4$ m and the 10-meter pole does not fit. Who's right?

Both. The doors are not closed "at the same instant" in the runner's frame — simultaneity is relative. In the barn's frame the back door opens as the front door closes, but in the runner's frame the back door opens first, the pole starts emerging, and only later does the front door close on the trailing end. Events that the barn calls simultaneous, the runner calls sequential. No paradox once you stop assuming a universal "now."

See also

10. Special relativity in code

Three quick primitives: a Lorentz factor, a boost applied to a four-vector, and a velocity-addition function. None is more than a handful of lines, and each is the exact formula you've been reading.

import numpy as np

C = 299792458.0   # speed of light in m/s

# ---------- 1. Lorentz factor ----------
def gamma(v):
    beta = v / C
    return 1.0 / np.sqrt(1.0 - beta * beta)

# ---------- 2. Lorentz boost along x ----------
def boost_x(four_vec, v):
    # four_vec = (ct, x, y, z) in some frame S.
    # Returns the same event's coordinates in S', moving at +v along x.
    g = gamma(v)
    beta = v / C
    ct, x, y, z = four_vec
    ct_p = g * (ct - beta * x)
    x_p  = g * (x - beta * ct)
    return np.array([ct_p, x_p, y, z])

# ---------- 3. Relativistic velocity addition ----------
def add_velocities(u_prime, v):
    return (u_prime + v) / (1.0 + (u_prime * v) / (C * C))

# Demo: two rockets approaching, each at 0.9 c
v_rel = add_velocities(0.9 * C, 0.9 * C)
print(f"relative speed / c = {v_rel / C:.6f}")  # ~0.994475

# Demo: muon lifetime dilated at gamma = 20
tau0 = 2.2e-6
v_mu = C * np.sqrt(1 - 1 / 400.0)        # gamma=20
print(f"dilated lifetime = {gamma(v_mu) * tau0 * 1e6:.2f} us")

import math

C = 299792458.0

def gamma(v):
    beta = v / C
    return 1.0 / math.sqrt(1.0 - beta * beta)

def time_dilation(tau_proper, v):
    # Return the duration in the lab frame for a proper time tau_proper
    return gamma(v) * tau_proper

def length_contraction(L0, v):
    return L0 / gamma(v)

def rel_energy(m_kg, v):
    return gamma(v) * m_kg * C * C

# A 6.5 TeV proton at the LHC
E_target = 6.5e12 * 1.602e-19               # joules
m_p      = 1.6726e-27                         # kg
g_lhc    = E_target / (m_p * C * C)
print(f"LHC gamma ~ {g_lhc:.0f}")
print(f"1 - beta ~ {1 / (2 * g_lhc * g_lhc):.2e}")

11. Cheat sheet