Postulates of Special Relativity

Two sentences, written by Einstein in 1905, quietly dismantled the Newtonian universe. The first is old and almost obvious: the laws of physics look the same to every observer who moves uniformly — no experiment done inside a sealed, smoothly cruising cabin can reveal its speed, because absolute rest is not a physical fact. Galileo had said as much for mechanics with his ship below decks; Einstein's only addition was the word all — electromagnetism must obey the relativity principle too. The second sentence is the detonator: the speed of light in vacuum, $c = 2.99792458\times10^{8}\,\mathrm{m/s}$ , is the same number in every inertial frame, no matter how fast the source moves or the observer chases it. Maxwell's equations already contained $c$ as a fixed constant, and the Michelson–Morley experiment had failed to find the aether wind that would let $c$ vary — Einstein took the hint and elevated invariance to a principle. Put the two together and something has to break. If a light pulse outruns you at $c$ whether you stand still or sprint toward it, then your ruler and your clock cannot be the same as everyone else's. The casualty is absolute, universal time: observers in relative motion disagree about whether two separated events happen at once, about how long a clock takes to tick, and about how long a moving rod is. The transformations that connect their measurements are not Galileo's but Hendrik Lorentz's, and they braid space and time into a single four-dimensional fabric — Minkowski spacetime — whose one invariant everyone agrees on is the spacetime interval. From these two postulates the whole kinematic cathedral follows by pure logic: relativity of simultaneity, time dilation, length contraction, velocity addition, and ultimately $E = mc^2$ . General relativity later keeps the postulates but makes them local, valid in the flat patch of spacetime at every point of a curved world.

Key takeaways

Special relativity rests on two postulates: the principle of relativity — physical laws take the same form in every inertial frame — and the invariance of the speed of light $c$ , independent of the motion of source or observer.
Together they overthrow Newtonian absolute time: simultaneity becomes frame-dependent and the Galilean velocity law $u' = u - v$ must be discarded.
The only transformations compatible with both postulates are the Lorentz transformations, which mix space and time and are governed by the factor $\gamma = 1/\sqrt{1-\beta^2}$ .
The quantity left invariant is no longer separate distance and duration but the spacetime interval $\Delta s^2 = -c^2\Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2$ .
Time dilation, length contraction, and simultaneity shifts are logical consequences, confirmed to extraordinary precision by muons, atomic clocks, and particle accelerators.
Key idea: demand that the laws of physics — including the value of $c$ — read identically in every inertial frame, and the only transformations consistent with that demand are the Lorentz transformations, which fuse space and time into one geometry with a single invariant interval.

The core insight

Demand that the laws of physics — including the value of $c$ — look identical in every inertial frame, and the only transformations consistent with that requirement are the Lorentz transformations, which unify space and time into a single geometric structure with an invariant interval.

Where this lives today

The postulates are tested continuously, and every test has passed. Particle accelerators are the most relentless: at the Large Hadron Collider protons circulate at $\beta = 0.999999991$ , with $\gamma \approx 7500$ , and the beams stay bunched only because the timing is computed relativistically — the machine would not steer under Newtonian rules. Optical-lattice atomic clocks, accurate to parts in $10^{18}$ , now register time dilation from a walking pace or a centimetre of height. Meanwhile physicists probe whether the postulates are exactly true: "Lorentz-invariance violation" searches — using gamma-ray bursts, resonant cavities, and trapped ions — hunt for any minuscule frame-dependence of $c$ , since some approaches to quantum gravity predict Lorentz symmetry might fail at the Planck scale. To date $c$ is invariant to the limits of measurement, and the two 1905 sentences remain unbroken.

The physical picture

Stand on a platform as a train rolls through at speed $v$ . In Newton's world, a ball thrown forward at $u'$ by a passenger reaches you at $u' + v$ : velocities simply add. Now swap the ball for a flash of light. The second postulate insists you still clock it at $c$ — not $c + v$ . The extra speed the train should have contributed has vanished. Since speed is distance over time, and distance and time are exactly what you and the passenger are measuring, the only escape is that your rulers and clocks disagree. The train's clock no longer ticks in step with yours, and neither of you can call the other wrong. That single refusal of light to add its speed to the source's is the seed of everything. The cleanest way to see why nothing else can change is geometric. A light pulse traces a world-line at $45^\circ$ on a spacetime diagram — one unit of space per unit of $ct$ — and invariant $c$ demands it stay at $45^\circ$ for every observer. When you boost to a moving frame, its time axis $ct'$ and space axis $x'$ tilt toward each other like closing scissors, but they close symmetrically about the light line so it keeps bisecting them. That symmetric scissoring is the Lorentz transformation: the only way to hold the $45^\circ$ light line fixed is to let space and time rotate into one another by a hyperbolic angle set by $\beta = v/c$ .

Figure

A light pulse traced in red at 45° has the same speed c in the lab frame S with blue axes and in the boosted frame S′ moving at β = v/c = 0.6 with amber axes ct′ and x′. The light world-line bisects the angle between each frame's time and space axes, so both observers measure c = 2.998×10⁸ m/s

The two postulates have very different personalities. The first is a symmetry statement — it constrains the form of the laws without naming any number. The second is a dynamical fact — it names a specific speed and declares it invariant. Remarkably, almost all of special relativity follows from the first postulate alone plus the mild assumption that some finite speed is invariant; electromagnetism then supplies the last piece — that the invariant speed equals the speed of light. The second figure shows the price of that invariance: the Lorentz factor $\gamma$ , which measures how strongly clocks slow and rods shrink, is nearly $1$ for everyday motion but rises without bound as $v \to c$ , walling off the speed of light from anything with mass.

Figure

The Lorentz factor γ = 1/√(1−β²) plotted against β = v/c. It sits at essentially 1 for ordinary speeds, reaches γ = 1.15 at 0.5c, climbs to γ = 2.29 at 0.9c and γ = 7.09 at 0.99c, and diverges as v → c — the quantitative reason no massive body can reach the speed of light

The core equations

The theory begins not with an equation but with two declarations. The first postulate is a statement about the form of all physical laws; the second fixes one number that appears in them. Everything downstream is a logical consequence of holding both at once.

\textbf{(P1)}\ \ \text{The laws of physics take the same form in all inertial frames.}\qquad \textbf{(P2)}\ \ c\ \text{is invariant.} \qquad (1)

where: an inertial frame is one in which a free particle moves at constant velocity; frames related by uniform relative motion are all inertial. $c = 2.99792458\times10^{8}\,\mathrm{m/s}$ is the vacuum speed of light — exact by the 1983 SI definition of the metre — and P2 asserts every inertial observer measures this same value regardless of source or detector motion. Physical reading: P1 forbids a preferred frame; P2 forbids light from ever adding the source's velocity. Impose both postulates on the map between two frames in standard configuration (relative motion along $x$ ) and the map is uniquely fixed. It is linear because space and time are homogeneous, and it reduces to the Lorentz boost.

t' = \gamma\left(t - \frac{v x}{c^2}\right),\qquad x' = \gamma\,(x - v t),\qquad y'=y,\ z'=z \qquad (2)

where: $v$ is the relative speed of frame $S'$ along $x$ ; $(t,x)$ are an event's coordinates in $S$ and $(t',x')$ the same event in $S'$ ; $\gamma$ is the Lorentz factor below. Physical reading: the mixing of $t$ into $x'$ and of $x$ into $t'$ is the mathematical face of "space and time are not separate." The single dimensionless factor $\gamma$ controls the strength of every relativistic effect. It equals $1$ at rest and diverges at $c$ , which is why the speed of light is a kinematic ceiling for massive bodies.

\gamma \equiv \frac{1}{\sqrt{1 - v^2/c^2}} = \frac{1}{\sqrt{1-\beta^2}},\qquad \beta \equiv \frac{v}{c} \qquad (3)

where: $\beta$ is the speed in units of $c$ , with $0 \le \beta < 1$ ; $\gamma \ge 1$ always. Physical reading: for $\beta \ll 1$ , $\gamma \approx 1 + \tfrac12\beta^2$ , so everyday relativistic corrections are of order $\beta^2$ — utterly tiny, which is why Newton sufficed for three centuries. The quantity all observers agree on is not distance or duration separately but a specific combination of them: the spacetime interval. Its invariance is the geometric heart of the theory, equivalent to invariant $c$ .

\Delta s^2 = -c^2\,\Delta t^2 + \Delta x^2 + \Delta y^2 + \Delta z^2 = \eta_{\mu\nu}\,\Delta x^\mu \Delta x^\nu \qquad (4)

where: $\Delta s^2$ is the squared interval between two events, identical in every inertial frame; $\eta_{\mu\nu} = \mathrm{diag}(-1,+1,+1,+1)$ is the Minkowski metric and $x^\mu = (ct, x, y, z)$ . Physical reading: $\Delta s^2 < 0$ is timelike (a clock can visit both events), $\Delta s^2 > 0$ spacelike (no signal connects them), $\Delta s^2 = 0$ the light cone — the invariant record of P2. Applying the boost to a clock or a rod yields the two most famous consequences: a moving clock runs slow and a moving rod is short, both by the factor $\gamma$ .

\Delta t = \gamma\,\Delta\tau \ \ge\ \Delta\tau,\qquad L = \frac{L_0}{\gamma}\ \le\ L_0 \qquad (5)

where: $\Delta\tau$ is the proper time on the moving clock (elapsed in its own rest frame); $\Delta t$ the coordinate time in the frame where it moves at $v$ ; $L_0$ the proper (rest) length and $L$ the length measured while it moves. Physical reading: "moving clocks tick slow, moving rods shrink along the motion" — but each observer says it of the other, and the interval reconciles them. Finally, velocities no longer add linearly. The relativistic law keeps every composed speed below $c$ and maps $c$ to $c$ , enforcing P2 automatically.

u = \frac{u' + v}{1 + u'v/c^2} \qquad (6)

where: $u'$ is a particle's velocity measured in $S'$ and $u$ its velocity in $S$ , with $S'$ moving at $v$ relative to $S$ . Physical reading: set $u' = c$ and the formula returns $u = c$ for any $v$ ; two sub-luminal speeds always compose to something still below $c$ .

Derivation AFrom the two postulates to the Lorentz transformation

The goal is to show that the Lorentz boost of equation 2 is not an assumption but the unique linear transformation forced by the two postulates. We assume only that space and time are homogeneous and space is isotropic — so the map between frames is linear — and then let a light pulse do the work: because it must travel at $c$ in both frames, the coefficients are pinned down, leaving a single scale factor that reciprocity fixes as $\gamma$ . Nothing about clocks or rods is put in by hand.

Step 1. Homogeneity of space and time forbids any quadratic dependence, so the most general map between the two frames is linear: $x' = Ax + Bt$ and $t' = Dx + Et$ . The spatial origin of $S'$ ( $x'=0$ ) moves as $x = vt$ in $S$ , which forces $B = -Av$ , giving $x' = A(x - vt)$ . Why: linearity is exactly the mathematical content of "no point or instant is special," and tracking the moving origin ties the coefficients to the physical relative velocity $v$ .

Step 2. Now demand invariant $c$ . A light pulse leaving the common origin satisfies $x = ct$ in $S$ and must satisfy $x' = ct'$ in $S'$ by P2. Substituting the linear forms and requiring consistency for the light pulse relates $D$ and $E$ to $A$ and $v$ , collapsing four unknowns down to the single overall scale $A$ . Why: this is the only place the second postulate enters, and it is decisive — invariance of one speed removes almost all the freedom in the map.

Step 3. Fix the scale by reciprocity: the inverse transformation must look identical but with $v \to -v$ , since $S$ moves at $-v$ relative to $S'$ . Composing the boost with its inverse and demanding the identity forces $A = \gamma = (1 - v^2/c^2)^{-1/2}$ , reproducing $t' = \gamma(t - vx/c^2)$ and $x' = \gamma(x - vt)$ exactly. Why: the relativity principle privileges neither frame, so the forward and backward maps are mirror images — the condition that seals the scale factor as $\gamma$ .

Go deeper · Derivation B · 1 minTime dilation from a light clock

Derivation BTime dilation from a light clock

The goal is to derive the time-dilation factor with no algebra beyond Pythagoras, exposing that "moving clocks run slow" is pure geometry once $c$ is the same on every path. We build the simplest clock — light bouncing between two mirrors a fixed distance apart — and compare one tick in its rest frame with one tick seen from a frame where it glides sideways. Because the light must cover a longer, diagonal path at the very same speed $c$ , the moving tick is stretched by exactly $\gamma$ .

Step 1. In the clock's rest frame, light travels straight up to a mirror a distance $D$ away and back, so one tick lasts $\Delta\tau = 2D/c$ — the proper time. Why: with the clock at rest the geometry is trivial, and this rest-frame tick is the benchmark every other frame is compared against.

Step 2. In a frame where the clock moves sideways at speed $v$ , during one tick the clock advances a distance $v\,\Delta t$ , so the light traces the two sloping sides of an isoceles triangle. Each leg is the hypotenuse of a right triangle with vertical side $D$ and horizontal side $v\,\Delta t/2$ , and light still covers it at $c$ , giving $(c\,\Delta t/2)^2 = D^2 + (v\,\Delta t/2)^2$ . Why: this is the crux — postulate P2 forbids the light from going any faster on the longer diagonal, so the only way to conserve $c$ is for the tick $\Delta t$ itself to grow.

Step 3. Solve: $\Delta t^2(c^2 - v^2) = 4D^2$ , hence $\Delta t = (2D/c)\,/\sqrt{1-v^2/c^2} = \gamma\,\Delta\tau$ . The moving clock's tick is longer by $\gamma \ge 1$ , so it runs slow. Why: the factor emerges from the same $\sqrt{1-\beta^2}$ that pinned the boost, confirming that time dilation and the Lorentz transformation are one phenomenon; muon lifetimes and flown atomic clocks verify it directly.

Go deeper · Derivation C · 1 minThe spacetime interval is Lorentz invariant

Derivation CThe spacetime interval is Lorentz invariant

The goal is to prove that although two observers disagree about $\Delta t$ and $\Delta x$ separately, they compute the identical value for $\Delta s^2 = -c^2\Delta t^2 + \Delta x^2$ . This is the deepest restatement of the postulates: invariance of the interval is fully equivalent to invariance of $c$ , and it is what turns relativity from a list of strange effects into a clean geometry, with the interval playing the role ordinary distance plays in Euclidean space. We substitute the boost and watch the cross terms cancel.

Step 1. Take two events and their separation $(\Delta t, \Delta x)$ in $S$ ; the boost gives $\Delta t' = \gamma(\Delta t - v\,\Delta x/c^2)$ and $\Delta x' = \gamma(\Delta x - v\,\Delta t)$ . Form the primed interval $-c^2\Delta t'^2 + \Delta x'^2$ by substituting these. Why: we test invariance by brute computation — plug the transformed coordinates into the interval and see what survives.

Step 2. Expand both squares. The mixed $\Delta x\,\Delta t$ terms appear with opposite signs — $+2\gamma^2 v\,\Delta x\,\Delta t$ from one square and $-2\gamma^2 v\,\Delta x\,\Delta t$ from the other — and cancel exactly. What remains is $\gamma^2(1 - v^2/c^2)\big(-c^2\Delta t^2 + \Delta x^2\big)$ . Why: the cancellation of the cross terms is the entire magic trick; it is precisely what a Galilean transformation fails to do, which is why Galilean physics has no invariant interval.

Step 3. Use $\gamma^2(1 - v^2/c^2) = 1$ by the definition of $\gamma$ . The prefactor collapses to $1$ , leaving $-c^2\Delta t'^2 + \Delta x'^2 = -c^2\Delta t^2 + \Delta x^2$ : the interval is unchanged. Why: the same $\gamma$ that stretches time and shrinks length is exactly the factor that makes the interval invariant — the effects conspire to protect the one quantity all observers share.

Limiting cases

The postulates reduce to familiar physics in the appropriate corners and sharpen into new statements in others.

Case	Result	Meaning
Low speed, $\beta \to 0$	$t' = t,\ \ x' = x - vt$	The Lorentz boost becomes the Galilean one; absolute time returns and velocities add linearly. Newtonian mechanics is the $c \to \infty$ idealization.
Lightlike separation	$\Delta s^2 = 0$	Emission and absorption of a light ray are separated by zero interval in every frame — the geometric encoding of invariant $c$ and the frame-independent light cone.
Rest frame of a clock, $v=0$	$\mathrm{d}\tau = \mathrm{d}t$	Coordinate time equals proper time only in the clock's own frame; proper time is the invariant elapsed reading everyone can compute from the interval.
Ultra-relativistic, $\beta \to 1$	$\gamma \to \infty$	Time dilation and length contraction grow without bound; a massive particle would need infinite energy to reach $c$ , which is therefore unattainable.
Local limit in curved spacetime	$g_{\mu\nu}(p) = \eta_{\mu\nu},\ \ \partial_\alpha g_{\mu\nu}(p) = 0$	At any point one can build a local inertial frame in which the metric is Minkowskian and special relativity — both postulates — governs infinitesimal physics. This is how the postulates survive into general relativity.

Common trapSpecial relativity says everything is relative, so there are no absolute facts

The name misleads. Special relativity is not a doctrine that "everything is relative" — it is a theory about what stays absolute when observers disagree about space and time. Frame-independent quantities abound: the speed of light $c$ , the spacetime interval $\Delta s^2$ , the proper time $\Delta\tau$ along any world-line, the proper length of an object, the rest mass, the causal order of timelike-separated events, and the very form of the physical laws. What is relative is only the split of the invariant interval into a "time part" and a "space part" — that partition depends on the observer's motion, the way a stick's shadow depends on where you shine the light. So two observers disagree about which of two lightning strikes came first only when the strikes are spacelike-separated (causally disconnected); for any pair of events that could influence one another, all observers agree on the order. Relativity replaces one universal time with one universal geometry.

Worked examples

1 · Cosmic-ray muons reach the ground (numeric)

A muon is created about $10\,\mathrm{km}$ up when a cosmic ray hits the atmosphere. Its rest-frame lifetime is $\tau = 2.2\,\mu\mathrm{s}$ and it descends at $v = 0.998\,c$ . Classically it should never reach sea level. Show how the postulates rescue the observed abundance, from both viewpoints.

Step 1. Classical estimate. Even at nearly $c$ , in one lifetime the muon travels only $v\tau \approx (0.998)(2.998\times10^{8})(2.2\times10^{-6}) \approx \mathbf{660\,m}$ — far short of $10\,\mathrm{km}$ , so a Newtonian calculation predicts essentially no muons at the ground, contradicting experiment.

Step 2. Lorentz factor. At $\beta = 0.998$ , $\gamma = 1/\sqrt{1 - 0.998^2} = 1/\sqrt{0.003996} = \mathbf{15.82}$ .

Step 3. Ground frame — time dilation. The muon's lifetime measured in the lab is stretched to $\Delta t = \gamma\tau = 15.82 \times 2.2\,\mu\mathrm{s} = \mathbf{34.8\,\mu s}$ . In that time it covers $v\,\Delta t = (0.998)(2.998\times10^{8})(34.8\times10^{-6}) \approx \mathbf{10.4\,km}$ — enough to reach the ground. The dilation earns the muon a factor of $\gamma$ more range.

Step 4. Muon frame — length contraction. The muon considers itself at rest and sees the atmosphere rushing up at $0.998\,c$ . The $10.4\,\mathrm{km}$ column is contracted to $L = L_0/\gamma = 10.4/15.82 \approx \mathbf{0.66\,km}$ , which it crosses in $0.66\times10^{3}/(2.992\times10^{8}) \approx 2.2\,\mu\mathrm{s}$ — one ordinary lifetime. Both descriptions agree the muon arrives, because they share the invariant interval; time dilation in one frame is length contraction in the other.

2 · Relativistic velocity addition (numeric)

A spaceship recedes from Earth at $v = 0.9\,c$ and fires a probe forward at $u' = 0.9\,c$ relative to the ship. Does an Earth observer see the probe exceed $c$ ? What if the ship fires a laser pulse instead?

Step 1. Galilean intuition says $0.9c + 0.9c = 1.8c$ — impossible under the postulates. Use equation 6 instead: $u = (u' + v)/(1 + u'v/c^2)$ .

Step 2. Substitute the probe speeds: $u = (0.9c + 0.9c)/(1 + 0.9\times0.9) = 1.8c/1.81 = \mathbf{0.9945\,c}$ . Two speeds each at $0.9c$ compose to just under $c$ , not above it. The denominator $1 + u'v/c^2$ is exactly the relativistic tax that keeps the sum sub-luminal.

Step 3. Now the laser: $u' = c$ . Then $u = (c + 0.9c)/(1 + 0.9) = 1.9c/1.9 = \mathbf{c}$ . The pulse is measured at $c$ from Earth, exactly as P2 demands — velocity addition maps $c$ to $c$ for any boost. No composition of sub-luminal speeds ever reaches $c$ , and light always travels at $c$ .

3 · The special-relativistic clock offset in GPS (numeric)

A GPS satellite orbits at $v \approx 3.87\,\mathrm{km/s}$ . Working from the second postulate's kinematics alone, estimate how much its onboard atomic clock drifts per day relative to a ground clock due to time dilation. (A gravitational blueshift, treated in general relativity, partly offsets this.)

Step 1. Compute $\beta = v/c = 3874/(2.998\times10^{8}) = 1.29\times10^{-5}$ . This is tiny, so use the low-speed expansion $\gamma \approx 1 + \tfrac12\beta^2$ , meaning the fractional slowing is $\tfrac12\beta^2$ .

Step 2. Evaluate: $\tfrac12\beta^2 = \tfrac12(1.29\times10^{-5})^2 = \mathbf{8.35\times10^{-11}}$ . The moving clock loses this fraction of every second.

Step 3. Multiply by a day: $8.35\times10^{-11}\times 86400\,\mathrm{s} \approx \mathbf{7.2\,\mu s/day}$ . The satellite clock ticks about $7\,\mu\mathrm{s}$ slow per day from special-relativistic time dilation. Untreated, the associated ranging error accrues at roughly $c\times7\,\mu\mathrm{s} \approx 2\,\mathrm{km}$ per day — GPS would fail within minutes. The postulates are wired into the navigation in your pocket.

Physics of the portrait

The postulates have one author. Others assembled the pieces — the mathematics, the null experiment, the transformation that bears another man's name — but only Einstein saw that the fix was to change space and time themselves.

Origins · 1 minAlbert Einstein and how this was discovered

Albert Einstein

1879–1955 · the 1905 special relativity paper

In 1905, his annus mirabilis, a 26-year-old patent examiner in Bern published "Zur Elektrodynamik bewegter Körper" ("On the Electrodynamics of Moving Bodies") and rebuilt kinematics from two axioms. What set Albert Einstein apart was not new mathematics — Lorentz had written the transformation equations, Poincaré had glimpsed the relativity principle, FitzGerald had proposed length contraction — but a change of stance. Where Lorentz treated contraction as a dynamical effect of motion through the aether, Einstein discarded the aether entirely and took the invariance of $c$ as a postulate, deriving the transformations as a consequence rather than a patch. The famous paper carries no references and no aether; a few months later a three-page sequel drew out $E = mc^2$ . Einstein initially dismissed Minkowski's 1908 geometric reformulation as "superfluous learnedness," yet it was precisely that four-dimensional picture he would need a decade later. Curving Minkowski's spacetime to encode gravity, he produced general relativity in 1915 — but its local scaffolding, at every point, is still the special relativity he built from two sentences in 1905.

History

1632

— Galileo argues in his Dialogue that mechanics is identical in a smoothly moving ship's cabin

1865

— Maxwell's electromagnetic theory fixes the speed of light c inside the laws themselves

1887

— Michelson and Morley detect no aether wind, finding c unchanged in all directions

1905

— Einstein derives relativistic kinematics from two postulates in "Zur Elektrodynamik bewegter Körper"

1908

— Minkowski recasts the theory as the geometry of a unified four-dimensional spacetime

Going further

Texts. Einstein, "Zur Elektrodynamik bewegter Körper," Annalen der Physik 17, 891 (1905) · Misner, Thorne & Wheeler, Gravitation (1973), Ch. 1–2 · Wald, General Relativity (1984), Ch. 1 · Carroll, Spacetime and Geometry (2004), Ch. 1 · Hartle, Gravity (2003), Ch. 4 · Schutz, A First Course in General Relativity (2009), Ch. 1. Key numerical anchors: $c = 2.99792458\times10^{8}\,\mathrm{m/s}$ (exact, SI); $\gamma = (1 - v^2/c^2)^{-1/2}$ with tabulated values $\gamma(0.9c)=2.294$ , $\gamma(0.99c)=7.089$ , $\gamma(0.998c)=15.82$ ; the boost, interval invariance (cross terms cancel), light-clock time dilation $\Delta t = \gamma\Delta\tau$ , and velocity addition mapping $c \to c$ are all standard results. The muon lifetime $2.2\,\mu\mathrm{s}$ gives a classical range of $660\,\mathrm{m}$ and a dilated range of $\sim10.4\,\mathrm{km}$ at $0.998c$ ; the velocity-addition result is $0.9945c$ ; and the GPS special-relativistic clock offset is $\sim7\,\mu\mathrm{s/day}$ .

Builds on: Galilean relativity and inertial frames · Maxwell's equations and the speed of light. Leads to: Lorentz Transformations · Relativity of Simultaneity · Time Dilation · Length Contraction · Spacetime Interval · Minkowski Spacetime.

Ask the tutor: Ask me to walk through the twin paradox, or to show how invariant $c$ forces the relativity of simultaneity in the train thought experiment.

Are the postulates proven, or just assumed?

They are assumed — that is what "postulate" means — but they are the most thoroughly tested assumptions in physics. Einstein did not derive them; he elevated two experimental hints (the failure to detect any preferred frame, and the appearance of $c$ as a fixed constant in Maxwell's equations) to the status of principles, then showed that an enormous body of consequences follows by logic alone. Those consequences — time dilation, length contraction, velocity addition, $E = mc^2$ — have been confirmed in muon decay, accelerators, atomic clocks, and GPS to extraordinary precision. A single clean violation would overturn the theory, and none has ever been found.

Do I really need both postulates, or does one imply the other?

You need both, but they do unequal work. Almost the entire structure of relativity — the linear form of the transformation, the existence of an invariant speed, time dilation, length contraction — follows from the first postulate plus the assumption that space is homogeneous and isotropic. What that argument leaves undetermined is the value of the invariant speed; it could in principle be infinite, which would give back Galilean physics with absolute time. The second postulate supplies the missing fact: the invariant speed is finite and equals the speed of light. So the first postulate builds the machine and the second sets its one free dial.

If motion is relative, who is really the one whose clock runs slow?

Both, symmetrically — and there is no contradiction. Each inertial observer sees the other's clock running slow by the same factor $\gamma$ , because "running slow" compares one moving clock against a whole array of synchronized clocks in your frame, and the two observers synchronize their arrays differently (relativity of simultaneity). It is like two people walking apart each seeing the other shrink with distance: no one is wrong, because "smaller" is a perspective, not an absolute. The apparent paradox becomes a real, asymmetric aging difference only when one observer turns around and returns — but that observer must accelerate, breaking the symmetry, and the invariant proper time along each world-line settles who is younger unambiguously.

Does the second postulate forbid anything from going faster than light?

Not directly — it is a statement about the speed of light, not a blanket speed limit. The limit is a consequence. Because $\gamma \to \infty$ as $v \to c$ , accelerating a massive particle to $c$ would require infinite energy, so no massive body can reach $c$ , while massless particles must travel at exactly $c$ . What is genuinely forbidden is any signal or causal influence outrunning $c$ , because that would let some observer see effects precede their causes. Certain things can appear to move faster than $c$ without violating anything — the sweep of a laser spot across a distant wall, the phase velocity of a wave, the recession of galaxies in cosmic expansion — but none carries information from point to point faster than light.

How can two events be simultaneous for me but not for someone moving past?

Because "simultaneous" means "at the same time," and the two of you slice spacetime into instants differently. In the Lorentz transformation $t' = \gamma(t - vx/c^2)$ : even when two events share the same $t$ for you, their different positions $x$ give them different $t'$ for a moving observer. Concretely, a lamp at the centre of a moving train car flashes; in the car frame the light reaches both ends at once, but a platform observer sees the rear rush toward the flash and the front recede, so for her the rear is struck first. Neither is mistaken — invariant $c$ forces it. Simultaneity of separated events is not absolute but a frame-dependent choice, which is precisely why absolute time had to go.

Where do the postulates go when gravity is present?

They become local. General relativity's equivalence principle says that in a small enough freely-falling laboratory, gravity is undetectable and physics looks exactly special-relativistic. Mathematically, at any spacetime point you can choose coordinates in which the metric equals $\eta_{\mu\nu}$ and its first derivatives vanish — a local inertial frame. Inside that patch both postulates hold: the laws take their special-relativistic form and $c$ is invariant. What changes globally is that these local frames cannot be stitched into one universal inertial frame, because spacetime is curved. Gravity is that curvature — the failure of local Minkowski patches to mesh — while special relativity remains exact in the infinitesimal.