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6.2 — The Two Postulates and the Lorentz Transformation
Einstein's 1905 paper begins by throwing away the ether and asserting two things. Everything in special relativity follows from them, and the derivation needs no other input at all.
Postulate 1 — the principle of relativity. The laws of physics take the same form in every inertial frame. No experiment of any kind, mechanical or electromagnetic, can distinguish one uniformly moving frame from another.
Postulate 2 — the invariance of c. Light travels in vacuum at the same speed c in every inertial frame, regardless of the motion of the source or the observer.
Postulate 1 is old, and reasonable, and Galileo would have signed it. What Einstein added was the phrase "of any kind", extending it from mechanics to electromagnetism.
Postulate 2 is the outrageous one. It says that if you chase a light beam at 0.99c, it still recedes from you at 3\times10^8 m/s — not at 3\times10^6. That is not a small correction to velocity addition; it flatly contradicts it.
Notice what postulate 2 does not say. It does not say nothing can go faster than light. It does not say time slows down. Those are consequences, and they are on their way.
Notice also that postulate 2 is not an extra assumption in disguise. If the laws of physics are the same in every frame, then Maxwell's equations are the same in every frame, and Maxwell's equations produce a definite speed 1/\sqrt{\mu_0\varepsilon_0} with no frame attached. So postulate 2 is very nearly forced by postulate 1 — which is exactly why Einstein was willing to take it seriously when others were not.
The light clock: where time dilation comes from
Before deriving the general transformation, here is the argument that shows why something has to give. It uses one imaginary device and Pythagoras, and it is the cleanest piece of reasoning in physics.
The device. Two mirrors a distance L apart, with a light pulse bouncing between them. One tick is one round trip. There is nothing special about it; any clock would do, and the light clock is chosen because its rate is fixed by c alone, which is the quantity postulate 2 pins down.
In the clock's own frame, the pulse goes straight up and straight back:
\Delta t_0 = \frac{2L}{c}
The subscript zero marks this as the proper time — the time measured by a clock at rest with respect to the events it is timing.
Now watch the same clock go past you at speed v. While the pulse travels from the bottom mirror to the top one, the whole clock has moved sideways. So in your frame the pulse follows a diagonal.
Let one tick take \Delta t in your frame. In half a tick the clock moves v\Delta t/2 sideways, and the pulse travels c\Delta t/2 along the diagonal, with the perpendicular distance still L. Pythagoras:
\left(\frac{c\Delta t}{2}\right)^2 = L^2 + \left(\frac{v\Delta t}{2}\right)^2
\frac{c^2\Delta t^2}{4} - \frac{v^2\Delta t^2}{4} = L^2
\Delta t^2(c^2-v^2) = 4L^2
\Delta t = \frac{2L}{\sqrt{c^2-v^2}} = \frac{2L}{c}\cdot\frac{1}{\sqrt{1-v^2/c^2}}
And the first factor is \Delta t_0:
\boxed{\Delta t = \frac{\Delta t_0}{\sqrt{1-v^2/c^2}} = \gamma\,\Delta t_0}
The moving clock ticks more slowly, by the factor:
\boxed{\gamma = \frac{1}{\sqrt{1-v^2/c^2}}}
Read aloud: gamma equals one over the square root of one minus v-squared over c-squared. This is the Lorentz factor, and it appears in every equation for the rest of this Part.
Where the argument is forced. The only step that used postulate 2 is writing the diagonal length as c\Delta t/2 — assuming the light travels at c in your frame too, even though the clock is moving. Under Galilean thinking the pulse would inherit the clock's sideways velocity and still take 2L/c. Postulate 2 forbids that, and the price is that \Delta t \neq \Delta t_0.
And it applies to every clock, not just light clocks. Suppose a mechanical watch strapped to the light clock kept the old time. Then by comparing them you could measure your absolute velocity, violating postulate 1. So all processes — springs, atoms, chemical reactions, radioactive decay, ageing — must slow by exactly the same factor. Time itself runs slow, not the mechanism.
The size of \gamma
| v | \gamma | Slowing |
|---|---|---|
| 100 km/h | 1.0000000000000043 | negligible |
| 7.7 km/s (ISS) | 1.00000000033 | 0.03 ppb |
| 0.1c | 1.005 | 0.5 % |
| 0.5c | 1.155 | 15 % |
| 0.9c | 2.294 | 129 % |
| 0.99c | 7.089 | |
| 0.999c | 22.37 | |
| 0.99999c | 223.6 |
This table explains why nobody noticed for three hundred years. At everyday speeds \gamma differs from 1 in the fifteenth decimal place. Relativity is not a correction that was overlooked through carelessness; it is a correction that was invisible until people started measuring fast particles and precise clocks.
Deriving the Lorentz transformation
The light clock gives one result. Now derive the full coordinate transformation from the two postulates.
Setup. Frame S is at rest; frame S' moves at speed v along the shared x axis. Origins coincide at t = t' = 0. An event has coordinates (x, t) in S and (x', t') in S'.
Step 1: the transformation must be linear.
Why? Because a free particle moves in a straight line at constant speed in any inertial frame — that is what "inertial" means. A straight line in (x,t) must map to a straight line in (x',t'), and only a linear transformation does that for every line. A nonlinear map would bend some straight worldlines into curves, meaning a free particle would appear to accelerate for no reason.
So write:
x' = Ax + Bt, \qquad t' = Dx + Et
with four constants to determine, all possibly depending on v.
Step 2: the origin of S'.
The point x' = 0 is, by definition, moving at x = vt as seen from S. Substituting:
0 = A(vt) + Bt \quad\Longrightarrow\quad B = -Av
So:
x' = A(x - vt)
Step 3: symmetry between the frames.
By postulate 1, neither frame is special. Seen from S', frame S moves at -v. So the inverse transformation must have the same form with v replaced by -v and the same constant A:
x = A(x' + vt')
Step 4: apply postulate 2.
Send a light pulse from the common origin at t = 0. In S it is at x = ct. In S', by postulate 2, it is at x' = ct' — the same c, not c adjusted for the motion.
Substitute both into the two equations above:
ct' = A(ct - vt) = At(c-v)
ct = A(ct' + vt') = At'(c+v)
Multiply them together:
c^2tt' = A^2tt'(c-v)(c+v) = A^2tt'(c^2-v^2)
Cancel tt':
c^2 = A^2(c^2-v^2) \quad\Longrightarrow\quad A^2 = \frac{c^2}{c^2-v^2} = \frac{1}{1-v^2/c^2}
\boxed{A = \gamma}
The Lorentz factor drops out of the algebra. It was not assumed anywhere.
Step 5: get t'.
Substitute x' = \gamma(x-vt) into x = \gamma(x'+vt'):
x = \gamma\left[\gamma(x-vt) + vt'\right] = \gamma^2x - \gamma^2vt + \gamma vt'
Solve for t':
\gamma vt' = x - \gamma^2x + \gamma^2vt = x(1-\gamma^2) + \gamma^2vt
t' = \frac{x(1-\gamma^2)}{\gamma v} + \gamma t
Now simplify 1-\gamma^2. Since \gamma^2 = 1/(1-\beta^2) with \beta = v/c:
1-\gamma^2 = 1 - \frac{1}{1-\beta^2} = \frac{(1-\beta^2)-1}{1-\beta^2} = \frac{-\beta^2}{1-\beta^2} = -\gamma^2\beta^2
So:
t' = \frac{-\gamma^2\beta^2 x}{\gamma v} + \gamma t = -\frac{\gamma\beta^2x}{v}+\gamma t
And \beta^2/v = v^2/c^2/v = v/c^2:
t' = \gamma\left(t - \frac{vx}{c^2}\right)
The result
\boxed{\begin{aligned} x' &= \gamma(x - vt)\\ y' &= y\\ z' &= z\\ t' &= \gamma\left(t - \frac{vx}{c^2}\right)\end{aligned}}
and the inverse, obtained by swapping primes and reversing v:
x = \gamma(x'+vt'), \qquad t = \gamma\left(t'+\frac{vx'}{c^2}\right)
These are the Lorentz transformations. Lorentz wrote them in 1904 as a mathematical property of Maxwell's equations and did not believe the time coordinate was real. Einstein derived them in 1905 from two physical postulates and said the time coordinate is exactly as real as the space one.
Reading the equations
The x equation is the Galilean x - vt with a stretch factor \gamma. That is length contraction in embryo.
The t equation is the revolutionary one. Compare it with the Galilean t' = t:
t' = \gamma\left(t - \frac{vx}{c^2}\right)
There are two separate departures.
The \gamma out front is time dilation — the moving clock runs slow, as the light clock showed.
The -vx/c^2 term is the deeper one, and it has no classical analogue at all. Read it: the time in the moving frame depends on where you are, not just when. Two events at the same t but different x have different t'.
That is the loss of absolute simultaneity, and it is the single most important idea in relativity. Chapter 6.3 takes it apart with a concrete example. For now, note that it is the term that makes everything else work: without it, the two frames would disagree about the speed of light.
The y and z coordinates are untouched
Why is there no contraction perpendicular to the motion? Because of a symmetry argument that is worth spelling out, since it is short and airtight.
Suppose transverse lengths did contract. Take two identical rings, one on each frame, moving towards each other along their common axis. From frame 1's viewpoint, ring 2 is moving, so ring 2 shrinks and passes inside ring 1. From frame 2's viewpoint, ring 1 is moving, so ring 1 shrinks and passes inside ring 2.
Both cannot be true, because "which ring is inside" is a physical fact everybody must agree on — you could paint the inner one as it passes. Postulate 1 says neither frame is right, so the only consistent answer is that neither contracts.
Recovering Galileo
A new theory must reproduce the old one wherever the old one was tested. Check it.
When v \ll c, \beta^2 = v^2/c^2 is minuscule, so \gamma \to 1, and vx/c^2 \to 0:
x' \to x - vt, \qquad t' \to t
Exactly the Galilean transformation. Newtonian mechanics is not wrong; it is the low-speed limit of relativity, and it is an outstandingly good approximation for everything slower than about a tenth of light speed.
How good? At orbital speed, 7.7 km/s, \gamma - 1 = 3.3\times10^{-10}. Over a day, that is 28 microseconds — negligible for a spacecraft's trajectory and, as Chapter 6.9 shows, absolutely not negligible for GPS.
Velocity addition, corrected
The Galilean rule u' = u - v must be wrong, because it would let two velocities add past c. Derive the right one.
A particle has velocity u = dx/dt in S. Its velocity in S' is u' = dx'/dt'. Differentiate the transformations:
dx' = \gamma(dx - v\,dt), \qquad dt' = \gamma\left(dt - \frac{v\,dx}{c^2}\right)
Divide, and the \gamma cancels:
u' = \frac{dx - v\,dt}{dt - v\,dx/c^2}
Divide top and bottom by dt:
\boxed{u' = \frac{u-v}{1-\frac{uv}{c^2}}}
The numerator is the Galilean answer. The denominator is entirely new, and it is what keeps everything below c.
Two checks
Slow speeds. With uv \ll c^2 the denominator is 1 and u' = u - v. A ball thrown at 20 m/s on a 30 m/s train: the correction factor is 1 - (20)(30)/(9\times10^{16}) = 1 - 6.7\times10^{-15}. Galileo is right to fourteen decimal places.
Light. Set u = c and see what any observer measures:
u' = \frac{c-v}{1-\frac{cv}{c^2}} = \frac{c-v}{1-v/c} = \frac{c-v}{\frac{c-v}{c}} = c
Exactly c, for any v whatsoever. Postulate 2 is not merely consistent with the transformation — it is built into its structure.
Worked example: two rockets
Two rockets approach each other, each at 0.8c relative to Earth. What speed does one measure for the other?
Galileo says 1.6c. Relativity says:
u' = \frac{0.8c-(-0.8c)}{1-\frac{(0.8c)(-0.8c)}{c^2}} = \frac{1.6c}{1+0.64} = \frac{1.6c}{1.64} = 0.9756c
Under c, and not by a small margin — 97.6 % rather than 160 %.
Try harder. Each at 0.99c:
u' = \frac{1.98c}{1+0.9801} = \frac{1.98c}{1.9801} = 0.99995c
You cannot get there by adding. The formula has c as a fixed point that no combination of sub-light speeds can exceed. This is not a rule imposed from outside; it is arithmetic.
Why nothing with mass reaches c
Three independent reasons, all visible already:
\gamma diverges. As v \to c, \gamma \to \infty, so time dilation and length contraction become infinite. Chapter 6.4 shows the energy required goes to infinity too.
Velocity addition cannot reach it. Boost and boost again and you approach c asymptotically.
Causality. Chapter 6.5 shows that a signal faster than c could be used to send information into the past, in some frame, which permits genuine paradoxes.
But things with no mass must travel at exactly c, always, in every frame, and cannot be slowed or stopped. Photons and gravitational waves do this. And "faster than light" is not forbidden for everything: the phase velocity of a wave in a medium can exceed c, and the expansion of the universe carries distant galaxies away faster than c (Chapter 12.5), because neither of those transmits information or moves an object through space.
What the next chapter fixes
The transformation is derived. What it means is not yet unpacked, and the meanings are strange enough that they need working through one at a time with concrete examples. Chapter 6.3 does that: it shows that two events simultaneous for one observer happen at different times for another, that a moving object is genuinely shorter, that the twin who travels comes back younger and exactly why that is not a contradiction, and it presents the measurement on cosmic-ray muons that settles the whole thing experimentally.