Einstein postulates and limits of classical mechanics

Classical mechanics rests on two tacit assumptions: time is absolute ($t' = t$) and velocities add simply, $u = u' + v$ (Galilean relativity). Maxwell's electromagnetism, however, predicts waves travelling in vacuum at $c = 1/\sqrt{\varepsilon_0\mu_0}$, a speed fixed by constants of nature rather than by the motion of the source; the emission spectrum shows that a moving source shifts the frequency (the lines move) but does not change the propagation speed. Einstein removed the conflict with two postulates: (1) the principle of relativity — all inertial frames are equivalent for every law of physics, not only for mechanics; (2) the constancy of the speed of light — every inertial observer measures the same vacuum value $c$, regardless of the motion of source or observer. Together these postulates demand the Lorentz transformation instead of the Galilean one, and hence time dilation, length contraction, and relativistic velocity addition $u = \dfrac{u' + v}{1 + u'v/c^2}$. Classical mechanics is not wrong: it is the low-speed limit $v \ll c$, where $u'v/c^2 \to 0$ and the Galilean formula is recovered. The limits appear wherever speeds approach $c$: fast particles, precise time synchronisation (GPS), and the propagation of light itself.

A spacecraft recedes from Earth at $v = 0.5c$ and launches a probe forward at $u' = 0.5c$ relative to the spacecraft. Find the probe's speed relative to Earth with the relativistic formula and compare it with the Galilean answer.

Two stages accelerate a craft: relative to Earth a stage moves at $v = 0.3c$, and a module separates from it at $u' = 0.4c$ in the same direction. Find the module's speed relative to Earth and the relative error that classical mechanics would give. May Galilean addition be used in this problem?

A star recedes from an observer at $v = 0.3c$ and emits light that reaches the observer. In the star's frame the light travels toward the observer at $u' = -c$. What does Galilean addition predict, and what does the relativistic formula give? Is the answer consistent with the spectrum: the star's spectral lines are shifted to longer wavelengths by about a factor 1.36 in wavelength?

1. Open an available interactive special-relativity simulation (a light clock or a relativistic velocity adder), or a spreadsheet.
2. Build a table: with $u' = 0.5c$ fixed and $v/c$ from $0.01$ to $0.99$ in steps of $0.01$, compute

$$u_{\text{рел}} = \frac{u' + v}{1 + \frac{u'v}{c^2}}, \qquad u_{\text{гал}} = u' + v, \qquad \delta = \frac{|u_{\text{гал}} - u_{\text{рел}}|}{u_{\text{рел}}}.$$

3. Plot $\delta$ against $v/c$; mark the point where $\delta = 1\,\%$ and the point where $u_{\text{гал}} = c$. Check that as $v \to c$ the value $u_{\text{рел}} \to c$, while $u_{\text{гал}} \to 1.5c$ (it grows linearly with $v$ and exceeds $c$ for $v > 0.5c$).
4. State a conclusion: up to which $v/c$ the classical formula keeps the error below 1 %, and why it cannot be used for light.

Safety. A numerical experiment and a simulation are sufficient. Do not run experiments with radiation sources, high-voltage vacuum tubes, or fast particles.

Show where Galilean velocity addition violates the second postulate and how the relativistic formula keeps every speed below the limit $c$.

A graph: horizontal axis $u'/c$ over the range $-1$ to $1$, vertical axis $u/c$ for fixed $v = 0.5c$. Two curves — the straight line $u_{\text{гал}} = u' + v$ (rising above the level $c$ for $u' > 0.5c$) and the curve $u = \frac{u' + v}{1 + u'v/c^2}$, which reaches $u = c$ at $u' = c$ (and $u = -c$ at $u' = -c$) and stays within the light-speed limit. Mark the example points: $u' = 0.5c$ gives $u = 0.8c$, while $u' = c$ gives $u = c$ for any $v$.

A GPS satellite orbits at an altitude of 20 200 km with a speed of about $3.9\times 10^3\ \mathrm{m/s}$ relative to the ground. Its radio signal covers the $2.02\times 10^7\ \mathrm{m}$ to your phone in $t = d/c \approx 0.067\ \mathrm{s}$, and the receiver turns that travel time into a distance to within a few metres.

The second postulate is at work here: in the receiver's frame the signal travels at exactly $c = 2.998\times 10^8\ \mathrm{m/s}$, whatever the satellite's motion. The Galilean rule $u' = c \pm 3.9\times 10^3\ \mathrm{m/s}$ would shift the range by $\pm 3.9\times 10^3 \times 0.067\ \mathrm{s} \approx 260\ \mathrm{m}$ — far more than the few metres the travel-time measurement actually delivers.

A train moves at $30\ \mathrm{m/s}$ (108 km/h) towards a horn sounding in still air. A passenger measures the sound arriving at $343 + 30 = 373\ \mathrm{m/s}$ — the ordinary Galilean sum, an $8.7\%$ change. The same passenger then measures the light from a lamp: classical addition would give $c + 30\ \mathrm{m/s}$, a shift of only $1\times 10^{-7}$ of $c$, too small to see. The decisive test used the Earth's orbital motion, $v = 3.0\times 10^4\ \mathrm{m/s}$, that is $v/c = 1.0\times 10^{-4}$, which a Michelson interferometer could resolve — and no change in the speed of light was found.

Classical mechanics is built on adding velocities, and for sound the rule holds: the passenger's $373\ \mathrm{m/s}$ is exactly what Galilean relativity predicts. The second postulate says light does not obey it: a lamp and a star give the same $c$ for every inertial observer, so no frame exists in which light is at rest — and with that frame the classical picture of an absolute space in which light propagates loses its footing.

Cosmic rays create muons about $10\ \mathrm{km}$ above the ground. A muon flies at $0.998c = 2.99\times 10^8\ \mathrm{m/s}$ and has a mean lifetime of $2.2\ \mu\mathrm{s}$. Classical kinematics gives a mean path of $2.99\times 10^8 \times 2.2\times 10^{-6} \approx 660\ \mathrm{m}$, so only a fraction $e^{-10000/660} \approx 3\times 10^{-7}$ should survive the trip. In fact about one muon crosses each square centimetre of ground every minute.

This is the limit of classical mechanics in a single measurement. Under Galilean transformations neither the $2.2\ \mu\mathrm{s}$ lifetime nor the $10\ \mathrm{km}$ of air changes between frames, so classical kinematics has no way to account for the observed flux; 'time' and 'length' themselves must depend on the frame, and the postulates are the minimal assumptions that deliver that dependence.