Throw a ball forward from a moving train and its speed adds to the train's — that's everyday life. Now shine a flashlight forward from that train. You'd expect the light to go "train speed + light speed." It doesn't. Every observer, no matter how fast they move, measures light travelling at the very same speed, about 300,000 km every second. That one stubborn fact is the whole story. If the speed of light can't change, then the things we use to measure speed — distance and time — must change instead.
Putting numbers on it
Write the speed as a fraction of light speed, $\beta = v/c$. Everything is controlled by one number, the Lorentz factor $\gamma = 1/\sqrt{1-\beta^2}$. When you barely move, $\beta\approx 0$ and $\gamma\approx 1$: nothing strange happens. As you approach light speed, $\beta\to 1$ and $\gamma$ shoots toward infinity. Two simple rules follow. A moving clock ticks slowly, $\Delta t = \gamma\,\Delta\tau$, and a moving object shrinks along its motion, $L = L_0/\gamma$. Try $v = 0.866c$: then $\beta^2 = 0.75$, so $\gamma = 1/\sqrt{0.25} = 2$. A passing clock ticks at half speed, and a 1 m rod measures just 0.5 m. Half as fast, half as long.
The precise picture
Mass joins in too. The energy of a moving object is $E = \gamma m_0 c^2$, so even at rest ($\gamma=1$) it carries $E_0 = m_0 c^2$ — Einstein's famous line. Its momentum is $p = \gamma m_0 v$, and the three quantities lock together as $E^2 = (pc)^2 + (m_0c^2)^2$. Because $\gamma\to\infty$ as $v\to c$, pushing any massive object to light speed would cost infinite energy — so nothing with mass ever gets there. Speeds don't simply add either: combine two velocities with $u = (v_1+v_2)/(1+v_1v_2/c^2)$, which can never exceed $c$. In the sim, the v/c slider sets $\beta$ (and therefore $\gamma$), while the proper-time and proper-length sliders set the "rest-frame" values that get dilated and contracted.
Try this in the sim above
Slide v/c from 0 up past 0.99 and watch $\gamma$ explode while the moving clock crawls — that runaway is why $c$ is a cosmic speed limit. Switch to Relativistic Mass and compare the orange (relativistic) and dashed-yellow (classical $\tfrac12 m v^2$) curves: they overlap at low speed but split dramatically near $c$. Finally set v/c back to 0 and confirm $\gamma = 1$, every effect vanishing — special relativity quietly contains ordinary physics inside it.
Section 03
Equations & Derivation
Lorentz Factor
$$\gamma=\frac{1}{\sqrt{1-(v/c)^2}},\quad\gamma\geq1,\quad\gamma\to\infty\text{ as }v\to c$$
$$K=(\gamma-1)m_0c^2,\quad K\approx\tfrac{1}{2}m_0v^2\text{ for }v\ll c$$
Key Constants
Symbol
Quantity
Unit/Value
$c=2.998\times10^8$
Speed of light
m s⁻¹
$\gamma=1/\sqrt{1-\beta^2}$
Lorentz factor
≥ 1
$\beta=v/c$
Velocity ratio
0 ≤ β < 1
$m_0$
Rest mass
kg or MeV/c²
1
Two postulates. Einstein (1905): (1) Physics laws identical in all inertial frames. (2) Speed of light $c$ is constant in all inertial frames, independent of source motion. These simple postulates, taken seriously, imply time dilation, length contraction, $E=mc^2$, and the relativity of simultaneity.
2
Time dilation verified. Muons created in upper atmosphere (15 km) travel at $v=0.998c$; their mean lifetime is 2.2 μs, classical range ~660 m. Yet they reach sea level because $\gamma\approx15$, extending the mean lifetime to ~33 μs in the lab frame. Also confirmed by atomic clocks on aircraft (Hafele-Keating 1971).
3
$E=mc^2$ implications. Rest mass is a form of energy. 1 kg converts to $9\times10^{16}$ J = 25 billion kWh. Nuclear reactions convert ~0.1% of mass to energy. Annihilation (positron+electron) converts 100%. Kinetic energy at $v\to c$: $\to\infty$, so nothing with mass can reach $c$.
4
Relativity of simultaneity. Two events simultaneous in one frame are generally not simultaneous in another. This is not an illusion — the order of spacelike-separated events depends on the observer. Only causal (timelike-separated) events have invariant order, protecting causality.
Ref: Taylor & Wheeler — Spacetime Physics (2nd Ed.); Halliday, Resnick & Walker 10th Ed., Ch. 37; Einstein — On the Electrodynamics of Moving Bodies (1905).
Section 04
Frequently Asked Questions
Moving clocks run slower. A clock moving at speed $v$ ticks at rate $1/\gamma$ compared to a stationary clock. At $v=0.866c$: $\gamma=2$, moving clock runs at half rate. This is not an optical illusion — it's measured directly by comparing clocks after a round trip (twins paradox). The moving twin genuinely ages less.
Key takeaway: Moving clocks tick slower by factor $1/\gamma$. At $v=0.866c$: half rate. Real, measurable effect.
GPS satellites: moving at ~14,000 km/h, their clocks run slow by 7 μs/day (special relativity) but also fast by 45 μs/day (general relativity, altitude). Net +38 μs/day correction applied. Without it, GPS positions drift ~10 km/day. PET scanners use $E=mc^2$ (positron annihilation). Particle accelerators need relativistic equations constantly.
Key takeaway: GPS, PET scanners, particle accelerators, and nuclear energy all require relativistic corrections.
Relativistic momentum $p=\gamma mv$ → ∞ as $v→c$ (since $\gamma→∞$). To accelerate to $c$ would require infinite energy. Every bit of energy added increases both speed and $\gamma$, so the speed approaches but never reaches $c$ asymptotically. Massless particles (photons) travel at exactly $c$ in vacuum — they have no rest frame.
Key takeaway: $\gamma→∞$ as $v→c$: infinite energy needed. Massless particles always travel at $c$.
$\gamma=1/\sqrt{1-0.99^2}=7.09$. $K=(\gamma-1)m_pc^2=(6.09)(938.3\text{ MeV})=5714$ MeV = 5.7 GeV. Classical: $K_{\rm cl}=\frac{1}{2}m_pv^2=0.5\times938.3\times0.98=460$ MeV — 12.4× smaller! The LHC accelerates protons to 6.5 TeV ($v=0.99999999c$, $\gamma=6928$).
Key takeaway: Proton at 0.99c: $K=5.7$ GeV relativistic vs 460 MeV classical — factor 12.4× difference.
Resources: Khan Academy; HyperPhysics; MIT OCW.
Section 05
Common Misconceptions
❌ Time dilation is just an optical illusion caused by light travel time.
✅ Time dilation is a genuine physical effect, not an illusion. After a round trip, the moving observer actually accumulates less proper time. The Hafele-Keating experiment (1971) flew atomic clocks around the world and measured exactly the predicted time difference. GPS systems correct for it. Muon lifetimes are extended.
❌ At low speeds, special relativity gives different results from classical mechanics.
✅ At $v\ll c$: $\gamma\approx1+v^2/(2c^2)$, so $K\approx\frac{1}{2}mv^2$ (classical KE), $p\approx mv$, and time dilation/length contraction are negligible. Special relativity is a superset of classical mechanics — it reduces to classical mechanics in the appropriate limit.
📖 HRW 10th Ed., §37-8.
❌ $E=mc^2$ means mass is converted entirely to energy in nuclear reactions.
✅ In nuclear reactions, only a small fraction of mass converts to energy — about 0.1% for fission, 0.7% for fusion. "Mass defect" is the binding energy difference. For complete conversion, particle-antiparticle annihilation is needed. The equation $E=mc^2$ says rest mass IS a form of energy; it does not say reactions must convert all mass.
📖 HRW 10th Ed., §37-8.
Misconception research: Driver et al. — Making Sense of Secondary Science; Scherr et al. (2001), Am. J. Phys.