Theory of Relativity Formula Calculator

Special relativity rests on two postulates — the laws of physics are identical in every inertial frame, and $c$ is the same in every frame — and almost every formula that follows is one dimensionless quantity times a classical expression. That quantity is the Lorentz factor:

$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}} = \frac{1}{\sqrt{1 - \beta^2}}, \qquad \beta = \frac{v}{c}$

with $v$ the relative speed in m/s and $c = 2.998 \times 10^8$ m/s. $\gamma \ge 1$ always, and $\gamma \to 1$ as $\beta \to 0$, which is why Newtonian mechanics survives at everyday speeds.

The three kinematic results:

$\Delta t = \gamma\, \Delta t_0, \qquad L = \frac{L_0}{\gamma}, \qquad u' = \frac{u - v}{1 - uv/c^2}$

The assumption people forget: $\Delta t_0$ is the proper time, measured by a clock at rest with respect to the event, and $L_0$ is the proper length, measured in the object's own rest frame. Get those two backwards and the effect comes out inverted.

Mass-energy equivalence is the famous line, but the version you actually compute with is the total energy:

$E_0 = mc^2, \qquad E = \gamma m c^2, \qquad K = (\gamma - 1)mc^2$

• $m$ — invariant (rest) mass, kg; the same in every frame
• $E_0$ — rest energy, joules; $E$ — total energy, joules
• $K$ — relativistic kinetic energy, joules

Momentum picks up the same factor, $p = \gamma m v$ in kg·m/s, and the three combine into the frame-independent relation

$E^2 = (pc)^2 + (mc^2)^2$

which is the most useful equation on this page: it lets you go between energy and momentum without ever computing $v$. For a photon $m = 0$ and it collapses to $E = pc$.

Practical units: particle physicists work in electronvolts, with $1\ \text{eV} = 1.602 \times 10^{-19}$ J. An electron's rest energy is $0.511$ MeV, a proton's is $938.3$ MeV.

• Treating "relativistic mass" $\gamma m$ as a real mass — it is an obsolete bookkeeping device. Modern practice keeps $m$ invariant and puts $\gamma$ on the energy and momentum instead.
• Using $K = \tfrac12 mv^2$ at high $\beta$ — at $0.99c$ the classical formula is off by a factor of twelve.
• Forgetting to square $\beta$ — the denominator is $\sqrt{1 - v^2/c^2}$, so $v = 0.8c$ gives $\sqrt{1 - 0.64}$, not $\sqrt{1 - 0.8}$.
• Adding velocities classically — $0.6c + 0.6c$ is $0.882c$, not $1.2c$. Nothing crosses $c$.
• Applying special relativity to accelerating or gravitating frames — that needs general relativity.
• Mixing eV and joules mid-calculation — convert once, at the start or at the end, never in the middle.