Last updated: 2026-06-16

Relativistic Energy

Relativistic energy is the energy an object carries when special relativity is taken into account. Unlike the Newtonian picture, where energy is purely kinetic and mass is just an inert label, relativity ties energy and mass together through the speed of light $c \approx 3 \times 10^8\ \text{m/s}$. Even a motionless object has energy locked in its mass, and a moving object gains energy that climbs without bound as its speed approaches $c$.

This calculator takes a rest mass $m_0$ and a velocity $v$, and returns the Lorentz factor $\gamma$, the rest energy $E_0$, the total energy $E$, and the relativistic kinetic energy $K$.

How it works

The cornerstone is the relation $E = \gamma m_0 c^2$, where the Lorentz factor $\gamma = 1/\sqrt{1 - v^2/c^2}$ captures how strongly relativistic effects build up with speed. When the object is at rest, $\gamma = 1$ and the energy reduces to the rest energy $E_0 = m_0 c^2$ — the famous mass–energy equivalence. The extra energy a moving object carries beyond its rest energy is the relativistic kinetic energy $K = (\gamma - 1)m_0 c^2$.

At everyday speeds $\gamma$ is barely above one, so the kinetic energy matches the Newtonian $\tfrac{1}{2}m_0 v^2$. As $v$ grows toward $c$, $\gamma$ diverges and the total energy soars, which is why no massive object can ever be accelerated to the speed of light.

Formula

As $v \to 0$, $\gamma \to 1$, $K \to 0$ and $E \to E_0$. As $v \to c$, $\gamma \to \infty$ and $E \to \infty$.

Worked example

A 1 kg object moves at v = 0.8c.

Step 1 — Lorentz factor:

Step 2 — Rest energy:

Step 3 — Total energy:

Step 4 — Kinetic energy:

Enter these values in the calculator to reproduce the result.

Energy at different speeds

Real-world relevance

Particle accelerators are built around relativistic energy. Protons in a large collider are pushed to within a fraction of a percent of the speed of light, where their total energy is thousands of times their rest energy. The rest energy of subatomic particles is routinely quoted in electronvolts — an electron's rest energy is about 0.511 MeV — and conservation of total relativistic energy governs how particles are created and destroyed in collisions.

Limitation: this calculator covers special relativity only

The formulas here describe a free object in flat spacetime, with mass treated as the invariant rest mass. They do not include potential energy, binding energy, or the curvature effects of general relativity near very massive bodies. For kinematics and particle physics in inertial frames, the special-relativistic result is sufficient.