relativistic energy
Relativistic energy is special relativity's replacement for the familiar '1/2 m v^2' kinetic energy of everyday physics — a formula that, like ordinary momentum, quietly fails as speeds approach that of light. The headline result is startling: an object has energy even when it is standing perfectly still, and its total energy grows without bound as it is pushed toward light-speed. Energy and mass, it turns out, are two faces of the same thing.
Precisely: the total relativistic energy of a free object is E = gamma m c^2, where m is its mass, c is the speed of light, and gamma = 1 / sqrt(1 - v^2/c^2) is the Lorentz factor. When the object is at rest, gamma = 1 and this becomes the rest energy E_0 = m c^2 — the energy locked up in mass itself. The rest is kinetic energy: KE = E - m c^2 = (gamma - 1) m c^2, the energy of motion. For slow speeds a little algebra shows (gamma - 1) m c^2 is almost exactly 1/2 m v^2, so the old Newtonian kinetic energy re-emerges as the low-speed approximation of this deeper formula. A compact and useful companion relation ties energy to momentum and mass: E^2 = (p c)^2 + (m c^2)^2.
This is where E = m c^2 lives, and it is not metaphor — it is measured. In nuclear reactions a tiny loss of mass reappears as an enormous release of energy, because c^2 is a huge multiplier. The energy-momentum relation E^2 = (p c)^2 + (m c^2)^2 also makes room for massless particles: set m = 0 and you get E = p c, exactly the energy-momentum link a photon obeys.
Push a proton to v = 0.9c (gamma = 2.29) and its total energy is 2.29 times its rest energy; only 1.29 rest-energies of that is kinetic. Try to reach c and gamma runs to infinity, so the energy needed grows without limit — the reason no massive particle ever quite gets there.
Total energy E = gamma m c^2 splits into rest energy m c^2 plus kinetic energy (gamma - 1) m c^2.
The full total energy is E = gamma m c^2; the rest energy m c^2 is just the v = 0 special case, and the kinetic part is the DIFFERENCE (gamma - 1) m c^2, not gamma m c^2. Don't mistake the total energy for the kinetic energy.