Relativistic Mass Calculator

Relativistic Mass

In special relativity, an object's apparent inertia grows with its velocity. The historical formula is m = m₀ / √(1 - v²/c²), where m₀ is the rest mass, v the velocity, and c the speed of light. The denominator is the inverse of the Lorentz factor γ = 1/√(1 - v²/c²), introduced by Einstein in his 1905 paper on the electrodynamics of moving bodies.

Modern physicists generally avoid the term "relativistic mass" and prefer to use the invariant rest mass together with the full energy-momentum relation E² = p²c² + m²c⁴. The "mass grows with speed" picture is kept mostly for pedagogical purposes when introducing γ and kinetic energy at high velocities.

Applications

Particle accelerators such as the LHC at CERN and the RHIC at Brookhaven push protons and heavy ions to within 0.999999991 of c, where γ reaches the thousands. The magnetic rigidity needed to bend these beams scales directly with γm₀, making the Lorentz factor a daily working number for accelerator physicists. Cosmic-ray muons reaching ground level are another classic example.

FAQ

Does the object really get heavier? Its rest mass stays the same; only the inertia measured by an external observer scales with γ. A spaceship crew weighing themselves onboard would see no change.

Why is the concept falling out of use? Because mass should be an invariant property of the object. Tying it to the observer's frame leads to confusion in problems involving energy and momentum.

What happens at v = c? The denominator goes to zero and the formula diverges, which is why massive particles can never reach the speed of light.