Relativistic Mass Calculator

Relativistic mass vs. rest mass: what this calculator does

In special relativity, the mass of an object at rest is called its rest mass or invariant mass, usually written \(m_0\). When the object moves at a significant fraction of the speed of light, its energy and momentum increase according to the Lorentz factor \(\gamma\).

Historically, some authors defined a relativistic mass \[ m = \gamma m_0 \] that grows with speed. Modern textbooks usually avoid this term and keep “mass” fixed (meaning \(m_0\)), letting energy and momentum carry the relativistic effects instead.

This calculator supports both viewpoints:

• It computes the Lorentz factor \(\gamma\) from the speed.
• It shows the corresponding relativistic mass \(m = \gamma m_0\) if you want to use that language.
• It also gives the total energy \(E\), kinetic energy \(K\), and momentum \(p\), which are the preferred modern quantities.

Relativistic mass formulas

Back-calculating speed from \(\gamma\)

If you know the Lorentz factor \(\gamma\) instead of the speed, you can invert the formula:

\[ \frac{v}{c} = \sqrt{1 - \frac{1}{\gamma^2}} \]

The calculator lets you enter \(\gamma\) directly and will compute the corresponding \(v/c\) and \(v\) in m/s for you.

Worked example: 1 kg at 0.8c

Suppose a 1 kg object moves at \(v = 0.8c\).

1. Compute \(\gamma\)
   \[ \gamma = \frac{1}{\sqrt{1 - (0.8)^2}} = \frac{1}{\sqrt{1 - 0.64}} = \frac{1}{\sqrt{0.36}} \approx 1.6667 \]
2. Relativistic mass
   \[ m = \gamma m_0 \approx 1.6667 \times 1\ \text{kg} = 1.6667\ \text{kg} \]
3. Total energy
   \[ E = \gamma m_0 c^2 \approx 1.6667 \times 1 \times c^2 \] Numerically, \(c^2 \approx 8.98755 \times 10^{16}\ \text{m}^2/\text{s}^2\), so \[ E \approx 1.50 \times 10^{17}\ \text{J} \]
4. Kinetic energy
   \[ K = (\gamma - 1)m_0 c^2 \approx 0.6667 \times c^2 \approx 5.99 \times 10^{16}\ \text{J} \]
5. Momentum
   \[ p = \gamma m_0 v = 1.6667 \times 1 \times 0.8c \approx 1.3333 c \approx 4.00 \times 10^8\ \text{kg·m/s} \]

These are exactly the values you see in the default state of the calculator.

Why many physicists avoid “relativistic mass”

The idea of relativistic mass was historically useful, but it has some drawbacks:

• It suggests that the intrinsic mass of an object depends on the observer’s frame.
• It can obscure the elegant symmetry of the energy–momentum relation \[ E^2 = (m_0 c^2)^2 + (pc)^2. \]
• It leads to ambiguous statements like “mass increases with speed,” which can confuse students.

Modern treatments of special relativity therefore use:

• Mass = invariant rest mass \(m_0\) (same in all inertial frames).
• Energy and momentum as the frame-dependent quantities.

This calculator exposes the relativistic mass \(m = \gamma m_0\) for users who still encounter the term, but it also highlights the more robust energy and momentum picture.

Common questions about relativistic mass

Does increasing relativistic mass cause gravity to increase?

In general relativity, gravity is sourced by the full stress–energy tensor, not just a scalar mass. Energy, momentum, pressure and stress all contribute to spacetime curvature. Saying “gravity increases because mass increases” is an oversimplification; it is more accurate to say that energy and momentum determine the gravitational field.

Can anything with mass reach the speed of light?

No. As \(v \to c\), the Lorentz factor \(\gamma\) grows without bound, so the required energy \(E = \gamma m_0 c^2\) diverges. A massive object can get arbitrarily close to \(c\), but never reach it with finite energy input.

What about photons — do they have relativistic mass?

Photons have zero rest mass (\(m_0 = 0\)) but nonzero energy and momentum: \[ E = pc = h\nu. \] Some older texts say photons have “relativistic mass” \(m = E/c^2\), but modern usage simply talks about their energy and momentum instead of assigning them a mass.