Lorentz Factor Calculator

The Lorentz factor is defined as:

$$\gamma = \frac{1}{\sqrt{1 - \beta^2}}$$

where $$\beta = v/c$$ is the velocity expressed as a fraction of the speed of light. The Lorentz factor appears throughout special relativity:

• Time dilation: $$t = \gamma t_0$$ — moving clocks run slow by factor γ. The percentage shown is $$(\gamma - 1) \times 100\%$$.
• Length contraction: $$L = L_0/\gamma$$ — moving objects are shortened by factor 1/γ. The percentage is $$(1 - 1/\gamma) \times 100\%$$.
• Relativistic momentum: $$p = \gamma m_0 v$$.
• Relativistic energy: $$E = \gamma m_0 c^2$$, with kinetic energy $$KE = (\gamma - 1)m_0 c^2$$.
• Lorentz transformation: $$x' = \gamma(x - vt)$$, $$t' = \gamma(t - vx/c^2)$$.

Behavior of γ at notable velocities:

The γ − 1 value is particularly useful because it directly gives the kinetic energy as a multiple of rest energy: $$KE = (\gamma - 1) m_0 c^2$$. When γ − 1 = 1, the kinetic energy equals the rest energy. For ultra-relativistic particles (γ ≫ 1), almost all the energy is kinetic.

For very small velocities, computing γ directly from $$1/\sqrt{1-\beta^2}$$ can suffer from floating-point precision loss. In such cases, the Taylor expansion $$\gamma \approx 1 + \beta^2/2 + 3\beta^4/8$$ gives better numerical precision.

The Lorentz factor γ is the master parameter of special relativity. γ = 1 means no relativistic effects; γ = 2 means time runs at half speed and lengths are halved; γ = 10 means time runs at 1/10 speed and lengths shrink to 10% of their rest values. The time dilation percentage shows how much extra time a stationary observer measures (e.g., 100% means double the proper time). The length contraction percentage shows how much shorter a moving object appears (e.g., 50% means half the rest length).