Special Relativity Calculator: Lorentz Factor, Time Dilation and E = mc2

The Lorentz factor, time dilation, length contraction, momentum and energy of a mass moving near the speed of light.

Calculated from standard textbook formulas, with every step shown. The formulas are fixed, so the working does not need checking: what matters is the numbers you type in. Use these calculators only if you are competent to choose the inputs (loads, material strength, dimensions, supports) and check your units and assumptions. For learning and checking your own working: design and sign-off are for a qualified engineer. Drafting Central does not store the numbers you type.

How this calculator works

Enter the speed as a percentage of the speed of light, the rest mass, a time on the moving clock and a rest length. The Lorentz factor is 1 over the square root of 1 minus v squared over c squared. A moving clock runs slow by that factor, and a moving length is shorter by it.

It also gives the rest energy E = m c squared, the kinetic energy (gamma minus 1) m c squared, the total energy and the momentum, and compares the relativistic kinetic energy with the classical half m v squared.

At everyday speeds the Lorentz factor is 1 to many decimal places, so none of this matters. At 80 percent of the speed of light it is 1.667.

Formulas used

  • \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}
  • E_0 = mc^2
  • K = (\gamma - 1)mc^2
  • p = \gamma m v

Worked example

With the default numbers in the calculator above:

  1. Speed: 2.3983e+8 m/s
  2. Lorentz factor: 1.6667
  3. Time on the stationary clock for the time on the moving clock (time dilation): 1.6667 s
  4. Length seen of the moving object (length contraction): 0.6 m
  5. Rest energy: 8.9876e+16 J (21481 kilotonnes of TNT)
  6. Kinetic energy: 5.9917e+16 J
  7. Total energy: 1.4979e+17 J
  8. Momentum: 3.9972e+8 kg·m/s
  9. Classical kinetic energy for comparison: 2.876e+16 J (the relativistic value is 2.08 times bigger)

Change the numbers in the calculator to see the full working, every step, for your own case.

Common questions

Can anything go faster than light?

Nothing with mass can reach it. The energy needed goes to infinity.

Is E = mc2 only for nuclear reactions?

It is true for all energy, but the mass change is too small to measure in chemical reactions.

Does time dilation matter in practice?

Yes. Satellite navigation (GPS) clocks have to be corrected for it.

Related calculators

Got a question about this? Ask an engineering question and get a plain-language answer with references.

Further reading: Einstein (1905); Halliday, Resnick and Walker, Fundamentals of Physics (relativity); CODATA speed of light.

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