Lorentz factor
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The Lorentz factor is a convenient term to define in special relativity.
It is usually defined
- <math>\gamma \equiv \frac{1}{\sqrt{1 - \beta^2}}<math>
where
- <math>\beta = u/c<math>
is the velocity u in units of c, the speed of light. Note that if tanh r = β, then γ = cosh r. Here r is known as the rapidity. Rapidity has the property that relative rapidities are additive, a useful property which velocity does not have in Special Relativity. Sometimes (especially in discussion of superluminal motion) γ is written as Γ rather than γ.
The Lorentz factor applies to time dilation, length contraction and relativistic mass relative to rest mass in Special Relativity. An object moving with respect to an observer will be seen to move in slow motion given by multiplying its actual elapsed time by gamma. Its length is measured shorter as though its local length were divided by gamma. All seeming paradoxes of special relativity are resolved by the proper visualization of desynchronization.
Table
| %c | Lorentz factor | reciprocal |
|---|---|---|
| 0 | 1.000 | 1.000 |
| 10 | 1.005 | 0.995 |
| 50 | 1.155 | 0.867 |
| 90 | 2.294 | 0.436 |
| 99 | 7.089 | 0.141 |
| 99.9 | 22.366 | 0.045 |
For large γ: <math>v \approx (1-\frac {1} {2} \gamma ^{-2})c<math>
Proof
First of all, one must realize that from every observer, light travels at the speed of light (which is why the speed of light is represented as <math>c<math>). Imagine two observers, the first, observer <math>A<math>, traveling at a speed <math>v<math> with a laser, and the other, observer <math>B<math>, in an inertial rest frame. <math>A<math> points his laser upward (perpendicular to the direction of travel). From <math>B<math>'s perspective, the light is traveling at an angle. After a period of time <math>t<math>, <math>A<math> has traveled (from <math>B<math>'s perspective) a distance <math>d = v t<math>; the light had traveled (also from <math>B<math> perspective) a distance <math>d = c t<math> at an angle. The upward compnent of the path <math>d_t<math> of the light can be solved by the Pythagorean theorem.
<math>d_t = \sqrt{(c t)^2 - (v t)^2}<math>
Factoring out <math>ct<math> gives us,
<math>d_t = c t\sqrt{1 - {\left(\frac{v}{c}\right)}^2}<math>
This distance is the same distance that <math>A<math> sees the light travel. Because the light must travel at <math>c<math>, <math>A<math>'s time, <math>t'<math>, will be equal to <math>\frac{d_u}{c}<math>. Therefore
<math>t' = \frac{c t\sqrt{1 - {\left(\frac{v}{c}\right)}^2}}{c}<math>
which simplifies to
<math>t' = t\sqrt{1 - {\left(\frac{v}{c}\right)}^2}<math>