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Encyclopedia > Lorentz factor

The Lorentz factor is a convenient term to define in special relativity. Image File history File links Please see the file description page for further information. ... In special relativity, many key relationships depart from both Newtonian mechanics and classical theory by a special velocity-dependent term that we can call a Lorentz term , due to its earlier appearance in Lorentzian electrodynamics. ... Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. It replaced Newtonian notions of space and time and incorporated electromagnetism as represented by Maxwells equations. ...

It is usually defined $gamma equiv frac{1}{sqrt{1 - beta^2}}$

where $beta = frac{u}{c}$

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. Cherenkov effect in a swimming pool nuclear reactor. ... In mathematics, the hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions. ... In astronomy, superluminal motion is the apparently faster-than-light motion seen in some radio galaxies, quasars and recently also in some galactic sources called microquasars. ...

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 γ. This article is in need of attention from an expert on the subject. ... Length contraction, according to Albert Einsteins special theory of relativity, is the decrease in length experienced by people or objects traveling at a substantial fraction of the speed of light. ... The term mass in special relativity can be used in different ways, occasionally leading to confusion. ...

%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: $v approx (1-frac {1} {2} gamma ^{-2})c$

## 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 c). Imagine two observers, the first, observer A, traveling at a speed v with a laser, and the other, observer B, in an inertial rest frame. A points his laser upward (perpendicular to the direction of travel). From B's perspective, the light is traveling at an angle. After a period of time t, A has traveled (from B's perspective) a distance d = vt; the light had traveled (also from B perspective) a distance d = ct at an angle. The upward component of the path dt of the light can be solved by the Pythagorean theorem. The Pythagorean theorem: The sum of the areas of the two squares on the legs (blue and red) equals the area of the square on the hypotenuse (purple). ... $d_t = sqrt{(c t)^2 - (v t)^2}$

Factoring out ct gives us, $d_t = c tsqrt{1 - {left(frac{v}{c}right)}^2}$

This distance is the same distance that A sees the light travel. Because the light must travel at c, A's time, t', will be equal to $frac{d_u}{c}$. Therefore $t' = frac{c tsqrt{1 - {left(frac{v}{c}right)}^2}}{c}$

which simplifies to $t' = tsqrt{1 - {left(frac{v}{c}right)}^2}$ Results from FactBites:

 A Question of Time: The Lorentz Transformation (1798 words) Lorentz thought that perhaps the molecules of a solid might be compressed in the direction of motion, and this could explain why the time of travel is the same in both directions. Lorentz conjectured as early as 1895 that perhaps the rod parallel to the earths movement might shrink because electrons would lose their spherical shape due to the earths motion. Lorentz quantified his conjecture in a paper in 1904.
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