Lorentz factor

In special relativity, the Lorentz factor or Lorentz term is a term that appears very often and is used to make writing equations easier. It is used in time dilation, length contraction, and to convert rest mass to relativistic mass, among others. It gets its name from its earlier appearance in Lorentzian electrodynamics.

It is usually defined

$\gamma \equiv {\frac {dt}{d\tau }}={\frac {1}{\sqrt {1-\beta ^{2}}}}$		where $\beta ={\frac {u}{c}}$ is the velocity in terms of the speed of light, u is the velocity as observed in the reference frame where time t is measured $\tau$ is the proper time, and c is 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 (uppercase-gamma) rather than γ (lowercase-gamma).

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.

γ may also (less often) refer to ${\frac {d\tau }{dt}}={\sqrt {1-\beta ^{2}}}$. This may make the symbol γ ambiguous, so many authors prefer to avoid possible confusion by writing out the Lorentz term in full.

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

Proof

First of all, one must realize that for every observer, light travels at the same speed of light (which is why the speed of light is represented as a constant ($c$)). Imagine two observers: the first, observer $A$, traveling at a constant speed $v$ with respect to a second inertial reference frame in which observer $B$ is stationary. $A$ points a laser “upward” (perpendicular to the direction of travel). From $B$'s perspective, the light is traveling at an angle. After a period of time $t_{B}$, $A$ has traveled (from $B$'s perspective) a distance $d=vt_{B}$; the light had traveled (also from $B$ perspective) a distance $d=ct_{B}$ at an angle. The upward component of the path $d_{t}$ of the light can be solved by the Pythagorean theorem.

$d_{t}={\sqrt {(ct_{B})^{2}-(vt_{B})^{2}}}$

Factoring out $ct_{B}$ gives us,

$d_{t}=ct{\sqrt {1-{\left({\frac {v}{c}}\right)}^{2}}}$

This distance is the same distance that $A$ sees the light travel. Because the light must travel at $c$, $A$'s time, $t_{A}$, will be equal to ${\frac {d_{u}}{c}}$. Therefore

$t_{A}={\frac {ct_{B}{\sqrt {1-{\left({\frac {v}{c}}\right)}^{2}}}}{c}}$

which simplifies to

$t_{A}=t_{B}{\sqrt {1-{\left({\frac {v}{c}}\right)}^{2}}}$

See also

• special relativity
• Lorentz transformation