The
Lorentz factor or
Lorentz
term appears in several equations in
special relativity, including
time dilation,
length contraction, and the
relativistic mass formula. Because of its
ubiquity,
physicists generally represent it
with the shorthand symbol
γ. It gets its name from its
earlier appearance in
Lorentzian
electrodynamics.
The Lorentz factor is named after the
Dutch physicist
Hendrik Lorentz.
It is defined as:
 \gamma \equiv \frac{c}{\sqrt{c^2  u^2}} = \frac{1}{\sqrt{1 
\beta^2}} = \frac{\mathrm{d}t}{\mathrm{d}\tau}
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
 τ is the proper time,
and
 c is the speed of light.
Approximations
The Lorentz factor has a
Maclaurin
series of:
 \gamma ( \beta ) = 1 + \frac{1}{2} \beta^2 + \frac{3}{8}
\beta^4 + \frac{5}{16} \beta^6 + \frac{35}{128} \beta^8 + ...
The approximation γ ≈ 1 +
^{1}/
_{2} β
^{2}
is occasionally used to calculate relativistic effects at low
speeds. It holds to within 1% error for v < 0.4 c (v <
120,000 km/s), and to within 0.1% error for v 0.22 c (v < 66,000
km/s).
The truncated versions of this series also allow
physicists to prove that
special relativity reduces to
Newtonian mechanics at low speeds. For
example, in special relativity, the following two equations
hold:
 \vec p = \gamma m \vec v
 E = \gamma m c^2 \,
For γ ≈ 1 and γ ≈ 1 +
^{1}/
_{2} β
^{2},
respectively, these reduce to their Newtonian equivalents:
 \vec p = m \vec v
 E = m c^2 + \frac{1}{2} m v^2
The Lorentz factor equation can also be inverted to yield:
 \beta = \sqrt{1  \frac{1}{\gamma^2}}
This has an asymptotic form of:
 \beta = 1  \frac{1}{2} \gamma^{2}  \frac{1}{8} \gamma^{4} 
\frac{1}{16} \gamma^{6}  \frac{5}{128} \gamma^{8} + ...
The first two terms are occasionally used to quickly calculate
velocities from large γ values. The approximation β ≈ 1 
^{1}/
_{2} γ
^{2} holds to within 1%
tolerance for γ > 2, and to within 0.1% tolerance for γ >
3.5.
Values
Lorentz factor as a function of
velocity.
It starts at value 1 and for v\to c it goes to infinity.
Speed 
Lorentz factor 
Reciprocal 
\beta = v/c 
\gamma 
1/\gamma 
0.000 
1.000 
1.000 
0.100 
1.005 
0.995 
0.200 
1.021 
0.980 
0.300 
1.048 
0.954 
0.400 
1.091 
0.917 
0.500 
1.155 
0.866 
0.600 
1.250 
0.800 
0.700 
1.400 
0.714 
0.800 
1.667 
0.600 
0.866 
2.000 
0.500 
0.900 
2.294 
0.436 
0.990 
7.089 
0.141 
0.999 
22.366 
0.045 

In the above chart, the lefthand column shows speeds as different
fractions of the speed of light (c). The middle column shows the
corresponding Lorentz factor.
Rapidity
Note that if
tanh r =
β, then
γ = cosh
r. Here the
hyperbolic angle r is known as the
rapidity. Using the property of
Lorentz transformation, it can be
shown that rapidity is additive, a useful property that velocity
does not have. Thus the rapidity parameter forms a
oneparameter group, a foundation for
physical models.Sometimes (especially in discussion of
superluminal motion) γ is written as
Γ (uppercasegamma) rather than
γ
(lowercasegamma).
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{\mathrm{d}\tau}{\mathrm{d}t}
= \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.
In
particle physics, rapidity is
usually defined as (For example, see )
 :y = \frac{1}{2} \ln \left(\frac{E+p_L}{Ep_L}\right)
Derivation
One of the fundamental postulates of Einstein's
special theory of relativity is
that all
inertial
observers will measure the same speed of light in vacuum regardless
of their relative motion with respect to each other or the source.
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 = v t_B; the light had traveled
(also from B perspective) a distance d = c t_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{(c t _B)^2  (v t_B)^2}
Factoring out ct_B gives,
 d_t = c t _B\sqrt{1  {\left(\frac{v}{c}\right)}^2}
The distance that A sees the light travel is d_t = c t_A and
equating this with d_t calculated from B reference frame
gives,
 ct_A = ct_B \sqrt{1  {\left(\frac{v}{c}\right)}^2}
which simplifies to
 t_A = t_B\sqrt{1  {\left(\frac{v}{c}\right)}^2}
See also
References