The
Lorentz factor or
Lorentz term appears in several equations in
special relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
, including
time dilationIn the theory of relativity, time dilation is an observed difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at...
,
length contractionIn physics, length contraction – according to Hendrik Lorentz – is the physical phenomenon of a decrease in length detected by an observer of objects that travel at any nonzero velocity relative to that observer...
, and the relativistic mass formula. Because of its ubiquity,
physicistsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
generally represent it with the shorthand symbol
γ (lowercase
gammaGamma is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. It was derived from the Phoenician letter Gimel . Letters that arose from Gamma include the Roman C and G and the Cyrillic letters Ge Г and Ghe Ґ.Greek:In Ancient Greek, gamma represented a...
). It gets its name from its earlier appearance in
Lorentzian electrodynamicsWhat is now often called Lorentz Ether theory has its roots in Hendrik Lorentz's "Theory of electrons", which was the final point in the development of the classical aether theories at the end of the 19th and at the beginning of the 20th century....
. The Lorentz factor is named after the
DutchThe Netherlands is a constituent country of the Kingdom of the Netherlands, located mainly in NorthWest Europe and with several islands in the Caribbean. Mainland Netherlands borders the North Sea to the north and west, Belgium to the south, and Germany to the east, and shares maritime borders...
physicist
Hendrik LorentzHendrik Antoon Lorentz was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect...
.
It is defined as:

where:
 is the velocity in terms of the speed of light
The speed of light in vacuum, usually denoted by c, is a physical constant important in many areas of physics. Its value is 299,792,458 metres per second, a figure that is exact since the length of the metre is defined from this constant and the international standard for time...
,
 v is the velocity as observed in the reference frame where time t is measured
 τ is the proper time
In relativity, proper time is the elapsed time between two events as measured by a clock that passes through both events. The proper time depends not only on the events but also on the motion of the clock between the events. An accelerated clock will measure a smaller elapsed time between two...
, and
 c is the speed of light.
Approximations
The Lorentz factor has a
Maclaurin seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
of:

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
physicistsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
to prove that
special relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
reduces to Newtonian mechanics at low speeds. For example, in special relativity, the following two equations hold:
For γ ≈ 1 and γ ≈ 1 +
^{1}/
_{2} β
^{2}, respectively, these reduce to their Newtonian equivalents:
The Lorentz factor equation can also be inverted to yield:

This has an asymptotic form of:

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 is γ > 2, and to within 0.1% tolerance for γ > 3.5.
Values
Speed  Lorentz factor  Reciprocal 

  
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 angleIn mathematics, a hyperbolic angle is a geometric figure that divides a hyperbola. The science of hyperbolic angle parallels the relation of an ordinary angle to a circle...
r is known as the
rapidity. Using the property of
Lorentz transformationIn physics, the Lorentz transformation or LorentzFitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
, it can be shown that rapidity is additive, a useful property that velocity does not have. Thus the rapidity parameter forms a
oneparameter groupIn mathematics, a oneparameter group or oneparameter subgroup usually means a continuous group homomorphismfrom the real line R to some other topological group G...
, a foundation for physical models.
Sometimes (especially in discussion of
superluminal motionIn astronomy, superluminal motion is the apparently fasterthanlight motion seen in someradio galaxies, quasars and recently also in some galactic sources called microquasars...
) γ is written as
Γ (uppercasegamma) rather than
γ (lowercasegamma).
The Lorentz factor applies to
time dilationIn the theory of relativity, time dilation is an observed difference of elapsed time between two events as measured by observers either moving relative to each other or differently situated from gravitational masses. An accurate clock at rest with respect to one observer may be measured to tick at...
,
length contractionIn physics, length contraction – according to Hendrik Lorentz – is the physical phenomenon of a decrease in length detected by an observer of objects that travel at any nonzero velocity relative to that observer...
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.
In
particle physicsParticle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...
, rapidity is usually defined as (For example, see )

Derivation
One of the fundamental postulates of Einstein's special theory of relativity is that all
inertialIn physics, an inertial frame of reference is a frame of reference that describes time homogeneously and space homogeneously, isotropically, and in a timeindependent manner.All inertial frames are in a state of constant, rectilinear motion with respect to one another; they are not...
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
, traveling at a constant speed
with respect to a second inertial reference frame in which observer
is stationary.
points a laser “upward” (perpendicular to the direction of travel). From
's perspective, the light is traveling at an angle. After a period of time
,
has traveled (from
's perspective) a distance
; the light had traveled (also from
perspective) a distance
at an angle. The upward component of the path
of the light can be solved by the
Pythagorean theoremIn mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a right triangle...
.
Factoring out
gives,
The distance that
sees the light travel is
and equating this with
calculated from
reference frame gives,
which simplifies to
See also
 Special relativity
Special relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
 Lorentz transformation
In physics, the Lorentz transformation or LorentzFitzgerald transformation describes how, according to the theory of special relativity, two observers' varying measurements of space and time can be converted into each other's frames of reference. It is named after the Dutch physicist Hendrik...
 Pseudorapidity
 Proper velocity
In relativity, propervelocity, also known as celerity, is an alternative to velocity for measuring motion. Whereas velocity relative to an observer is distance per unit time where both distance and time are measured by the observer, proper velocity relative to an observer divides observermeasured...