Lorentz factor

The Lorentz factor or Lorentz period is a quantity expressing the measure of changes in time, length, and other physical properties for an object that the object is moving. The expression appears in many equations in special relativity , and arises in the derivations of Lorentz transformations . The name originated from its earlier appearance in Lorentzian electrodynamics – named after the Dutch physicist Hendrik Lorentz . [1]

It is usually represented as (Greek lowercase letter gamma ). Sometimes (especially in discussions of superluminal motion ) the factor is written as (Greek uppercase -gamma ) instead of .


The Lorentz factor is defined as

{\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}={\frac {1}{\sqrt {1-\beta ^{2}}}}={\frac {dt}{d\tau }}},

Where from:

  • v is the relative velocity between the inertial reference frames,
  • c is the speed of light in vacuum ,
  • The ratio of β v to c is,
  • T is the time coordinate ,
  • is the proper time of an observer (to measure the time interval in the observer’s own frame).

This is the most frequently used form in practice, though not the only one (see below for alternative forms).

To complement the definition, some authors define reciprocal

{\displaystyle \alpha ={\frac {1}{\gamma }}={\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\ ={\sqrt {1-{\beta }^{2}}};}


Following is a list of formulas from special relativity which use as a shorthand :

  • Lorentz Transformation : Boost in the simplest case is the x -direction (arbitrary directions and more general forms including rotation not listed here) which describes the transformation from an inertial frame using space-time coordinates ( x , y ) , z , t ) to another ( x ‘ , y ‘ , z ‘ , t ‘ ) with relative velocity v :
{\displaystyle t'=\gamma \left(t-{\frac {vx}{c^{2}}}\right),}
{\displaystyle x'=\gamma \left(x-vt\right).}

The results of the above changes are:

  • Time dilation :  time (Δ t ‘ ) between two ticks as measured in the frame in which the clock is moving, is longer than the time (Δ t ) between these ticks as measured in the clock’s rest frame:
{\displaystyle \Delta t'=\gamma \Delta t.}

Length contraction : The length (Δ x ‘ ) of an object as measured in the frame in which it is moving, is less than its length (Δ x ) in its own rest frame:

{\displaystyle \Delta x'=\Delta x/\gamma .}

Applying conservation of momentum and energy gives these results:

  • Relativistic mass : The mass m is dependent on the motion of an object in motionand the rest mass
m_0 :\gamma 

Relativistic motion : The relative motion relation takes the same form as classical motion, but using the above relativistic mass:

{\displaystyle {\vec {p}} = m {\vec {v}} = \gamma m_ {0} {\vec {v}}.}
  • Relative Kinetic Energy : The relative kinetic energy relation takes a slightly modified form:
E_k = E - E_0 = (\gamma - 1) m_0 c^2

As a function , gives the non-relativistic limit , as expected from Newton’s ideas.

\gamma{\frac {v} {c}}{\displaystyle \lim _ {c \to \infty} E_ {k} = {\frac {1} {2}} m_ {0} v ^ {2}}

Numerical values

Lorentz factor
Lorentz factor

In the table below, the left-hand column shows the speed as different degrees of the speed of light (that is , in units of c ). The middle column represents the corresponding Lorentz factor, the last being the reciprocal. Values ​​in bold are accurate.

speed (units of c),
\beta = v/c
Lorentz factor,

alternative representation

There are other ways to write factors. Above all, velocity V was used, but related variables such as speed and intensity may also be convenient.


To solve the previous relativistic motion equation is taken to

\gamma = \sqrt{1+\left ( \frac{p}{m_0 c} \right )^2 } .

This form is rarely used, although it appears in the Maxwell–Juttner distribution .


Applying the definition of rapidity as a hyperbolic angle

{\displaystyle \tanh \varphi =\beta }

also leads to ( using the hyperbolic identity ):

{\displaystyle \gamma =\cosh \varphi ={\frac {1}{\sqrt {1-\tanh ^{2}\varphi }}}={\frac {1}{\sqrt {1-\beta ^{2}}}}.}

Using the property of the Lorentz transformation , it can be shown that momentum is additive, a useful property that velocity does not possess. Thus the rapidity parameter forms a one-parameter group , which is a basis for the physical model.

chain extension (velocity)

The Maclaurin series in the Lorentz factor is :

{\displaystyle {\begin{aligned}\gamma &={\dfrac {1}{\sqrt {1-\beta ^{2}}}}\\&=\sum _{n=0}^{\infty }\beta ^{2n}\prod _{k=1}^{n}\left({\dfrac {2k-1}{2k}}\right)\\&=1+{\tfrac {1}{2}}\beta ^{2}+{\tfrac {3}{8}}\beta ^{4}+{\tfrac {5}{16}}\beta ^{6}+{\tfrac {35}{128}}\beta ^{8}+{\tfrac {63}{256}}\beta ^{10}+\cdots ,\\\end{aligned}}}

which is a special case of a binomial series .

Approximation Gamma 1 +1/2 β 2 can be used to calculate relativistic effects at low speeds . This places it within a 1% error for v < 0.4 c ( v < 120,000 km/s) and within a 0.1% error for v < 0.22  c ( v < 66,000 km/s).

The truncated version of this series also allows physicists to prove that special relativity reduces to Newtonian mechanics at low speeds. For example, in special relativity, the following two equations hold:

{\displaystyle {\begin{aligned}{\vec {p}}&=\gamma m{\vec {v}},\\E&=\gamma mc^{2}.\end{aligned}}}

for Gamma 1 and Gamma 1 +1/2 β 2 , respectively, these are reduced to their Newtonian counterparts:

{\displaystyle {\begin{aligned}{\vec {p}}&=m{\vec {v}},\\E&=mc^{2}+{\tfrac {1}{2}}mv^{2}.\end{aligned}}}

The Lorentz factor equation can also be inverted to yield

{\displaystyle \beta ={\sqrt {1-{\frac {1}{\gamma ^{2}}}}}.}

it has an asymptomatic form

\beta = 1 - \tfrac12 \gamma^{-2} - \tfrac18 \gamma^{-4} - \tfrac{1}{16} \gamma^{-6} - \tfrac{5}{128} \gamma^{-8} + \cdots.

The first two terms are sometimes used to quickly calculate velocities from large values Approximation β 1 −1/2 The tolerance for -2 holds to within 1% for > 2, and the tolerance for > 3.5to within 0.1%

applications in astronomy

The Standard Model of long-period gamma-ray bursts (GRBs) considers these bursts to be super-relativistic (early). more than about 100), which is invoked to explain the so-called “compactness” problem: in the absence of this hyper-relativistic extension, the ejecta at a typical peak spectral energy of some 100 keV would be optically thick for pair production. , while the early emission is assumed to be non-thermal.


Subatomic particles called muons have a relatively high Lorentz factor and therefore experience extreme time dilation . As an example, the typically average lifetime of a muon is about . It happens2.2 μs which means that muons produced by cosmic ray collisions about 10 km above in the atmosphere should be non-detectable on the ground due to their decay rates. However, it has been found that ~10% of muons are still found on the surface, proving that their decay rate to be detectable has slowed relative to our inertial frame of reference.