Lorentz factor

The Lorentz factor or Lorentz term is a dimensionless quantity expressing how much the measurements of time, length, and other physical properties change for an object while it moves. The expression appears in several equations in special relativity, and it arises in derivations of the Lorentz transformations. The name originates from its earlier appearance in Lorentzian electrodynamics – named after the Dutch physicist Hendrik Lorentz.
It is generally denoted . Sometimes the factor is written as rather than.

Definition

The Lorentz factor is defined as
where:

is the relative velocity between inertial reference frames,
is the speed of light in vacuum,
is the ratio of to,
is coordinate time,
is the proper time for an observer.

This is the most frequently used form in practice, though not the only one.
To complement the definition, some authors define the reciprocal
see velocity addition formula.

Occurrence

Following is a list of formulae from Special relativity which use as a shorthand:

The Lorentz transformation: The simplest case is a boost in the -direction, which describes how spacetime coordinates change from one inertial frame using coordinates to another with relative velocity :

Corollaries of the above transformations are the results:Time dilation: The time between two ticks as measured in the frame in which the clock is moving, is longer than the time between these ticks as measured in the rest frame of the clock: Length contraction: The length of an object as measured in the frame in which it is moving, is shorter than its length in its own rest frame:
Applying conservation of momentum and energy leads to these results:Relativistic mass: The relativistic mass of an object in motion is dependent on and the rest mass : Relativistic momentum: The relativistic momentum relation takes the same form as for classical momentum, but using the above relativistic mass: Kinetic energy#Relativistic [kinetic energy of rigid bodies|Relativistic kinetic energy]: The relativistic kinetic energy relation takes the slightly modified form: As is a function of, the non-relativistic limit gives, as expected from Newtonian considerations.

Numerical values

In the table below, the left-hand column shows speeds as different fractions of the speed of light. The middle column shows the corresponding Lorentz factor, the final is the reciprocal. Values in bold are exact.

Speed,	Lorentz factor,	Reciprocal,
0	1	1
0.050	1.001	0.999
0.100	1.005	0.995
0.150	1.011	0.989
0.200	1.021	0.980
0.250	1.033	0.968
0.300	1.048	0.954
0.400	1.091	0.917
0.500	1.155	0.866
0.600	1.25	0.8
0.700	1.400	0.714
0.750	1.512	0.661
0.800	1.667	0.6
0.866	2	0.5
0.900	2.294	0.436
0.990	7.089	0.141
0.999	22.366	0.045
0.99995	100.00	0.010

Alternative representations

There are other ways to write the factor. Above, velocity was used, but related variables such as momentum and rapidity may also be convenient.

Momentum

Solving the previous relativistic momentum equation for leads to
This form is rarely used, although it does appear in the Maxwell–Jüttner distribution.

Rapidity

Applying the definition of rapidity as the hyperbolic angle :
also leads to :
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 one-parameter group, a foundation for physical models.

Bessel function

The Bunney identity represents the Lorentz factor in terms of an infinite series of Bessel functions:

Series expansion (velocity)

The Lorentz factor has the following Maclaurin series:
which is a special case of a binomial series.
The approximation may be used to calculate relativistic effects at low speeds. It holds to within 1% error for < 0.4 , and to within 0.1% error for < 0.22 .
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:
For and, respectively, these reduce to their Newtonian equivalents:
The Lorentz factor equation can also be inverted to yield
This has an asymptotic form
The first two terms are occasionally used to quickly calculate velocities from large values. The approximation holds to within 1% tolerance for and to within 0.1% tolerance for

Applications in astronomy

The standard model of long-duration gamma-ray bursts holds that these explosions are ultra-relativistic, which is invoked to explain the so-called "compactness" problem: absent this ultra-relativistic expansion, the ejecta would be optically thick to pair production at typical peak spectral energies of a few 100 keV, whereas the prompt emission is observed to be non-thermal.
Muons, a subatomic particle, travel at a speed such that they have a relatively high Lorentz factor and therefore experience extreme time dilation. Since muons have a mean lifetime of just 2.2 μs, muons generated from cosmic-ray collisions high in Earth's atmosphere should be nondetectable on the ground due to their decay rate. However, roughly 10% of muons from these collisions are still detectable on the surface, thereby demonstrating the effects of time dilation on their decay rate.