The Lorentz factor or Lorentz term is the factor by which time, length, and relativistic mass change for an object while that object is moving. 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 Γ rather than.
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.
The Lorentz transformation: The simplest case is a boost in the x-direction, which describes how spacetime coordinates change from one inertial frame using coordinates to another with relative velocity v:
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:
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
0.000
1.000
1.000
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
1.250
0.800
0.700
1.400
0.714
0.750
1.512
0.661
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
0.99995
100.00
0.010
Alternative representations
There are other ways to write the factor. Above, velocity v 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.
The Lorentz factor has the Maclaurin series: which is a special case of a binomial series. The approximation ≈ 1 + β2 may be used to calculate relativistic effects at low speeds. It holds to within 1% error for v < 0.4 c, and to within 0.1% error for v < 0.22 c. 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 ≈ 1 and ≈ 1 + β2, 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 β ≈ 1 − −2 holds to within 1% tolerance for > 2, and to within 0.1% tolerance for > 3.5.
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. Subatomic particles called muons, have a relatively high Lorentz factor and therefore experience extreme time dilation. As an example, muons generally have a mean lifetime of about which means muons generated from cosmic ray collisions at about 10 km up in the atmosphere should be non-detectable on the ground due to their decay rate. However, it has been found that ~10% of muons are still detected on the surface, thereby proving that to be detectable they have had their decay rates slow down relative to our inertial frame of reference.
