"Gamma factor" redirects here. Not to be confused with
gamma function.
Definition of the Lorentz factor γ
The Lorentz factor or Lorentz term (also known as the gamma factor[1]) is a
quantity that expresses how much the measurements of time, length, and other physical properties 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.[2]
It is generally denoted γ (the Greek lowercase letter
gamma). Sometimes (especially in discussion of
superluminal motion) the factor is written as Γ (Greek uppercase-gamma) rather than γ.
Following is a list of formulae from Special relativity which use γ as a shorthand:[3][5]
The Lorentz transformation: The simplest case is a boost in the x-direction (more general forms including arbitrary directions and rotations not listed here), which describes how spacetime coordinates change from one inertial frame using coordinates (x, y, z, t) to another (x′, y′, z′, t′) with relative velocity v:
Corollaries of the above transformations are the results:
Time dilation: The time (∆t′) between two ticks as measured in the frame in which the clock is moving, is longer than the time (∆t) between these ticks as measured in the rest frame of the clock:
Length contraction: The length (∆x′) of an object as measured in the frame in which it is moving, is shorter than its length (∆x) in its own rest frame:
Relativistic momentum: The relativistic
momentum relation takes the same form as for classical momentum, but using the above relativistic mass:
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
Lorentz factor γ as a function of fraction of given velocity and speed of light. Its initial value is 1 (when v = 0); and as velocity approaches the speed of light (v → c)γ increases without bound (γ → ∞).α (Lorentz factor inverse) as a function of velocity—a circular arc
In the table below, the left-hand column shows speeds as different fractions of the speed of light (i.e. in units of c). The middle column shows the corresponding Lorentz factor, the final is the reciprocal. Values in bold are exact.
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 γ leads to
This form is rarely used, although it does appear in the
Maxwell–Jüttner distribution.[6]
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:[8]
The approximation may be 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:
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 γ values. The approximation holds to within 1% tolerance for γ > 2, and to within 0.1% tolerance for γ > 3.5.
Applications in astronomy
The standard model of long-duration gamma-ray bursts (GRBs) holds that these explosions are ultra-relativistic (initial γ greater than approximately 100), 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.[9]
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 10 km (6.2 mi) 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.[10]