TheLorentz factor orLorentz term (also known as thegamma factor^[1]) is adimensionless 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 inspecial relativity, and it arises in derivations of theLorentz transformations. The name originates from its earlier appearance inLorentzian electrodynamics – named after theDutch physicistHendrik Lorentz.^[2]

It is generally denotedγ (the Greek lowercase lettergamma). Sometimes (especially in discussion ofsuperluminal motion) the factor is written asΓ (Greek uppercase-gamma) rather thanγ.

The Lorentz factorγ is defined as^[3]$${\displaystyle \gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}=\sqrt{\frac{c^2}{c^2-v^2}}=\frac{c}{\sqrt{c^2-v^2}}=\frac{1}{\sqrt{1-{\beta }^2}}=\frac{dt}{d\tau },}$$ where:

• v is therelative velocity between inertial reference frames,
• c is thespeed of light in vacuum,
• β is the ratio ofv toc,
• t iscoordinate time,
• τ is theproper time for an observer (measuring time intervals 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 the reciprocal^[4]$${\displaystyle \alpha =\frac{1}{\gamma }=\sqrt{1-\frac{v^2}{c^2}}=\sqrt{1-{\beta }^2};}$$ seevelocity addition formula.

Following is a list of formulae fromspecial relativity which useγ as a shorthand:^[3][5]

• TheLorentz transformation: The simplest case is a boost in thex-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 velocityv: 

The above transformations engender the following effects:

• Time dilation: The time∆t′ between two ticks of a spaceship's onboard clock is longer than the time∆t between two ticks of a terrestrial clock:$${\displaystyle \mathrm{\Delta }{t}^{\prime }=\gamma \mathrm{\Delta }t.}$$  Consequently, the distance∆x′ travelled by a spaceship per one tick∆t′ of a terrestrial clock is shorter than the distance∆x travelled by the spaceship per one tick∆t of an onboard clock:$${\displaystyle \mathrm{\Delta }{x}^{\prime }=\mathrm{\Delta }x/\gamma .}$$ 
• Length contraction: The lengthL′ of a spaceship from the viewpoint of a terrestrial observer is shorter than lengthL of the same spaceship from the viewpoint of its passenger:$${\displaystyle L^{\prime }=L/\gamma .}$$ 

Applyingconservation ofmomentum and energy leads to these results:

• Relativistic mass: Themassm of an object in motion is dependent on${\displaystyle \gamma }$  and therest massm₀:$${\displaystyle m=\gamma m_0.}$$ 
• Relativistic momentum: The relativisticmomentum relation takes the same form as for classical momentum, but using the above relativistic mass:$${\displaystyle \overrightarrow{p}=m\overrightarrow{v}=\gamma m_0\overrightarrow{v}.}$$ 
• Relativistic kinetic energy: The relativistic kineticenergy relation takes the slightly modified form:$${\displaystyle E_k=E-E_0=(\gamma -1)m_0{c}^2}$$ As${\displaystyle \gamma }$  is a function of${\displaystyle \frac{v}{c}}$ , the non-relativistic limit gives$\underset{v/c\to 0}{lim}{E}_k=\frac{1}{2}{m}_0{v}^2$ , as expected from Newtonian considerations.

In the table below, the left-hand column shows speeds as different fractions of the speed of light (i.e. in units ofc). 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, velocityv was used, but related variables such asmomentum andrapidity may also be convenient.

Solving the previous relativistic momentum equation forγ leads to$${\displaystyle \gamma =\sqrt{1+{\left(\frac{p}{m_{0}c}\right)}^2}.}$$ This form is rarely used, although it does appear in theMaxwell–Jüttner distribution.^[6]

Applying the definition ofrapidity as thehyperbolic angle${\displaystyle \phi }$ :^[7]$${\displaystyle \tanh \phi =\beta }$$ also leads toγ (by use ofhyperbolic identities):$${\displaystyle \gamma =\cosh \phi =\frac{1}{\sqrt{1-{\tanh }^2\phi }}=\frac{1}{\sqrt{1-{\beta }^2}}.}$$ 

Using the property ofLorentz transformation, it can be shown that rapidity is additive, a useful property that velocity does not have. Thus the rapidity parameter forms aone-parameter group, a foundation for physical models.

The Bunney identity represents the Lorentz factor in terms of an infinite series ofBessel functions:^[8]$${\displaystyle \sum _{m=1}^{\mathrm{\infty }}\left(J_{m-1}^2(m\beta )+J_{m+1}^2(m\beta )\right)=\frac{1}{\sqrt{1-{\beta }^2}}.}$$ 

The Lorentz factor has theMaclaurin series: which is a special case of abinomial series.

The approximation$\gamma \approx 1+\frac{1}{2}{\beta }^2$  may be used to calculate relativistic effects at low speeds. It holds to within 1% error forv < 0.4 c (v < 120,000 km/s), and to within 0.1% error forv < 0.22 c (v < 66,000 km/s).

The truncated versions of this series also allowphysicists to prove thatspecial relativity reduces toNewtonian mechanics at low speeds. For example, in special relativity, the following two equations hold:

 

For${\displaystyle \gamma \approx 1}$  and$\gamma \approx 1+\frac{1}{2}{\beta }^2$ , respectively, these reduce to their Newtonian equivalents:

 

The Lorentz factor equation can also be inverted to yield$${\displaystyle \beta =\sqrt{1-\frac{1}{{\gamma }^2}}.}$$ This has an asymptotic form$${\displaystyle \beta =1-\frac{1}{2}{\gamma }^{-2}-\frac{1}{8}{\gamma }^{-4}-\frac{1}{16}{\gamma }^{-6}-\frac{5}{128}{\gamma }^{-8}+\cdots .}$$ 

The first two terms are occasionally used to quickly calculate velocities from largeγ values. The approximation$\beta \approx 1-\frac{1}{2}{\gamma }^{-2}$  holds to within 1% tolerance forγ > 2, and to within 0.1% tolerance forγ > 3.5.

The standard model of long-duration gamma-ray bursts (GRBs) holds that these explosions are ultra-relativistic (initialγ 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 extremetime dilation. Since muons have a mean lifetime of just 2.2 μs, muons generated fromcosmic-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]

• Inertial frame of reference
• Proper velocity
• Pseudorapidity

• Merrifield, Michael."γ – Lorentz Factor (and time dilation)".Sixty Symbols.Brady Haran for theUniversity of Nottingham.
• Merrifield, Michael."γ2 – Gamma Reloaded".Sixty Symbols.Brady Haran for theUniversity of Nottingham.