Lorentz factor

File:RelativeFactor.png

Definition of the Lorentz factor γ

The Lorentz factor or Lorentz term (also known as the gamma factor<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>) 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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

It is generally denoted Template:Math (the Greek lowercase letter gamma). Sometimes (especially in discussion of superluminal motion) the factor is written as Template:Math (Greek uppercase-gamma) rather than Template:Math.

Definition

The Lorentz factor Template:Math is defined as<ref name="Forshaw 2014">Template:Cite book</ref> <math display="block">\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}} = \frac{1}{\sqrt{1 - \beta^2}} = \frac{dt}{d\tau} ,</math> where:

• Template:Mvar is the relative velocity between inertial reference frames,
• Template:Mvar is the speed of light in vacuum,
• Template:Mvar is the ratio of Template:Mvar to Template:Mvar,
• Template:Mvar is coordinate time,
• Template:Mvar is the proper 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<ref>Yaakov Friedman, Physical Applications of Homogeneous Balls, Progress in Mathematical Physics 40 Birkhäuser, Boston, 2004, pages 1-21.</ref> <math display="block">\alpha = \frac{1}{\gamma} = \sqrt{1- \frac{v^2}{c^2}} \ = \sqrt{1- {\beta}^2} ;</math> see velocity addition formula.

Occurrence

Following is a list of formulae from Special relativity which use Template:Math as a shorthand:<ref name="Forshaw 2014" /><ref>Template:Cite book</ref>

• The Lorentz transformation: The simplest case is a boost in the Template:Mvar-direction (more general forms including arbitrary directions and rotations not listed here), which describes how spacetime coordinates change from one inertial frame using coordinates Template:Math to another Template:Math with relative velocity Template:Mvar: <math display="block">\begin{align}

 t' &= \gamma \left( t - \tfrac{vx}{c^2} \right ), \\[1ex]
 x' &= \gamma \left( x - vt \right ).

\end{align}</math> Corollaries of the above transformations are the results:

• Time dilation: The time (Template:Math) between two ticks as measured in the frame in which the clock is moving, is longer than the time (Template:Math) between these ticks as measured in the rest frame of the clock: <math display="block">\Delta t' = \gamma \Delta t.</math>
• Length contraction: The length (Template:Math) of an object as measured in the frame in which it is moving, is shorter than its length (Template:Math) in its own rest frame: <math display="block">\Delta x' = \Delta x/\gamma.</math>

Applying conservation of momentum and energy leads to these results:

• Relativistic mass: The mass Template:Mvar of an object in motion is dependent on <math>\gamma</math> and the rest mass Template:Math: <math display="block">m = \gamma m_0.</math>
• Relativistic momentum: The relativistic momentum relation takes the same form as for classical momentum, but using the above relativistic mass: <math display="block">\vec p = m \vec v = \gamma m_0 \vec v.</math>
• Relativistic kinetic energy: The relativistic kinetic energy relation takes the slightly modified form: <math display="block">E_k = E - E_0 = (\gamma - 1) m_0 c^2</math>As <math>\gamma</math> is a function of <math>\tfrac{v}{c}</math>, the non-relativistic limit gives <math display="inline">\lim_{v/c\to 0}E_k=\tfrac{1}{2}m_0v^2</math>, as expected from Newtonian considerations.

Numerical values

File:Lorentz factor 2.png

Lorentz factor Template:Mvar as a function of fraction of given velocity and speed of light. Its initial value is 1 (when Template:Math); and as velocity approaches the speed of light Template:Nobr Template:Mvar increases without bound Template:Nobr

File:Lorentz factor inverse.svg

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 Template:Mvar). The middle column shows the corresponding Lorentz factor, the final is the reciprocal. Values in bold are exact.

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

File:Lorentz factor log log.svg

Log-log plot of Lorentz factor γ (left) and 1/γ (right) vs fraction of speed of light β (bottom) and 1−β (top)

Alternative representations

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

Momentum

Solving the previous relativistic momentum equation for Template:Math leads to <math display="block">\gamma = \sqrt{1+\left ( \frac{p}{m_0 c} \right )^2 } \,.</math> This form is rarely used, although it does appear in the Maxwell–Jüttner distribution.<ref>Synge, J.L (1957). The Relativistic Gas. Series in physics. North-Holland. LCCN 57-003567</ref>

Rapidity

Applying the definition of rapidity as the hyperbolic angle <math>\varphi</math>:<ref>Kinematics Template:Webarchive, by J.D. Jackson, See page 7 for definition of rapidity.</ref> <math display="block"> \tanh \varphi = \beta</math> also leads to Template:Math (by use of hyperbolic identities): <math display="block"> \gamma = \cosh \varphi = \frac{1}{\sqrt{1 - \tanh^2 \varphi}} = \frac{1}{\sqrt{1 - \beta^2}}.</math>

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:<ref>Cameron R D Bunney and Jorma Louko 2023 Class. Quantum Grav. 40 155001</ref> <math display="block"> \sum_{m=1}^\infty \left(J^2_{m-1}(m\beta)+J^2_{m+1}(m\beta)\right)=\frac{1}{\sqrt{1-\beta^2}}.</math>

Series expansion (velocity)

The Lorentz factor has the Maclaurin series: <math display="block">\begin{align} \gamma & = \dfrac{1}{\sqrt{1 - \beta^2}} \\[1ex] & = \sum_{n=0}^{\infty} \beta^{2n}\prod_{k=1}^n \left(\dfrac{2k - 1}{2k}\right) \\[1ex] & = 1 + \tfrac12 \beta^2 + \tfrac38 \beta^4 + \tfrac{5}{16} \beta^6 + \tfrac{35}{128} \beta^8 + \tfrac{63}{256} \beta^{10} + \cdots , \end{align}</math> which is a special case of a binomial series.

The approximation <math display="inline">\gamma \approx 1 + \frac{1}{2}\beta^2</math> may be used to calculate relativistic effects at low speeds. It holds to within 1% error for Template:Mvar < 0.4 Template:Mvar (Template:Mvar < 120,000 km/s), and to within 0.1% error for Template:Mvar < 0.22 Template:Mvar (Template:Mvar < 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:

<math display="block">\begin{align} \mathbf p & = \gamma m \mathbf v, \\ E & = \gamma m c^2. \end{align}</math>

For <math>\gamma \approx 1</math> and <math display="inline">\gamma \approx 1 + \frac{1}{2}\beta^2</math>, respectively, these reduce to their Newtonian equivalents:

<math display="block">\begin{align} \mathbf p & = m \mathbf v, \\ E & = m c^2 + \tfrac12 m v^2. \end{align}</math>

The Lorentz factor equation can also be inverted to yield <math display="block">\beta = \sqrt{1 - \frac{1}{\gamma^2}} .</math> This has an asymptotic form <math display="block">\beta = 1 - \tfrac12 \gamma^{-2} - \tfrac18 \gamma^{-4} - \tfrac{1}{16} \gamma^{-6} - \tfrac{5}{128} \gamma^{-8} + \cdots\,.</math>

The first two terms are occasionally used to quickly calculate velocities from large Template:Mvar values. The approximation <math display="inline">\beta \approx 1 - \frac{1}{2}\gamma^{-2}</math> holds to within 1% tolerance for Template:Nobr and to within 0.1% tolerance for Template:Nobr

Applications in astronomy

The standard model of long-duration gamma-ray bursts (GRBs) holds that these explosions are ultra-relativistic (initial Template:Mvar 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.<ref>Template:Cite journal</ref>

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 Template:Cvt 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.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

See also

• Inertial frame of reference
• Proper velocity
• Pseudorapidity

External links

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