Editing Lorentz factor

{{Short description|Quantity in relativistic physics}}
{{Redirect-distinguish|Gamma factor|gamma function}}
[[File:RelativeFactor.png|thumb|Definition of the Lorentz factor γ|upright=1]]
The '''Lorentz factor''' or '''Lorentz term''' (also known as the '''gamma factor'''<ref>{{Cite web |title=The Gamma Factor |url=https://webs.morningside.edu/slaven/Physics/relativity/relativity6.html |access-date=2024-01-14 |website=webs.morningside.edu}}</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 transformation]]s. The name originates from its earlier appearance in [[Lorentz ether theory|Lorentzian electrodynamics]] – named after the [[Netherlands|Dutch]] physicist [[Hendrik Lorentz]].<ref>{{cite web |last1=Tyson |first1=Neil deGrasse |author1-link=Neil deGrasse Tyson |last2=Liu |first2=Charles Tsun-Chu |author2-link=Charles Liu |first3=Robert |last3=Irion |title=The Special Theory of Relativity |url=http://www.nap.edu/html/oneuniverse/motion_knowledge_concept_12.html |website=One Universe |publisher=[[National Academies of Sciences, Engineering, and Medicine]] |archive-url=https://web.archive.org/web/20210725095818/http://www.nap.edu/html/oneuniverse/motion_knowledge_concept_12.html |archive-date=2021-07-25 |access-date=2024-01-06 }}</ref>

It is generally denoted {{math|''γ''}} (the Greek lowercase letter [[gamma]]). Sometimes (especially in discussion of [[superluminal motion]]) the factor is written as {{math|Γ}} (Greek uppercase-gamma) rather than {{math|''γ''}}.

==Definition==
The Lorentz factor {{math|''γ''}} is defined as<ref name="Forshaw 2014">{{cite book |last1=Forshaw |first1=Jeffrey |last2=Smith |first2=Gavin |title=Dynamics and Relativity |publisher=[[John Wiley & Sons]] |date=2014 |isbn=978-1-118-93329-9 |p= 118 |url=https://books.google.com/books?id=5TaiAwAAQBAJ }}</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:
* {{mvar|v}} is the [[relative velocity]] between inertial reference frames,
* {{mvar|c}} is the [[speed of light]] in vacuum,
* {{mvar|β}} is the ratio of {{mvar|v}} to {{mvar|c}},
* {{mvar|t}} is [[coordinate time]],
* {{mvar|τ}} 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 {{math|γ}} as a shorthand:<ref name="Forshaw 2014" /><ref>{{cite book |last2=Freedman |last1=Young |date=2008 |title=Sears' and Zemansky's University Physics |edition=12th |publisher=Pearson Ed. & Addison-Wesley |isbn=978-0-321-50130-1 }}</ref>
* The '''[[Lorentz transformation]]:''' The simplest case is a boost in the {{mvar|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 {{math|(''x'', ''y'', ''z'', ''t'')}} to another {{math|1=(''x''{{′}}, ''y''{{′}}, ''z''{{′}}, ''t''{{′}})}} with relative velocity {{mvar|v}}: <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 ({{math|∆''t''{{′}}}}) between two ticks as measured in the frame in which the clock is moving, is longer than the time ({{math|∆''t''}}) 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 ({{math|∆''x''{{′}}}}) of an object as measured in the frame in which it is moving, is shorter than its length ({{math|∆''x''}}) in its own rest frame: <math display="block">\Delta x' = \Delta x/\gamma.</math>

Applying [[Conservation law (physics)|conservation]] of [[Conservation of linear momentum|momentum]] and energy leads to these results:
* '''[[Relativistic mass]]:''' The [[mass]] {{mvar|m}} of an object in motion is dependent on <math>\gamma</math> and the [[invariant mass|rest mass]] {{math|''m''<sub>0</sub>}}: <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>
* '''[[Kinetic energy#Relativistic kinetic energy of rigid bodies|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|thumb|Lorentz factor {{mvar|γ}} as a function of fraction of given velocity and speed of light. Its initial value is 1 (when {{math|''v'' {{=}} 0}}); and as velocity approaches the speed of light {{nobr|(''v'' → ''c'')}} {{mvar|γ}} increases without bound {{nobr|({{mvar|γ}} → ∞).}}]]
[[File:Lorentz factor inverse.svg|thumb|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 {{mvar|c}}). The middle column shows the corresponding Lorentz factor, the final is the reciprocal. Values in bold are exact.

{| class="wikitable" style="text-align:center;"
|-
! Speed (units of {{mvar|c}}), <br/>{{math|''β'' {{=}} ''v''/''c''}}
! Lorentz factor, <br/>{{mvar|γ}}
! Reciprocal, <br/>{{math|1/''γ''}}
|-
| '''0''' || '''1''' || '''1'''
|-
| '''0.050{{fsp|2}}''' || 1.001 || 0.999
|-
| '''0.100{{fsp|2}}''' || 1.005 || 0.995
|-
| '''0.150{{fsp|2}}''' || 1.011 || 0.989
|-
| '''0.200{{fsp|2}}''' || 1.021 || 0.980
|-
| '''0.250{{fsp|2}}''' || 1.033 || 0.968
|-
| '''0.300{{fsp|2}}''' || 1.048 || 0.954
|-
| '''0.400{{fsp|2}}''' || 1.091 || 0.917
|-
| '''0.500{{fsp|2}}''' || 1.155 || 0.866
|-
| '''0.600{{fsp|2}}''' || '''1.25''' || '''0.8{{fsp|2}}'''
|-
| '''0.700{{fsp|2}}''' || 1.400 || 0.714
|-
| '''0.750{{fsp|2}}''' || 1.512 || 0.661
|-
| '''0.800{{fsp|2}}''' || 1.667 || '''0.6{{fsp|2}}'''
|-
| 0.866{{fsp|2}} || '''2''' || '''0.5{{fsp|2}}'''
|-
| '''0.900{{fsp|2}}''' || 2.294 || 0.436
|-
| '''0.990{{fsp|2}}''' || 7.089 || 0.141
|-
| '''0.999{{fsp|2}}''' || 22.366 || 0.045
|-
| '''0.99995''' || 100.00 || 0.010
|}
[[File:Lorentz_factor_log_log.svg|thumb|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==
{{Main|Momentum|Rapidity}}

There are other ways to write the factor. Above, velocity {{mvar|v}} was used, but related variables such as [[momentum]] and [[rapidity]] may also be convenient.

===Momentum===
Solving the previous relativistic momentum equation for {{math|γ}} 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>[http://pdg.lbl.gov/2005/reviews/kinemarpp.pdf Kinematics] {{Webarchive|url=https://web.archive.org/web/20141121115205/http://pdg.lbl.gov/2005/reviews/kinemarpp.pdf |date=2014-11-21 }}, by [[John David Jackson (physicist)|J.D. Jackson]], See page 7 for definition of rapidity.</ref>
<math display="block"> \tanh \varphi = \beta</math>
also leads to {{math|γ}} (by use of [[Hyperbolic function#Useful relations|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 [[Taylor series|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 {{mvar|v}}&nbsp;&lt; 0.4&nbsp;{{mvar|c}} ({{mvar|v}}&nbsp;&lt; 120,000&nbsp;km/s), and to within 0.1% error for {{mvar|v}}&nbsp;&lt; 0.22&nbsp;{{mvar|c}} ({{mvar|v}}&nbsp;&lt; 66,000&nbsp;km/s).

The truncated versions of this series also allow [[physicist]]s 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 {{mvar|γ}} values. The approximation <math display="inline">\beta \approx 1 - \frac{1}{2}\gamma^{-2}</math> holds to within 1% tolerance for {{nobr|{{math|''γ'' > 2}},}} and to within 0.1% tolerance for {{nobr|{{math|''γ'' > 3.5}}.}}

==Applications in astronomy==
The standard model of long-duration gamma-ray bursts (GRBs) holds that these explosions are ultra-relativistic (initial {{mvar|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&nbsp;keV, whereas the prompt emission is observed to be non-thermal.<ref>{{cite journal |last1=Cenko |first1=S. B. |display-authors=etal |title=iPTF14yb: The First Discovery of a Gamma-Ray Burst Afterglow Independent of a High-Energy Trigger |journal=Astrophysical Journal Letters |page=803 |date=2015 |volume=803 |issue=L24 |doi=10.1088/2041-8205/803/2/L24 |arxiv=1504.00673 |bibcode=2015ApJ...803L..24C }}</ref>

[[Muon]]s, 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&nbsp;[[Microsecond|μs]], muons generated from [[Cosmic ray|cosmic-ray]] collisions {{cvt|10|km|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.<ref>{{cite web |url=http://hyperphysics.phy-astr.gsu.edu/hbase/Relativ/muon.html |title=Muon Experiment in Relativity |website=HyperPhysics.Phy-Astr.GSU.edu |access-date=2024-01-06 }}</ref>

==See also==
* [[Inertial frame of reference]]
* [[Proper velocity]]
* [[Pseudorapidity]]

==References==
{{Reflist}}

==External links==
* {{cite web |last=Merrifield |first=Michael |title=γ – Lorentz Factor (and time dilation)|url=http://www.sixtysymbols.com/videos/lorentz.htm |work=Sixty Symbols |publisher=[[Brady Haran]] for the [[University of Nottingham]] }}
* {{cite web |last=Merrifield |first=Michael |title=γ2 – Gamma Reloaded |url=http://www.sixtysymbols.com/videos/gamma_reloaded.htm |work=Sixty Symbols |publisher=[[Brady Haran]] for the [[University of Nottingham]] }}

[[Category:Doppler effects]]
[[Category:Equations]]
[[Category:Hendrik Lorentz]]
[[Category:Minkowski spacetime]]
[[Category:Special relativity]]
[[Category:Dimensionless quantities]]