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| {{redirect-distinguish|Gamma factor|gamma function}}
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| The '''Lorentz factor''' or '''Lorentz term''' is an expression which appears in several equations in [[special relativity]]. It arises from deriving 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>[http://www.nap.edu/html/oneuniverse/motion_knowledge_concept_12.html One universe], by [[Neil deGrasse Tyson]], Charles Tsun-Chu Liu, and Robert Irion.</ref>
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| Due to its ubiquity, it is generally denoted with the symbol ''γ'' (Greek lowercase [[gamma]]). Sometimes (especially in discussion of [[superluminal motion]]) the factor is written as ''Γ'' (Greek uppercase-gamma) rather than ''γ''.
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| ==Definition==
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| The Lorentz factor is defined as:<ref name=Forshaw>Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 978-0-470-01460-8</ref>
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| :<math>\gamma = \frac{1}{\sqrt{1 - v^2/c^2}} = \frac{1}{\sqrt{1 - \beta^2}} = \frac{\mathrm{d}t}{\mathrm{d}\tau} </math> | |
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| where:
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| * ''v'' is the [[relative velocity]] between inertial reference frames,
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| * β is the ratio of ''v'' to the speed of light ''c''.
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| * ''τ'' is the [[proper time]] for an observer (measuring time intervals in the observer's own frame),
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| * ''c'' is the ''[[speed of light]] in a vacuum''.
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| This is the most frequently used form in practice, though not the only one (see below for alternative forms).
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| 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>
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| :<math>\alpha = \frac{1}{\gamma} = \sqrt{1- v^2/c^2} \ , </math>
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| see [[velocity addition formula]].
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| ==Occurrence== | |
| Following is a list of formulae from Special relativity which use ''γ'' as a shorthand:<ref name=Forshaw/><ref>{{cite book |title=Sears' and Zemansky's University Physics |last=Young |last2=Freedman |edition=12th |publisher=Pearson Ed. & Addison-Wesley |year=2008 |isbn=978-0-321-50130-1 }}</ref>
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| * 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'':
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| ::<math>t' = \gamma \left( t - \frac{vx}{c^2} \right ) </math>
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| ::<math>x' = \gamma \left( x - vt \right ) </math>
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| Corollaries of the above transformations are the results:
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| * '''[[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:
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| ::<math>\Delta t' = \gamma \Delta t. \,</math>
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| * '''[[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:
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| ::<math>\Delta x' = \Delta x/\gamma. \,\!</math>
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| Applying [[Conservation law|conservation]] of [[Conservation of linear momentum|momentum]] and energy leads to these results:
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| * '''[[Relativistic mass]]:''' The [[mass]] of an object ''m'' in motion is dependent on <math>\gamma</math> and the [[Invariant mass|rest mass]] ''m''<sub>0</sub>:
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| ::<math>m = \gamma m_0. \,</math>
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| * '''[[Relativistic momentum]]:''' The relativistic [[momentum]] relation takes the same form as for classical momentum, but using the above relativistic mass:
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| ::<math>\vec p = m \vec v = \gamma m_0 \vec v. \,</math>
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| == Numerical values ==
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| [[Image:Lorentz factor.svg|thumb|right|Lorentz factor ''γ'' as a function of velocity. Its initial value is 1 (when ''v'' = 0); and as velocity approaches the speed of light (''v'' → ''c'') ''γ'' increases without bound (''γ'' → ∞).]]
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| In the chart 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.
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| {| class="wikitable"
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| ! Speed (units of c) !! Lorentz factor !! Reciprocal
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| |-
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| ! <math>\beta = v/c \,\!</math> !! <math>\gamma \,\!</math> !! <math>1/\gamma \,\!</math>
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| |-
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| | 0.000 || 1.000 || 1.000
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| |-
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| | 0.100 || 1.005 || 0.995
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| |-
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| | 0.200 || 1.021 || 0.980
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| |-
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| | 0.250 || 1.033 || 0.968
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| |-]]
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| | 0.300 || 1.048 || 0.954
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| |-
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| | 0.400 || 1.091 || 0.917
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| |-
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| | 0.500 || 1.155 || 0.866
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| |-
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| | 0.600 || 1.250 || 0.800
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| |-
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| | 0.700 || 1.400 || 0.714
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| |-
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| | 0.750 || 1.512 || 0.661
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| |-
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| | 0.800 || 1.667 || 0.600
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| |-
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| | 0.866 || 2.000 || 0.500
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| |-
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| | 0.900 || 2.294 || 0.436
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| |-
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| | 0.990 || 7.089 || 0.141
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| |-
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| | 0.999 || 22.366 || 0.045
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| |-
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| |}
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| ==Alternative representations==
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| {{Main|Momentum|Rapidity}}
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| 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.
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| ===Momentum===
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| Solving the previous relativistic momentum equation for ''γ'' leads to:
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| :<math>\gamma = \sqrt{1+\left ( \frac{p}{m_0 c} \right )^2 } </math>
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| This form is rarely used, it does however appear in the [[Maxwell–Boltzmann distribution#Distribution for relativistic speeds|Maxwell–Juttner distribution]].<ref>Synge, J.L (1957). The Relativistic Gas. Series in physics. North-Holland. LCCN 57-003567</ref>
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| ===Rapidity===
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| Applying the definition of [[rapidity]] as the following [[hyperbolic angle]] ''φ'':<ref>[http://pdg.lbl.gov/2005/reviews/kinemarpp.pdf Kinematics], by [[J.D. Jackson]], See page 7 for definition of rapidity.</ref>
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| :<math> \tanh \varphi = \beta \,\!</math>
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| also leads to ''γ'' (by use of [[Hyperbolic function#Useful relations|hyperbolic identities]]):
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| :<math> \gamma = \cosh \varphi = \frac{1}{\sqrt{1 - \tanh^2 \varphi}} = \frac{1}{\sqrt{1 - \beta^2}} \,\!</math>
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| 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.
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| ===Series expansion (velocity)===
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| The Lorentz factor has a [[Taylor series|Maclaurin series]] of:
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| :<math>\begin{align}
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| \gamma & = \dfrac{1}{\sqrt{1 - \beta^2}} \\
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| & = \sum_{n=0}^{\infty} \beta^{2n}\prod_{k=1}^n \left(\dfrac{2k - 1}{2k}\right) \\
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| & = 1 + \tfrac12 \beta^2 + \tfrac38 \beta^4 + \tfrac{5}{16} \beta^6 + \tfrac{35}{128} \beta^8 + \cdots \\
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| \end{align}</math>
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| The approximation ''γ'' ≈ 1 + <sup>1</sup>/<sub>2</sub> ''β''<sup>2</sup> 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).
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| The truncated versions of this series also allow [[physics|physicists]] to prove that [[special relativity]] reduces to [[Newtonian mechanics]] at low speeds. For example, in special relativity, the following two equations hold:
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| :<math>\vec p = \gamma m \vec v </math>
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| :<math>E = \gamma m c^2 \,</math>
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| For ''γ'' ≈ 1 and ''γ'' ≈ 1 + <sup>1</sup>/<sub>2</sub> ''β''<sup>2</sup>, respectively, these reduce to their Newtonian equivalents:
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| :<math>\vec p = m \vec v </math>
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| :<math> E = m c^2 + \tfrac12 m v^2 </math>
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| The Lorentz factor equation can also be inverted to yield:
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| : <math>\beta = \sqrt{1 - \frac{1}{\gamma^2}} </math>
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| This has an asymptotic form of:
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| : <math>\beta = 1 - \tfrac12 \gamma^{-2} - \tfrac18 \gamma^{-4} - \tfrac{1}{16} \gamma^{-6} - \tfrac{5}{128} \gamma^{-8} + \cdots</math>
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| The first two terms are occasionally used to quickly calculate velocities from large ''γ'' values. The approximation ''β'' ≈ 1 - <sup>1</sup>/<sub>2</sub> ''γ''<sup>−2</sup> holds to within 1% tolerance for ''γ'' > 2, and to within 0.1% tolerance for ''γ'' > 3.5.
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| ==See also==
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| * [[Inertial frame of reference]]
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| * [[Pseudorapidity]]
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| * [[Proper velocity]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| * {{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]]}}
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| * {{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]]}}
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| {{DEFAULTSORT:Lorentz Factor}}
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| [[Category:Doppler effects]]
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| [[Category:Equations]]
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| [[Category:Minkowski spacetime]]
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| [[Category:Special relativity]]
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