Lorentz factor
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The Lorentz factor or Lorentz term (also known as the gamma factor) is a
dimensionless quantity Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into unit of measurement, units of measurement. ISBN 978-92-822-2272-0. Typically expressed as ratios that a ...
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 In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
, and it arises in derivations of the
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant vel ...
s. The name originates from its earlier appearance in Lorentzian electrodynamics – named after the Dutch physicist
Hendrik Lorentz Hendrik Antoon Lorentz ( ; ; 18 July 1853 – 4 February 1928) was a Dutch theoretical physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for their discovery and theoretical explanation of the Zeeman effect. He derive ...
. It is generally denoted (the Greek lowercase letter
gamma Gamma (; uppercase , lowercase ; ) is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter normally repr ...
). Sometimes (especially in discussion of
superluminal motion In astronomy, superluminal motion is the apparently faster-than-light motion seen in some radio galaxies, BL Lac objects, quasars, blazars and recently also in some galactic sources called microquasars. Bursts of energy moving out along the ...
) the factor is written as (Greek uppercase-gamma) rather than .


Definition

The Lorentz factor is defined as \gamma = \frac = \frac = \frac , where: * is the
relative velocity The relative velocity of an object ''B'' relative to an observer ''A'', denoted \mathbf v_ (also \mathbf v_ or \mathbf v_), is the velocity vector of ''B'' measured in the rest frame of ''A''. The relative speed v_ = \, \mathbf v_\, is the v ...
between inertial reference frames, * is the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant exactly equal to ). It is exact because, by international agreement, a metre is defined as the length of the path travelled by light in vacuum during a time i ...
in vacuum, * is the ratio of to , * is coordinate time, * is the
proper time In relativity, proper time (from Latin, meaning ''own time'') along a timelike world line is defined as the time as measured by a clock following that line. The proper time interval between two events on a world line is the change in 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 \alpha = \frac = \sqrt \ = \sqrt ; see velocity addition formula.


Occurrence

Following is a list of formulae from Special relativity which use as a shorthand: * The
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant vel ...
: The simplest case is a boost in the -direction (more general forms including arbitrary directions and rotations not listed here), which describes how spacetime coordinates change from one inertial frame using coordinates to another with relative velocity : \begin t' &= \gamma \left( t - \tfrac \right ), \\ ex x' &= \gamma \left( x - vt \right ). \end Corollaries of the above transformations are the results: *
Time dilation Time dilation is the difference in elapsed time as measured by two clocks, either because of a relative velocity between them (special relativity), or a difference in gravitational potential between their locations (general relativity). When unsp ...
: The time () between two ticks as measured in the frame in which the clock is moving, is longer than the time () between these ticks as measured in the rest frame of the clock: \Delta t' = \gamma \Delta t. *
Length contraction Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame. It is also known as Lorentz contraction or Lorentz–FitzGerald ...
: The length () of an object as measured in the frame in which it is moving, is shorter than its length () in its own rest frame: \Delta x' = \Delta x/\gamma. Applying conservation of
momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
and energy leads to these results: *
Relativistic mass The word "mass" has two meanings in special relativity: ''invariant mass'' (also called rest mass) is an invariant quantity which is the same for all observers in all reference frames, while the relativistic mass is dependent on the velocity ...
: The
mass Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
of an object in motion is dependent on \gamma and the
rest mass The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an object or system of objects that is independent of the overall motion of the system. More precisely, ...
: m = \gamma m_0. * Relativistic momentum: The relativistic
momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
relation takes the same form as for classical momentum, but using the above relativistic mass: \vec p = m \vec v = \gamma m_0 \vec v. * Relativistic kinetic energy: The relativistic kinetic
energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...
relation takes the slightly modified form: E_k = E - E_0 = (\gamma - 1) m_0 c^2As \gamma is a function of \tfrac, the non-relativistic limit gives \lim_E_k=\tfracm_0v^2, as expected from Newtonian considerations.


Numerical values

In the table below, the left-hand column shows speeds as different fractions of the speed of light (i.e. in units of ). The middle column shows the corresponding Lorentz factor, the final is the reciprocal. Values in bold are exact.


Alternative representations

There are other ways to write the factor. Above, velocity was used, but related variables such as
momentum In Newtonian mechanics, momentum (: momenta or momentums; more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. ...
and
rapidity In special relativity, the classical concept of velocity is converted to rapidity to accommodate the limit determined by the speed of light. Velocities must be combined by Einstein's velocity-addition formula. For low speeds, rapidity and velo ...
may also be convenient.


Momentum

Solving the previous relativistic momentum equation for leads to \gamma = \sqrt \,. This form is rarely used, although it does appear in the Maxwell–Jüttner distribution.


Rapidity

Applying the definition of
rapidity In special relativity, the classical concept of velocity is converted to rapidity to accommodate the limit determined by the speed of light. Velocities must be combined by Einstein's velocity-addition formula. For low speeds, rapidity and velo ...
as the hyperbolic angle \varphi: \tanh \varphi = \beta also leads to (by use of hyperbolic identities): \gamma = \cosh \varphi = \frac = \frac. Using the property of
Lorentz transformation In physics, the Lorentz transformations are a six-parameter family of Linear transformation, linear coordinate transformation, transformations from a Frame of Reference, coordinate frame in spacetime to another frame that moves at a constant vel ...
, 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 In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism :\varphi : \mathbb \rightarrow G from the real line \mathbb (as an additive group) to some other topological group G. If \varphi is in ...
, a foundation for physical models.


Bessel function

The Bunney identity represents the Lorentz factor in terms of an infinite series of
Bessel functions Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex ...
: \sum_^\infty \left(J^2_(m\beta)+J^2_(m\beta)\right)=\frac.


Series expansion (velocity)

The Lorentz factor has the
Maclaurin series Maclaurin or MacLaurin is a surname. Notable people with the surname include: * Colin Maclaurin (1698–1746), Scottish mathematician * Normand MacLaurin (1835–1914), Australian politician and university administrator * Henry Normand MacLaurin ...
: \begin \gamma & = \dfrac \\ ex& = \sum_^ \beta^\prod_^n \left(\dfrac\right) \\ ex& = 1 + \tfrac12 \beta^2 + \tfrac38 \beta^4 + \tfrac \beta^6 + \tfrac \beta^8 + \tfrac \beta^ + \cdots , \end which is a special case of a
binomial series In mathematics, the binomial series is a generalization of the binomial formula to cases where the exponent is not a positive integer: where \alpha is any complex number, and the power series on the right-hand side is expressed in terms of the ...
. The approximation \gamma \approx 1 + \frac\beta^2 may be used to calculate relativistic effects at low speeds. It holds to within 1% error for  < 0.4  ( < 120,000 km/s), and to within 0.1% error for  < 0.22  ( < 66,000 km/s). The truncated versions of this series also allow
physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...
s to prove that
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Ele ...
reduces to
Newtonian mechanics Newton's laws of motion are three physical laws that describe the relationship between the motion of an object and the forces acting on it. These laws, which provide the basis for Newtonian mechanics, can be paraphrased as follows: # A body r ...
at low speeds. For example, in special relativity, the following two equations hold: \begin \mathbf p & = \gamma m \mathbf v, \\ E & = \gamma m c^2. \end For \gamma \approx 1 and \gamma \approx 1 + \frac\beta^2, respectively, these reduce to their Newtonian equivalents: \begin \mathbf p & = m \mathbf v, \\ E & = m c^2 + \tfrac12 m v^2. \end The Lorentz factor equation can also be inverted to yield \beta = \sqrt . This has an asymptotic form \beta = 1 - \tfrac12 \gamma^ - \tfrac18 \gamma^ - \tfrac \gamma^ - \tfrac \gamma^ + \cdots\,. The first two terms are occasionally used to quickly calculate velocities from large values. The approximation \beta \approx 1 - \frac\gamma^ holds to within 1% tolerance for and to within 0.1% tolerance for


Applications in astronomy

The standard model of long-duration gamma-ray bursts (GRBs) holds that these explosions are ultra-relativistic (initial 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.
Muon A muon ( ; from the Greek letter mu (μ) used to represent it) is an elementary particle similar to the electron, with an electric charge of −1 '' e'' and a spin of  ''ħ'', but with a much greater mass. It is classified as a ...
s, a subatomic particle, travel at a speed such that they have a relatively high Lorentz factor and therefore experience extreme
time dilation Time dilation is the difference in elapsed time as measured by two clocks, either because of a relative velocity between them (special relativity), or a difference in gravitational potential between their locations (general relativity). When unsp ...
. Since muons have a mean lifetime of just 2.2 
μs A microsecond is a unit of time in the International System of Units (SI) equal to one millionth (0.000001 or 10−6 or ) of a second. Its symbol is μs, sometimes simplified to us when Unicode is not available. A microsecond is to one second, ...
, muons generated from cosmic-ray collisions 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.


See also

*
Inertial frame of reference In classical physics and special relativity, an inertial frame of reference (also called an inertial space or a Galilean reference frame) is a frame of reference in which objects exhibit inertia: they remain at rest or in uniform motion relative ...
* Proper velocity * Pseudorapidity


References


External links

* * {{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 The University of Nottingham is a public research university in Nottingham, England. It was founded as University College Nottingham in 1881, and was granted a royal charter in 1948. Nottingham's main campus (University Park Campus, Nottingh ...
Doppler effects Equations Hendrik Lorentz Minkowski spacetime Special relativity Dimensionless quantities