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Lorentz factor
 
 
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The Lorentz factor or Lorentz term appears in several equations in special relativity
Special relativity

Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "Annus Mirabilis Papers#Special relativity"....
, including time dilation
Time dilation

Time dilation is the phenomenon whereby an observer finds that another's clock, which is physically identical to their own, is ticking at a slower rate as measured by their own clock....
, length contraction
Length contraction

Length contraction, according to Hendrik Lorentz, is the physical phenomenon of a decrease in length detected by an observer in objects that travel at any non-zero velocity relative to that observer....
, and the relativistic mass formula. Because of its ubiquity, physicists
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
 generally represent it with the shorthand symbol ?. It gets its name from its earlier appearance in Lorentzian electrodynamics
Lorentz ether theory

What is now called Lorentz Ether theory has its roots in Hendrik Lorentz's "Theory of electrons", which was the final point in the development of the classical aether theories at the end of the 19th and at the beginning of the 20th century....
. The Lorentz factor is named after the Dutch
Netherlands

The Netherlands is a country that is part of the Kingdom of the Netherlands. It is a parliamentary democratic constitutional monarchy. The Netherlands is located in North-West Europe, and bordered by the North Sea to the north and west, Belgium to the south, and Germany to the east....
 physicist Hendrik Lorentz
Hendrik Lorentz

Hendrik Antoon Lorentz was a Netherlands physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect....
.

It is defined as:



where:

is the velocity in terms of the speed of light
Speed of light

The speed of light in an free space is an important physical constant usually written as c, with a value of 299,792,458 metres per second....
,
u is the velocity as observed in the reference frame where time t is measured
τ is the proper time
Proper time

In theory of relativity, proper time is time measured by a single clock between events that occur at the same place as the clock. It depends not only on the events but also on the motion of the clock between the events....
, and
c is the speed of light.


Lorentz factor has a Maclaurin series
Taylor series

In mathematics, the Taylor series is a representation of a function as an Series of terms calculated from the values of its derivatives at a single point....
 of:



The approximation ? ˜ 1 + 1/2 ฿2 is occasionally used to calculate relativistic effects at low speeds.






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Encyclopedia


The Lorentz factor or Lorentz term appears in several equations in special relativity
Special relativity

Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "Annus Mirabilis Papers#Special relativity"....
, including time dilation
Time dilation

Time dilation is the phenomenon whereby an observer finds that another's clock, which is physically identical to their own, is ticking at a slower rate as measured by their own clock....
, length contraction
Length contraction

Length contraction, according to Hendrik Lorentz, is the physical phenomenon of a decrease in length detected by an observer in objects that travel at any non-zero velocity relative to that observer....
, and the relativistic mass formula. Because of its ubiquity, physicists
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
 generally represent it with the shorthand symbol ?. It gets its name from its earlier appearance in Lorentzian electrodynamics
Lorentz ether theory

What is now called Lorentz Ether theory has its roots in Hendrik Lorentz's "Theory of electrons", which was the final point in the development of the classical aether theories at the end of the 19th and at the beginning of the 20th century....
. The Lorentz factor is named after the Dutch
Netherlands

The Netherlands is a country that is part of the Kingdom of the Netherlands. It is a parliamentary democratic constitutional monarchy. The Netherlands is located in North-West Europe, and bordered by the North Sea to the north and west, Belgium to the south, and Germany to the east....
 physicist Hendrik Lorentz
Hendrik Lorentz

Hendrik Antoon Lorentz was a Netherlands physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect....
.

It is defined as:



where:

is the velocity in terms of the speed of light
Speed of light

The speed of light in an free space is an important physical constant usually written as c, with a value of 299,792,458 metres per second....
,
u is the velocity as observed in the reference frame where time t is measured
τ is the proper time
Proper time

In theory of relativity, proper time is time measured by a single clock between events that occur at the same place as the clock. It depends not only on the events but also on the motion of the clock between the events....
, and
c is the speed of light.


Approximations

The Lorentz factor has a Maclaurin series
Taylor series

In mathematics, the Taylor series is a representation of a function as an Series of terms calculated from the values of its derivatives at a single point....
 of:



The approximation ? ˜ 1 + 1/2 ฿2 is occasionally 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).

The truncated versions of this series also allow physicists
Physics

Physics is the natural science which examines basic concepts such as energy, force, and spacetime and all that derives from these, such as mass, charge, matter and its Motion ....
 to prove that special relativity
Special relativity

Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "Annus Mirabilis Papers#Special relativity"....
 reduces to Newtonian mechanics at low speeds. For example, in special relativity, the following two equations hold:

For ? ˜ 1 and ? ˜ 1 + 1/2 ฿2, respectively, these reduce to their Newtonian equivalents:

The Lorentz factor equation can also be inverted to yield:



This has an asymptotic form of:



The first two terms are occasionally used to quickly calculate velocities from large ? values. The approximation ฿ ˜ 1 - 1/2 ?-2 holds to within 1% tolerance for ? > 2, and to within 0.1% tolerance for ? > 3.5.

Values


Speed Lorentz factor Reciprocal
0.000 1.000 1.000
0.100 1.005 0.995
0.200 1.021 0.980
0.300 1.048 0.954
0.400 1.091 0.917
0.500 1.155 0.866
0.600 1.250 0.800
0.700 1.400 0.714
0.800 1.667 0.600
0.866 2.000 0.500
0.900 2.294 0.436
0.990 7.089 0.141
0.999 22.366 0.045


Rapidity

Note that if tanh r = ฿, then ? = cosh r. Here the hyperbolic angle
Hyperbolic angle

A hyperbolic angle in standard position is the angle at between the ray to and the ray to where x > 1.The magnitude of the hyperbolic angle is the area of the corresponding hyperbolic sector which is loge x....
 r is known as the rapidity. Using the property of Lorentz transformation
Lorentz transformation

In physics, the Lorentz transformation converts between two different observers' measurements of space and time, where one observer is in constant motion with respect to the other....
, 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
One-parameter group

In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphismfrom the real line R to some other topological group G....
, a foundation for physical models. Sometimes (especially in discussion of superluminal motion
Superluminal motion

In astronomy, superluminal motion is the apparently faster-than-light motion seen in someradio galaxy, quasars and recently also in some galactic sources called microquasars....
) ? is written as G (uppercase-gamma) rather than ? (lowercase-gamma).

The Lorentz factor applies to time dilation
Time dilation

Time dilation is the phenomenon whereby an observer finds that another's clock, which is physically identical to their own, is ticking at a slower rate as measured by their own clock....
, length contraction
Length contraction

Length contraction, according to Hendrik Lorentz, is the physical phenomenon of a decrease in length detected by an observer in objects that travel at any non-zero velocity relative to that observer....
 and relativistic mass relative to rest mass in Special Relativity. An object moving with respect to an observer will be seen to move in slow motion given by multiplying its actual elapsed time by gamma. Its length is measured shorter as though its local length were divided by ?.

? may also (less often) refer to . This may make the symbol ? ambiguous, so many authors prefer to avoid possible confusion by writing out the Lorentz term in full.

In particle physics
Particle physics

Particle physics is a branch of physics that studies the elementary particle constituents of matter and radiation, and the interactions between them....
, rapidity is usually defined as (For example, see )


Derivation

One of the fundamental postulates of Einstein's special theory of relativity is that all inertial
Inertial frame of reference

In physics, an inertial frame of reference is a frame of reference, tied to the state of motion of an Observer , with the property that each physical law portrays itself in the same form in every inertial frame....
 observers will measure the same speed of light in vacuum regardless of their relative motion with respect to each other or the source. Imagine two observers: the first, observer , traveling at a constant speed with respect to a second inertial reference frame in which observer is stationary. points a laser “upward” (perpendicular to the direction of travel). From 's perspective, the light is traveling at an angle. After a period of time , has traveled (from 's perspective) a distance ; the light had traveled (also from perspective) a distance at an angle. The upward component of the path of the light can be solved by the Pythagorean theorem
Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' theorem is a relation in Euclidean geometry among the three sides of a triangle#Types of triangles....
.

Factoring out gives,

The distance that sees the light travel is and equating this with calculated from reference frame gives,

which simplifies to

See also

  • Special relativity
    Special relativity

    Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "Annus Mirabilis Papers#Special relativity"....
  • Lorentz transformation
    Lorentz transformation

    In physics, the Lorentz transformation converts between two different observers' measurements of space and time, where one observer is in constant motion with respect to the other....
  • Pseudorapidity
    Pseudorapidity

    In experimental particle physics, Pseudorapidity, , is a commonly used spatial coordinate describing the angle of a particle relative to the beam axis....