The Lorentz factor or Lorentz term is a
quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. 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 regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The law ...
, and it arises in derivations of the
Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
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 physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect. He also derived the Lorent ...
.
It is generally denoted (the Greek lowercase letter
gamma). Sometimes (especially in discussion of
superluminal motion) the factor is written as (Greek uppercase-gamma) rather than .
Definition
The Lorentz factor is defined as
:
,
where:
*''v'' is the
relative velocity
The relative velocity \vec_ (also \vec_ or \vec_) is the velocity of an object or observer B in the rest frame of another object or observer A.
Classical mechanics
In one dimension (non-relativistic)
We begin with relative motion in the classi ...
between inertial reference frames,
*''c'' is the ''
speed of light
The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
in a vacuum'',
* is the ratio of ''v'' to ''c'',
*''t'' is
coordinate time,
* 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
:
see
velocity addition formula
In relativistic physics, a velocity-addition formula is a three-dimensional equation that relates the velocities of objects in different reference frames. Such formulas apply to successive Lorentz transformations, so they also relate different fra ...
.
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 transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
: 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'':
Corollaries of the above transformations are the results:
* Time dilation
In physics and relativity, time dilation is the difference in the elapsed time as measured by two clocks. It is either due to a relative velocity between them ( special relativistic "kinetic" time dilation) or to a difference in gravitational ...
: 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:
* 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–FitzGera ...
: 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:
Applying conservation of momentum
In Newtonian mechanics, momentum (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. If is an object's mass ...
and energy leads to these results:
* Relativistic mass: The mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different element ...
''m'' of an object in motion is dependent on 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''0:
* Relativistic momentum: The relativistic momentum
In Newtonian mechanics, momentum (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. If is an object's mass ...
relation takes the same form as for classical momentum, but using the above relativistic mass:
* Relativistic kinetic energy: The relativistic kinetic energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of hea ...
relation takes the slightly modified form: As is a function of , the non-relativistic limit gives , 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 ''c''). 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 ''v'' was used, but related variables such as momentum
In Newtonian mechanics, momentum (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. If is an object's mass ...
and rapidity may also be convenient.
Momentum
Solving the previous relativistic momentum equation for leads to
:.
This form is rarely used, although it does appear in the Maxwell–Jüttner distribution.
Rapidity
Applying the definition of rapidity as the hyperbolic angle
In geometry, hyperbolic angle is a real number determined by the area of the corresponding hyperbolic sector of ''xy'' = 1 in Quadrant I of the Cartesian plane. The hyperbolic angle parametrises the unit hyperbola, which has hyperbolic functio ...
:
:
also leads to (by use of hyperbolic identities
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the un ...
):
:
Using the property of Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
, 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.
Series expansion (velocity)
The Lorentz factor has the Maclaurin series:
:
which is a special case of a binomial series.
The approximation ≈ 1 + 2 may be used to calculate relativistic effects at low speeds. It holds to within 1% error for , and to within 0.1% error for .
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 ca ...
s to prove that special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The law ...
reduces to Newtonian mechanics
Newton's laws of motion are three basic Scientific law, laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at re ...
at low speeds. For example, in special relativity, the following two equations hold:
:
For ≈ 1 and ≈ 1 + 2, respectively, these reduce to their Newtonian equivalents:
:
The Lorentz factor equation can also be inverted to yield
:
This has an asymptotic form
:.
The first two terms are occasionally used to quickly calculate velocities from large values. The approximation 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.
Subatomic particles called 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 , but with a much greater mass. It is classified as a lepton. As wi ...
s travel at a speed such that they have a relatively high Lorentz factor and therefore experience extreme time dilation
In physics and relativity, time dilation is the difference in the elapsed time as measured by two clocks. It is either due to a relative velocity between them ( special relativistic "kinetic" time dilation) or to a difference in gravitational ...
. As an example, muons generally have a mean lifetime of about which means muons generated from cosmic ray collisions at about 10 km up in the atmosphere should be non-detectable on the ground due to their decay rate. However, it has been found that ~10% of muons are still detected on the surface, thereby proving that to be detectable they have had their decay rates slow down relative to our inertial frame of reference.
See also
*Inertial frame of reference
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. ...
* Pseudorapidity
* Proper velocity
References
External links
*
*
{{DEFAULTSORT:Lorentz Factor
Doppler effects
Equations
Minkowski spacetime
Special relativity
Hendrik Lorentz