The Lorentz factor or Lorentz term is the factor by which time, length, and
relativistic mass change for an object while that object is moving. The
expression appears in several equations in special relativity, and it arises in
derivations of the Lorentz transformations. The name originates
from its earlier appearance in Lorentzian electrodynamics – named after
the Dutch physicist Hendrik Lorentz. Due to its ubiquity, it is generally
denoted with the symbol γ. Sometimes the factor is written as Γ rather than γ.
Definition The Lorentz factor is defined as:
where: v is the relative velocity between
inertial reference frames, β is the ratio of v to the speed of
light c. τ is the proper time for an observer,
t is coordinate time c is the speed of light in a vacuum.
This is the most frequently used form in practice, though not the only one.
To complement the definition, some authors define the reciprocal:
see velocity addition formula. Occurrence
Following is a list of formulae from Special relativity which use γ as a
shorthand: The Lorentz transformation: The simplest
case is a boost in the x-direction, which describes how spacetime
coordinates change from one inertial frame using coordinates to another with
relative velocity v: Corollaries of the above transformations
are the results: Time dilation: 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:
Length contraction: 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:
Applying conservation of momentum and energy leads to these results:
Relativistic mass: The mass of an object m in motion is dependent on and the
rest mass m0: Relativistic momentum: The relativistic
momentum relation takes the same form as for classical momentum, but using the
above relativistic mass: Relativistic kinetic energy: The
relativistic kinetic energy relation takes the slightly modified form:
Numerical values In the table below, the left-hand column
shows speeds as different fractions of the speed of light. 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 and rapidity may also be convenient.
= Momentum= Solving the previous relativistic
momentum equation for γ leads to: This form is rarely used, it does
however appear in the Maxwell–Jüttner distribution.
= Rapidity= Applying the definition of rapidity as
the following hyperbolic angle φ: also leads to γ:
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.
= Series expansion= The Lorentz factor has the Maclaurin
series: which is a special case of a binomial
series. The approximation γ ≈ 1 + 1/2 β2 may be
used to calculate relativistic effects at low speeds. It holds to within 1%
error for v 2, and to within 0.1% tolerance for γ > 3.5.
Applications in astronomy The standard model of long-duration
gamma-ray bursts holds that these explosions are ultra-relativistic, 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.
See also Inertial frame of reference
Pseudorapidity Proper velocity
References External links
Merrifield, Michael. "γ – Lorentz Factor". Sixty Symbols. Brady Haran for
the University of Nottingham. Merrifield, Michael. "γ2 – Gamma
Reloaded". Sixty Symbols. Brady Haran for the University of Nottingham.