# Dictionary Definition

# User Contributed Dictionary

# Extensive Definition

The Lorentz factor or Lorentz term appears in
several equations in special
relativity, including time
dilation, length
contraction, and the relativistic
mass formula. Because of its ubiquity, physicists generally represent
it with the shorthand symbol γ. It gets its name from its
earlier appearance in Lorentzian
electrodynamics. The Lorentz factor is named after Hendrik
Lorentz.

It is defined as:

- \gamma \equiv \frac = \frac = \frac

...where:

- \beta = \frac is the velocity in terms of the speed of light,
- u is the velocity as observed in the reference frame where time t is measured
- τ is the proper time, and
- c is the speed of light.

## Approximations

The Lorentz factor has a Maclaurin series of:- \gamma ( \beta ) = 1 + \frac \beta^2 + \frac \beta^4 + \frac \beta^6 + \frac \beta^8 + ...

The approximation γ ≈ 1 + 1/2
β2 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 to prove that
special
relativity reduces to Newtonian
mechanics at low speeds. For example, in special relativity,
the following two equations hold:

- \vec p = \gamma m \vec v
- E = \gamma m c^2 \,

For γ ≈ 1 and γ ≈ 1
+ 1/2 β2, respectively, these reduce to their Newtonian
equivalents:

- \vec p = m \vec v
- E = m c^2 + \frac m v^2

The Lorentz factor equation can also be inverted
to yield:

- \beta = \sqrt

This has an asymptotic form of:

- \beta = 1 - \frac \gamma^ - \frac \gamma^ - \frac \gamma^ - \frac \gamma^ + ...

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

## Values

## Rapidity

Note that if tanh r = β, then γ = cosh r. Here the hyperbolic angle r is known as the rapidity. Rapidity has the property that relative rapidities are additive, a useful property which velocity does not have. Thus the rapidity parameter forms a one-parameter group, a foundation for physical models. Sometimes (especially in discussion of superluminal motion) γ is written as Γ (uppercase-gamma) rather than γ (lowercase-gamma).The Lorentz factor applies to time
dilation, length
contraction 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 γ.

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

## Derivation

One of the fundamental postulates of Einstein's special theory of relativity is that all inertial 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 A, traveling at a constant speed v with respect to a second inertial reference frame in which observer B is stationary. A points a laser “upward” (perpendicular to the direction of travel). From B's perspective, the light is traveling at an angle. After a period of time t_B, A has traveled (from B's perspective) a distance d = v t_B; the light had traveled (also from B perspective) a distance d = c t_B at an angle. The upward component of the path d_t of the light can be solved by the Pythagorean theorem.- d_t = \sqrt

Factoring out ct_B gives,

- d_t = c t _B\sqrt

The distance that A sees the light travel is d_t
= c t_A and equating this with d_t calculated from B reference
frame gives,

- ct_A = ct_B \sqrt

which simplifies to

- t_A = t_B\sqrt

## References

- - See page 7 for definition of rapidity.

rapidity in Bosnian: Lorentzov faktor

rapidity in Czech: Lorentzův faktor

rapidity in Spanish: Factor de Lorentz

rapidity in German: Lorentzfaktor

rapidity in French: Facteur de Lorentz

rapidity in Croatian: Lorentzov faktor

rapidity in Italian: Fattore di Lorentz

rapidity in Polish: Czynnik Lorentza

rapidity in Romanian: Factor Lorentz

rapidity in Russian: Быстрота

rapidity in Slovenian: Lorentzov faktor

rapidity in Swedish: Lorentzfaktorn

rapidity in Chinese: 洛侖茲因子

# Synonyms, Antonyms and Related Words

air speed, alacrity, bat, briskness, ceaselessness, celerity, chattering, constancy, constant flow,
continualness,
continuity, decisiveness, dispatch, expedition, expeditiousness,
fastness, flight, flit, flurry, gait, ground speed, haste, hurry, immediacy, immediateness, incessancy, instantaneousness,
knots, lightning speed,
miles per hour, noninterruption,
oscillation,
pace, perpetuity, precipitation, promptitude, promptness, pulsation, punctuality, punctualness, quick fire,
quickness, rapid fire,
rapid recurrence, rapid succession, readiness, regularity, repetition, round pace,
rpm, rush, snappiness, speed, speediness, staccato, steadiness, stuttering, summariness, sustainment, swift rate,
swiftness, tattoo, unintermission, uninterruption, velocity, vibration