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 (lowercase
gamma). It gets its name from its earlier appearance in
Lorentzian electrodynamics. The Lorentz factor is named after the
Dutch physicist
Hendrik Lorentz.
One universe, by
Neil deGrasse Tyson, Charles Tsun-Chu Liu, and Robert Irion.
It is defined as:
\gamma \equiv \frac{c}{\sqrt{c^2 - u^2}} = \frac{1}{\sqrt{1 - \beta^2}} = \frac{\mathrm{d}t}{\mathrm{d}\tau}
where:
\beta = \frac{u}{c} 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{1}{2} \beta^2 + \frac{3}{8} \beta^4 + \frac{5}{16} \beta^6 + \frac{35}{128} \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{1}{2} m v^2
The Lorentz factor equation can also be inverted to yield:
\beta = \sqrt{1 - \frac{1}{\gamma^2}}
This has an asymptotic form of:
\beta = 1 - \frac{1}{2} \gamma^{-2} - \frac{1}{8} \gamma^{-4} - \frac{1}{16} \gamma^{-6} - \frac{5}{128} \gamma^{-8} + ...
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
In the above chart, the lefthand column shows speeds as different fractions of the speed of light (c). The middle column shows the corresponding Lorentz factor.
Rapidity
Note that if
tanh r =
β, then
γ = cosh
r. Here the
hyperbolic angle r is known as the
rapidity Kinematics, by
J.D. Jackson, See page 7 for definition of rapidity.. 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.
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{\mathrm{d}\tau}{\mathrm{d}t} = \sqrt{1 - \beta^2}. This may make the symbol γ ambiguous, so many authors prefer to avoid possible confusion by writing out the Lorentz term in full.
In
particle physics, rapidity is usually defined as (For example, see Introduction to High-Energy Heavy-Ion Collisions, by
Cheuk-Yin Wong, See page 17 for definition of rapidity.)
:y = \frac{1}{2} \ln \left(\frac{E+p_L}{E-p_L}\right)
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{(c t _B)^2 - (v t_B)^2}
Factoring out ct_B gives,
d_t = c t _B\sqrt{1 - {\left(\frac{v}{c}\right)}^2}
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{1 - {\left(\frac{v}{c}\right)}^2}
which simplifies to
t_A = t_B\sqrt{1 - {\left(\frac{v}{c}\right)}^2}
See also
References