Lorentz factor

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.^[1] It is defined as:

<math>\gamma \equiv \frac{c}{\sqrt{c^2 - u^2}} = \frac{1}{\sqrt{1 - \beta^2}} = \frac{\mathrm{d}t}{\mathrm{d}\tau} </math>

...where:

<math>\beta = \frac{u}{c}</math> 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:

<math>\gamma ( \beta ) = 1 + \frac{1}{2} \beta^2 + \frac{3}{8} \beta^4 + \frac{5}{16} \beta^6 + \frac{35}{128} \beta^8 + ...</math>

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:

<math>\vec p = \gamma m \vec v </math>

<math>E = \gamma m c^2 \,</math>

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

<math>\vec p = m \vec v </math>

<math> E = m c^2 + \frac{1}{2} m v^2 </math>

The Lorentz factor equation can also be inverted to yield:

<math>\beta = \sqrt{1 - \frac{1}{\gamma^2}} </math>

This has an asymptotic form of:

<math>\beta = 1 - \frac{1}{2} \gamma^{-2} - \frac{1}{8} \gamma^{-4} - \frac{1}{16} \gamma^{-6} - \frac{1}{128} \gamma^{-8} + ...</math>

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

Lorentz factor as a function of velocity. It starts at value 1 and for <math>v\to c</math> it goes to infinity.

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 <math>\frac{\mathrm{d}\tau}{\mathrm{d}t} = \sqrt{1 - \beta^2}</math>. 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 <math>A</math>, traveling at a constant speed <math>v</math> with respect to a second inertial reference frame in which observer <math>B</math> is stationary. <math>A</math> points a laser “upward” (perpendicular to the direction of travel). From <math>B</math>'s perspective, the light is traveling at an angle. After a period of time <math>t_B</math>, <math>A</math> has traveled (from <math>B</math>'s perspective) a distance <math>d = v t_B</math>; the light had traveled (also from <math>B</math> perspective) a distance <math>d = c t_B</math> at an angle. The upward component of the path <math>d_t</math> of the light can be solved by the Pythagorean theorem.

<math>d_t = \sqrt{(c t _B)^2 - (v t_B)^2}</math>

Factoring out <math>ct_B</math> gives us,

<math>d_t = c t _B\sqrt{1 - {\left(\frac{v}{c}\right)}^2} </math>

This distance is the same distance that <math>A</math> sees the light travel. Because the light must travel at <math>c</math>, <math>A</math>'s time, <math>t_A</math>, will be equal to <math>\frac{d_t}{c}</math>. Therefore

<math>t_A = \frac{c t_B \sqrt{1 - {\left(\frac{v}{c}\right)}^2}}{c}</math>

which simplifies to

<math>t_A = t_B\sqrt{1 - {\left(\frac{v}{c}\right)}^2}</math>

See also

• special relativity
• Lorentz transformation
• pseudorapidity

References

• J.D. Jackson (2004). "Kinematics". Particle Data Group. - See page 7 for definition of rapidity.

1. ^ One universe, by Neil deGrasse Tyson, Charles Tsun-Chu Liu, and Robert Irion.