Lorentz factor

The Lorentz factor or Lorentz term (also known as the gamma factor) is a dimensionless quantity expressing how much the measurements of time, length, and other physical properties change for an object while it moves. 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. It is generally denoted γ (the Greek lowercase letter gamma). Sometimes (especially in discussion of superluminal motion) the factor is written as Γ (Greek uppercase-gamma) rather than γ.

Definition The Lorentz factor γ is defined as

γ = 1 1 − v 2 c 2 = 1 1 − β 2 = d t d τ , {\displaystyle \gamma ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}={\frac {1}{\sqrt {1-\beta ^{2}}}}={\frac {dt}{d\tau }},}

where:

v is the relative velocity between inertial reference frames, c is the speed of light in vacuum, β is the ratio of v to c, t is coordinate time, τ 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

α = 1 γ = 1 − v 2 c 2 = 1 − β 2 ; {\displaystyle \alpha ={\frac {1}{\gamma }}={\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\ ={\sqrt {1-{\beta }^{2}}};}

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 (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: t ′ = γ ( t − v x c 2 ) , x ′ = γ ( x − v t ) . {\displaystyle {\begin{aligned}t'&=\gamma \left(t-{\tfrac {vx}{c^{2}}}\right),\\[1ex]x'&=\gamma \left(x-vt\right).\end{aligned}}}

Corollaries of the above transformations are the results:

Time dilation: 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: Δ t ′ = γ Δ t . {\displaystyle \Delta t'=\gamma \Delta t.}

Length contraction: 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: Δ x ′ = Δ x / γ . {\displaystyle \Delta x'=\Delta x/\gamma .}

Applying conservation of momentum and energy leads to these results:

Relativistic mass: The relativistic mass m of an object in motion is dependent on γ {\displaystyle \gamma } and the rest mass m0: m = γ m 0 . {\displaystyle m=\gamma m_{0}.}

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