Lorentz factor

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 (Greek lowercase gamma). Sometimes (especially in discussion of superluminal motion) the factor is written as Γ (Greek uppercase-gamma) rather than γ.

The Lorentz factor is defined as:

where:

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:

see velocity addition formula.

Following is a list of formulae from Special relativity which use γ as a shorthand:

Corollaries of the above transformations are the results:

Applying conservation of momentum and energy leads to these results:

In the table below, the left-hand column shows speeds as different fractions of the speed of light (i.e. in units of c). The middle column shows the corresponding Lorentz factor, the final is the reciprocal. Values in bold are exact.

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.

Solving the previous relativistic momentum equation for γ leads to:

This form is rarely used, it does however appear in the Maxwell–Jüttner distribution.

...
Wikipedia