• 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:

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:
• 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:

Applying conservation of momentum and energy leads to these results:

• Relativistic mass: The mass m of an object in motion is dependent on and the rest mass m0:
• Relativistic momentum: The relativistic momentum relation takes the same form as for classical momentum, but using the above relativistic mass:
• Relativistic kinetic energy: The relativistic kinetic energy relation takes the slightly modified form:

Numerical values

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.

Alternative representations

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, although it does appear in the Maxwell–Jüttner distribution.^[5]

also leads to γ (by use of hyperbolic identities):

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.

The Lorentz factor has the Maclaurin series:

which is a special case of a binomial series.

The approximation γ ≈ 1 + 1/2 β2 may be 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:

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

The Lorentz factor equation can also be inverted to yield

This has an asymptotic form

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.

Applications in astronomy

Subatomic particles called muons, have a relatively high lorentz factor and therefore experience extreme time dilation. As an example, muons generally have a mean lifetime of about 2.2 μs which means muons generated from cosmic ray collisions at about 10 km up in the atmosphere should be non-detectable on the ground due to their decay rate. However, it has been found that ~10% of muons are still detected on the surface, thereby proving that to be detectable they have had their decay rates slow down relative to our inertial frame of reference.^[8]

See also

• Inertial frame of reference
• Pseudorapidity
• Proper velocity

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