Lorentz factor

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Lorentz factor as a function of in units of , i.e. H. as a function of

In the special theory of relativity, the dimensionless Lorentz factor (gamma) describes the time dilation as well as the reciprocal of the length contraction in the coordinate transformation between inertial systems moving relative to one another . It was developed by Hendrik Antoon Lorentz as part of the Lorentz transformation he worked out , which forms the mathematical basis of the special theory of relativity.

The Lorentz factor is defined as:

For reference systems at rest relative to one another, the following applies

Is , but still small compared to the speed of light

so becomes by a Taylor expansion

In which order the development in classical physics can be broken off cannot be answered in general. For most applications the constant one can be assumed, for the kinetic energy the first order is proportional to decisive.

Lorentz factor for accelerations

The time derivative of is interesting in order to formulate the relativistic form of Newton's second law for accelerations in the direction of motion, since the relativistic correct relationship is via the momentum . The following applies: .

It follows directly:

and one obtains for the time derivative of the Lorentz factor:

and thus for the correct relationship between force and acceleration:

Lorentz factor as a function of momentum p

The Lorentz factor can also be specified as:

With

This notation can mainly be found in theoretical physics .

The proof of equivalence can be provided by equating it with the “normal” Lorentz factor, which results in the relativistic momentum.

Lorentz factor as a function of the kinetic energy

The Lorentz factor can also be specified as:

With

Individual evidence

  1. Thorsten Fließbach: Mechanics . 6th edition. Spectrum, Heidelberg 2013, p. 327 .