Lorentz invariants are physical parameters that remain the same in all reference frames. They are useful in problems that involve multiple reference frames.

## Phase Space Volume Edit

Phase space volume is the product of real space volume and volume in momentum space (sometimes called reciprocal space). In a boosted frame the real space volume decreases by a Lorentz factor $ \gamma $, while the momentum space volume increases by the same amount, so both changes cancel out.

$ d^3 x' \cdot d^3 p' = \frac{d^3 x}{\gamma} \cdot d^3 p \gamma = d^3 x \cdot d^3 p $

As a corollary, the distribution function i.e. the number of particles per unit phase space, (usually denoted as $ f $) is also constant because the number of particles does not change between reference frames.

## Specific Radiative Intensity Edit

The specific radiative intensity can be related to the distribution function through

$ I_{\nu} = \nu^3 f $

Hence $ I_{\nu}/\nu^3 $ is a Lorentz invariant.

## Specific Emissivity Edit

The emissivity can be related to the specific radiative intensity through

$ \frac{d I_{\nu}}{ds} = \varepsilon_{\nu} $

where $ s $ is the optical path. The number of times a certain wavelength fits into the optical path is a Lorentz invariant, hence $ \frac{s}{\lambda} \propto s \nu $ is also a Loretnz invariant. Hence, $ \varepsilon_{\nu} / \nu^2 $ is also a Lorentz invariant.