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.