# Boltzmann's equation and Liouville's theorem in curved spacetime

I have two related questions:

How Boltzmann equation can be written in a covariant (using differential forms, connections, and etc.) way in a classical (not quantum) but curved system?

How does the Liouville's theorem change in a curved background?

Any answers, suggestions, clarifications, or references are welcome.

• The derivation is in Straumann. – Slereah Dec 16 '17 at 20:17
• @Slereah I didn't find it! Can you tell me which chapter? – Kiarash Dec 17 '17 at 8:53

The volume $\mathscr{V}$ occupied by a given swarm of $N$ particles is independent of location along the world line of the swarm.
The Boltzmann equation is more complicated. For example, if we just have a relativistic gas of particles with the same mass (and we choose units where that mass is 1 unit), and the gas at some point has momentum $p^\alpha$, then Boltzmann's equation for the distribution function $f$ looks like $$p^\alpha \frac{\partial f}{\partial x^\alpha} - \Gamma^\gamma_{\alpha \beta} p^\alpha p^\beta \frac{\partial f}{\partial p^\gamma} = Q(f, f),$$ where $Q$ is a complicated collision operator and $\Gamma$ are Christoffel's symbols. An original reference for this is Bichteler's paper, but a more modern discussion can be found here, for example.