If you pour sand into a glass, it settles in a flat layer. But instead of sand, take the tiniest particles of gas: in a gravitational field they don't just fall: each particle races along its own trajectory, collides with neighbors, and all this commotion creates pressure. Add the effects of relativity, and the familiar picture changes slightly.
The authors started with the equation that Ludwig Boltzmann wrote a hundred years ago for ordinary gas and carefully incorporated gravity into it, as required by general relativity. The result is something like a recipe: take the distribution of particle velocities (how many slow, how many fast), add a correction for time running slower at the bottom than at the top, and another for the curvature of space. From this recipe follow the rules by which mass and energy flow in such a "nearly relativistic" gas.
The most interesting part is the conservation of energy. In ordinary physics, energy is neither created nor destroyed. Here, this law also holds, but to derive it, they had to look one step further than usual. It's like balancing a family budget: first you account for salary and expenses, then you remember the small interest on deposits. Here, too, they needed to know the structure of spacetime in a bit more detail than for calculating the fluxes themselves — only then did the energy balance close with the required precision.
The result is a set of equations — hydrodynamics with relativistic corrections. It will be useful when you need to understand how matter swirls around a black hole or a neutron star: there, velocities are enormous, gravity is strong, and such corrections become indispensable.
The distribution of particle velocities in a gas was first written for the ideal case without external forces. When gravity was added, it turned out that at each point the velocity "bell" is slightly shifted and distorted, as if it were slightly flattened and pushed downward.
In Poul Anderson's novel "Tau Zero," a starship accelerates so hard that the crew sees the entire universe contract and change — similar corrections to the motion of matter are at work there.