A galaxy such as our Milky Way contains ∼10¹² stars—easily enough to permit a kinetic-theory description of their distribution; each star contains so many atoms (∼10⁵⁶) that the masses of the stars can be regarded as continuously distributed, not discrete. Almost everywhere in a galaxy, the stars move with speeds small compared to light, but deep in the cores of most galaxies there resides a massive black hole near which the stars move with relativistic speeds. In this exercise we explore the foundations for treating such a system: “particles” with continuously distributed rest masses and relativistic speeds.(a) For a subset S of particles like that of Fig. 3.3 and associated discussion, but with a range of rest masses dm centered on some value m, introduce the phase-space volume

that the particles S occupy. Explain why this occupied volume is frame invariant.(b) Show that this invariant occupied volume can be rewritten as 

Explain the physical meaning of each term in parentheses, and show that each is frame invariant. If the number of particles in the set S is dN, then we define the frame-invariant distribution function by

This is a function of location P in 4-dimensional spacetime and location p(vector) in 4- dimensional momentum space (not confined to the mass hyperboloid), and thus a function of location in 8-dimensional phase space. We explore the evolution of this distribution function in Box 3.2

Fig 3.3

Transcribed Image Text:

upd^P*AP=A₂P