Radiation Transport Around Kerr Black Holes Jeremy David ...

52 CHAPTER 2. RAY-TRACING IN THE KERR METRIC
e x ^
l
θ θ’
l
K
Figure 2-3: Two reference frames for a finite volume of matter flowing parallel
to the x-axis. On the left is the “lab” frame K, and on the right is the material’s
local rest frame K ′ . A ray propagates through the medium at respective angles θ
and θ ′ in the two frames. Reproduced from Rybicki & Lightman (1979).
K’
number and therefore also invariant. The angular spectral energy density U ν (Ω) =
I ν /c can be expressed in terms of the phase space density f:
Writing p = hν/c, we have
U ν (Ω)dΩdν = hνfd 3 p = hνfp 2 dpdΩ. (2.58)
I ν
ν = h4
f = Lorentz invariant. (2.59)
3 c2 Since the source function S ν appears in equations (2.52) and (2.53) as the difference
I ν − S ν , it must have the same transformation properties as I ν , so we can write
S ν
= Lorentz invariant. (2.60)
ν3 Another Lorentz invariant is the optical depth, since the fraction of photons passing
through a finite medium is given by e −τ , which is just a number, and thus the
same in any reference frame. From this feature, we can calculate the absorption coefficient
in a relativistic medium. Consider a small volume of matter flowing in the
eˆx direction with respect to the lab frame K, as in Figure 2-3. The temperature and
density and thus the emissivity j ′ ν is typically given in the rest frame of the material
K ′ . Since the motion is in the x direction, the slab thickness l is the same in both
reference frames. The optical depth τ ν can be written
τ ν = lα ν
sin θ =
l
ν sin θ να ν = Lorentz invariant. (2.61)
Since ν sin θ is proportional to the p y component of the photon 4-momentum, it must
be the same in both frames because the boost is in a perpendicular direction. Thus