Radiation Transport Around Kerr Black Holes Jeremy David ...

54 CHAPTER 2. RAY-TRACING IN THE KERR METRIC
between tabulated points are given by
dx µ i+1/2 = xµ i+1 − xµ i . (2.65)
These are the distances between solid circles in Figure 2-4. Then we can define a
differential path length around x µ i by the average
dx µ i = 1 2 (dxµ i−1/2 + dxµ i+1/2
), (2.66)
which is the distance between the empty circles in Figure 2-4.
Next, we transform from the coordinate basis to the ZAMO basis defined at x µ i ,
giving the differential dx µ i → dxˆµ i and the momentum p µ,i → pˆµ,i . In the ZAMO frame
(here it can be thought of as the lab frame), the photon spatial path length is given
by
ds 2 i = η ĵˆk dxĵdxˆk
i i . (2.67)
In principle, we know the fluid velocity at a collection of fixed points in spacetime from
another tabulated set of data produced by an independent hydrodynamics simulation.
Using multi-linear extrapolation, the fluid variables (4-velocity, density, temperature)
can be determined at the point x µ i . The 4-velocity of the fluid in the ZAMO basis uˆµ i
gives the angles θ and θ ′ from Figure 2-3, and the tabulated density and temperature
(and thus absorption α ν ′ and emissivity j′ ν ) are typically given in the rest frame of the
fluid.
To calculate the special relativistic redshift between the photons in the ZAMO
frame and fluid frame, we define a null 4-vector parallel to the photon momentum in
the ZAMO frame:
nˆµ = [−1,⃗n], (2.68)
where ⃗n = nĵ is a normalized 3-vector in standard Cartesian coordinates. Writing
the fluid velocity
uˆµ = [γ, γ⃗v], (2.69)
with ⃗v = vĵ having magnitude |⃗v| = β = v/c, the frequency ratio is then given as
ν
ν ′ = γ(1 + β cos θ ′ ) =
1
γ(1 − ⃗v · ⃗n) , (2.70)
where γ ≡ 1/ √ 1 − β 2 as usual. Now we have enough information to solve the
radiative transfer equation in a relativistic flow (Rybicki & Lightman, 1979):
dI
(
ν ν
) ( )
2
ds = j
′ ν
′
ν ′ ν − α ν ′ ν
I ν. (2.71)