To be submitted to ApJ

A Formalism for Covariant Polarized Radiative Transport by Ray

Tracing

Charles F. Gammie1,2 and Po Kin Leung3

1 Astronomy Department, University of Illinois, 1002 West Green Street, Urbana, IL

61801, USA

2 Department of Physics, University of Illinois, 1110 West Green Street, Urbana, IL 61801,

USA

3 Department of Physics, University of California, Santa Barbara, CA 93106, USA

gammie@illinois.edu

ABSTRACT

We write down a covariant formalism for polarized radiative transfer appro-

priate for ray-tracing through a turbulent plasma. The polarized radiation field is

represented by the polarization tensor (coherency matrix) Nαβ ≡ 〈aαka∗βk 〉, whereak is a Fourier coefficient for the vector potential. Using Maxwell’s equations,

the Liouville-Vlasov equation, and the WKB approximation we show that the

transport equation in vacuo is kµ∇µNαβ = 0. We show that this is equivalent to

Broderick & Blandford (2004)’s formalism based on invariant Stokes parameters

and a rotation coefficient, and suggest a modification that may reduce truncation

error in some situations. Finally we write down several alternative approaches

to integrating the transfer equation.

Subject headings: Black hole physics — Plasmas — Polarization — Radiative

transfer — Relativistic processes

1. Introduction

Polarization data is now available at many wavelengths for Sgr A*, the radio, millimeter,

infrared, and X-ray source at the galactic center. Polarization characteristics have already

provided interesting constraints on models that site the source in a hot plasma surrounding

– 2 –

a 4 × 106M⊙ black hole (Aitken et al. 2000; Marrone et al. 2006, 2007), and may provide

more constraints with the aid of appropriate models. We set out to model the radio through

submillimeter polarization of Sgr A*, and in the process have developed the argument below

that describes a procedure for solving the polarized radiative transfer equation in a curved

spacetime. Other possible applications include calculation of X-ray polarization that may be

relevant to future X-ray polarimetry missions such as GEMS and IXO, and more generally

to polarized radiative transfer in neutron star atmospheres, pulsar magnetospheres, other

galactic nuclei, and even cosmological settings.

Work on covariant unpolarized radiative transport began with Lindquist (1966), al-

though there were earlier studies of the Boltzmann equation in covariant form. Later work

by Anderson & Spiegel (1972), and then by Thorne (1981), extended this to a formalism

in which the angular (momentum space) structure of the radiation field is described by a

moment formalism, again for unpolarized radiation.

Work by Connors, Piran & Stark (1980) transported polarized radiation from its origin

on the surface of a thin disk near a black hole through vacuum to an observer at large

radius by parallel transporting the polarization vector along a geodesic (more recent works

by, e.g., Schnittman & Krolik (2009), Dovciak et al. (2008) and Li et al. (2005), use a similar

procedure, although Schnittman & Krolik (2009) include Compton scattering).

The first clear description of fully relativistic polarized radiative transport that we are

aware of is by Bildhauer (1989a,b); this work built on earlier work by Dautcourt & Rose

(1978). Later polarized transport equations were written down in a cosmological context by

Kosowsky (1994, 1996), Challinor (2000), and Weinberg (2008). More recently, Broderick &

Blandford (2004, hereafter BB04) have developed an elegant formalism for treating transport

along a ray, which was discussed further by Shcherbakov & Huang (2010), and applied by,

e.g., Broderick, Loeb & Narayan (2009); Huang et al. (2009) and Shcherbakov et al. (2010)

to models of Sgr A*.

If a covariant polarized transport formalism exists, why are we revisiting the issue?

Most earlier work describes the polarized radiation field in terms of dependent variables,

like the invariant Stokes parameters I/ν3, Q/ν3, U/ν3, V/ν3, that depend on the frame in

which they are observed: they are only defined up to a rotation that interchages Q/ν3 and

U/ν3 and a coordinate inversion that determines the sign of V/ν3. The way the Stokes

parameters change along a geodesic therefore depends on how the observer frame changes

along the geodesic. In many applications this is unobjectionable because there is a natural

choice of frames (e.g., in cosmology) that varies slowly along the geodesic. In accretion flow

problems, however, the natural (plasma) frame fluctuates rapidly along the geodesic because

of turbulence in the underlying flow. The Stokes parameters can then fluctuate, even if there

– 3 –

is no interaction between the plasma and the radiation field. This seems unsatisfactory. The

procedure we describe below is manifestly frame independent.

A second motivation is that the widely used BB04 formalism is written down from

physical arguments but not derived. It was not clear (to us) that all relativistic effects were

properly included in BB04’s treatment. We derive BB04’s equations starting from Maxwell’s

equations and the Liouville-Vlasov equation.

A third motivation is that the use of preferred observers seems inelegant. It ought to

be possible to define a tensor quantity to represent the polarized radiation field and then

write the basic equations without reference to a special frame. One manifestly covariant

description of a polarized radiation field is

Qαβγδ = 〈FαβF∗γδ〉, (1)

where Fαβ is the electromagnetic field tensor and ∗ denotes complex conjugate. This unwieldy

rank 4 polarization tensor has 21 degrees of freedom, most of which are redundant due to

the radiative character of the electromagnetic field. Qαβγδ also does not satisfy a simple

transport equation. A simpler but still covariant description of the radiation field is

Nαβ = 〈AαA∗β〉, (2)

where Aµ is the four-vector potential, with Fαβ = ∂αAβ − ∂βAα. This is the description of

the radiation field used by Bildhauer (1989a,b). Some components of Nαβ have a natural

interpretation as a photon phase space density and obey a simple equation along photon

trajectories. Other components represent extra (gauge) degrees of freedom that can be

clearly identified and that are eliminated when a final physical measurement is made by

projecting Nαβ onto an appropriate tetrad basis, into the plane that is perpendicular to the

wavevector and the observer four-velocity.

A final motivation is pedagogical: we want to make the transition from familiar territory—

Maxwell’s equations—to a covariant polarized transfer equation with a minimum of technical

overhead.

We adopt the standard notation of Misner, Thorne & Wheeler (1973, hereafter MTW)

including a signature for the metric of −,+,+,+. We set c = 1 except where specifically

noted otherwise. It may help to recall that, if xµ is a set of coordinates and kµ is a wave four-

vector then the following are coordinate invariant: d3k/(√−gkt) (here g is the determinant of

gµν); d3x√−gkt; d3xd3k (phase space volume). Here d3k = dk1dk2dk3 and d3x = dx1dx2dx3,

where 1, 2, 3 are spacelike coordinates.

Our paper is organized as follows. In §2 we review the properties of WKB solutions

for electromagnetic waves in a vacuum spacetime. In §3 we make the transition from a

– 4 –

single wave (wave equation) to an ensemble of waves (transport equation). In §4 we define

the polarization tensor, or coherency matrix, Nµν , relate it to the Stokes parameters, and

write down a transport equation. In §5 we describe interaction of the radiation with the

plasma and relate emission and absorption coefficients in the plasma frame to those in the

coordinate frame. We also explicitly demonstrate that the Stokes parameters do not depend

on the gauge freedom aµkµ. In §6 we write the transport equation in several forms that may

be suitable for numerical integration, and we show that one of these forms is equivalent to

BB04’s. In §7 we give a brief summary.

2. WKB wave review

This review largely follows MTW, §22, but is included to fix notation and identify

approximations. Consider a single electromagnetic wave given by

Aµ(xν) = aµ(x

ν) exp(ikαxα) (3)

where aµ is an amplitude and kα = ∂αθ (θ is the phase) is, as usual in WKB, the wavevector.1

The Lorenz gauge condition ∇µAµ = 0 implies to leading order in WKB that

kµaµ = 0. (4)

The Lorenz gauge does not fix Aµ uniquely, since we can always send Aµ → Aµ + φkµ.

Maxwell’s equations in the Lorenz gauge imply

∇α∇αAµ +Rµ

αAα = 4πJµ. (5)

where Jµ is the current and Rµν is the Ricci tensor. Usually one is concerned with radiative

transport in a tenuous plasma surrounding a self-gravitating body (only a cosmological scale

self-gravitating plasma can be both relativistic and Thomson thin), so the stress-energy

tensor T µν ≈ 0 (“test plasma” approximation) and therefore Einstein’s equations imply

Rµν ≈ 0.

Suppose for now that the wave is propagating in vacuum and Jµ = 0. Then

∇α∇αAµ = 0. (6)

1The sign convention for kµ is opposite that in Melrose (2008, 2009); we use signature − + ++ for the

metric instead of Melrose’s +−−−.

– 5 –

In the WKB approximation this yields to zeroth order

kµkµ = 0 (7)

or, in nonrelativistic language, ω2 = c2k2. Rewriting Eq. (7) as ∇µθ∇µθ, taking the gradient,

and interchanging indices implies

kµ∇µkν = 0, (8)

which is the geodesic equation.

To first order in WKB one obtains an evolution equation for the vector Fourier ampli-

tudes 2:

kµ∇µaν +

1

2aν∇µk

µ = 0. (9)

The amplitude evolution equation can be decomposed into an equation for the scalar ampli-

tude a = (aµaµ)1/2 and for the polarization unit vector fµ = aµ/a:

kµ∇µa+1

2a∇µk

µ = 0. (10)

and

kµ∇µfν = 0. (11)

The scalar amplitude equation can be rewritten

∇µ(kµa2) = 0 (12)

i.e. as a conservation equation for a2.

How are physical measurements made from aµ? First, the electromagnetic field tensor

is

Fµν = ∂µAν − ∂νAµ (13)

which to zeroth order in WKB is

Fµν = i(kµaν − kνaµ). (14)

F can be separated into an E and B field given an observer four-velocity uµ. The electric

field four-vector is

Eµ ≡ uνFµν = i(kµ(aνu

ν)− aµ(kνuν)), (15)

2Second order and higher terms in the WKB expansion are smaller by the ratio of the wavelength to a

characteristic radius for the spacetime, and are usually negligible.

– 6 –

This definition is consistent with the Lorentz force uβ∇βuα = (q/m)uβF

αβ. The magnetic

field four-vector is

Bµ ≡ uν∗F νµ (16)

where∗F µν =

1

2ǫµνκλFκλ, (17)

is the dual of the field tensor, the Levi-Civita tensor

ǫµνκλ = − 1√−g[µνκλ], (18)

and [µνκλ] is the permutation symbol (1 for even permutations of 0123, −1 for odd permu-

tations, zero otherwise). Eµ and Bµ reduce to the usual E and B fields in an orthonormal

tetrad that is at rest in the uµ frame. For a radiative electromagnetic field they are both

orthogonal to

Kµ ≡ (gµν + uµuν)kν = kµ + uµ(uνk

ν) (19)

which is the spatial part of kµ.

The stress-energy tensor for a wave is

T µν =1

8πa2kµkν (20)

from which we see that in a particular coordinate frame the energy density

T tt =1√−g

dE

d3x=

1

8πktkta2 (21)

and, since photon number dE = ~ktdN the photon number density is

1√−g

dN

d3x=

1

8π~kta2, (22)

which implies that the invariant photon number density is

1√−gkt

dN

d3x=

a2

8π~, (23)

since√−gktd3x is invariant.

Now suppose the wave is propagating in a test plasma, so the wave equation is

∇µ∇µAα = 4πJα (24)

– 7 –

where Jα is the 4-current induced by the wave train. The linear response tensor Π is defined

by

Jα ≡ ΠαβA

β, (25)

and depends on the field strength, direction, etc. The response tensor is gauge independent;

Eµ = i(kµuν − (kβuβ)gµν)aν , and the induced current is uniquely related to the electric field.

The response tensor incorporates the effects of Faraday rotation and absorption.

Using the response tensor, the wave equation becomes

−2ikβ∇βaα − iaα∇βk

β = 4πΠαβa

β (26)

or

kβ∇βaα = −1

2aα∇βk

β + 2πiΠαβa

β. (27)

Now expand ∇µ(kµaαa∗β)

∇µ(kµaαa∗β) = a∗βkµ∇µa

α + aαkµ∇µa∗β + aαa∗β∇µk

µ (28)

and evaluate it using the wave equation:

∇µ(kµaαa∗β) = 2πi

(

a∗βΠαµa

µ − aαΠ∗βµ a∗µ

)

. (29)

Since this will become the polarized transport equation, we rewrite the right hand side

compactly as

∇µ(kµaαa∗β) = Hαβκλaκa

∗λ (30)

where

Hαβκλ ≡ 2πi(

gβλΠακ − gακΠ∗βλ)

(31)

is the tensor that describes Faraday rotation and absorption. Contracting,

∇µ(kµa2) = 2πi(Πλκ − Π∗κλ)aκa

∗λ. (32)

Since kµa2 is a “flux density” of photon number along the ray that is conserved in vacuo,

the anti-hermitian part of Παβ encodes the effects of absorption.

3. Ensemble of waves

We now want to make the transition from WKB wave packets, which are approximately

a δ function in momentum space, to a transport equation for an ensemble of photons in

momentum space.

– 8 –

Consider a small, invariant spatial volume ∆V = ∆3x√−gkt and a small, invariant mo-

mentum space volume ∆Vp = ~3∆Vk = ~

3∆3k/(√−gkt). Populate the phase space volume

∆V∆Vp with an ensemble of wave packets labeled by an index i with definite amplitude ai.

The photon distribution function

f ≡ dN

d3xd3p≈ 1

∆Vp

∑

i

1√−gkt

dNi

d3x. (33)

This is invariant since d3k/(√−gkt) is invariant (here d3k = dk1dk2dk3, i.e. indices are

down). The distribution function is recovered in the limit ∆Vp → 0.

The photon number density is quadratic in Aµ. Recall that

Aµ =∑

j

ajµeikjνx

ν

(34)

so

AµA∗ν =

∑

ij

aiµa∗jνe

i(kiν−kjν)xν

. (35)

If the phases of the wave packets are independent (radiation is incoherent) the cross terms

vanish and

〈AµA∗ν〉 =

∑

i

〈aiµa∗iν〉 (36)

where the 〈〉 is a suitable average (e.g. Bildhauer (1989a)) From now on we drop the explicit

〈〉.

The wave equation analysis implies that

1

∆Vk

∑

i

1√−gkt

dNi

d3x=

1

∆Vk

∑

i

a2i8π~

. (37)

This motivates the definition of power spectrum

a2k ≡1

∆Vk

∑

i

a2i , (38)

so

f =a2k

8π~4. (39)

Writing factors of c explicitly, the usual specific intensity Iν of radiative transfer theory is

Iν =h4ν3

c2dN

d3xd3p=

2π3ν3

c3a2k (40)

– 9 –

where ν is the frequency.

Now we invoke the Liouville-Vlasov equation, which for photons in vacuo implies

df

dλ= 0 ⇒ d(Iν/ν

3)

dλ= 0, (41)

where λ is the affine parameter along a ray, or

da2kdλ

= (kµ∇µ) a2k = 0 (42)

and demand consistency with Maxwell’s equations. Recasting in terms of the wave ampli-

tudes

kµ∇µ

(

1

∆Vk

∑

i

a2i

)

= 0, (43)

and expanding,∑

i

(

1

∆Vk

(kµ∇µ)a2i −

a2i∆V 2

k

(kµ∇µ)∆Vk

)

= 0. (44)

Applying the wave equation to this gives

−a2k (∇µkµ + kµ∇µ ln∆Vk) = 0. (45)

The first term in parentheses is the fractional rate of change of the invariant three-volume

∆V = ∆3x√−gkt occupied by a group of photons in the wave (exercise 22.1 of MTW), i.e.

it is d ln∆V/dλ = (kµ∇µ) ln∆V . Then

kµ∇µ ln∆V + kµ∇µ ln∆Vk ∝d

dλ(∆V∆Vk) = 0, (46)

that is, along a photon trajectory the phase space volume ∆V∆Vk occupied by a group of

photons is constant.

4. Polarization tensors

We now need a mathematical description of a polarized radiation field. The usual

approach is to use either Stokes parameters I,Q, U, V or the coherency matrix

P ij = 〈EiE∗j〉 (47)

where Ei is the electric field 3-vector. Pij is nonzero in the two-dimensional space (x, y)

perpendicular to the wave 3-vector. We will assume the wavevector is oriented along the +z

– 10 –

axis, and that x, y, z form a right-handed coordinate system. Then the Stokes parameters

and coherency matrix are related by

P ij = C

(

I +Q U + iV

U − iV I −Q

)

. (48)

where C is a constant. This relationship is consistent with the IEEE and IAU conventions

for the definition of Q, U , and V (see Hamaker & Bregman 1996, for a helpful discussion):

Q > 0 for a wave linearly polarized along the x axis; Q < 0 for a wave linearly polarized

along the y axis; U > 0 for a wave linearly polarized at 45deg to the x axis, and U < 0 for a

wave linearly polarized at 135deg to the x axis; V > 0 for a wave whose polarization vector

rotates in a right-handed sense with respect to the z axis in the plane z = const..

What is the covariant generalization of (47)? The most straightforward generalization

is

P µν = 〈EµE∗ν〉. (49)

where Eµ is the electric field four-vector. For a radiative electromagnetic field kµEµ = 0,

where kµ is the wave four-vector, and uµEµ = 0 by (15). The only nonzero components of

P µν are therefore in the two dimensional subspace perpendicular to kµ and uµ.

It is possible to recast the relationship between P µν and the Stokes parameters in terms

of tensor operations by requiring that components in the two-dimensional subspace of P µν

perpendicular to kµ and uµ reduce to (48). Thus

I = P µµ /(2C), (50)

I2 +Q2 + U2 = P (µν)P(µν)/(2C2), (51)

(P (µν) ≡ (1/2)(P µν + P νµ)) and

V 2 = −P [µν]P[µν]/(2C2) (52)

(P [µν] ≡ (1/2)(P µν − P νµ)). In Minkowski space with Cartesian coordinates t, x, y, z we can

also write Q = (P xx − P yy)/(2C), U = (P xy + P yx)/(2C) and V = −i(P xy − P yx)/(2C),

but these are not tensor operations: Q and U are interchanged under a 45deg rotation in

the x − y plane (Chandrasekhar 1960, §15.5) and V changes sign if the handedness of the

coordinate system is inverted.

The polarization tensor P is an appealing description of the polarized radiation field

because it is the covariant analog of the usual coherence matrix, but it does not satisfy a

simple transport equation (e.g., Portsmouth & Bertschinger 2004) because it contains an

explicit frame dependence on uµ.

– 11 –

A simple transport equation is obtained with a polarization tensor based on the vector

potential. Given the correspondence between photon phase space density and a2, it is natural

to define

Nµν ≡ 〈aµa∗ν〉. (53)

What is the physical interpretation of Nµν and what is its relation to P µν? From §2 it is

apparent that Nµµ is proportional to the photon phase space density. The definition of F µν ,

Eµ, and aµ imply

Eµ = −iωsµνaν . (54)

where sµν is a “screen projection” operator that projects into the plane perpendicular to uµ

and kµ:

sµν ≡ gµν + uµuν − eµ(K)eν(K) = gµν − kµkν

ω2+

uµkν

ω+

kµuν

ω(55)

where

eµ(K) =kµ

ω− uµ (56)

is a unit vector along the spatial part of kµ and ω ≡ −kµuµ. Thus

P µν = ω2sµαsνβN

αβ ≡ ω2Nαβ. (57)

where˜denotes a screen-projected version of a tensor (which depends on the frame uµ). A

little algebra shows that Nµµ = Nµ

µ , N(µν)N(µν) = N (µν)N(µν) and N [µν]N[µν] = N [µν]N[µν], and

therefore I = ω2N ii/(2C), I2+Q2+U2 = ω4N (µν)N(µν)/(2C

2) and V 2 = −ω4N [µν]N[µν]/(2C2).

That is, none of the invariants are affected by screen projection, as one would expect.

We are now in a position to write down the transport equation for Nµν . The Liouville

equation permitted us to show that the phase space volume ∆V∆Vk occupied by the group

of photons is constant along a photon trajector, d(∆V∆Vk)/dλ = 0. Maxwell’s equations

implied (30). Combining these and using the definition of Nµν yields

kα∇αNµν = Jµν +HµνκλNκλ, (58)

the polarized radiative transport equation, where H is the plasma response tensor (capturing

Faraday rotation and absorption), and we have also introduced the emissivity tensor Jµν . H

can be modified to incorporate the effects of scattering.

5. Interaction with Plasma

5.1. Tetrad basis

It is natural to introduce a tetrad basis to relate quantities such as the emission and

absorption coefficients that are most readily calculated in a Cartesian frame comoving with

– 12 –

the plasma to their values in a coordinate basis. The timelike basis vector is eα(t) = uα and

a second, spacelike basis vector is eα(K). The other two basis vectors are then fixed up to

a rotation. For radiative transfer in a magnetized plasma it is natural to use the magnetic

field four-vector bµ to uniquely specify the orientation of the remaining basis vectors (as in

BB04), but any trial spacelike four-vector tµ not degenerate with eα(K) will do.

Gram-Schmidt orthogonalization yields the following explicit expressions for the spatial

basis vectors:

eα(K) =kα

ω− uα (59)

eα(‖) =tα − kνtν

ωeα(K)

(t2 − ((kλtλ)/ω)2)1/2(60)

eα(⊥) = ǫαβγδe(t)β e(K)

γ e(‖)δ =

ǫαβγδuβkγtδ(t2ω2 − (kλtλ)2)1/2

. (61)

Notice that eα(t)aα = aβuβ and eα(K)aα = −aβu

β. It is straightforward to confirm that eα(a)e(b)α =

δ(b)(a). For t

µ = bµ this basis is identical to that given in BB04 (if their kµkµ = 0; they consider

the more general case where kµ is not necessarily null, i.e. where ω2 = c2k2+ a non-negligible

correction due to the plasma frequency).

To fix a handedness for the basis we must order the basis vectors. If we identify t, ‖,⊥, K

with t, x, y, z then x, y, z form a right-handed coordinate system and the relations between

the coherency matrix in the tetrad basis and the Stokes parameters will follow the IEEE/IAU

conventions.

5.2. Response tensor

The response tensor Παβ can be constructed from the response 3-tensor by identifying

the corresponding terms in the tensor and using the charge-continuity and gauge-invariance

conditions (Melrose 2008, §1.5.8). Another way to proceed is to decompose the response

tensor into the hermitian (h) and antihermitian (a) parts, such that

Παβ = Παβh +Παβ

a (62)

where

Παβh =

1

2(Παβ +Π∗βα) , Παβ

a =1

2(Παβ − Π∗βα). (63)

The hermitian part conserves total energy whereas the antihermitian part causes dissipation.

It is then natural to rewrite Eq. (31) as

Hαβκλ = Aαβκλ +Rαβκλ (64)

– 13 –

where the absorption part

Aαβκλ ≡ 2πi(gβλΠακa + gακΠλβ

a ) ≡ 1

2(gβλAακ + gακAλβ) (65)

contains the dissipative terms, and the generalized Faraday rotation part

Rαβκλ ≡ 2πi(gβλΠακh − gακΠλβ

h ) ≡ 1

2(gβλRακ − gακRλβ) (66)

contains the non-dissipative terms. Due to symmetries Aαβκλ has 4 degrees of freedom while

Rαβκλ has 3 degrees of freedom.

In flat space, one can write the radiative transfer equation as

d

dsIS = JS −MST IT , (67)

where IS = I,Q, U, V T contains the Stokes parameters, JS = jI , jQ, jU , jV T contains the

emission coefficients, which have units of dE/dtdV dνdΩ, and the Mueller Matrix MST is

MST ≡

αI αQ αU αV

αQ αI rV −rUαU −rV αI rQαV rU −rQ αI

. (68)

The parameters αi are the absorption coefficients and rQ, rU and rV are the Faraday mixing

coefficients. By comparing the terms in Eqs. (67) and (58) in a tetrad basis eµ(t), eµ(‖), e

µ(⊥)

and eµ(K), one can write

A(a)(b) ≡ ǫ

0 0 0 0

0 αI + αQ αU + iαV 0

0 αU − iαV αI − αQ 0

0 0 0 0

(69)

and

R(a)(b) ≡ −iǫ

0 0 0 0

0 rQ rU + irV 0

0 rU − irV −rQ 0

0 0 0 0

. (70)

whre ǫ ≡ hν is the photon energy. Expressions for magnetobremsstrahlung absorption

coefficients α can be found in Eq. (48) of Leung, Gammie & Noble (2011), and formulae for

the Faraday mixing coefficients in Eq. (33) 3 of Shcherbakov (2008), or Eq. (58) of Huang &

Shcherbakov (2011).

3Note the sign difference in Eq. (32) of Shcherbakov (2008) compared to our equations.

– 14 –

5.3. Emissivity tensor

The emissivity tensor can be written in Stokes basis as

J (i)(j) ≡ 2c3

ǫ2

0 0 0 0

0 jI + jQ jU + ijV 0

0 jU − ijV jI − jQ 0

0 0 0 0

. (71)

Expressions for the magnetobremsstrahlung emissivities j are given by Eq. (28) of Leung,

Gammie & Noble (2011).

5.4. Gauge Invariance

The interaction with the plasma is most easily calculated by setting the remaining gauge

freedom φ to zero in the plasma frame. Here we show that this procedure does not affect

the Stokes parameters.

First construct an explicit expression for the polarization tensor in a tetrad basis, N (a)(b).

Consider a Cartesian tetrad attached to an observer with velocity vµ. The tetrad components

e(K) and e(t) are defined just as for the plasma frame; the components e(x) and e(y) are only

defined up to a rotation.

Let a(a) = φ, ax, ay, φ and k(a) = K1, 0, 0, 1, consistent with the Lorenz gauge

condition kµaµ = 0. In this frame

N (a)(b) =

〈φφ∗〉 〈φa∗x〉 〈φa∗y〉 〈φφ∗〉〈axφ∗〉 〈axa∗x〉 〈axa∗y〉 〈axφ∗〉〈ayφ∗〉 〈aya∗x〉 〈aya∗y〉 〈ayφ∗〉〈φφ∗〉 〈φa∗x〉 〈φa∗y〉 〈φφ∗〉

, (72)

Since individual wave packets can have different φ, the correlations involving φ are nontrivial.

For example, 〈φφ∗〉〈axa∗y〉 need not equal 〈φa∗y〉〈axφ∗〉, just as 〈aya∗x〉〈axa∗y〉 need not equal

〈axa∗x〉〈aya∗y〉 and I2 need not equal Q2 + U2 + V 2.

Nµν is Hermitian and so has 16 real degrees of freedom. Four of these are the Stokes

parameters, contained in Nµν . There are four additional degrees of freedom in 〈φa∗x〉 and

〈φa∗y〉. The remaining eight degrees of freedom are eliminated by the four complex conditions

kµNµν = 0.

– 15 –

The effect of the projection tensor on N (a)(b) is to zero all components of the tensor

containing φ:

N (c)(d) =

0 0 0 0

0 〈axa∗x〉 〈axa∗y〉 0

0 〈aya∗x〉 〈aya∗y〉 0

0 0 0 0

. (73)

Recall that the projection projection tensor depends on the frame vµ.

How does Nµν transform to another tetrad? The most general possible transformation

consists of a boost followed by a rotation. At the outset we know that this cannot result in

anything but a rotation in the x-y plane (interchange of U and V ) because I, Q2 + U2, and

V are invariant (the latter only if the handedness of the coordinate system is fixed).

To be explicit, consider the transformation into a tetrad basis attached to an observer

with four-velocity uµ, with basis vectors eµ(t′) eµ(x′) e

µ(y′), and eµ(K′). Recall that e

µ(t′) = uµ and

eµ(K′) = −[kµ/(kαuα) + uµ]. Then

N(a′)(b′) = eµ(a′)eν(b′)e

(c)µ e(d)ν N(c)(d). (74)

Now N(c)(d) is constructed from a(a), and

eµ(x′)e(a)µ a(a) = eµ(x′)

(

e(t)µ a(t) + e(x)µ a(x) + e(y)µ a(y) + e(K)µ a(K)

)

. (75)

But a(t)eµ(t) + a(K)e

µ(K) = −φkµ/ω and therefore has no component along eµ(x′), which is by

construction perpendicular to the wave four-vector. Hence

eµ(x′)e(a)µ a(a) = eµ(x′)

(

e(x)µ a(x) + e(y)µ a(y))

(76)

therefore the transformation to the new frame simply mixes component in the (x), (y) plane

into the (x′), (y′) plane. None of the new screen-projected (measured) components depend

on φ. Therefore at a single event we can change frames at will, resetting φ to zero in the

fluid frame when calculating the emission coefficients.

For completeness, the transformation of the Stokes parameters can be written in terms

of eµ(a) and eµ(a′) as

Q = Q cos 2θ − U sin 2θ (77)

U = U cos 2θ +Q sin 2θ (78)

where cos θ ≡ eµ(x′)e(x)µ and sin θ ≡ eµ(x′)e

(y)µ . This is the usual transformation of Q and U by

rotation.

– 16 –

6. Polarized transport formalisms

Strategies for integrating the transport equation numerically can be classified accord-

ing to the choice of dependent variables. Almost every integration strategy will require

calculating the emissivity tensor J and the absorptivity tensor H in the plasma basis and

transforming it to whatever basis is being used for N ; this requires constructing the plasma

basis at each point along the ray.

6.1. Tetrad basis

What is the transport equation in a tetrad basis? Project the right hand side of Eq. (58)

onto the tetrad basis:

J (a)(b) +H(a)(b)(c)(d)N(c)(d). (79)

The left side is

e(a)µ e(b)ν kα∇α

(

N (c)(d)eµ(c)eν(d)

)

=dN (a)(b)

dλ+ e(a)µ e(b)ν N (c)(d)kα∇α

(

eµ(c)eν(d)

)

. (80)

If we define the rotation coefficients

c(a)(b) ≡ e(a)ν (kα∇α) e

ν(b) (81)

then the full transport equation in a tetrad basis is

dN (a)(b)

dλ+N (a)(d)c

(b)(d) +N (c)(b)c

(a)(c) = J (a)(b) +H(a)(b)(c)(d)N(c)(d). (82)

Differentiating e(a)µ eµ(b) = δ

(a)(b) , we conclude that e

(a)ν (kα∇α)e

ν(b) = −e

(b)ν (kα∇α)e

ν(a), so c

(a)(b) is

antisymmetric. For a tetrad basis that is parallel transported along the ray, kα∇αeν(a) = 0

and the rotation coefficients vanish.

6.2. Plasma tetrad; equivalence to BB04

In the plasma tetrad basis the basic equation is (82). Furthermore there are four degrees

of freedom because one integrates the four screen-projected components of the Hermitian

N (a)(b) rather than the sixteen complex components of the Hermitian N (a)(b).

BB04 write down the transport equation in an elegant way by defining the occupation

numbers NI , NQ, NU , NV where

NI ∝I

ω3= C(N (‖)(‖) +N (⊥)(⊥)) (83)

– 17 –

NQ ∝ Q

ω3= C(N (‖)(‖) −N (⊥)(⊥)) (84)

NU ∝ U

ω3= C(N (‖)(⊥) +N (⊥)(‖)) (85)

NV ∝ V

ω3= −iC(N (‖)(⊥) −N (⊥)(‖)) (86)

and C is a new constant that is independent of ω.

What are the transport equations for the Ni? The equations for NI and NV do not

involve any rotation coefficients, since NI and NV are invariant if the handedness of the

coordinate system is defined consistently. Thus, in vacuo,

dNI

dλ= 0 (87)

dNV

dλ= 0. (88)

On the other hand NQ, NU are only defined up to rotation so one must somehow connect

the orientation of the coordinate systems at points along the ray. Beginning with NQ,

dNQ

dλ= C

(

dN (‖)(‖)

dλ− dN (⊥)(⊥)

dλ

)

(89)

and, assuming J = H = 0 (vacuum),

dNQ

dλ= C

(

−N (‖)(a)c(‖)(a) −N (a)(‖)c

(‖)(a) +N (⊥)(a)c

(⊥)(a) +N (a)(⊥)c

(⊥)(a)

)

. (90)

Transport along the ray, like transformation to a new frame discussed in the last section,

only mixes NQ and NU . To be explicit, expand the first term in (90),

N (‖)(a)c(‖)(a) = N (‖)(t)c

(‖)(t) +N (‖)(‖)c

(‖)(‖) +N (‖)(⊥)c

(‖)(⊥) +N (‖)(K)c

(‖)(K). (91)

Antisymmetry implies c(‖)(‖) = 0. Then

N (‖)(t) = N (‖)(K) (92)

because k(t) = −K, k(K) = K, k(‖) = k(⊥) = 0 and k(a)N(a)(b) = 0, hence

N (‖)(a)c(‖)(a) = N (‖)(t)

(

c(‖)(t) + c

(‖)(K)

)

+N (‖)(⊥)c(‖)(⊥). (93)

The term in parentheses is

e(‖)µ (kα∇α)(

eµ(t) + eµ(K)

)

= e(‖)µ (kα∇α)

(

− kµ

kνuν

)

= e(‖)µ kµ(kα∇α)

(

− 1

kνuν

)

= 0 (94)

– 18 –

where the penultimate equality follows from (kα∇α)kµ = 0 and the final equality follows

from e(‖)µ kµ = 0. Then N (‖)(a)c

(‖)(a) = N (‖)(⊥)c

(‖)(⊥).

Similar considerations for the other terms in Eq. (90) yield

dNQ

dλ= C

(

−N (‖)(⊥)c(‖)(⊥) −N (⊥)(‖)c

(‖)(⊥) +N (⊥)(‖)c

(⊥)(‖) +N (‖)(⊥)c

(⊥)(‖)

)

= −2NUc(‖)(⊥) (95)

where the last equality follows from the antisymmetry of c(a)(b) and the definition of NU . Using

the definition of c(a)(b) ,

dNQ

dλ= −2NU

(

e(‖)ν (kα∇α)eν(⊥)

)

(96)

which differs by a sign from BB04’s eq. (17) and (18). A similar calculation gives

dNU

dλ= 2NQ

(

e(‖)ν (kα∇α)eν(⊥)

)

. (97)

The difference in sign between our equation and BB04’s is due to a different definition of

the relation between the Stokes parameters and the coherency matrix.

6.3. Alternative approaches

BB04’s formalism ties Stokes U and V to the plasma tetrad. If the magnetic field has

significant small scale structure– for example, in ray tracing through a numerical model of

a turbulent disk– the plasma tetrad may rotate rapidly along the ray. This implies rapid

interconversion of Q and U even when there is negligible interaction of radiation with the

plasma, and, possibly, accumulation of truncation error.

A first alternative to BB04 is to build a tetrad using a trial vector tµ that is not based on

bµ but rather on a coordinate direction (e.g. radius). The rotation coefficients will generally

be small and Stokes U and V will vary only gradually along the line of sight. This approach

will require that the emissivity tensor and response tensor be calculated in the plasma tetrad

and then projected onto the new tetrad.

A second alternative that is conceptually simple is to directly integrate Nµν (16 real

valued degrees of freedom) in a coordinate frame.

A third alternative is to integrate N (a)(b) in a parallel transported tetrad. Here we need

only integrate the screen-projected components and again there are just four dependent

variables for the radiation field. The orthonormal tetrad itself, however, must be parallel

transported along the ray. In practice only one basis vector is needed, so there are a total

– 19 –

of 8 dependent variables. This is equivalent to the approach taken by Connors et al. (1980)

and papers based on their approach such as Schnittman & Krolik (2010), which parallel

transport a polarization vector along the ray.

7. Summary

We have described a framework for covariant polarized radiative transport. The basic

object that describes the polarization is the polarization tensor Nµν , defined in Eq. (53).

Given absorption, Faraday rotation, and emission coefficients in the plasma frame, one can

then calculate absorption and emission tensors in the coordinate frame (see §7). The trans-

port equation (58) can then be integrated directly in the coordinate basis, in an orthonormal

tetrad, or using a Stokes-based approach. Along the way we explained the connection be-

tween the approach to relativistic polarized radiative transport given in BB04 and the earlier

approach based on parallel transport of the polarization vector pioneered by Connors, Piran

& Stark (1980).

This work was supported by the National Science Foundation under grants PHY 02-

05155 and AST 07-09246, and by a Richard and Margaret Romano Professorial scholarship

and a University Scholar appointment to CFG. Part of this work was completed during a

visit by CFG to Max-Planck-Institut fur Astrophysik, and he would like to thank the Henk

Spruit and Rashid Sunyaev for their hospitality. We thank Avery Broderick and Stu Shapiro

for comments.

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This preprint was prepared with the AAS LATEX macros v5.0.