Baumgarte T., Shapiro S. Numerical Relativity. Solving Einstein’s Equations on the Computer

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5.2 Hydrodynamics 141

Here b

is a constant that depends on the type of radiation: for thermal photons it is the usual

radiation constant b;

for each ﬂavor of nondegenerate thermal neutrino or antineutrino

(chemical potential =0) the constant is (7/16)b. The quantity ¯χ is the Rosseland mean

opacity of the interacting gas particles.

Radiation transport may be treated in the diffusion

approximation whenever the mean-free path of the energy carriers (e.g., photons) due to

interactions with the gas particles is short compared to the system scale size, in which case

we say that the gas is “optically thick” to radiation.

5.2.3 Radiation hydrodynamics

In general, a gas is neither optically thick nor optically thin (i.e., transparent) everywhere,

nor is the radiation always in thermal equilibrium with the matter. In such cases we cannot

treat radiation transport in the diffusion approximation as discussed above. To handle

the more general case, the radiation ﬁeld can be described by a radiation stress-energy

tensor,

rad



dν d I

, (5.75)

where I

= I (x

; N

; ν) is the speciﬁc intensity of radiation at x

with frequency ν and

4-momentum p

moving in the direction N

≡ p

/ hν,whereh is Planck’s constant and

the integral is over all frequencies and solid angles. Here ν, I

and d are all measured in

the local Lorentz frame of a ﬁducial observer with 4-velocity u

,wherebyhν =−p

The speciﬁc intensity is related to the radiation phase-space distribution function f deﬁned

by equation (5.191) below according to

f =





. (5.76)

Like f , the ratio I

/ν

is a Lorentz invariant.

To ascertain the physical meaning of the components of T

rad

, consider a spherically

symmetric spacetime. In spherical symmetry, the direction of radiation at an arbitrary

point in space (r,θ,φ) is speciﬁed by polar angle , measured with respect to the outward

radial direction. In other words, the intensity at any instant of time is independent of the

azimuthal direction  at that spatial point, so that I

= I

(t, r; µ), where µ = cos .We

use this information in the next exercise to compute the components of T

rad

Exercise 5.15 At each point (t, r,θ,φ) in a spherical spacetime set up an orthonor-

mal set of basis vectors {e

(

, e

(

, e

(

θ)

, e

(

φ)

}where e

(

points along the outward radial

direction from the center of the spatial coordinate system. Show that in such a frame

b = 8π

/(15c

) = 7.56464 × 10

−15

erg/cm

/deg

See Shapiro and Teukolsky (1983), equation (I.28), with χ

≡ κ

, for the deﬁnition of Rosseland mean opacity.

142 Chapter 5 Matter sources

the components of T

rad

are

rad







EF 00

FP 00

(E − P)0

00 0

(E − P)







. (5.77)

The quantities appearing in equation (5.77) are the frequency-integrated radiation

moments constructed at each point from the frequency-integrated intensity I =

I (t, r ; µ), measured in the orthonormal frame. The intensity I is

I =



dν I

, (5.78)

the radiation energy density is

E(t, r) = 2π



dν dµ I

, (5.79)

the radiation energy ﬂux is

F(t, r) = 2π



dν dµµI

, (5.80)

and the radiation pressure is

P(t, r) = 2π



dν dµµ

. (5.81)

In the above expressions, the frequency ν is integrated from 0 to ∞, while the cosine

µ is integrated from −1to1.

The dynamical equations for the radiation moments are given by

∇

rad

=−G

(5.82)

where G

is the radiation 4-force density expressing the interaction of the matter with the

radiation,



dν d

[

− η

]

. (5.83)

Here χ

is the total opacity and η

is the emissivity of the gas. The hydrodynamical

equations for the ﬂuid then become ∇

ﬂuid

=−∇

rad

= G

, which must be solved

simultaneously with equations (5.82) for the radiation moments and the equation of radia-

tive transport (the “photon Boltzmann equation” or the “radiation kinetic equation”) for

. The contribution of the radiation must also be included in the gravitational ﬁeld source

terms, equations (5.1)–(5.2).

Exercise 5.16 Assume that the scattering opacity (superscript “s”) is elastic and

forward-backward symmetric and that the absorption opacity and thermal emissivity

In this section we use the symbol P for radiation pressure alone, as deﬁned by equation (5.81). Throughout the rest of

the book P is used to denote the total pressure.

5.2 Hydrodynamics 143

(superscript “a”) are related by Kirchoff’s law, η

/χ

= B

(T ), where B

(T )isthe

Planck function

and T is the matter temperature.

(a) Show that in the rest frame of the ﬂuid, the orthonormal components of G

are

given by



dνdχ

− B

) =



dνχ

− 4π B

), (5.84)



dν(χ

+ χ

(5.85)

where



dI

(5.86)

is the speciﬁc radiation energy density and F

is the speciﬁc radiation ﬂux 4-vector

deﬁned by

= 0, F



dI

. (5.87)

Thus the ﬂux satisﬁes F

= 0, where u

is the ﬂuid 4-velocity.

(b) Adopt “gray-body” scattering and absorption opacities, whereby χ

= κρ

, with

κ a constant independent of frequency. Show that

= κ

(E − 4π B), (5.88)

= (κ

+ κ

)ρ

(5.89)

gives the radiation 4-force density, where 4π B = bT

E =



dνdI



dν E

(5.90)

is the total integrated radiation energy density, and

= 0, F



dνdI



dν F

(5.91)

is the total integrated radiation ﬂux 4-vector.

= κ

(E − 4π B)u

+ (κ

+ κ

)ρ

. (5.92)

As derived in exercise 5.38, the equation of radiative transfer takes on the general form

D(I

/ν

)



δ(I

/ν

)

δλ



coll

(

− χ

)

/ν

, (5.93)

Actually, B

is the Planck function for photons, but it is the thermal intensity corresponding to a Fermi–Dirac

distribution in the case of neutrinos; we will use the same symbol for both cases. In many problems involving neutrino

transport in stars, the neutrino energies are much larger than their masses, in which case the neutrinos can be treated

as massless.

144 Chapter 5 Matter sources

where the left-hand side is deﬁned by equation (5.196) below and the right-hand side

accounts for the gain or loss of photons due to interactions with the matter: η

is the

emissivity and χ

is the absorption coefﬁcent.

Consider the special case of a spherically

symmetric medium in comoving coordinates, for which the metric may be cast in the form

=−e

2

+ e

2

+ R

(dθ

+ sin

θdφ

), (5.94)

where r is a Lagrangian radial coordinate and ,  and R are functions of r and t only. In

the comoving frame, the equation of radiation transfer is then

/ν

) + µD

/ν

) − ν



µD

 + µ

 + (1 − µ

)(U/R)



(∂(I

/ν

)/∂ν)

+(1 − µ

){(/R) − D

 + µ

[

(U/R) − D



]

}(∂(I

/ν

)/∂µ) = (η

− I

)/ν

(5.95)

where we have introduced two operators

= e

−ψ

(∂/∂t), D

= e

−

(∂/∂r), (5.96)

and two auxiliary variables

U =

R,= D

R. (5.97)

Note that for radiation problems it is always convenient to work in the rest frame of the

ﬂuid, since that is the frame in which details of the radiation and matter interactions are

most easily speciﬁed and where the intensity in local thermodynamic equilibrium is close

to a Planck function at the local gas temperature. However, it is not always convenient or

possible to work in the comoving frame, which is also the Lagrangian ﬂuid frame, when

solving the coupled 3 + 1 equations for the combined matter–radiation–ﬂuid system,

particularly when the ﬂow deviates from spherical symmetry.

Exercise 5.17 Consider the collapse of a homogeneous dust sphere from rest at a

ﬁnite radius (i.e., Oppenheimer–Snyder collapse, as described in Chapter 1.4), for

which the interior metric may be written in the familiar closed-Friedmann form

=−dτ

+ a

(τ )[dχ

+ sin

χ(dθ

+ sin

θdφ

)]. (5.98)

Show that in this background spacetime the comoving equation of radiation transfer

becomes

∂ I

∂τ

+ 3

+ µ

∂ I

∂χ

−

∂ I

∂ν

+ (1 −µ

)

cotχ

∂ I

∂µ

= η

− χ

, (5.99)

where

a = da/dτ .

The radiation hydrodynamics problem is very difﬁcult to solve in general, since the

intensity is a function of 6-dimensional phase space plus time, and the radiation transport

See exercises 5.37 and 5.38 and the related discussion and references.

Lindquist (1966); Mihalas and Mihalas (1984), p. 443.

5.2 Hydrodynamics 145

equation (5.93) with complicated radiation-ﬂuid interaction terms (including scattering)

has a nontrivial integrodifferential character. Even in spherical symmetry, the intensity

varies with frequency, radial position and angular direction, plus time, so that even the

frequency-integrated transport equation is 2 +1 dimensional (cf. equation 5.95). Approxi-

mation schemes are often used to simplify the problem. They sometimes involve by-passing

the full radiation transport equation and only solving coupled, partial differential equations

for the lowest moments of the transport equation, integrated over frequency and subject to

approximate, physically plausible, “closure” relations to truncate the inﬁnite set of moment

equations.

Exercise 5.18 Deﬁne the “variable Eddington factor” f

Edd

according to P = f

Edd

where E and P are the radiation moments given by equations (5.79)and(5.81). In

general, the variation of f

Edd

in space and time can only be determined by solving

the full transport equation. Show, however, that it is quite simple to calculate f

Edd

the extreme opposite limits of (a) isotropic radiation and (b) outward radial beaming

of the radiation, in a spherically symmetric spacetime. How is the ﬂux F given by

equation (5.80) related to E in the two opposite limits?

Exercise 5.19 Consider a system which is sufﬁciently optically thick that in the

ﬂuid rest frame the radiation ﬁeld is nearly isotropic everywhere. However, allow for

a net ﬂux of radiation at each point. This is the physical situation that applies to the

interior of a star, either in equilibrium or during collapse. Deﬁne the local radiation

energy density E as in equation (5.90) and the local radiation ﬂux four-vector F

in equation (5.91).

(1) Argue that for an isotropic radiation ﬁeld, the orthonormal components of the

stress-energy tensor in the rest-frame of the ﬂuid are given by

rad

= Pδ

, P = E /3. (5.100)

(2) Show that for such a system the radiation stress-energy tensor in a general frame

may be written

rad

= Eu

+ F

+ P(g

+ u

), (5.101)

where u

is the ﬂuid 4-velocity. Thus the equation of motion for the radiation ﬁeld,

equation (5.82), together with equation (5.92)forG

, provides four equations for

the four radiation variables E and F

, with P given by equation (5.100)andF

obtained by solving F

= 0.

The truncated moment equations are essentially equivalent to the equations of motion

for the radiation ﬁeld, equations (5.82). The matter proﬁle enters the source terms in the

moment equations through the radiation 4-force density G

, and the moments appear in

the equations of motion of the ﬂuid, so the combined radiation-hydrodynamical system

must be evolved simultaneously.

Thorne

has constructed a “projected, symmetric trace-free tensor” (PSTF) formalism

for handling radiative transfer in relativistic systems. It consists of an inﬁnite hierarchy

Thorne (1981).

146 Chapter 5 Matter sources

of partial differential equations corresponding to an inﬁnite number of moments of the

radiative transfer equation. The formalism is particularly straightforward to implement for

systems with spherical or planar symmetry, in which case the tensor moments reduce to

scalar functions analogous to E, F and P deﬁned in exercise 5.15.

The simple collapse scenario described in exercise 5.20 provides one test of a relativistic

radiative hydrodynamics code in curved spacetime. It is speciﬁcally designed to check how

well a numerical scheme that solves the coupled equations of relativistic hydrodynamics,

gravitational ﬁeld evolution and radiative transport handles collapse to a black hole. As

shown in the exercise, an analytic solution can be found

when the matter and metric

follow the Oppenheimer–Snyder (OS) solution (see equation 5.98 and Chapter 1.4)and

the radiation can be treated in the thermal diffusion approximation. The resulting scenario

may be called “thermal OS collapse”.

Exercise 5.20 Consider the thermal radiation ﬂux emitted during collapse to a

black hole of a spherical, homogeneous, stellar dust-ball initially subjected to a

small isothermal temperature ﬂuctuation. Take the mass of the star to be M and

the initial radius to be R

and assume that the matter density, velocity and the

gravitational ﬁeld are described by the OS solution. Assume further that the internal

energy and pressure generated by the temperature perturbation are much smaller

than ρ

∗

, the total mass-energy density, and are dominated by thermal radiation.

Treat the radiation in the diffusion approximation.

(a) Use equation (5.72) to write the total stress-energy tensor T

= ρ

∗

+ PP

+ q

, (5.102)

where q

is given by equation (5.73) with λ

given by equation (5.74)and ¯χ ≡ κρ

where κ is a constant. Evaluate u

∇

= 0 to obtain the law of local energy

conservation with radiation,

dρ

∗

dτ

∗

+ P

dρ

dτ

−∇

− a

. (5.103)

(b) Substitute 3P = E = bT

,whereE is the thermal radiation energy density in

the ﬂuid rest frame and evaluate equation (5.103) for OS collapse to show

∂

˜τ

sin

(χ

∂

sin

(χ

z)∂

, (5.104)

where E

(˜τ,z) ≡ EQ

is the radiation energy density “corrected for adiabatic col-

lapse”, Q(τ ) is given by equation (1.97), z ≡ χ/χ

,0≤ z ≤ 1, is a nondimensional

See Thorne et al. (1981); Rezzolla and Miller (1994, 1996); Zampieri et al. (1996); Balberg et al. (2000), and references

therein, for applications of the PSTF moment formalism to some astrophysical problems involving relativistic radiative

hydrodynamics in spherical symmetry.

Shapiro (1989).

Shapiro (1996) has numerically integrated the relativistic Boltzmann equation coupled to the radiation moment

equations to obtain the interior radiation ﬁeld without approximation for this problem. Optically thin as well as thick

cases were considered. The analytic solution obtained in exercise 5.20 is in good agreement with numerical result in

the optically thick case.

5.2 Hydrodynamics 147

Lagrangian (comoving) radius, and ˜τ is a nondimensional time given by

˜τ =



dτ



Q(τ



)

3κρ

(0)R

sinχ

4κρ

(0)R

1/2



η +

sin η +

sin 2η

&

sinχ

. (5.105)

In equation (5.105) η is the OS conformal time parameter. The quantity ˜τ measures

time in units of the radiation diffusion time scale across R

; it is related to proper

time τ through η (see equation 1.93).

at the origin,

(˜τ,1) = 0 = ∂

(˜τ,0), ˜τ ≥ 0. (5.106)

Then solve equation (5.104) subject to the isothermal initial condition

(0, z) = E

, 0 ≤ z < 1. (5.107)

Hint: Make the substitution

(˜τ,z) =

sin(χ

f (˜τ,z), (5.108)

in which case equation (5.104) becomes

∂

˜τ

f = ∂

f + χ

f. (5.109)

Then expand f in a Fourier sine series consistent with the boundary conditions,

f (˜τ,z) =

∞



n=1

(˜τ )sin(nπ z), (5.110)

substitute into equation (5.109) to get each time-dependent coefﬁcient B

to a constant factor, and use the initial data to determine the constant. Obtain

ﬁnally

(˜τ,z) = 2E

sinχ

sin(χ

˜τ

∞



n=1

(−1)

n+1

nπ

−n

˜τ

sin(nπ z)



− χ

'

(5.111)

(d) Show that the radiation ﬂux F ≡ (q

)

1/2

in the ﬂuid rest frame is given by

F(˜τ,z) =−

3κρ

(0)R

sinχ

∂

(˜τ,z), (5.112)

and evaluate it at the surface to get the emergent ﬂux,

F(˜τ,1) =

κρ

(0)R

sinχ

˜τ

∞



n=1

−n

˜τ



− χ



. (5.113)

148 Chapter 5 Matter sources

Equations (5.82) have been cast in conservative form for the case of a nearly isotropic

radiation ﬁeld obeying the conditions set forth in exercise 5.19.

It is then possible to unite

the matter and radiation equations in a single HRSC scheme that evolves both the ﬂuid and

the radiation ﬁeld simultaneously. Test simulations with such a scheme involving radiation

shocks and nonlinear waves propagating in Minkowski spacetime yield good agreement

with analytic results. When used in conjuntion with a 3 + 1 scheme for the gravitational

ﬁeld, the method can reproduce the “thermal OS collapse” solution derived in exercise 5.20

quite well.

Important applications of radiative hydrodynamics in relativistic spacetimes include

neutrino transport during stellar core collapse and supernovae explosions, photon emission

from gas accretion onto black holes and neutron stars, and photon propagation, decoupling

and re-ionization in the Big Bang Universe. In many of these examples, the radiation ﬁeld

can play an important dynamical role in inﬂuencing the ﬂow of gas, as well as contributing

a source of observable energy ﬂux.

5.2.4 M agnetohydrodynamics

Magnetic ﬁelds play a crucial role in determining the evolution of many relativistic objects.

In any highly conducting astrophysical plasma, a frozen-in magnetic ﬁeld can be ampliﬁed

appreciably by gas compression or shear. Even when an initial seed ﬁeld is weak, the ﬁeld

can grow in the course of time to signiﬁcantly inﬂuence the gas-dynamical behavior of

the system. In problems where the self-gravitation of the magnetized gas can be ignored,

calculations can be performed in a ﬁxed, stationary background spacetime. In this case the

metric does not have to be evolved numerically. Some important gas accretion problems

fall into this category, including accretion onto neutron stars and black holes. In many

other problems, the effect of the magnetized gas on the metric cannot be ignored, and

the gas, the magnetic ﬁelds and the metric must be evolved self-consistently. The ﬁnal

fate of many of these astrophysical systems, which often involve compact objects and

their distinguishing observational signatures, may hinge on the role that magnetic ﬁelds

play during the evolution. Some of these systems are promising sources of gravitational

radiation for detection by laser interferometers. Others may be responsible for gamma-ray

bursts. Examples of astrophysical scenarios involving strong-ﬁeld, dynamical spacetimes

in which magnetic ﬁelds may play a decisive role include core collapse in supernovae, mag-

netorotational collapse of hypermassive neutron stars and supermassive stars, the merger

of neutron star–neutron star and black hole–neutron star binaries, and the suppression of

r-mode instabilities in rotating neutron stars.

In many astrophysical applications involving magnetic ﬁelds, the gas is highly ion-

ized and an excellent conductor of current. The ideal magnetohydrodynamic (MHD)

Farris et al. (2008).

See, e.g., Baumgarte and Shapiro (2003b) for a brief discussion and references. Many of these applications will be

discussed in Chapters 14, 16 and 17.

5.2 Hydrodynamics 149

approximation applies in the limit of inﬁnite conductivity and it is precisely that regime

on which we shall focus below.

Electromagnetic ﬁeld equations

We decompose the Faraday tensor F

= n

− n

+ n



dabc

, (5.114)

where 

abcd

√

−g [abcd] is the Levi-Civita tensor, [abcd] is the completely antisym-

metric symbol and where E

and B

are the electric and magnetic ﬁelds observed by a

normal observer n

. Both ﬁelds are purely spatial (E

= B

= 0), and one can easily

show that

= F

, B



abcd

. (5.115)

The electromagnetic stress-energy tensor

4π T

= F

−

(5.116)

may be written in terms of E

and B

according to

4π T

+ γ

)(E

+ B

)

+2n



b)cd

− (E

+ B

(5.117)

Here 

abc

= n



dabc

is the familiar 3-dimensional, spatial Levi-Civita tensor. Along with

the electromagnetic ﬁeld, we shall assume the presence of a perfect ﬂuid, so that the total

stress-energy tensor is given by

= ρ

+ Pg

+ T

. (5.118)

Exercise 5.21 Insert the electromagnetic stress-energy tensor into equations (5.1)–

(5.2) to obtain the electromagnetic contributions to the source terms in the 3 + 1

gravitational ﬁeld equations:

4πρ

+ B

) =

+ B

), (5.119)

4π S

= 

ijk

= (E × B)

, (5.120)

4π S

=−E

− B

+ B

), (5.121)

4π S

+ B

). (5.122)

The above results are not surprising: expressed in terms of the electromagnetic ﬁeld

components as measured by a normal observer, n

, i.e., an observer who is at rest with

150 Chapter 5 Matter sources

respect to the slices , the 3 + 1 source terms have the same form as in ﬂat space.

Thus

is the familiar energy-density for an electromagnetic ﬁeld, S

is the Poynting vector,

etc.

In many astrophysical applications, we can assume perfect conductivity. In the limit of

inﬁnite conductivity, Ohm’s law yields the ideal MHD condition:

= 0. (5.123)

To see this, rewrite the Faraday tensor in terms of E

(u)

and B

(u)

, the magnetic and electric

ﬁelds measured by an observer u

at rest with respect to the ﬂuid, to obtain

= u

(u)

− u

(u)

+ u



dabc

(u)

, (5.124)

where B

(u)

= 0 = E

(u)

. Then the ideal MHD condition (5.123) simply states that

the electric ﬁeld E

(u)

= F

vanishes in the ﬂuid rest frame, as required for a perfect

conductor.

Exercise 5.22 Show that the covariant generalization of Ohm’s law is

− ρ

= σ F

, (5.125)

where J

is the electromagnetic current 4-vector, ρ

=−J

is the charge density

as seen by an observer comoving with the ﬂuid 4-velocity u

, F

is the Faraday

tensor and σ is the electrical conductivity. Argue that the perfect conductivity con-

dition (5.123) follows immediately from equation (5.125) in the limit of inﬁnite

conductivity.

In the ideal MHD case, F

is completely determined by B

(u)

= 

abcd

(u)

, (5.126)

(u)



abcd

. (5.127)

Exercise 5.23 Show that the condition that the electric ﬁeld vanish in the ﬂuid rest

frame may be written u

= 0, or

αE

=−

ijk

+ β

, (5.128)

where v

= u

. Note that when evaluated in a Minkowski spacetime, the last

equation reduces to the familiar ﬂat spacetime expression E

=−

ijk

E =−v × B.

It might appear that the ﬂuid rest-frame components of the magnetic ﬁeld are the most

convenient choice for integrating the evolution equations for the electromagnetic ﬁeld,

due to the vanishing of the electric ﬁeld in that frame. However, as we shall see below,

the magnetic equations are easiest to express in terms of the normal components of the

magnetic ﬁeld. Of course, the two sets of components are easily related.

cf. exercise 5.1 in Misner et al. (1973).