# The general relativistic equations of radiation hydrodynamics in the viscous limit

###### Abstract

We present an analysis of the general relativistic Boltzmann equation for radiation, appropriate to the case where particles and photons interact through Thomson scattering, and derive the radiation energy-momentum tensor in the diffusion limit, with viscous terms included. Contrary to relativistic generalizations of the viscous stress tensor that appear in the literature, we find that the stress tensor should contain a correction to the comoving energy density proportional to the divergence of the four-velocity, as well as a finite bulk viscosity. These modifications are consistent with the framework of radiation hydrodynamics in the limit of large optical depth, and do not depend on thermodynamic arguments such as the assignment of a temperature to the zeroth-order photon distribution. We perform a perturbation analysis on our equations and demonstrate that, as long as the wave numbers do not probe scales smaller than the mean free path of the radiation, the viscosity contributes only decaying, i.e., stable, corrections to the dispersion relations. The astrophysical applications of our equations, including jets launched from super-Eddington tidal disruption events and those from collapsars, are discussed and will be considered further in future papers.

###### Subject headings:

radiation: dynamics – radiative transfer – relativistic processes## 1. Introduction

Radiation contributes substantially to the dynamics of many astrophysical systems. The most natural way to analyze the mechanics of such systems is through the formalism of radiation hydrodynamics, wherein one treats the radiation as a fluid that interacts with matter. With the covariant derivative denoted , the equations of radiation hydrodynamics can be generalized to include both gravitational fields and relativistic motions by writing them in the manifestly covariant form

(1) |

where Greek indices range from to 3, repeated upper and lower indices imply summation,

(2) |

is the energy-momentum tensor of the massive constituents of the fluid ( is the enthalpy in the fluid rest-frame, is the four-velocity, is the pressure in the fluid rest-frame, and is the inverse of the metric associated with the background geometry) and

(3) |

is the energy-momentum tensor of the radiation (see, e.g., Mihalas & Mihalas 1984 for a more thorough discussion of the origin of this tensor). In equation (3), is the four-momentum of a photon, is the distribution function of the radiation that describes the density of quanta in both momentum and position space, and is the Lorentz-invariant phase-space volume. Because is a scalar, such a “moment" formalism, i.e., proceeding by taking integrals over the distribution function, is a natural way to analyze the dynamics of the radiation in a covariant fashion. Note, however, that equation (3) is only valid in a locally-flat frame – one in which , being the Minkowski metric – as otherwise one must include factors that depend on the metric (see, e.g., Debbasch & van Leeuwen 2009a). To ensure that the radiation energy-momentum tensor (and derivatives thereof) transforms correctly at all points in the spacetime under consideration, one can explicitly insert the metric-dependent factors that enter into the phase-space volume element. Another, equally valid manner by which one can obtain the general-relativistic form of , however, is by evaluating the integrals in equation (3) and writing the results in a manifestly-covariant form, guaranteeing their frame independence. Because we will be working directly with the distribution function, the latter route is the simpler one to follow and is the one that we will pursue in our ensuing analyses (see sections 3 and 4).

For many astrophysical applications of the equations of radiation hydrodynamics, the medium under consideration is optically thick, meaning that the radiative flux observed at any location within the fluid is very nearly zero. This means, equivalently, that a photon is scattered a large number of times as it propagates through the medium, and that the radiation field as seen by a moving fluid element within the medium is approximately isotropic. It has been known for some time, however, that the finite mean free path of a photon leads to the presence of viscous-like terms in the radiation energy-momentum tensor (Thomas, 1930). The viscous nature of some fluids, especially in applications for which photons dominate the pressure of a system, can therefore be attributed in part to their interactions with radiation.

Eckart (1940) analyzed the origin of generic, viscous effects in relativistic fluids, whether due to radiation or to other phenomena (see also Landau & Lifshitz 1958). He exploited the fact that the four-velocity of the fluid, , and the projection tensor, , are time-like and space-like tensors, respectively, that can be used to decompose any arbitrary tensor. Eckart realized that each of the terms in his decomposition had a physical interpretation, related to the dynamic viscosity or heat conduction of the fluid, which allowed him to postulate a form for the viscous stress tensor of a relativistic fluid that reduced correctly to its non-relativistic counterpart.

Thomas (1930) (and others after him, e.g., Lindquist 1966; Castor 1972; Buchler 1979; Munier 1986; Chen & Spiegel 2000) used the special relativistic Boltzmann equation, which describes changes to the distribution function due to particle-particle collisions, to derive the correction to the radiation energy-momentum tensor for the special case of Thomson scattering. Eckart’s approach, on the other hand, was more phenomenological in nature, employing thermodynamic arguments and an understanding of the Newtonian limit of the viscous stress tensor to derive its relativistic generalization. Weinberg (1971), with the intent of using the results to analyze entropy production in the early universe, compared the two approaches and showed that one of Eckart’s assumptions, namely that the viscous stress tensor be trace free, led to the incorrect conclusion that the bulk viscosity of the fluid vanish (see also Misner & Sharp (1965), where a similar approach is used to evaluate radiative effects during core-collapse supernovae). In his analysis, the part of the relativistic stress tensor expressing viscosity and heat conduction, a slightly generalized version of Eckart’s, was of the form

(4) |

where , , and are the coefficients of dynamic viscosity, bulk viscosity, and heat conduction, respectively, and is the temperature of the gas. In the special-relativistic limit, is just the partial derivative, but is the covariant derivative if curvilinear coordinates are being used (or if the fluid is in a gravitational field). His comparison between the two theories enabled him to calculate , , and for a radiating fluid, confirming the notion that small-scale anisotropies in the radiation field, i.e., on the order of the mean free path of a photon, generate viscous-like effects.

Although they yield similar results, the fluid/thermodynamic approach of Eckart (1940), and correspondingly that of Weinberg (1971), is fundamentally different from the kinetic theory approach of Thomas (1930). The first difference arises from the fact that the former is a single-fluid analysis, meaning that , as given by equation (4), is the viscous correction to “the fluid." The second disparity comes about because Eckart defines thermodynamic quantities in terms of the energy-momentum tensor of the fluid; for example, Eckart (1940) and Weinberg (1971) both *define* the comoving energy density as , meaning that there is no correction to the observed energy. It is for this reason that there is no term proportional to in equation (4); however, the Eckart decomposition for an arbitrary tensor, one that has no restriction imposed upon it concerning the comoving energy density, will have such a term.

On the other hand, the kinetic theory approach considers the radiation and the matter to be two separate, interacting fluids – the same viewpoint that underlies all of radiation hydrodynamics – meaning that the viscous terms are understood as corrections to the radiation energy-momentum tensor, rather than that of the matter. Also, the kinetic theory description uses the Boltzmann equation to determine the distribution function, the viscous stress tensor then being written in terms of integrals over that function, viz., equation (3). There is thus no need to define bulk physical parameters, such as the temperature or the comoving energy density , in terms of the stress tensor. Indeed, one could verify the validity of Eckart’s assumption that if one knew the distribution function.

Our goal here is to analyze the Boltzmann equation and thereby evaluate the stress tensor for a relativistic, radiating fluid in the limit that the finite mean free path of the radiation provides the source of the viscosity. In section 2 we present the general relativistic Boltzmann equation, the formalism of general relativity being necessary because of the fact that scattering is handled most easily in the comoving (accelerating) frame of the fluid. In section 3 we restrict our attention to the case where the scattering is dominated by Thomson scattering and we solve the resultant equation for the distribution function to first order in the mean free path. Section 4 presents the equations of radiation hydrodynamics in the viscous limit and we demonstrate that the stress tensor departs from equation (4) in a few important ways, the first being that our equation has an additional correction to the comoving energy density, and the second being that our result is independent of thermodynamic considerations, such as the assignment of a temperature, and is therefore applicable to non-equilibrium radiation fields. Our coefficient of bulk viscosity also differs from that derived by Weinberg (1971). We perform a Fourier analysis of the perturbed equations in section 5 and show that they are indeed stable to perturbations of the fluid on scales larger than the mean free path of the radiation. A discussion of future applications, some comments on current radiation magnetohydrodynamic codes and conclusions are presented in section 6.

## 2. Relativistic Boltzmann equation

The distribution function for the radiation must satisfy a transfer equation – some statement of the conservation of photon number. As mentioned in the introduction, a natural choice for this equation is the Boltzmann equation, which describes the changes to the distribution function owing to emission, absorption, and scattering processes. However, because photons are massless, we must use a relativistic version of the Boltzmann equation; an obvious generalization to the relativistic regime is

(5) |

where Latin indices adopt the range 1 – 3, Greek indices range from 0 – 3, repeated upper and lower indices imply summation, and is the four-momentum of a photon. The right-hand sides represent changes to the distribution function through interactions with the surrounding medium. Equation (5) is the relativistic Boltzmann equation often encountered in the literature (e.g., Mihalas & Mihalas 1984). There are, however, two important subtleties associated with equation (5) that are often not mentioned explicitly.

The first issue is that the sum over on the left-hand side of (5) must incorporate the constraint that the four-momentum of the photon lie on the null cone, viz., (this problem vanishes in flat space because photons travel in straight lines, and hence ), which raises the question of whether or not the sum should occur over all four components of the momentum or just three. Furthermore, if we take the latter route, which three should we choose? Comparing equation (5) to the non-relativistic expression suggests taking the three spatial components; relativistic covariance, however, demands that the time component should not be treated differently from the spatial momenta.

The second subtlety stems from the fact that, in general relativity, there are two different momenta from which we can choose to describe the system: the covariant, , and the contravariant, , components, related by where is the metric. The distribution function treats the spatial variables, , as independent of the momentum. Therefore, should we consider and as independent coordinates, or and ? It is apparent that, depending on which one we choose, the results will differ as the metric is a function of the spatial coordinates.

In our ensuing treatment of equation (5), we will be analyzing interactions in the fluid frame – the one comoving with a given fluid parcel. Such a frame will, in general, be accelerating, meaning that and there will be an acceleration-induced metric , forcing us to confront each of the previously-raised questions. Recently, Debbasch & van Leeuwen (2009a, b) addressed these issues directly by returning to the most general definition of the distribution function, a sum of Dirac delta functions in position and momentum space. They demonstrated (Debbasch & van Leeuwen, 2009a) that one may consider either the covariant, spatial components or the contravariant components as the momentum independent from . However, as we mentioned previously, the Boltzmann equations that result from these choices are not identical, meaning that one must also define two different distribution functions, one dependent on the covariant components and the other on the contravariant components, to proceed unambiguously. Even though the routes were shown to be equivalent, one must be careful to use the components appropriate to a given Boltzmann equation.

In Debbasch & van Leeuwen (2009b), the authors showed that equation (5) is correct if 1) the sum involving in the second term is performed over the three spatial components, 2) the distribution function is considered to be a function of the contravariant, spatial components , and 3) any appearance of (or ) is replaced by . The dependence of on the spatial momenta can be determined by solving the equation for . As we stated, the frame of interest is the comoving frame of the fluid, and we will denote the components of any tensor in this frame with a prime on the index; e.g., is the th component of the momentum in the comoving frame. With this convention, the relativistic Boltzmann equation becomes

(6) |

where we have used the geodesic equation,

(7) |

to replace , and

(8) |

are the Christoffel symbols associated with the metric which, for our purposes, is induced by the acceleration associated with the comoving frame.

Equation (6) was provided by Lindquist (1966) and Castor (1972). Since they were considering spherically symmetric flows, they opted to change the form of equation (6) by using an orthonormal tetrad adapted to spherical coordinates. For our purposes, however, we will not be in the position to take advantage of any specific coordinate symmetries, so equation (6) will suffice.

Finally, it should also be mentioned that the right-hand side is evaluated at a specific location – the point in space and time where the collision occurs. The left-hand side, therefore, must also be evaluated at that spacetime point.

## 3. Diffusion approach to the transport equation

Here we will first make the approximation that the temperature of the gas is high enough such that all species, considered to comprise a single, massive fluid, are ionized, meaning that the right-hand side of equation (6) incorporates effects due only to scattering. In this case, the collisional term can be written (Hsieh & Spiegel, 1976)

(9) |

where is the redistribution function (not to be confused with the stress-energy tensor) and is the rest-frame number density of scatterers. The first term represents scatterings into the state from any initial momentum state , while the second embodies scatterings out of state into any other state . As long as photon wavelengths are long compared to the Compton wavelength and the gas is non-relativistic in the rest frame of the fluid, two suppositions we will make here, the redistribution function (in the comoving frame of the fluid) is that appropriate to Thomson scattering:

(10) |

where is the Thomson cross section (or, more generally, the cross section relevant to the scatterer). The equation for is identical to that for but with . Inserting equation (10) into equation (9) and the result of that substitution into equation (6) gives our final form for the transfer equation.

It should be noted that the integrals in equation (9) are performed under the restriction that the photon four-momenta lie on the null cone. Since collisions occur instantaneously at fixed locations in space, we can approximate the metric to be locally that of flat space, i.e., , and we can use the null cone condition () to write . Every appearance of in equation (9) can thus be replaced by .

Our goal here is to discern how the radiation field responds to gradients in the flow velocity. In the next section, we will use the equations of radiation hydrodynamics to deduce how the flow couples to changes in the radiation field, thereby completing the picture. To achieve this goal we will adopt a diffusion approximation, asserting that the distribution function may be written as , where is a small correction to . The “smallness" of is encoded in the mean free path and the gradients of the flow velocity, , across the mean free path, meaning that , where is the optical depth.

Because the right-hand side of equation (6) is proportional to the optical depth (per unit length), the consistency of our diffusion approach demands that . It can be verified that the right-hand side vanishes for any function that depends only on the magnitude of the momentum, meaning that is isotropic in the comoving frame. This result is consistent with the expectation that, in an optically-thick medium, the flux of radiation observed in a frame comoving with a fluid parcel is very nearly zero. Since the collision operator acts at a point in spacetime, however, the zeroth-order distribution function can also contain any other secular variation in space and time, meaning that its most general form is . The first-order transfer equation, to be solved for , is then

(11) |

To make more progress on this relation, we must determine the metric associated with the comoving frame. To do so, we will first assume a planar configuration of the fluid, with neither velocity nor variation in the direction. We will then use the fact that the coordinate transformation to move into the comoving frame of a given fluid parcel is a local Lorentz transformation, the inverse of which is given by

(12) |

(13) |

(14) |

(15) |

where , is the Lorentz factor (not the Christoffel symbol), and we have, without loss of generality, chosen the origin of the primed coordinate system to coincide with that of the lab frame. The integrals are necessary here because the velocities are all dependent on the coordinates , , and , but because we will ultimately be evaluating our expressions at the origin, one would obtain the same answer by letting , etc. Double-primed coordinates are simply dummy variables where for each integrand we let , etc. The line element, which in flat space is given by

(16) |

is invariant with respect to our choice of coordinates; the metric can therefore be determined by differentiating equations (12) – (15), inserting the results into equation (16) and grouping terms (see Castor 1972 for a similar, but non-relativistic, approach).

Calculating the Christoffel symbols, using the chain rule to determine and , inserting the expressions into equation (6) and evaluating the result at the origin (as this is the location of the fluid parcel – where the scattering occurs and where spacetime is locally Minkowskian), we find

(17) |

where, by using the definitions of the Christoffel symbols and using , it can be shown that

(18) |

(19) |

(20) |

(21) |

(22) |

We broke up the derivative into two separate components: , taken such that all appearances of not contained in the definition of , through the metric, are kept constant, and , taken such that all spatial coordinates that appear through are held fixed. Note that we must use the relativistically-correct version of the magnitude of the photon momentum, i.e., one involving the metric, because we are taking derivatives of the distribution function before evaluating the result at the location of the fluid parcel. Thus, even though spacetime is flat exactly at the point of interest, deviations exist at neighboring locations – the derivative requiring that we evaluate the distribution function at those locations. Though we did not explicitly denote it, the derivatives in equations (18) – (22) are to be evaluated at the origin.

The right-hand side of equation (6) involves an integral over (recall that is given by equation (10)), and the most direct means of evaluating would be to expand both the left-hand side of the transfer equation and the function in terms of spherical harmonics. Instead of pursuing this route, however, we will make the educated guess

(23) |

where the ’s are functions of and (since the ’s are just linear combinations of spherical harmonics, this method yields the same result as proceeding in the more rigorous fashion of expanding the functions in terms of spherical harmonics). Inserting this ansatz into , performing the integrals and comparing powers of on both sides, we find

(24) |

(25) |

(26) |

(27) |

(28) |

In addition, however, we find that there are isotropic terms, i.e., only dependent on , that arise from the integrations over and ; it is also apparent that , the first term in the sum in equation (17), represents an isotropic contribution to the left-hand side. Since the collision integral is zero for any isotropic term, however, these additional terms cannot be accounted for with our correction to the distribution function. (Equivalently, there are terms proportional to the spherical harmonic, a constant, that cannot be balanced by adding more terms to the distribution function.) We are therefore forced to equate these extraneous collision terms and the time derivative, yielding an extra constraint that the zeroth-order distribution function must satisfy:

(29) |

What does this condition mean physically? As an illustrative example, let us take the case where the zeroth-order distribution function is given by that for blackbody radiation:

(30) |

where is a constant, the value of which is unimportant, and we have taken Boltzmann’s constant to be one. With this form for , equation (29) becomes

(31) |

This, however, is just the gas energy equation for an isentropic, relativistic gas in the frame comoving with the fluid. Equation (29), therefore, is equivalent to the statement that the zeroth-order distribution function be isentropic.

(32) |

where is the opacity. Since we used a relativistic approach to derive this expression, we should be able to write it in a covariant fashion. This is indeed the case, the covariant form being

(33) |

where is the projection tensor introduced by Eckart (1940) (see section 1). We introduced the quantity to signify the derivative with respect to coordinate holding constant. This is an important distinction if we want to calculate derivatives of as we must use the relativistically-correct definition and depends on .

The preceding analysis only considered a single fluid element. However, the location of our origin, tantamount to the position of the fluid element under consideration, is arbitrary, meaning that equation (33) is applicable to the entire fluid. Also, even though we only considered planar flows, we can easily generalize the approach to include three-dimensional motion and variations, and (33) still holds. Greek indices therefore range from to 3 in equation (33).

In the next section we will use equation (33) and integrals thereof to write the equations of radiation hydrodynamics, equation (1), in terms of the four-velocity of the fluid, the mass density, and the radiation energy density, to which we will add the continuity equation for the scatterers to close the system. How do we reconcile these with equation (29), which seems to be an additional constraint? Recall that equation (29) was derived by equating the “extra," isotropic terms arising from both the collision integral and the derivatives of . If we were to attempt to derive the *second* order correction to the distribution function, , we would doubtless encounter more isotropic terms arising from the derivatives of and collision integrals of , this time to first order in the optical depth, which we would have to add to equation (29). Equation (29) therefore only captures effects to zeroth order in the mean free path. Although it constrains the spatial and temporal derivatives of appearing in equation (33), it has no effect on the form of this equation. However, the gas energy equation (see equation (50)) should reduce to equation (29) in the limit that .

## 4. Relativistic, diffusive equations of radiation hydrodynamics

The equations of radiation hydrodynamics, valid in any frame, are given by equation (1). Because we now have the distribution function to the requisite order in the optical depth, we can simplify those equations by writing the radiation energy-momentum tensor as , where each is given by equation (3) with the appropriate distribution function. We will derive each of these tensors in the comoving frame but write them in a manifestly covariant form, the results then being applicable in any coordinate system.

The comoving, isotropic energy-momentum tensor is given by

which we can show is equivalent to

(34) |

where

(35) |

is the isotropic radiation energy density. Since the projection tensor is orthogonal to the four-velocity, a time-like vector that selects the energy component of a tensor, it is reasonable to associate with the pressure, or momentum density, exhibited by the fluid. With this association, equation (34), not surprisingly, demonstrates that radiation acts like a gas with a relativistic equation of state, i.e., one with an adiabatic index of .

The correction to the energy-momentum tensor,

(36) |

will have a number of terms, as evidenced by equation (33). Integrating by parts and performing some simple manipulations, we can show that is given by

(37) |

Written in this manner, it is evident that transforms like a tensor. It should also be noted that each term in the tensor contains a derivative of a quantity, either the fluid velocity or the energy of the radiation field, with respect to the optical depth, which is what we expected.

Equation (37) differs from equation (4), the viscous stress tensor proposed by Weinberg (1971), in two notable ways. The first is that we have not postulated the existence of a temperature; instead we just used the fact that the zeroth-order distribution function is isotropic in the comoving frame to leave the comoving energy density, given by equation (35), as an unknown. The temperature has in fact been a difficult quantity to define in past treatments (see Weinberg’s discussion of the reconciliation between the results of Thomas (1930) and Eckart’s general form for a relativistic viscous stress tensor; see also Lima & Waga 1990), and it is reassuring to find that the physics is perfectly well-described without invoking such a quantity.

The second difference is contained in the presence of the first term of our stress tensor, proportional to , which shows that there is a correction to the comoving radiation energy density – *defined* to be zero by Eckart (1940) and Weinberg (1971) – given by

(38) |

This expression can be understood as follows: imagine that we take a spherical volume of fluid and contract it by some amount, i.e., such that is negative. Because of the nature of the Thomson cross section, any radiation intersected by the contracting fluid will be scattered preferentially in the direction of motion. Therefore, the radiation energy in this volume will be increased owing to the in-scattering of photons, which is reflected in equation (38). Furthermore, if we recall that the term in the stress tensor proportional to can be interpreted as the pressure exerted by the radiation on a fluid element, we see that

(39) |

This demonstrates that the change in pressure is 1/3 the change in energy, which is what we expect – the relativistic nature of the photon gas is preserved independent of the manner in which we expand the distribution function. It can also be shown that the trace of is zero, which is another familiar property of a relativistic gas.

Comparing the other terms in equation (37) to the form of an arbitrary viscous stress tensor given by (4), we find

(40) |

for the coefficient of dynamic viscosity, which agrees with the findings of Loeb & Laor (1992). Thomas (1930) used an incorrect form for the Thomson cross section, so Weinberg (1971) did not have the factor of 10/9. The proportionality to is sensible, as the viscous effect is mediated by radiation; therefore, a higher radiation energy density permits a higher transfer of energy and momentum to neighboring fluid elements. The viscous effect is also proportional to the mean free path of the radiation, which is also a reasonable result: smaller mean free paths mean that the observed velocity difference across a mean free path is smaller for a given shear, implying less transfer of momentum per scattering. The coefficient of bulk viscosity is found to be

(41) |

and is not zero, as predicted by Eckart (1940) and Weinberg (1971) for a radiation-dominated fluid. Because we did not introduce a temperature, we cannot define a coefficient of heat conduction in a manner analogous to that of Weinberg (1971). However, for the case where the zeroth-order distribution function is that of blackbody radiation, it can be verified that , which agrees with his findings.

We can also calculate the correction to the flux of photons, where the flux four-vector is given by

(42) |

As expected, the zeroth-order flux only has a non-vanishing number density in the comoving frame, given by

(43) |

which can be written covariantly as

(44) |

As for the energy-momentum tensor, we can use equation (33) and integrate by parts to write the correction to the flux vector in terms of . We find

(45) |

We see that this expression yields

(46) |

as the correction to the comoving number density of photons, a result in contrast to the analysis of Eckart (1940), who *defined* to be zero. However, equation (46) has a similar interpretation to equation (38): by noting that a contracting gas preferentially scatters photons in the direction of motion of the scatterers, one would expect an increased amount of radiation within that contracting volume. We also find

(47) |

reaffirming the notion that radiation behaves as a relativistic gas and demonstrating that it is these extra photons, , that add to the energy of the contracting fluid.

For the case of a cold gas, where , we will, for completeness, write down the full set of equations:

(48) |

(49) |

The first of these is just the continuity equation. We will also derive the gas energy equation, obtained by contracting equation (49) with the four velocity, which gives

(50) |

The left-hand side is just the change in energy for an adiabatic, gas, where is the adiabatic index. Note that if and we let , the left-hand side equals equation (29). The right-hand side therefore represents the energy added to the radiation during interactions with the scatterers (see section 6 for a discussion concerning the entropy generated by this heat addition).

## 5. Perturbation analysis

Hiscock & Lindblom (1985) showed that the general viscous tensor proposed by Eckart (1940) is unstable to small perturbations in a fluid. An interesting question is whether or not these instabilities appear in our set of equations.

To answer this question, consider an equilibrium solution where all of the variables are constants in space and time, the fluid is motionless and the space is flat. On top of this equilibrium solution we will impose perturbations on our variables small enough such that their products are negligible. With this configuration, the zeroth-order fluid equations are trivially satisfied. The first-order perturbations to the energy-momentum tensors are

(51) |

(52) |

(53) |

We assumed here that the perturbations are small enough such that gravitational corrections can be ignored, i.e., . The first-order conservation equations that must be satisfied are now

(54) |

These four equations must also be coupled to the mass continuity equation, the first-order correction to which is

(55) |

The normalization of the four-velocity, , demonstrates that .

For the present analysis we will restrict our attention to planar flows, such that and any perturbations in the direction are exactly zero. In this case, only the and components of equation (54) are non-trivial. Carrying out the derivatives, we find that they become, respectively,

(56) |