Abstract
Three applications of covariant fluid theory are discussed in this chapter: a covariant form of cold-plasma theory, covariant MHD theory, and a covariant form of quantum fluid theory.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Three applications of covariant fluid theory are discussed in this chapter: a covariant form of cold-plasma theory, covariant MHD theory, and a covariant form of quantum fluid theory.
In a cold plasma model each species in the plasma is described as a fluid. The fluid equations are used to calculate the response of each species, with the response of the plasma obtained by summing over the contributions of all species. Magnetohydrodynamics (MHD) is the conventional fluid description of a plasma. It differs from a cold plasma model in that the plasma is treated as a single fluid, rather than separate fluids for each species, and through the inclusion of a pressure. In quantum fluid theory, additional quantum effects are included in the fluid equation: quantum mechanical diffusion (the Bohm term), degeneracy and spin.
A covariant form of the fluid equations for a magnetized cold plasma requires a covariant description of a magnetostatic field. The Maxwell 4-tensor, \({F}^{\mu \nu }\), is used to construct basis 4-vectors in § 1.1. The covariant description of the orbit of a particle is identified, and used in § 1.2 to find the response of a cold plasma using both the forward-scattering and the fluid methods. The cold plasma model is generalized to include the effect of motions along the field lines in § 1.3. A covariant form of MHD theory for a relativistic plasma is introduced in § 1.4 and used to derive the properties of the MHD modes. Quantum fluid theory is discussed in § 1.5. SI units are used in introducing the theory, before reverting to natural units for the formal development of the theory. Except where indicated otherwise, formulae are in natural units.
1.1 Covariant Description of a Magnetostatic Field
A static electromagnetic field is said to be a magnetostatic field if there exists an inertial frame in which there is a magnetic field but no electric field. In this section, the Maxwell 4-tensor is written down for an arbitrary static electromagnetic field, and then specialized to a magnetostatic field. The Maxwell tensor is used to separate space-time into two 2-dimensional subspaces, one containing the time-axis and the direction of the magnetic field, and the other perpendicular to the magnetic field.
1.1.1 Maxwell 4-Tensor
The Maxwell tensor for any electromagnetic field, corresponding to an electric field, \({E}\) and a magnetic field, \({B}\), has components (SI units)
where argument x denotes the components of the 4-vector \({x}^{\mu } = [ct,{x}]\). As in volume 1, the 4-tensor indices, μ, ν run over 0, 1, 2, 3, and the metric tensor, g μν, is diagonal, \((+1,-1,-1,-1)\), such that one has \({x}_{\mu } = [ct,-{x}]\). Thus, in cartesian coordinates, the contravariant components with μ = 0, 1, 2, 3 are x 0 = ct, x 1 = x, x 2 = y, x 3 = z, and the covariant components are x 0 = ct, \({x}_{1} = -x\), \({x}_{2} = -y\), \({x}_{3} = -z\). The contravariant components E i(x), B i(x), with i = 1, 2, 3 correspond to the respective cartesian components of the corresponding 3-vectors.
The dual of the Maxwell tensor is defined by
with εμναβ the completely antisymmetric tensor with \({\epsilon }^{0123} = +1\). The dual has components (SI units)
Maxwell’s equations in covariant form are
with \({\partial }_{\mu } = \partial /\partial {x}^{\mu }\).
1.1.1.1 Classification of Static Fields
An arbitrary static homogeneous electromagnetic field may be classified as a magnetostatic field, an electrostatic field, or an electromagnetic wrench. This classification is based on the fact that there are two independent electromagnetic invariants (SI units)
where the argument x is omitted. An arbitrary static electromagnetic field is (a) a magnetostatic field for S < 0, P = 0, (b) an electrostatic field for S > 0, P = 0, or (c) an electromagnetic wrench for P≠0. This classification follows from the fact that in these three cases, one can always choose an inertial frame such that (a) there is no electric field and the magnetic field is along a chosen axis, (b) there is no magnetic field and the electric field is along a chosen axis, and (c) the electric and magnetic fields are parallel and along a chosen axis.
1.1.1.2 Magnetostatic Field
It is convenient to denote the magnetostatic field by \({F}_{0}^{\mu \nu }\), and to write
Equation (1.1.6) defines the invariant B, interpreted as the magnetic field strength, and introduces a dimensionless 4-tensor \({f}^{\mu \nu }\). A second dimensionless 4-tensor is the dual of \({f}^{\mu \nu }\):
The tensors \({f}^{\mu \nu }\) and \({\phi }^{\mu \nu }\) are both antisymmetric and they are orthogonal to each other:
1.1.1.3 Alternative Form for the Maxwell 4-Tensor
The most general form for a static electromagnetic field is an electromagnetic wrench. Such a field may be written in the form (SI units)
The tensors \({f}^{\mu \nu }\), \({\phi }^{\mu \nu }\) have the forms (1.1.16) in the frame in which the electric and magnetic fields are parallel and along the 3 axis. For E = 0 (1.1.9) describes a magnetostatic field along the 3 axis, and for B = 0 (1.1.9) describes an electrostatic field along the 3 axis.
1.1.1.4 4-Tensor \({B}^{\mu }\)
The description of the magnetostatic field in terms of \({F}_{0}^{\mu \nu }\) applies in any frame. There is an alternative description of a magnetostatic field in terms of a 4-vector, \({B}^{\mu }\), that is available if there is some preferred frame. In the presence of a medium there is a preferred frame: the rest frame of the medium. Let \({\tilde{u}}^{\mu }\) be the 4-velocity of the rest frame, with \({\tilde{u}}^{2} ={ \tilde{u}}^{\mu }{\tilde{u}}_{\mu } = 1\). One has \(\tilde{u} = [1,{\bf 0}]\) in the rest frame. The 4-tensor is
One has \({B}^{\mu }{B}_{\mu } = -{B}^{2}\) and hence \({b}^{\mu }{b}_{\mu } = -1\). In the rest frame one may write \({b}^{\mu } = [0,{b}]\), where \({b}\) is a unit vector along the direction of the magnetic field.
1.1.2 Projection Tensors \({g}_{\perp }^{\mu \nu }\) and \({g}_{\parallel }^{\mu \nu }\)
The tensors \({f}^{\mu \nu }\) and \({\phi }^{\mu \nu }\) allow one to construct projection tensors \({g}_{\perp }^{\mu \nu }\) and \({g}_{\parallel }^{\mu \nu }\):
The tensors \({g}_{\perp }^{\mu \nu }\) and \({g}_{\parallel }^{\mu \nu }\) span the 2-dimensional perpendicular and time-parallel subspaces, respectively. They correspond to a separation of the metric tensor into metric tensors for these two subspaces:
In a frame in which the magnetostatic field is along the 3-axis, \({g}_{\parallel }^{\mu \nu }\) is diagonal (1, 0, 0, − 1) and \({g}_{\perp }^{\mu \nu }\) is diagonal \((0,-1,-1,0)\).
An alternative definition of \({g}_{\parallel }^{\mu \nu }\) assumes the existence of a frame in which (1.1.10) applies. This frame is described by its 4-velocity \({\tilde{u}}^{\mu }\), and by the 4-vector b μ along the direction of the magnetic field. One has
Despite appearances, \({g}_{\parallel }^{\mu \nu }\) is independent of \(\tilde{u}\), that is, it is not dependent on the choice of a preferred frame.
1.1.2.1 Parallel and Perpendicular Invariant Components
The projection tensors allow one to separate any 4-vector, \({a}^{\mu }\) say, into a sum of two orthogonal 4-vectors, \({a}^{\mu } = {a}_{\perp }^{\mu } + {a}_{\parallel }^{\mu }\);
Similarly, for a single 4-vector, \({a}^{\mu }\), the invariant a 2 is written in the form \({a}^{2} = {({a}^{2})}_{\perp } + {({a}^{2})}_{\parallel }\), and for two 4-vectors, \({a}^{\mu }\) and \({b}^{\mu }\), the invariant ab is written in the form \(ab = {(ab)}_{\perp } + {(ab)}_{\parallel }\). These separations are made by writing
The components of the wave 4-vector, \({k}^{\mu } = [\omega /c,{\bf{ k}}]\), with { k} = (k x , k y , k z ), give \({({k}^{2})}_{\perp } = -{k}_{\perp }^{2} = -({k}_{x}^{2} + {k}_{y}^{2})\), \({({k}^{2})}_{\parallel } = {\omega }^{2}/{c}^{2} - {k}_{z}^{2}\).
1.1.2.2 Components for \({B}\) Along the 3-Axis
In any frame in which \({F}_{0}^{\mu \nu }\) corresponds to \({E} = 0\), one is free to write \({B} = B{b}\), and to orient the axies such that \({b} = (0,0,1)\) is along the 3-axis. The tensors \({f}^{\mu \nu }\) and \({\phi }^{\mu \nu }\) then have components
The projection tensors (1.1.11) have components
1.1.3 Basis 4-Vectors
Consider the response of a medium described by the response tensor \({\Pi }^{\mu \nu }(k)\). The tensor indices can depend only on the available 4-vectors and 4-tensors. In a magnetized medium the available 4-tensors and 4-vectors are \({f}^{\mu \nu }\), \({k}^{\mu }\), and the 4-velocity, \({\tilde{u}}^{\mu }\), of the rest frame of the medium. It is possible to construct several different sets of basis 4-vectors that may be used to represent \({\Pi }^{\mu \nu }(k)\). For the magnetized vacuum, \({\tilde{u}}^{\mu }\) is undefined, and one needs a set that does not involve it. A set that involves \({\tilde{u}}^{\mu }\) is more convenient for a material medium.
1.1.3.1 Basis 4-Vectors for the Magnetized Vacuum
One can construct four mutually orthogonal 4-vectors from \({k}^{\mu }\) and the Maxwell tensor for the magnetostatic field. A convenient choice is
where the components apply in a frame in which the magnetic field is along the 3-axis, and \({k}\) is in the 1–3 plane. In terms of this choice of basis 4-vectors, the 4-tensors (1.1.16) and (1.1.17) become
with \({({k}^{2})}_{\parallel } = {k}_{\parallel }^{\mu }{k}_{\parallel \mu } = {\omega }^{2} - {k}_{z}^{2}\), \({({k}^{2})}_{\perp } = {k}_{\perp }^{\mu }{k}_{\perp \mu } = -{k}_{\perp }^{2}\).
Writing the response 4-tensor in terms of invariant components along the 4-vectors (1.1.18) gives
with A, B = ∥ , ⊥ , D, G. The charge-continuity and gauge-invariance relations, \({k}_{\mu }{\Pi }^{\mu \nu }(k) = 0\), \({k}_{\nu }{\Pi }^{\mu \nu }(k) = 0\), imply that the invariant components Π AB(k) in (1.1.20) satisfy
with A, B = ∥ , ⊥ , D, G.
1.1.3.2 Shabad’s Basis 4-Vectors
A related choice of basis 4-vectors, made by Shabad [24], is
In a form analogous to (1.1.20), the response tensor,
involves only six invariant components, with the counterpart of (1.1.21) becoming \({\Pi }_{4B}(k) = 0 = {\Pi }_{A4}(k)\).
The set (1.1.22) of basic vectors \({b}_{A}^{\mu }\) with A = 1, 2, 3 may be replaced by any linear combination of them that preserves the orthogonality condition. In particular, one can choose a linear combination that separates longitudinal and transverse parts. Let this choice be denoted \({b}_{A^{\prime}}^{\mu }\), with A′ = 1′, 2′, 3′. One requires that two of these satisfy \({{b}}_{A^{\prime}} \cdot {k} = 0\), and that the third has its 3-vector part parallel to \({k}\). The combination \({k}_{z}{b}_{2}^{\mu } + \omega {b}_{3}^{\mu }\) has its 3-vector part proportional to \({k}\). Thus the choice of (unnormalized) basis 4-vectors,
allows one to project onto the transverse plane, due to \({b}_{1^{\prime}}^{\mu }\), \({b}_{2^{\prime}}^{\mu }\) being transverse and \({b}_{3^{\prime}}^{\mu }\) being longitudinal. (This procedure breaks down for the special case k 2 = 0.)
The normalization of these 4-vectors is problematic. Only one of the 4-vectors can be time-like, and this 4-vector is to be normalized to unity with the others three 4-vectors normalized to minus unity. However, which of them is time-like depends on the signs of the invariants k 2 and \({({k}^{2})}_{\parallel } = {\omega }^{2} - {k}_{z}^{2}\). The choice (1.1.24) is not used in the following.
1.1.3.3 Basis 4-Vectors for a Magnetized Medium
In the presence of a medium, the 4-vector, \({\tilde{u}}^{\mu }\), of the rest frame of the medium may be chosen as one of the basis 4-vectors, and it is then the only time-like basis 4-vector. One needs three independent space-like 3-vectors. Given the unit 3-vectors \(\kappa = {k}/\vert {k}\vert \) and \({b}\) along the wave 3-vector and the magnetostatic field, respectively, an orthonormal set of 3-vectors in the rest frame is κμ = [0, κ], \({a}^{\mu } = [0,{a}]\), \({t}^{\mu } = [0,{t}]\), with
A covariant form of these definitions is
with the fourth basis 4-vector being \({\tilde{u}}^{\mu }\).
To interpret these basis vectors further, it is helpful to express other quantities that appear in the rest frame in terms of invariants. Let the angle between the 3-vector \({k}\) and \({B}\) in the rest frame be θ. One has
Using these relations one may rewrite many of the formulae in the noncovariant theory in a covariant form.
The 4-vectors (1.1.26) may be written in terms of the angles (1.1.27) in the rest frame of the medium. With the 3-axis along the magnetic field and \({k}\) in the 1–3 plane, one finds
The 4-vectors \({t}^{\mu }\), \({a}^{\mu }\) span the 2-dimensional transverse space, and may be chosen as basis vectors for the representation of transverse polarization states.
An alternative choice of basis 4-vectors, closely related to the set (1.1.28), is the set
Covariant definitions of the last three of these correspond to \({e}_{1}^{\mu } = {k}_{\perp }^{\mu }/{k}_{\perp }\), \({e}_{2}^{\mu } = {k}_{G}^{\mu }/{k}_{\perp }\), \({b}^{\mu } = {B}^{\mu }/B\), with \({B}^{\mu }\) defined by (1.1.10).
1.1.4 Linear and Nonlinear Response Tensor
The response of a plasma is described in terms of the linear response tensor and a hierarchy of nonlinear response tensors. The definition of these response tensors follow from an expansion of the induced current in the plasma in powers of a fluctuating electromagnetic field. This is referred to as the weak-turbulence expansion. The linear response tensor is defined by the leading term in this expansion.
1.1.4.1 Weak-Turbulence Expansion
The expansion of the induced current density, \({J}_{\mathrm{ind}}^{\mu }(x)\), in powers of the electromagnetic field, described by \({A}^{\mu }(x)\), is made in terms of their Fourier transforms in time and space. The Fourier transform and its inverse for the induced current is
The weak-turbulence expansion is
where the n-fold convolution integral is defined by
with δ4(k) = δ(k 0)δ(k 1)δ(k 2)δ(k 3). The expansion (1.1.31) defines the linear response tensor Π μν(k) and a hierarchy of nonlinear response tensors, of which only the quadratic response tensor Π (2)μνρ(k 0, k 1, k 2), with \({k}_{0} + {k}_{1} + {k}_{2} = 0\), and the cubic response tensor Π (3)μνρσ(k 0, k 1, k 2, k 3), with \({k}_{0} + {k}_{1} + {k}_{2} + {k}_{3} = 0\), are usually considered when discussing specific weak-turbulence processes.
1.1.4.2 General Properties of \({\Pi }^{\mu \nu }(k)\)
The linear response tensor satisfies several general conditions.
The reality condition on Fourier transforms is that reversing the sign of \({k}^{\mu }\) and complex conjugating has no net effect. Hence one has
where ∗ denotes the complex conjugate. The Fourier transform of the charge-continuity condition, \({\partial }_{\mu }{J}_{\mathrm{ind}}^{\mu }(x) = 0\) implies the first of
The second of (1.1.34) is the gauge-invariance condition. It is desirable that the description of the response be independent of the choice of gauge for \({A}^{\nu }(k)\), and an arbitrary gauge transformation involves adding a term proportional to \({k}^{\nu }\) to \({A}^{\nu }(k)\). Imposing the second of the conditions (1.1.34) ensures that \({\Pi }^{\mu \nu }(k)\) is independent of the choice of gauge.
A third general condition is that a separation of \({\Pi }^{\mu \nu }(k)\) into hermitian (H) and antihermitian (A) parts is equivalent to a separation into the time-reversible or non-dissipative part of the response, and the time-irreversible or dissipative part of the response, respectively. These two parts are
respectively.
1.1.4.3 Onsager Relations
The Onsager relations follow from the time-reversal invariance properties of the equations of motion used in any calculation of the response tensor. The choice of basis 4-vectors (1.1.28) is particularly convenient for expressing the Onsager relations:
where the reversal of the sign of any external magnetostatic field is noted explicitly. With the choice of coordinate axes in (1.1.28), the Onsager relations (1.1.36) imply
In terms of the invariant components introduced in (1.1.20), the Onsager relations imply
with \({k}_{G}^{\mu } = [0,{k}_{\perp }{a}]\) in the frame defined by (1.1.28).
An important implication for the polarization vector of a wave in a magnetized medium is that its component along the \({a}\)-axis (the 2-axis or y-axis for { k} in the 1–3 plane) is out of phase with the other components. Thus, one can choose an overall phase factor such that the 2-component (\({a}\)-component) is imaginary and the 0-, 1-, 3-components (\(\tilde{u}\)-, κ-, \({t}\)-components) are all real.
1.2 Covariant Cold Plasma Model
A covariant formulation of the cold plasma model is used in this section to calculate the linear response tensor for a cold magnetized plasma. Cold plasma theory generalizes the magnetoionic theory, also called the Appleton-Hartree theory, which was developed in the 1930s [4, 13 ] to describe radio wave propagation in the ionosphere. The magnetoionic theory describes the response of a cold electron gas, and the generalization to include the motion of the ions was made in the 1950s [26, 27].
Natural units (\(\hslash = c = 1\)) are used in this an subsequent sections, except where stated otherwise.
1.2.1 Fluid Description of a Cold Plasma
A cold plasma, in which thermal or other random motions are neglected, can be described using fluid equations, with one fluid for each species of particle. Consider particles of species α, with charge \({q}_{\alpha }\) and mass \({m}_{\alpha }\). For simplicity in writing, the affix α, denoting the species, is suppressed for the present, with the charge and mass denoted by q and m, respectively.
The fluid equations in covariant form involve the proper number density, \({n}_{\mathrm{pr}}(x)\), and the fluid 4-velocity, \({u}^{\mu }(x)\). The equation of continuity is
The equation of motion for the fluid is
where the contributions of the static field \({F}_{0}^{\mu \nu }\) and of a fluctuating field \({F}^{\mu \nu }(x) = {\partial }^{\mu }{A}^{\nu }(x) - {\partial }^{\nu }{A}^{\mu }(x)\) are included separately in the 4-force on the right hand side. The operator \({u}^{\sigma }(x){\partial }_{\sigma }\) in (1.2.2) may be interpreted as the total derivative ∂ ∕ ∂τ(x), where τ(x) is the proper time along the flow lines of the fluid. The derivative ∂ ∕ ∂τ(x) arises naturally in a covariant (Lagrangian or Hamiltonian) treatment of particle dynamics, and in a fluid (Eulerian) theory it is interpreted as the convective derivative \({u}^{\sigma }(x){\partial }_{\sigma }\).
The response of the plasma is identified as the induced 4-current density:
The objective is to expand \({n}_{\mathrm{pr}}(x)\,{u}^{\mu }(x)\) in powers of the perturbing electromagnetic field, described by \({A}^{\mu }(x)\), and to identify the response tensor from the Fourier transform of (1.2.3).
1.2.1.1 Fourier Transform of the Fluid Equations
The first steps in evaluating the induced current for a cold plasma are to Fourier transform the fluid Eqs. (1.2.1) and (1.2.2) and the induced current (1.2.3), and to carry out a perturbation expansion in powers of \({A}^{\mu }\).
The Fourier transformed form of the continuity Eq. (1.2.1) is
where the convolution integral is defined by (1.1.32) with n = 2. The Fourier transform of the equation of fluid motion (1.2.2) is
with
The induced 4-current (1.2.3) becomes
The 4-current has contributions from each species, only one of which is included in (1.2.7).
1.2.1.2 Expansion in Powers of the 4-Potential
In the weak-turbulence expansion for a fluid, the proper number density and the fluid 4-velocity are expanded in powers of A(k).
The expansion of the proper number density is
where the zeroth order proper number density is denoted \({\bar{n}}_{\mathrm{pr}}\). In the rest frame of a cold fluid, all the particle are at rest, and in this frame the proper number density is equal to the actual number density, \(\bar{n}\) say. In the following, \(\bar{n}\) is used to describe the number density of the fluid, where \(\bar{n}\) is the actual number density in the rest frame of the fluid. Where no confusion should result, \(\bar{n}\) is written as n. The expansion of the fluid 4-velocity gives
The zeroth order fluid 4-velocity is necessarily nonzero, being \({\tilde{u}}^{\mu } = (1,0,0,0)\) in the rest frame of the fluid. More generally, the fluid has a nonzero velocity along the direction of the magnetic field, and the associated 4-velocity may be interpreted as that of the rest frame of the fluid, and written
Fluid flow (for a charged fluid) across the magnetic field is inconsistent with the assumption that there is no static electric field. The expansion of the induced 4-current (1.2.7) gives
The zeroth order current density, \(qn{\tilde{u}}^{\mu }\), associated with a given fluid is nonzero, but is usually ignored. The justification for this is either that the contributions to the 4-current from different species sum to zero, or that the static fields generated by this current density are negligible.
1.2.2 Linear Response Tensor: Cold Plasma
The derivation of \({\Pi }^{\mu \nu }(k)\) using the fluid model involves only the linear terms in the expansion of the fluid equations. The space components, Π ij(k), are simply related to the components of the dielectric tensor K i j (k), used in a conventional 3-tensor description.
1.2.2.1 First Order Current
The first order term in the expansion (1.2.11) of the current determines the linear response. On substituting (1.2.8) and (1.2.9) into (1.2.11), for N = 1 one has
where the subscript ‘ind’ is omitted. To first order, the equation of continuity (1.2.4) gives
which determines the first order number density in terms of the first order fluid velocity. The first order fluid velocity is determined by the first order terms in (1.2.5):
Let the solution of (1.2.14) be written
An an explicit form for the tensor τμν(ω) is derived below.
On inserting (1.2.13) and (1.2.15) into (1.2.12), one identifies the linear response tensor by writing \({J}^{(1)\mu }(k) = {\Pi }^{\mu \nu }(k){A}_{\nu }(k)\). The resulting cold plasma response tensor is
Note that, by hypothesis, any zeroth order motion is along the magnetic field lines, so that \(\tilde{u}\) is equal to \({\tilde{u}}_{\parallel }\), implying that the response tensor (1.2.16) is independent of k ⊥ .
1.2.3 Tensor τμν(ω)
The tensor \({\tau }^{\mu \nu }(\omega )\), introduced in (1.2.15), is constructed as follows. Write (1.2.14) in the form
with \({F}_{0}^{\mu \nu } = B{f}^{\mu \nu }\), \(\epsilon = -q/\vert q\vert \), \({\Omega }_{0} = \vert q\vert B/m\). In the rest frame of the plasma, \(k\tilde{u}\) is the frequency ω, and \({\tau }^{\mu \nu }(\omega )\) is defined as the inverse of the tensor \(\omega {g}^{\mu \nu } + i\epsilon {\Omega }_{0}{f}^{\mu \nu }\) on the left hand side of (1.2.17). Specifically, the definition is
Solving (1.2.18) gives
The matrix representation of \({\tau }^{\mu \nu }(\omega )\) is
In the unmagnetized limit, Ω 0 → 0, \({\tau }^{\mu \nu }(\omega )\) reduces to \({g}^{\mu \nu }\).
Note the unconventional choice of the sign o\(\epsilon = -q/\vert q\vert \), which is positive, \(\epsilon = +1\), for electrons and negative, \(\epsilon = -1\), for positively charged ions. This choice is made for convenience with comparison with the relativistic quantum case, where ε = ± 1 is a quantum number labeling electron and positron states.
1.2.3.1 Alternative Forms for the \({\Pi }^{\mu \nu }(k)\)
The explicit form (1.2.16) for the cold plasma response tensor is written in a concise notation, and a more explicit form is obtained by using the explicit form (1.2.6) for \({G}^{\mu \nu }(k,u)\) and (1.2.19) for \({\tau }^{\mu \nu }(\omega )\). This form is
where \({k}_{\perp }^{\mu }\), \({k}_{G}^{\mu }\) are defined by (1.1.18). Note that because \(\tilde{u}\) is restricted to the 0–3 plane, one has \({(k\tilde{u})}_{\perp } = 0\) and \(k\tilde{u} = {(k\tilde{u})}_{\parallel }\).
An alternative way of writing (1.2.21) is in terms of the 4-vector \({k}_{D}^{\mu } = ({k}_{z},0,0,\omega )\), rather than \({\tilde{u}}^{\mu }\). One has
The result (1.2.21) can be rewritten so that \({\tilde{u}}^{\mu }\) does not appear explicitly using \(k\tilde{u} = \omega \),
so that the tensors in (1.2.21) become
The forms on the right hand sides of (1.2.24) appear naturally when the relativistic quantum form of the response tensor, derived in § 9.2, is use to rederive the cold-plasma form in § 9.4.
1.2.4 Cold Plasma Dielectric Tensor
The dielectric tensor, K i j (k), is related to the mixed 3-tensor components Π i j (k), which are numerically equal to − Π ij(k). The contravariant space components of (1.2.21), with \({\tilde{u}}^{\mu } \rightarrow \gamma (1,0,0,\beta )\), are
where \({b}\) is a unit vector along the 3-axis, \({{k}}_{\perp }\) and \({{k}}_{G}\) are vectors of magnitude k ⊥ along the 1- and 2-axes, respectively, with \({g}_{\parallel }^{ij} = -{b}^{i}{b}^{j}\), g ⊥ ij diagonal \(-1,-1,0\), f ij nonzero only for \({f}^{12} = -{f}^{21} = -1\), and with \(\Omega = {\Omega }_{0}/\gamma \). In the absence of a streaming motion, (1.2.25) simplifies to
The dielectric tensor is
with \({\omega }_{p}^{2} = {q}^{2}n/{\epsilon }_{0}m\), and with
The cold plasma dielectric 3-tensor follows from (1.2.27) with (1.2.28) by adding a label, α say, for each species and summing over the species. This gives the standard form [26, 27]
with each of S, D, P involving a sum over the species
with
where \(-{\epsilon }_{\alpha }\) is the sign of the charge of species α. The dependence of K i j and of \(S,D,P,{R}_{\pm }\) on ω is implicit. The ± labeling of R ± is chosen such that the resonance, specifically an infinite value at \(\omega = {\Omega }_{\alpha }\), occurs in R + for charges that spiral in a right hand sense (electrons, ε e = 1) and in R − for charges that spiral in a left hand sense (ions, \({\epsilon }_{i} = -1\)).
No damping is included in the version of cold plasma theory discussed here; this is reflected in the dielectric tensor (1.2.29) being hermitian. It implies that the waves are undamped. Collisional damping can be included by including a frictional term in the fluid equations, but this is not done here.
Further quantities constructed from K i j appear in the derivation of the dispersion equation and the construction of the polarization vectors. These include the trace and the longitudinal part of the tensor,
the determinant of the tensor,
the square of the tensor,
and the trace and the longitudinal part of the square
1.2.4.1 Dielectric Tensor for a Cold Electron Gas
At frequencies much higher than the ion plasma frequency and the ion gyrofrequency, the contribution of the ions can be neglected. The dielectric tensor (1.2.29) then describes the response of a cold electron gas, with S, D, P given by
with ε = 1 for electrons. In the presence of an admixture of positrons, the parameter ε can be re-interpreted as the ratio of the difference to the sum of the number densities of the cold electrons and positrons.
The response tensor can be written in terms of two dimensionless magnetoionic parameters
In terms of these parameters, S, D, P in (1.2.36) become
1.3 Inclusion of Streaming Motions
A covariant formulation of the cold plasma response facilitates the inclusion of streaming motions. The response 4-tensor may be calculated in the frame in which there is no streaming motion, and transformed to the frame where the plasma has the specified streaming motion. In the presence of a magnetic field this applies only to streaming motions along the magnetic field lines. This procedure is used here in three ways. First, it is applied to the plasma as a whole. Second, different streaming motions are introduced for different components in the plasma, allowing the existence of instabilities due to counterstreaming motions. Third, a multi-fluid model is used to write down the response tensor for an arbitrary one-dimensional (1D), strictly-parallel (\({p}_{\perp } = 0\)) distribution of particles by summing over infinitesimal distributions with different streaming velocities.
1.3.1 Lorentz Transformation to Streaming Frame
Given the response tensor, \({\Pi }^{\mu \nu }(k)\), in one inertial frame, one can write it down in any other frame by making the appropriate Lorentz transformation. Let K and K′ be two inertial frames, and let \({L{}^{\mu ^{\prime}}}_{\mu }\) and its (matrix) inverse \({L{}^{\mu }}_{\mu ^{\prime}}\) be the Lorentz transform matrices between K and K′. These matrices are defined such that any 4-vector with contravariant components \({a}^{\mu }\) in K and \({a}^{\mu ^{\prime}}\) in K′ satisfies
with
Given Π μν(k) in K, the response tensor in K′ is Π μ′ν′(L − 1 k′), with
For the purpose of including a streaming motion in a magnetized plasma one is concerned with a boost in which the axes in K and K′ are parallel, and K′ is moving along the 3-axis of K at velocity − β. In K′ the streaming velocity of a plasma at rest in K is then β. The explicit forms for the transformation matrices in this case are, denoting the dependence of β explicitly,
with \(\gamma = {(1 - {\beta }^{2})}^{-1/2}\).
It is sometimes useful to write the dependence of \({\Pi }^{\mu \nu }(k)\) on k, in terms of invariants that involve k. In any other inertial frame the components of the response are given by applying a Lorentz transformation to \({\Pi }^{\mu \nu }(k)\), and by expressing the invariants that involve k in terms of k′.
1.3.1.1 Streaming Cold Distribution
The contribution to the response 4-tensor from a single cold distribution of particles may be written in the concise form
For a streaming distribution, the 4-velocity, u μ, is interpreted as the streaming 4-velocity of a streaming distribution of cold particles. The proper number density (for a cold plasma) is related to the actual number density, n, in the chosen frame in which the plasma is streaming with Lorentz factor γ by \({n}_{\mathrm{pr}} = n/\gamma \).
An alternative way of writing (1.3.5) is
A further alternative way of writing (1.3.5) is
where the 4-vectors introduced in (1.3.7) are defined by (1.1.18). With the form (1.3.7), the gyrotropic terms are those that depend on the sign, − ε, of the charge.
The space components of (1.3.7), for streaming at velocity β, are
where \({b}\) is a unit vector along the 3-axis, \({{k}}_{\perp }\) and \({{k}}_{G}\) are vectors of magnitude k ⊥ along the 1- and 2-axes, respectively, with \({g}_{\parallel }^{ij} = -{b}^{i}{b}^{j}\), g ⊥ ij diagonal \((-1,-1,0)\), f ij nonzero only for \({f}^{12} = -{f}^{21} = -1\), and with \(\Omega = {\Omega }_{0}/\gamma \).
1.3.1.2 Multiple Streaming Cold Components
The generalization of a single cold streaming distribution to a plasma consisting of several cold species in relative motion to each other follows by summing over the relevant contributions to the response 4-tensor. Let α label an arbitrary species, which has charge \({q}_{\alpha } = -{\epsilon }_{\alpha }\vert {q}_{\alpha }\vert \), mass m α, number density \({n}_{\alpha }\), cyclotron frequency \({\Omega }_{\alpha }\) and 4-velocity \({u}_{\alpha }^{\mu } = [{\gamma }_{\alpha },{\gamma }_{\alpha }{v}_{\alpha }{b}]\). With this generalization (1.3.5) implies
with (1.2.19) translating into
The alternative form (1.3.6) gives
The sum over species α may be interpreted as including a sum over different components of a single species with different streaming motions. For example, in the case of an electron gas that consists of two cold counterstreaming electron beams, one can interpret the sum over α as the sum over these two components.
1.3.2 Dielectric Tensor for a Streaming Distribution
In 3-tensor notation the response tensor is related to the dielectric tensor by \({K{}^{i}}_{j}(k) = {\delta }_{j}^{i} + {\Pi {}^{i}}_{j}(k)/{\epsilon }_{0}{\omega }^{2}\). In particular, the sum over components in (1.3.7), rewritten in the form (1.3.6), translates into the dielectric tensor
where \({\omega }_{p\alpha }^{2} = {q}_{\alpha }^{2}{n}_{\alpha }/{\epsilon }_{0}{m}_{\alpha }\) defines the plasma frequency for species α. In matrix form one has
The resulting matrix form for the dielectric tensor (1.3.12) can be written
with the components identified as
1.3.3 Cold Counterstreaming Electrons and Positrons
In an oscillating model for a pulsar magnetosphere [15], a parallel electric field, E z ≠0, accelerates the electrons and positrons, causing their streaming velocities to be different. Unlike the case where there is a single streaming velocity, the effect of relative streaming motions cannot be removed by a Lorentz transformation. However, one can transform to a frame in which the electrons and positrons are streaming in opposite directions with the same speed.
Consider the frame in which the streaming velocities, ± β, of electrons (ε = 1) and positrons (\(\epsilon = -1\)) are equal and opposite. Let their number densities be (note + for electrons, − for positrons) written \({n}^{\pm } = n(1 \pm \bar{\epsilon })\), \(n = {n}^{+} + {n}^{-}\), \(\bar{\epsilon } = ({n}^{+} - {n}^{-})/({n}^{+} + {n}^{-})\). One can identify two sources of gyrotropy is counterstreaming pair plasma. One is due to a charge imbalance, \(\bar{\epsilon }\neq 0\). The other is due to a nonzero current: the current density is \({J} = -e({n}^{+}{\beta }^{+} - {n}^{-}{\beta }^{-})\), with \({\beta }^{\pm } = \pm \beta {b}\) here. The sign of the current-induced gyrotropy is determined by the sign of the velocity, β.
One can write the response tensor in the form [30]
with \({K{}^{i}}_{j}(k) = {\delta }_{j}^{i} + {\mu }_{0}{\Pi {}^{i}}_{j}(k)/{\omega }^{2}\). It is convenient to introduce the notation
The components of the response tensor that do not depend on \(\bar{\epsilon }\) are
The components proportional to \(-\bar{\epsilon }\) are
The response of a counterstreaming pair plasma is gyrotropic even when the number densities are equal (\(\bar{\epsilon } = 0\)). This is due to the current from the oppositely directed flows of oppositely charged particles; the sign of the current is determined by the sign of β, which is included in \({\omega }_{\parallel }\), \({\omega }_{\perp }\) in (1.3.18).
1.4 Relativistic Magnetohydrodynamics
The fluid description of a magnetized plasma is referred to as magnetohydrodynamics (MHD). A relativistic generalization of MHD is presented here. The theory is then used to derive the properties of MHD waves in relativistic plasmas. SI units with c included explicitly are used in this section
1.4.1 Covariant Form of the MHD Equations
The relativistic MHD equations can either be postulated [3, 16, 29] or derived from kinetic (Vlasov) theory by taking moments. The zeroth and first moments lead to the continuity equation and the equation of motion for the fluid, respectively. Closing the expansion requires the introduction of another relation, which is identified as a generalized Ohm’s law. Apart from Maxwell’s equations, the MHD also include an equation of state for the fluid.
1.4.1.1 Basic Fluid Equations
Let the proper mass density be η(x) and the fluid 4-velocity be \({u}^{\mu }(x)\). The equation of mass continuity is
The equation of fluid motion depends on the assumed forces on the fluid. Assuming that the only forces are those internal to the system, including electromagnetic forces, the equation of motion can be written in the form of a conservation equation for the energy-momentum tensor:
where \({T}_{\mathrm{M}}^{\mu \nu }(x)\) is the energy-momentum tensor for the matter and \({T}_{\mathrm{EM}}^{\mu \nu }(x)\) is the energy-momentum tensor for the electromagnetic field. The terms in (1.4.2) can be rearranged into the rate of change of the 4-momentum density in the fluid, which arises from the kinetic energy contribution to \({T}_{\mathrm{M}}^{\mu \nu }(x)\), and thermal and electromagnetic forces that arise from the thermal contribution to \({T}_{\mathrm{M}}^{\mu \nu }(x)\) and from \({T}_{\mathrm{EM}}^{\mu \nu }(x)\), respectively.
1.4.1.2 Energy-Momentum Tensor for the Fluid
The energy-momentum tensor for the matter is identified as
where ηc 2 is the proper rest energy density, \(\mathcal{E} = U/V\), where V is the volume of the system, is the internal energy density and P is the pressure, and where the dependence on x is implicit. The combined first and second laws of thermodynamics imply the familiar relation \(dU = TdS - PdV\), where U is the internal energy, T is the temperature and S is the entropy. Regarding U(S, V ) as the state function, with independent variables S, V, implies \(T = {(\partial U/\partial S)}_{V }\), \(P = -{(\partial U/\partial V )}_{S}\). In the present case one has \(U = V \mathcal{E}\) and V ∝ 1 ∕ η. Making the physical assumption that all changes are adiabatic, that is, at constant entropy, the relation \(P = -{(\partial U/\partial V )}_{S}\) translates into
where constant entropy is implicit. The right hand side is the enthalpy, U + PV. Assuming an adiabatic law with an adiabatic index Γ one has
with \(\Gamma = 5/3\) for a monatomic nonrelativistic ideal gas, \(\Gamma = 4/3\) for a highly relativistic gas.
1.4.1.3 Electromagnetic Energy-Momentum Tensor
The electromagnetic energy-momentum tensor, which satisfies \({\partial }_{\mu }{T}_{\mathrm{EM}}^{\mu \nu } = {J}_{\mu }{F}^{\mu \nu }\), has the canonical form
The Maxwell tensor, \({F}^{\mu \nu }\), may be written in terms of the 4-vectors \({E}^{\mu } = {F}^{\mu \nu }{u}_{\nu }\), \({B}^{\mu } = {\mathcal{F}}^{\mu \nu }{u}_{\nu }\), defined for an arbitrary 4-velocity u, identified here as the fluid 4-velocity. One has
with Maxwell’s equations taking the form (1.1.4). The energy-momentum tensor (1.4.6) becomes
Combining (1.4.3) and (1.4.8), the total energy-momentum tensor for the system of fluid and electromagnetic field is
1.4.1.4 Lagrangian Density for Relativistic MHD
Relativistic MHD is amenable to a Lagrangian formulation [1, 11]. The action principle is
where η is the proper mass density and \(\mathcal{E}\) is the thermal energy density of the fluid. One may regard the form (1.4.10) as a postulate that defines relativistic MHD. The final term in (1.4.10) arises from the Lagrangian for the electromagnetic field, \({\mathcal{L}}_{\mathrm{EM}} = -{F}^{\alpha \beta }{F}_{\alpha \beta }/4{\mu }_{0}\), with \({F}^{\mu \nu } = {B}^{\mu }{u}^{\nu } - {B}^{\nu }{u}^{\mu }\) for E μ = 0.
The derivation of the equation of motion in the form (1.4.2) follows from the fact that the Lagrangian (1.4.10) may be regarded as a functional of η, \({u}^{\mu }\) and \({B}^{\mu }\): the dependence on x is implicit in this functional dependence. The energy-momentum tensor calculated from the Lagrangian (1.4.10) reproduces (1.4.9) for \({E}^{\mu } = 0\):
with \(\vert\text{ B}{\vert }^{2} = -{B}^{\sigma }{B}_{\sigma }\) and where \({h}^{\mu \nu }\) projects onto the 3-dimensional hypersurface orthogonal to the fluid 4-velocity. The equation of motion in the form (1.4.2) corresponds to the conservation law \({\partial }_{\mu }{T}^{\mu \nu } = 0\).
1.4.2 Derivation from Kinetic Theory
Fluid equations may be postulated or derived from kinetic theory. The latter approach is outlined here.
Consider a species α, with rest mass \({m}_{\alpha }\), charge \({q}_{\alpha }\), and with distribution function \({F}_{\alpha }(x,p)\). Fluid equations are obtained by considering moments of the distribution function. For simplicity in writing, the x dependences of all quantities are suppressed in the following equations.
The zeroth order moment defines the proper number density, \({n}_{\alpha \mathrm{pr}}\), for species α and the corresponding proper mass density is
The first moment defines the fluid 4-velocity, \({u}_{\alpha }^{\mu }\):
Each species satisfies a continuity equation of the form (1.2.1), specifically
The fluid is described by its proper mass density, η, and a fluid 4-velocity, \({u}^{\mu }\), given by
respectively. The continuity Eq. (1.4.1) for the fluid is then satisfied as a consequence of (1.4.14) with (1.4.15).
The second moment of the distribution defines the energy-momentum tensor for species α. This includes a contribution, \({(\eta )}_{\alpha }{u}_{\alpha }^{\mu }{u}_{\alpha }^{\nu }\), that corresponds to the rest mass energy in the rest frame of the fluid. It is convenient to separate the energy-momentum tensor into a part corresponding to its rest energy and a part due to internal motions in the fluid:
where ‘rm’ denotes rest mass and ‘th’ denotes thermal motions. One has
where \({\mathcal{E}}_{\alpha }\) is the internal energy density and \({P}_{\alpha }\) is the partial pressure for species α. Assuming that the only force is electromagnetic, the equation of fluid motion for species α is
with \({J}_{\alpha }^{\mu } = {q}_{\alpha }{n}_{\alpha \mathrm{pr}}{u}_{\alpha }^{\mu }\). On summing (1.4.18) over all species α, the net 4-force, \({F{}^{\nu }}_{\rho }{J}^{\rho }\), on the right hand side may be written as a 4-gradient and included in the left hand side. From Maxwell’s equations one has \({J}^{\rho } = {\partial }_{\mu }{F}^{\mu \rho }/{\mu }_{0}\), and hence this 4-force becomes \({F{}^{\nu }}_{\rho }{\partial }_{\mu }{F}^{\mu \rho }/{\mu }_{0}\), which may be rewritten as \(-{\partial }_{\mu }{T}_{\mathrm{EM}}^{\mu \nu }\), in terms of the energy-momentum tensor (1.4.6). The equation of motion then reduces to (1.4.2).
1.4.3 Generalized Ohm’s Law
A characteristic difference between MHD and kinetic theory is the appeal in MHD to some form of Ohm’s law to place a restriction on the electromagnetic field. Two examples of Ohm’s law are discussed briefly here: that for a nonrelativistic, collisional, electron-ion plasma, and that for a relativistic, collisionless pair plasma. A two-fluid model is assumed in both cases, with the fluids being electrons and ions, and electrons and positrons, respectively.
In a nonrelativistic electron-ion plasma, the ratio, m e ∕ m i , of the mass of the electron to the mass of the ion is a small parameter in which one can expand. The fluid velocity is equal to the velocity of the ions to lowest order in m e ∕ m i , and the current density is determined by the flow of the electrons relative to the ions. In the presence of collisions, there is a drag on the electrons that may be represented by a frictional force equal to − ν e times the momentum of the electrons, where ν e is the electron collision frequency. In an isotropic plasma, the effect of the collisions is described by a conductivity \({\sigma }_{0} = {\omega }_{p}^{2}/{\epsilon }_{0}{\nu }_{e}\). In a quasi-neutral plasma, when the charge density is assumed negligible, the static response may be written in the 4-tensor form
The limit of infinite conductivity corresponds to σ0 → ∞, ν e → 0, that is, to the collisionless limit. In this limit, for the current to remain finite one requires \({E}^{\mu } = 0\).
In the presence of a magnetic field the conductivity is anisotropic. The conductivity tensor may be obtained from the response tensor for a cold, magnetized electron gas by replacing the frequency, ω, in the rest frame by ω + iν e to take account of the collisions. This corresponds to identifying \({\sigma }^{\mu \nu }\) as \(i{\Pi }^{\mu \nu }/ku\) with \({\Pi }^{\mu \nu }\) given by the cold plasma form (1.3.5). In (1.3.5) one makes the replacement \(k\tilde{u} \rightarrow ku + i{\nu }_{e}\), and projects onto the 3-dimensional hyperplane orthogonal to \({u}^{\mu }\). This gives
with \({\tau }^{\mu \nu }(\omega )\) given by (1.2.19). In the static limit, ku → 0, one has
In the limit ν e → 0 only the component along \({b}^{\mu }{b}^{\nu }\) becomes infinite, and this requires the condition \(Eb = -{E}_{z} = 0\); the Hall term (\(\propto {f}^{\mu \nu }\)) remains finite, and the Pedersen terms ( ∝ g ⊥ μν) tends to zero in this limit.
1.4.4 Two-Fluid Model for a Pair Plasma
A two-fluid model for a relativistic electron-positron plasma with no thermal motions enables one to calculate \({E}^{\mu }\) and to discuss the assumption that it is zero in relativistic MHD. In this case there is no obvious small parameter, such as the mass ratio, that allows one to justify a simple approximation to Ohm’s law.
One may rearrange the fluid Eqs. (1.4.14) and (1.4.18) for \(\alpha = \pm \), \({m}_{\pm } = {m}_{e}\), \({\mathcal{E}}_{\pm } = 0 = {P}_{\pm }\), for electrons and positrons, respectively, into equations for the variables
The first two of the variables (1.4.22) are the proper number density and the proper charge density, respectively. The equations of continuity (1.4.14) for α = ± imply continuity equations \({\partial }_{\mu }(n{u}^{\mu }) = 0\) for mass and \({\partial }_{\mu }{J}^{\mu } = 0\) for charge. The equations of motion (1.4.18) for α = ± imply an equation of motion for the fluid of the form (1.4.2) and an equation for the current. The equation of motion for the fluid is
The term \(nm{u}^{\mu }{u}^{\nu }\) is the conventional energy-momentum tensor for a cold fluid, and conventional MHD is justified only if the additional term in (1.4.23) can be neglected.
The generalized Ohm’s law is identified by calculating \({E}^{\nu } = {F{}^{\nu }}_{\beta }{u}^{\beta }\) with u given by (1.4.22). Using the equations of motion (1.4.18) together with (1.4.22) implies
which gives the electric field in the rest frame of the fluid. The assumption \({E}^{\mu } = 0\) in conventional MHD applies to a pair plasma only if one can justify neglecting the right hand side of (1.4.24). If the assumption cannot be justified, relativistic MHD is not valid and should not be used. Relativistic MHD can break down for a variety of reasons [19], including that there are too few charges to carry the required current density.
It might be remarked that the ideal MHD assumption that the electric field is zero in the rest frame of the plasma is usually justified by arguing that in the limit of infinite conductivity a nonzero { E} would imply an infinite { J}, which is unphysical. However, in the collisionless limit, ν e → 0, only the parallel component of the conductivity becomes infinite in a magnetized plasma. The argument for the perpendicular component of { E} is different. A nonzero { E} ⊥ causes all particles to drift across the field lines with a velocity \(-\bf{ E} \times \bf{ B}/\vert \bf{ B}{\vert }^{2}\). The { E} ⊥ is removed by Lorentz transforming to the frame in which the plasma is stationary. The plasma is at rest in the frame { E} ⊥ = 0.
1.4.5 MHD Wave Modes
The properties of (small amplitude) waves in relativistic MHD are determined by a wave equation that may be derived from the Lagrangian. Suppose that there is an oscillating part of the fluid displacement, such that one has
where \({x}_{0}^{\mu }\) is the fluid displacement in the absence of the fluctuations. (The subscript 0 is omitted below after making the expansion.) The phase, or eikonal, Φ, satisfies \({k}_{\mu } = {\partial }_{\mu }\Phi \). The 4-velocity has a small perturbation, given by the derivative of (1.4.25) with respect to proper time. This gives \({u}^{\mu } \rightarrow {u}_{0}^{\mu } + \delta {u}^{\mu }\), with \(\delta {u}^{\mu } = -i\omega {\xi }^{\mu }\) to first order in the perturbation. The normalization u 2 = 1 must be preserved, and to lowest order, this requires ξu = 0. In the following it is assumed that \({\xi }^{\mu }\), like \({B}^{\mu }\), has no component along \({u}^{\mu }\).
On averaging the action (1.4.10) over the phase, only terms of even power in ξ remain. The first two terms implied by the phase average are [1]
respectively, with \({\tilde{A}}^{\mu } = {h}^{\mu \nu }{A}_{\nu }\) for any 4-vector \({A}^{\mu }\).
The wave equation for MHD waves follows from \(\partial {\mathcal{L}}^{(2)}/\partial {\xi }_{\mu }^{{_\ast}} = 0\) with \({\mathcal{L}}^{(2)}\) given by (1.4.27). The resulting equation is of the form
with \({B}^{\sigma }{B}_{\sigma } = -\vert {B}{\vert }^{2}\).
There are only three independent (orthogonal to u) components of \({\xi }^{\mu }\). A set of basis vectors that span the 3-dimensional space, orthogonal to \({u}^{\mu }\), consists of the direction of the magnetic field, b μ, the component of the wave 4-vector orthogonal to both the magnetic field and the fluid velocity, \({\kappa }_{\perp }^{\mu } \propto {k}^{\mu } - ku\,{u}^{\mu } + kb\,{b}^{\mu }\), and the direction orthogonal to these, \({a}^{\mu } = {\epsilon {}^{\mu }}_{\nu \rho \sigma }{u}^{\nu }{b}^{\rho }{\kappa }_{\perp }^{\sigma } = {f{}^{\mu }}_{\nu }{\kappa }_{\perp }^{\nu }\). The perpendicular and parallel components of k are introduced by writing \({k}_{\perp } = -k{\kappa }_{\perp }\), \({k}_{z} = -kb\), so that one has \({k}^{\mu } = ku\,{u}^{\mu } + {k}_{\perp }{\kappa }_{\perp }^{\mu } + {k}_{z}{b}^{\mu }\), and \({g}^{\mu \nu } = {u}^{\mu }{u}^{\nu } - {\kappa }_{\perp }^{\mu }{\kappa }_{\perp }^{\nu } - {a}^{\mu }{a}^{\nu } - {b}^{\mu }{b}^{\nu }\). Equation (1.4.28) may be written as three simultaneous equations for the components \({\xi }_{\perp } = -\xi {\kappa }_{\perp }\), \({\xi }_{a} = -\xi a\), \({\xi }_{b} = -\xi b\). The matrix form of these equations is
with the matrix components given by
with \(\Gamma = \partial (\ln P)/\partial (\ln \eta )\) for adiabatic changes. The resulting dispersion equation for relativistic MHD is
The dispersion Eq. (1.4.32) factorizes into D = 0, which gives the dispersion relation for the Alfvén mode, and \(AC - {B}^{2} = 0\), which has two solutions corresponding to the fast and slow magnetoacoustic modes.
1.4.5.1 Alfvén and Sound Speeds
It is convenient to define the Alfvén speed and the (adiabatic) sound speed by
The dispersion relation for the Alfvén (A) mode becomes
where \({v}_{\phi } = \omega /\vert {k}\vert \) is the phase speed. The fluid displacement in Alfvén waves is along \({a}^{\mu }\). The dispersion equation for the fast and slow modes is
Solving for the phase speed, the dispersion relations for the two modes are of the form \({v}_{\phi }^{2} = {v}_{\pm }^{2}\), with
The solution for the fluid displacement in the two modes is
For either a very low density or a very strong magnetic field, satisfying \(\vert {B}{\vert }^{2}/{\mu }_{0} \gg \eta {c}^{2} + \mathcal{E} + P\), the conventional Alfvén speed exceeds the speed of light, v A ≫ c, and the MHD speed becomes \({v}_{A}/{(1 + {v}_{A}^{2}/{c}^{2})}^{1/2}\). At sufficiently high (relativistic) temperature, the adiabatic index is \(\Gamma = 4/3\), the pressure satisfies \(P = \mathcal{E}/3 \gg \eta {c}^{2}\), implying that the sound speed approaches the limit \({c}_{\mathrm{s}} \rightarrow c/\sqrt{3}\).
1.5 Quantum Fluid Theory
A fluid approach to quantum plasmas, referred to here as quantum fluid theory (QFT), has generated an extensive literature since about 2000 [2, 12, 18]. QFT may be derived from moments of a kinetic equation, with the Vlasov equation replaced by an appropriate quantum counterpart. Although a derivation of fluid equations from the Dirac equation had been developed in the 1950s by Takabayasi [28], the later development of QFT started from simpler assumptions. In its simplest form (nonrelativistic, spinless, unmagnetized and longitudinal) the approach is to take moments of the Wigner-Moyal equations that describe such a quantum system. The subsequent extension of QFT involved generalizing it to include other effects in a piecemeal fashion.
1.5.1 Early QFT Theories
It was pointed out by Bohm [6], that a fluid-like description was implicit in alternative interpretations of Schrödinger’s theory, discussed by de Broglie and Madelung in 1926. Madelung [17] wrote the wavefunction in the form
where A and S are real functions. The Madelung equations are a continuity equation for \(n({x},t)\), which may be interpreted as a probability density for the electrons, and a Hamiltonian-Jacobi-like equation for \(S({x},t)\), which may be reinterpreted as an equation of motion for the fluid momentum \({p}({x},t)\). (Madelung’s equations are sometimes regarded as equivalent to Schrödinger’s equation, but this seems not to be the case [31].) In QFT, Madelung’s equations become the equation of continuity
with \({v} = {p}/m\), and the equation of motion
with \(A({x},t) = {[n({x},t)]}^{1/2}\). The final term in (1.5.3), referred to as the Bohm term in QFT, is an intrinsically quantum mechanical term that describes the effect of quantum mechanical diffusion and tunneling.
The QFT Eqs. (1.5.2) and (1.5.3) can be derived from moments of kinetic equations that include quantum effects. A quantum counterpart of the classical distribution function is the Wigner function, which satisfies a kinetic equation similar to the Boltzmann equation, re-interpreted as the Vlasov equation in plasma kinetic theory. The Wigner function is defined in terms of the outer product of the Schrödinger wavefunction and its complex conjugate, and it includes neither spin nor relativistic effects. In the generalization to Dirac’s theory, the outer product of the wave function and its adjoint is a 4 ×4 Dirac matrix, referred to as the Wigner matrix in § 8.4.2 of volume 1. This generalization leads to substantial increase in algebraic complexity. Existing versions of QFT that include spin and/or relativistic effects are based on various simplifying approximations.
1.5.1.1 Wigner-Moyal Equations
The Wigner function is defined in § 8.4 of volume 1. Let \(\psi ({x},t)\) be the one-dimensional wavefunction satisfying the one-dimensional Schrödinger equation. The Wigner function is defined by (ordinary units)
The notation used for the Wigner function, \(f({p},{x},t)\) in (1.5.4), is the same as for the classical distribution function, but the interpretation is different. One cannot interpret the Wigner function as a probability distribution, as is the case for its classical counterpart, in particular because it can be negative. The wavefunction and its adjoint in (1.5.4) satisfy the Schrödinger equation and its adjoint, respectively. For the case of an electron in a longitudinal field, described by an electrostatic potential \(\Phi ({x},t)\), the wavefunction satisfies (ordinary units)
The Wigner function satisfies (ordinary units)
with Poisson’s equation identified as
where n 0 is a constant positive background charge. Equations (1.5.6) and (1.5.7) are sometimes referred to a the Wigner-Moyal equations. The QFT equations may be derived by taking moments of (1.5.6).
A simple example of the implications of Eqs. (1.5.6) and (1.5.7) follows by linearizing and Fourier transforming them, and then solving for the dispersion relation for longitudinal waves. For any isotropic distribution this gives (ordinary units)
with ⟨v 2⟩ = 3V e 2 for a Maxwellian distribution and \(\langle {v}^{2}\rangle = 3{v}_{F}^{2}/5\) for a completely degenerate electron distribution. The final term in (1.5.8) arises from the Bohm term. This term has an obvious interpretation in terms of the quantum recoil. This implies a relation between quantum mechanical diffusion in coordinate space and the quantum recoil in momentum space [20].
1.5.2 Generalizations of QFT
The form of QFT outlined above applies only to nonrelativistic plasmas and longitudinal fields. As already mentioned, the generalization to include relativistic effects is discussed briefly in § 8.4 of volume 1, with the Wigner function generalized to a Wigner matrix whose evolution is determined by Dirac’s equation in place of Schrödinger’s equation. The derivation of quantum fluid equations from relativistic quantum theory [28] has become of renewed interest more recently [7, 9, 25].
1.5.3 Quasi-classical Models for Spin
The effect of the spin of the electron in a magnetic field has been included in QFT using several different quasi-classical approaches. In the simplest approach, the magnetic moment of the electron is identified as \({m} = g{\mu }_{B}{s}\), with (SI units)
where μ B is the Bohr magneton, and the gyromagnetic ratio, g, differs from 2 due to radiative corrections in QED. For an electron at rest, a classical form for the equation of motion of the spin is
A quasi-classical way of including spin dependence in a kinetic theory [8, 23] is based on an earlier theory in a different context [10, 14]: generalize phase space from the 6-dimensional { x}–{ p} space to a 9-dimensional { x}–{ p}–{ s} space (the restriction | s | = 1 formally reduces the dimensionality to 8), and introduce a distribution function, f({ x}, { p}, { s}), in this space. The Vlasov equation is then generalized to include the evolution of the spin, described by (1.5.10) in the simplest approximation.
Generalization of QFT to include the magnetic field leads to quantum MHD theory. The generalization of (1.5.3) involves including the Lorentz force and the force that results from the gradient of the energy associated with the magnetic moment. This gives
where arguments \(({x},t)\) are omitted. The inclusion of the spin in (1.5.11) is not rigorously justified. The use of fluid theory to describe a (classical or quantum) plasma imposes an intrinsic limitation that cannot be avoided in the magnetized case: the spiraling motion of particles cannot be taken into account. Magnetized fluid theory is reproduced by kinetic theory only when the gyroradii of the particles are assumed negligibly small. The gyroradius is a classical concept, and the quantum counterpart of the small gyroradius limit has not been identified in the context of QFT. (The relevant limit is referred to as the small-x approximation in § 9.4.2.)
1.5.3.1 Covariant Model for Spin: BMT Equation
A covariant generalization of the equation of motion (1.5.10) for the spin leads to the Bargmann-Michel-Telegdi (BMT) equation [5]. The spin vector, \({s}\), is interpreted as the space components of a 4-vector in the frame in which the electron is at rest. Writing \({s}^{\mu } = [{s}^{0},{s}]\) in an arbitrary frame, one assumes s 0 = 0 in the rest frame, and then su = 0 in the rest frame implies \(\gamma ({s}^{0} -{s} \cdot {v}) = 0\), and hence \({s}^{0} = {s} \cdot {v}\) in an arbitrary frame. For an accelerated particle, in its instantaneous rest frame, one has \(d{s}^{0}/dt = {s} \cdot d{v}/dt\), and together with (1.5.10) this determines \(d{s}^{\mu }/dt\) in the instantaneous rest frame. It is straightforward to rewrite the resulting equation in a covariant form. Assuming an equation of motion that omits the final two terms in (1.5.11), this becomes the BMT equation
with \(d\tau = dt/\gamma \), where τ is the proper time. In this model, the spin does not affect the dynamics, in the sense that there is no term corresponding to the force associated with the gradient of the magnetic energy, \(-\frac{1} {2}g{\mu }_{B}{s} \cdot {B}\).
An alternative covariant form of the magnetic moment is in terms of the second rank 4-tensor
In the rest frame, \({u}^{\beta } = [1,{\bf 0}]\), in the case where the spin is along the direction of \({B}\), assumed to be the 3-axis, one has \({m}^{12} = -{m}^{21} = \frac{1} {2}g{\mu }_{B}s\). Equations (1.5.12) and (1.5.13) imply, \(d{m}^{\mu \nu }/d\tau = 0\), and hence that the magnetic moment in this sense is conserved. This conservation law also applies when the radiative correction g − 2≠0 is included.
1.5.4 Spin-Dependent Cold Plasma Response
The classical covariant form of the response of a cold electron gas is calculated in § 1.2, and use of (1.5.12) and (1.5.13) facilitates generalizing that calculation of the response tensor to include the contribution due to the magnetic moments of the electrons in a magnetized electron gas. For simplicity the radiative correction is neglected by setting g = 2.
The 4-magnetization of the electron gas is \({M}^{\mu \nu } = {n}_{e}{m}^{\mu \nu }\). The assumption that the electron gas is magnetized implies that there is a nonzero mean spin, denoted \({\bar{s}}^{\mu }\). Let the average magnetization be \({M}^{\mu \nu } = {n}_{e}{\bar{m}}^{\mu \nu }\), with \({\bar{m}}^{\mu \nu } = {\mu }_{B}{\epsilon }^{\mu \nu \alpha \beta }{\bar{s}}_{\alpha }{\bar{u}}_{\beta }\), where an overbar denotes an average value. In the rest frame of the cold electron gas, one has \({\bar{u}}^{\mu } = [1,{\bf 0}]\), \({\bar{s}}^{\mu } = [0,{s}]\), implying a 3-magnetization \({M} = {\mu }_{B}{n}_{e}\bar{{s}}\).
The linear response tensor associated with the perturbation of the magnetic moments of the electrons can be evaluated in terms of the linear perturbation, \({M}^{(1)\mu \nu }(k)\), in the magnetization. The associated 4-current is \({J}^{(1)\mu }(k) = -i{k}_{\nu }{M}^{(1)\mu \nu }(k)\), and writing this in the form
defines the relevant contribution \({\Pi }_{\mathrm{m}}^{\mu \nu }(k)\) to the response tensor. In the model used here, the spin does not affect the dynamics, and hence there is assumed to be no perturbation in n e . The linear perturbation in the magnetization is
The perturbation in the 4-velocity is given by (1.2.15), and for electrons this becomes
with, from (1.2.19) for electrons,
The analogous perturbation in the spin 4-vector follows from (1.5.12), with g = 2 here:
Explicit evaluation gives
with τμν given by (1.5.17).
The covariant form (1.5.19) applies to a collection of electrons at rest in the frame moving with 4-velocity \({\bar{u}}^{\mu }\). As in the case of the cold plasma response, one can reinterpret (1.5.19) in a way that allows one to include an arbitrary distribution of particles in parallel velocity β. One replaces \(\bar{u}\) by u, with \({u}^{\mu } = \gamma [1,0,0,\beta ]\), \(\gamma = 1/{(1 - {\beta }^{2})}^{1/2}\), and replaces n e by the differential proper number density, \(d\beta \,{g}^{\epsilon }(\beta )/\gamma \), where \({g}^{\epsilon }(\beta )\) is the distribution function for electrons, ε = 1, or positrons, \(\epsilon = -1\). After integrating over β, this generalization of (1.5.19) gives the magnetic moment contribution to the response tensor for the distribution of electrons plus positrons.
This model does not include the spiraling motion of the electrons, and the response tensor (1.5.19) is valid only in the small-gyroradius limit.
1.5.4.1 Spin-Dependent Response in the Rest Frame
The spin-dependent contribution (1.5.19) to the response tensor simplifies considerably in the rest frame of the (cold) electron gas, when one has \({\bar{u}}^{\mu } = [1,{\bf 0}]\), \({\bar{s}}^{\mu } = [0,\bar{s}{b}]\), where \({b} = (0,0,1)\) is a unit vector along the magnetic field. One then has \(k\bar{u} = \omega \), \(k\bar{s} = -{k}_{z}\bar{s}\), and the spin-dependent contribution to the response tensor is proportional to the magnetization \(M = {\mu }_{B}{n}_{e}\bar{s}\).
For cold electrons in their rest frame, (1.5.20) reduces to [21]
The spin-dependent contribution to the dielectric tensor is (ordinary units)
where the frequency associated with the magnetization is
The ratio \({\Omega }_{\mathrm{m}}/{\omega }_{p}\) is small except in dense, strongly magnetized plasmas, where the plasmon energy, \(\hslash {\omega }_{p}\), is a significant fraction of the rest energy, m e c 2, and \(\bar{s}\) is of order unity.
The derivation of the response tensor (1.5.20) involves two different assumptions: the BMT equation for the spin evolution and the cold-plasma approximation. An analogous result derived using a nonrelativistic theory [22] differs from (1.5.20) in that \({\omega }^{2}/{c}^{2} - {k}_{z}^{2}\) is replaced by − k z 2. A similar result has been derived for a special case using kinetic theory [23]. The validity of the quasi-classical approach for including the spin in the dispersion is questioned in § 9.6, where it is argued that a rigorous theory does not reproduce the result (1.5.20) in any obvious way.
References
A. Achterberg, Phys. Rev. A 28, 2449 (1983)
D. Anderson, B. Hall, M. Lisak, M. Marklund, Phys. Rev. E 65, 046417 (2002)
A.M. Anile, Relativistic Fluids and Magneto-Fluids: With Applications in Astrophysics and Plasma Physics (Cambridge University Press, Cambridge, 1989)
E.V. Appleton, J. Inst. Electr. Eng. 71, 642 (1932)
V. Bargmann, L. Michel, V.L. Telegdi, Phys. Rev. Lett. 2, 435 (1959)
D. Bohm, Phys. Rev. 85, 166 (1952)
G. Brodin, M. Marklund, New J. Phys. 9, 277 (2008)
G. Brodin, M. Marklund, G. Manfredi, Phys. Rev. Lett. 100, 175001 (2008)
G. Brodin, M. Marklund, J. Zamanian, A. Ericsson, P.L. Mana, Phys. Rev. Lett. 101, 245002 (2008)
S.C. Cowley, R.M. Kulsrud, E. Valeo, Phys. Fluid 29, 430 (1986)
R.L. Dewar, Aust. J. Phys. 30, 533 (1977)
F. Haas, G. Manfredi, M.R. Feix, Phys. Rev. E 62, 2763 (2000)
D.R. Hartree, Proc. Camb. Philos. Soc. 27, 143 (1931)
R.M. Kulsrud, E.J. Valeo, S.C. Cowley, Nucl. Fusion 26, 1443 (1986)
A. Levinson, D. Melrose, A. Judge, Q. Luo, Astrophys. J. 31, 456 (2005)
A. Lichnerowicz, Relativistic Magnetohydrodynamics (Benjamin, New York, 1967)
E. Madelung, Z. Phys. 40, 32 (1926)
M. Marklund, P.K. Shukla, Rev. Mod. Phys. 78, 591 (2006)
A. Melatos, D.B. Melrose, Mon. Not. R. Astron. Soc. 279, 1168 (1996)
D.B. Melrose, A. Mushtaq, Phys. Plasmas 16, 094508 (2009)
D.B. Melrose, A. Mushtaq, Phys. Rev. E 83, 056404 (2011)
A.P. Misra, G. Brodin, M. Marklund, P.K. Shukla, J. Plasma Phys. 76, 875 (2010)
V.N. Oraevsky, V.B. Semikoz, Phys. At. Nucl. 66, 466 (2003)
A.E. Shabad, Ann. Phys. 90, 166 (1975)
P.K. Shukla, B. Eliasson, Physica (Utrecht) 53, 51 (2010)
T.H. Stix, The Theory of Plasma Waves (McGraw-Hill, New York, 1962)
T.H. Stix, Waves in Plasmas (Springer, New York, 1992)
T. Takabayasi, Prog. Theor. Phys. Suppl. 4, 1 (1957)
T. Uchida, Phys. Rev. E 56, 2181 (1997)
M.W. Verdon, D.B. Melrose, Phys. Rev. E 77, 046403 (2008)
T.C. Wallstrom, Phys. Rev. A 49, 1613 (1994)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Melrose, D. (2013). Covariant Fluid Models for Magnetized Plasmas. In: Quantum Plasmadynamics. Lecture Notes in Physics, vol 854. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4045-1_1
Download citation
DOI: https://doi.org/10.1007/978-1-4614-4045-1_1
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-4044-4
Online ISBN: 978-1-4614-4045-1
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)