1 Introduction
The nonlinear dynamics of relativistic plasmas is presently under extensive theoretical and experimental investigation, both in the context of laboratory plasmas such as laser-produced plasmas and in the context of high-energy astrophysics. While the description of the relativistic plasma dynamics would require a fully kinetic treatment involving the relativistic Vlasov equation coupled to Maxwell’s equation, on the large spatial and temporal scales that are of interest for astrophysical plasmas, fluid-type approximations can be usefully adopted. This is in particular the case in physically complex settings such as high-energy plasmas in curved space–time, see e.g. in the recent article of Del Zanna et al. (Reference Del Zanna, Pili, Olmi, Bucciantini and Amato2018) for the case of magnetized neutron stars and pulsar winds. Relativistic fluid descriptions have also been used in a simplified modelling of magnetic reconnection in high-energy plasmas, see e.g. Zenitani, Hesse & Klimas (Reference Zenitani, Hesse and Klimas2010).
A common feature of these descriptions, in the limit where dissipation and microscopic effects are disregarded, is the occurrence of topological invariants, such as the conservation of the magnetic flux through a surface comoving with the plasma (Alfvén theorem) in ideal magnetohydrodynamics, that restrict the plasma dynamics. In fact the process of magnetic reconnection mentioned above arises from the local violation of these constraints due to the local violation of the ideal Ohm’s law, that is of the condition
$\boldsymbol{E}+(\boldsymbol{v}/c)\times \boldsymbol{B}=0$
, where
$\boldsymbol{E}$
and
$\boldsymbol{B}$
are the electric and magnetic field and
$\boldsymbol{v}$
is the fluid plasma velocity and
$c$
is the velocity of light.
In general, even for relativistic plasmas, these topological constraints are formulated in a form that is not explicitly invariant under Lorentz transformations even if the ideal plasma condition
$\boldsymbol{E}+(\boldsymbol{v}/c)\times \boldsymbol{B}=0$
is fully relativistic and covariant, as can be seen explicitly by rewriting it in the four-dimensional form
$\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}\boldsymbol{u}^{\unicode[STIX]{x1D708}}=0$
where
$\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}$
is the electromagnetic field tensor and
$\boldsymbol{u}^{\unicode[STIX]{x1D708}}$
is the plasma fluid 4-velocity.
In order to bypass this limitation in two previous papers (Pegoraro Reference Pegoraro2012, Reference Pegoraro2016) the concept of covariant magnetic connections between fluid elements has been introduced. A basic ingredient in the definition of invariant connections is a procedure of time resetting along the fluid trajectories that is compatible with the ideal Ohm’s law, and that is required in order to restore the simultaneity between two fluid elements at different spatial locations as simultaneity is not preserved by a Lorentz transformation.
Following a different angle of approach, but under the same ideal plasma assumption, Hide defined (Hide Reference Hide1983, Reference Hide1996), by analogy with the potential vorticity introduced by Ertel (Reference Ertel1942) for fluids, a potential magnetic field that is advected by the plasma velocity field.
In the present article we provide a relativistic definition of the potential magnetic field that is explicitly Lorentz invariant. This generalization turns out to be rather convenient as it allows us on the one hand to recover the well-known Cauchy conditionsFootnote 1 for the magnetic field components in the mapping between Eulerian and Lagrangian variables (see Newcomb Reference Newcomb1962) and on the other to prove a Lorentz invariant form of the Alfvén theorem that, as in the case of the covariant connections in Pegoraro (Reference Pegoraro2012, Reference Pegoraro2016), requires a time resetting procedure in order to restore simultaneity among the different points of the comoving surface through which the magnetic flux is computed.
This article is organized as follows. In § 2, following closely Pegoraro (Reference Pegoraro2016), we recall for the sake of self-containedness the basic features of the Lichnerowicz–Anile (LA) representation (Lichnerowicz Reference Lichnerowicz1967; Anile Reference Anile1989) of the electromagnetic field tensor
$\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}$
, the definition of the magnetic 4-vector
$\boldsymbol{b}^{\unicode[STIX]{x1D707}}$
and of the electric 4-vector
$\boldsymbol{e}_{\unicode[STIX]{x1D707}}$
, the ideal plasma limit where the electric 4-vector
$\boldsymbol{e}_{\unicode[STIX]{x1D707}}$
vanishes and the expression for the divergentless dual electromagnetic tensor
$\unicode[STIX]{x1D642}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}$
in this ideal limit in terms of the 4-vectors
$\boldsymbol{b}^{\unicode[STIX]{x1D707}}$
and
$\boldsymbol{e}^{\unicode[STIX]{x1D707}}$
. Then we recall the gauge freedom (see also D’Avignon, Morrison & Pegoraro Reference D’Avignon, Morrison and Pegoraro2015) in the definition of the magnetic 4-vector. This gauge freedom plays a very important role in the derivation of the potential magnetic field and of the Alfvén theorem as it allows us to move easily from a Lorentz invariant four-dimensional formulation to a
$3+1$
-dimensional formulation in a chosen frame and back, whichever formulation is more convenient through the different steps of the proofs. In § 3 we derive the explicit relativistic invariant expression of the potential magnetic field. In § 4, using the gauge freedom mentioned above, we show that the relativistic expression of the potential magnetic field reduces to Hide’s definition when expressed in a
$3+1$
-dimensional form. In § 5, by introducing appropriately conserved ‘charges’ that are closely related to the potential magnetic field defined in § 3 and by using the gauge freedom explicitly, we recover in a Lorentz invariant form the Alfvén theorem for the conservation of the magnetic flux through a surface comoving with the plasma. Finally in § 6 the main results of this article are summarized and possible extensions are indicated.
2 Electric and magnetic 4-vectors
Following Pegoraro (Reference Pegoraro2015, Reference Pegoraro2016) we adopt the so called Lichnerowicz–Anile (LA) representation (Lichnerowicz Reference Lichnerowicz1967; Anile Reference Anile1989) (see also D’Avignon et al.
Reference D’Avignon, Morrison and Pegoraro2015) of the relativistic field tensor
$\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}$
$$\begin{eqnarray}\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}=\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}\unicode[STIX]{x1D706}\unicode[STIX]{x1D70E}}\boldsymbol{b}^{\unicode[STIX]{x1D706}}\boldsymbol{u}^{\unicode[STIX]{x1D70E}}+[\boldsymbol{u}_{\unicode[STIX]{x1D707}}\boldsymbol{e}_{\unicode[STIX]{x1D708}}-\boldsymbol{u}_{\unicode[STIX]{x1D708}}\boldsymbol{e}_{\unicode[STIX]{x1D707}}],\end{eqnarray}$$
where
$\boldsymbol{b}^{\unicode[STIX]{x1D707}}$
is the 4-vector magnetic field and
$\boldsymbol{e}_{\unicode[STIX]{x1D707}}$
is the 4-vector electric field, with
$\boldsymbol{u}^{\unicode[STIX]{x1D707}}\boldsymbol{e}_{\unicode[STIX]{x1D707}}=0$
and
$\boldsymbol{u}_{\unicode[STIX]{x1D707}}\boldsymbol{b}^{\unicode[STIX]{x1D707}}=0$
. The 4-vectors
$\boldsymbol{e}_{\unicode[STIX]{x1D707}}$
and
$\boldsymbol{b}^{\unicode[STIX]{x1D707}}$
are related to the standard electric and magnetic fields
$\boldsymbol{E}$
and
$\boldsymbol{B}$
in three-dimensional (3-D) space by
$$\begin{eqnarray}\boldsymbol{b}^{\unicode[STIX]{x1D707}}=\unicode[STIX]{x1D6FE}(\boldsymbol{B}+\boldsymbol{E}\times \boldsymbol{v},\boldsymbol{B}\boldsymbol{\cdot }\boldsymbol{v}),\end{eqnarray}$$
and
$$\begin{eqnarray}\boldsymbol{e}_{\unicode[STIX]{x1D707}}=\unicode[STIX]{x1D6FE}(\boldsymbol{E}+\boldsymbol{v}\times \boldsymbol{B},-\boldsymbol{E}\boldsymbol{\cdot }\boldsymbol{v}),\end{eqnarray}$$
with
$\boldsymbol{e}_{\unicode[STIX]{x1D707}}\boldsymbol{b}^{\unicode[STIX]{x1D707}}=\boldsymbol{E}\boldsymbol{\cdot }\boldsymbol{B}$
. We have adopted the Minkowski metric tensor
$\unicode[STIX]{x1D702}_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}$
defined by
$(+,+,+,-)$
and normalized 3-D velocities
$\boldsymbol{v}$
to the speed of light:
$\unicode[STIX]{x1D6FE}$
is the relativistic Lorentz factor and we have used
$\boldsymbol{u}^{\unicode[STIX]{x1D707}}=\unicode[STIX]{x1D6FE}(\boldsymbol{v},1)$
and
$\boldsymbol{u}_{\unicode[STIX]{x1D707}}\boldsymbol{u}^{\unicode[STIX]{x1D707}}=-1$
. The orthogonality conditions
$\boldsymbol{u}^{\unicode[STIX]{x1D707}}\boldsymbol{e}_{\unicode[STIX]{x1D707}}=\boldsymbol{u}_{\unicode[STIX]{x1D707}}\boldsymbol{b}^{\unicode[STIX]{x1D707}}=0$
make the LA representation unique.
The LA representation allows us to separate covariantly the magnetic and the electric parts of the e.m. field tensor relative to a given plasma element moving with 4-velocity
$\boldsymbol{u}^{\unicode[STIX]{x1D707}}$
. In the local rest frame of this plasma element the time components of
$\boldsymbol{e}_{\unicode[STIX]{x1D707}}$
and of
$\boldsymbol{b}^{\unicode[STIX]{x1D707}}$
vanish, while their space components reduce to the standard 3-D electric and magnetic fields.
A corresponding representation holds for the dual tensor
$\unicode[STIX]{x1D642}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}\equiv \unicode[STIX]{x1D700}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}/2$
with
$\boldsymbol{e}_{\unicode[STIX]{x1D707}}$
and
$\boldsymbol{b}^{\unicode[STIX]{x1D707}}$
interchanged. Thus:
$$\begin{eqnarray}\unicode[STIX]{x1D642}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}=\unicode[STIX]{x1D700}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}\unicode[STIX]{x1D706}\unicode[STIX]{x1D70E}}\boldsymbol{u}_{\unicode[STIX]{x1D706}}\boldsymbol{e}_{\unicode[STIX]{x1D70E}}+[\boldsymbol{u}^{\unicode[STIX]{x1D707}}\boldsymbol{b}^{\unicode[STIX]{x1D708}}-\boldsymbol{u}^{\unicode[STIX]{x1D708}}\boldsymbol{b}^{\unicode[STIX]{x1D707}}],\quad \text{with }\boldsymbol{e}_{\unicode[STIX]{x1D707}}=\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}\boldsymbol{u}^{\unicode[STIX]{x1D708}}\;\text{and}\;\boldsymbol{b}^{\unicode[STIX]{x1D707}}=\unicode[STIX]{x1D642}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}\boldsymbol{u}_{\unicode[STIX]{x1D708}}.\end{eqnarray}$$
If the ideal Ohm’s law
$\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}\boldsymbol{u}^{\unicode[STIX]{x1D708}}=0$
holds, the electric 4-vector
$\boldsymbol{e}_{\unicode[STIX]{x1D707}}$
vanishes, the tensors
$\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}$
and
$\unicode[STIX]{x1D642}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}$
have rank two and can be written as
$$\begin{eqnarray}\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}=\unicode[STIX]{x1D700}_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}\unicode[STIX]{x1D706}\unicode[STIX]{x1D70E}}\boldsymbol{b}^{\unicode[STIX]{x1D706}}\boldsymbol{u}^{\unicode[STIX]{x1D70E}},\quad \unicode[STIX]{x1D642}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}=[\boldsymbol{u}^{\unicode[STIX]{x1D707}}\boldsymbol{b}^{\unicode[STIX]{x1D708}}-\boldsymbol{u}^{\unicode[STIX]{x1D708}}\boldsymbol{b}^{\unicode[STIX]{x1D707}}],\end{eqnarray}$$
with
$$\begin{eqnarray}\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}\boldsymbol{b}^{\unicode[STIX]{x1D708}}=\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}\boldsymbol{u}^{\unicode[STIX]{x1D708}}=0,\end{eqnarray}$$
$$\begin{eqnarray}\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}\unicode[STIX]{x1D642}^{\unicode[STIX]{x1D708}\unicode[STIX]{x1D707}}=0\rightarrow \boldsymbol{E}\boldsymbol{\cdot }\boldsymbol{B}=0,\quad \text{and}\quad \boldsymbol{b}_{\unicode[STIX]{x1D707}}\boldsymbol{b}^{\unicode[STIX]{x1D707}}=\unicode[STIX]{x1D642}_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}\unicode[STIX]{x1D642}^{\unicode[STIX]{x1D708}\unicode[STIX]{x1D707}}/2=\unicode[STIX]{x1D641}_{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}\unicode[STIX]{x1D641}^{\unicode[STIX]{x1D708}\unicode[STIX]{x1D707}}/2.\end{eqnarray}$$
In this case we can use
$\boldsymbol{e}_{\unicode[STIX]{x1D707}}=0$
in order to express
$\boldsymbol{b}^{\unicode[STIX]{x1D707}}$
in terms of
$\boldsymbol{B}$
and
$\boldsymbol{v}$
only as
$$\begin{eqnarray}\boldsymbol{b}^{\unicode[STIX]{x1D707}}=\unicode[STIX]{x1D6FE}(\boldsymbol{B}/\unicode[STIX]{x1D6FE}^{2}+\boldsymbol{v}(\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{B}),\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{B}).\end{eqnarray}$$
Note that in general
$\unicode[STIX]{x2202}_{\unicode[STIX]{x1D707}}\boldsymbol{b}^{\unicode[STIX]{x1D707}}\not =0$
while from Maxwell’s equations we have
$$\begin{eqnarray}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D707}}\unicode[STIX]{x1D642}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}=0.\end{eqnarray}$$
2.1 Gauge freedom
As shown in detail in Pegoraro (Reference Pegoraro2016), D’Avignon et al. (Reference D’Avignon, Morrison and Pegoraro2015) a gauge freedom is allowed in the definition of the magnetic 4-vector field
$\boldsymbol{b}^{\unicode[STIX]{x1D707}}$
in the LA representation provided we relax the orthogonality condition
$\boldsymbol{b}^{\unicode[STIX]{x1D707}}\boldsymbol{u}_{\unicode[STIX]{x1D707}}=0$
:
$$\begin{eqnarray}\boldsymbol{b}^{\unicode[STIX]{x1D707}}\rightarrow \boldsymbol{h}^{\unicode[STIX]{x1D707}}\equiv \boldsymbol{b}^{\unicode[STIX]{x1D707}}+g\boldsymbol{u}^{\unicode[STIX]{x1D707}},\end{eqnarray}$$
where
$g$
is a free scalar field and the velocity 4-vector
$\boldsymbol{u}^{\unicode[STIX]{x1D707}}$
satisfies the continuity equation
$$\begin{eqnarray}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D707}}(N\boldsymbol{u}^{\unicode[STIX]{x1D707}})=0,\end{eqnarray}$$
with
$N$
the proper density of the plasma element and
$N\boldsymbol{u}^{\unicode[STIX]{x1D707}}$
the density 4-vector. Different choices of the gauge field
$g$
allow us to impose specific conditions on
$\boldsymbol{h}^{\unicode[STIX]{x1D707}}$
. If we take in a given frameFootnote
2
the gauge
$$\begin{eqnarray}g=-\boldsymbol{v}\boldsymbol{\cdot }\boldsymbol{B},\end{eqnarray}$$
we can make the time component of
$\boldsymbol{h}^{\unicode[STIX]{x1D707}}$
vanish and
$\boldsymbol{h}||\boldsymbol{B}$
in that frame i.e.
$$\begin{eqnarray}\boldsymbol{h}^{\unicode[STIX]{x1D707}}=(\boldsymbol{B}/\unicode[STIX]{x1D6FE},0).\end{eqnarray}$$
Note that the expression for
$\unicode[STIX]{x1D642}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}$
in (2.5) is unchanged if we insert
$\boldsymbol{h}^{\unicode[STIX]{x1D707}}$
for
$\boldsymbol{b}^{\unicode[STIX]{x1D707}}$
in (2.5).
3 Advected relativistic ‘potential magnetic field’
Let
$S$
be a scalar function in Minkowski space–time, then
$$\begin{eqnarray}(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D708}}S)\unicode[STIX]{x2202}_{\unicode[STIX]{x1D707}}\unicode[STIX]{x1D642}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}=\unicode[STIX]{x2202}_{\unicode[STIX]{x1D707}}[(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D708}}S)\unicode[STIX]{x1D642}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}]-(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D707}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D708}}S)\unicode[STIX]{x1D642}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}=\unicode[STIX]{x2202}_{\unicode[STIX]{x1D707}}[(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D708}}S)\unicode[STIX]{x1D642}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}]=0.\end{eqnarray}$$
This is a generalization of the solenoidal property of the magnetic field (for
$S=t$
, (3.1) reduces to
$\unicode[STIX]{x1D6FB}\boldsymbol{\cdot }\boldsymbol{B}=0$
) and includes the induction equation whose components are recovered for
$S=x,y,z$
, respectively.
In the ideal Ohm’s law case from (2.5) we obtain
$$\begin{eqnarray}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D707}}[(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D708}}S)\unicode[STIX]{x1D642}^{\unicode[STIX]{x1D707}\unicode[STIX]{x1D708}}]=-\unicode[STIX]{x2202}_{\unicode[STIX]{x1D707}}[\boldsymbol{b}^{\unicode[STIX]{x1D707}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}S-\boldsymbol{u}^{\unicode[STIX]{x1D707}}\boldsymbol{b}^{\unicode[STIX]{x1D708}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D708}}S]=0.\end{eqnarray}$$
We find it convenient to choose
$S$
such that it is advected by the 4-velocity field
$u^{\unicode[STIX]{x1D707}}$
(e.g. any function of the initial spatial positions
$\boldsymbol{a}$
at
$t=0$
). Then
$$\begin{eqnarray}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}S=\boldsymbol{u}^{\unicode[STIX]{x1D707}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D707}}S=0,\end{eqnarray}$$
and
$$\begin{eqnarray}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D707}}[\boldsymbol{u}^{\unicode[STIX]{x1D707}}\boldsymbol{b}^{\unicode[STIX]{x1D708}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D708}}S]=0.\end{eqnarray}$$
Using (2.11), from (3.4) we obtain
$$\begin{eqnarray}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D707}}[(N/N)\boldsymbol{u}^{\unicode[STIX]{x1D707}}\boldsymbol{b}^{\unicode[STIX]{x1D708}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D708}}S]=N\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}[(\boldsymbol{b}^{\unicode[STIX]{x1D708}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D708}}S)/N]=0.\end{eqnarray}$$
Equation (3.5) is gauge invariant under
$\boldsymbol{b}^{\unicode[STIX]{x1D707}}\rightarrow \boldsymbol{b}^{\unicode[STIX]{x1D707}}+g\boldsymbol{u}^{\unicode[STIX]{x1D707}}$
for any scalar function
$g$
since
$\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}S=0.$
The scalar quantity
$(\boldsymbol{b}^{\unicode[STIX]{x1D708}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D708}}S)/N$
is a relativistic Lorentz invariant generalization of the potential magnetic field introduced by Hide (Reference Hide1983, Reference Hide1996) along the lines of the potential vorticity and of Ertel’s theorem (Ertel Reference Ertel1942) in fluid dynamics.
4 The
$3+1$
formulation in a chosen reference frame
From
$\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}[(\boldsymbol{b}^{\unicode[STIX]{x1D708}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D708}}S)/N]=0$
and
$\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}S=0$
, we obtain
$$\begin{eqnarray}(\boldsymbol{b}^{\unicode[STIX]{x1D708}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D708}}S)/N={\mathcal{K}}(S,\boldsymbol{a}),\end{eqnarray}$$
where
$\boldsymbol{a}$
gives in the chosen reference frame the spatial initial conditions (at
$t=t_{o}$
) of the trajectories of the plasma fluid element, i.e. the 3-D Lagrangian coordinates, and
${\mathcal{K}}(S,\boldsymbol{a})$
is a function of the initial conditions and of the advected scalar
$S$
that we will fix as convenient.
Using the gauge freedom (2.10) in the form given by (2.13) we can rewrite (4.1) as
$$\begin{eqnarray}\frac{\boldsymbol{B}^{i}}{\unicode[STIX]{x1D6FE}N}\frac{\unicode[STIX]{x2202}\boldsymbol{a}^{j}}{\unicode[STIX]{x2202}\boldsymbol{x}^{i}}\frac{\unicode[STIX]{x2202}S}{\unicode[STIX]{x2202}\boldsymbol{a}^{j}}={\mathcal{K}}(S,\boldsymbol{a}).\end{eqnarray}$$
Choosing
$S=a^{k}$
with
$k=1,2,3$
, respectively, and recalling that
$\unicode[STIX]{x1D6FE}N$
is the density
$n(x,t)=n_{o}(\boldsymbol{a})/J(\boldsymbol{a},t)$
in 3-D space where
$J(\boldsymbol{a},t)=\text{det}|\unicode[STIX]{x2202}\boldsymbol{x}/\unicode[STIX]{x2202}\boldsymbol{a}|$
is the 3-D Jacobian determinant, with self-explanatory notation, for each choice
$S=a^{k}$
we obtain
$$\begin{eqnarray}\frac{\boldsymbol{B}^{i}}{\unicode[STIX]{x1D6FE}N}\frac{\unicode[STIX]{x2202}\boldsymbol{a}^{k}}{\unicode[STIX]{x2202}\boldsymbol{x}^{i}}={\mathcal{K}}^{k}(\boldsymbol{a}),\end{eqnarray}$$
i.e.
$$\begin{eqnarray}\boldsymbol{B}^{i}(\boldsymbol{x},t)=\frac{\unicode[STIX]{x2202}\boldsymbol{x}^{i}}{\unicode[STIX]{x2202}\boldsymbol{a}^{k}}\frac{n_{o}(\boldsymbol{a}){\mathcal{K}}^{k}(\boldsymbol{a})}{J(\boldsymbol{a},t)}=\frac{\unicode[STIX]{x2202}\boldsymbol{x}^{i}}{\unicode[STIX]{x2202}\boldsymbol{a}^{k}}\frac{\boldsymbol{B}_{o}^{k}(\boldsymbol{a})}{J(\boldsymbol{a},t)},\end{eqnarray}$$
which corresponds to the ‘Cauchy condition’ given by equation (2.22) of Newcomb (Reference Newcomb1962) (see also Andreussi, Morrison & Pegoraro (Reference Andreussi, Morrison and Pegoraro2013, (19))) that can be derived from the conservation of the flux of the 3-D magnetic field
$\boldsymbol{B}$
through a surface comoving with an ideal plasma.
Note that, knowing
$\boldsymbol{u}^{\unicode[STIX]{x1D707}}(\boldsymbol{x},t)$
in terms of
$\boldsymbol{u}^{\unicode[STIX]{x1D707}}(\boldsymbol{a},t_{o})$
and imposing
$\boldsymbol{b}^{\unicode[STIX]{x1D707}}\boldsymbol{u}_{\unicode[STIX]{x1D707}}=0$
for all
$t$
, it is possible to reverse the gauge transformation both on the right-hand side (using (2.1) with
$\boldsymbol{e}_{\unicode[STIX]{x1D707}}=0$
) and on the left-hand side of (4.4) (using (2.8)) and obtain the Cauchy condition for the 4-vector
$\boldsymbol{b}^{\unicode[STIX]{x1D707}}$
in agreement with (4.1).
5 Invariant Alfvén flux theorem
5.1 Preliminary proposition
We will make use of the following result (see e.g. Fitzpatrick Reference Fitzpatrick2008): if a 4-vector field
$\boldsymbol{Q}^{\unicode[STIX]{x1D707}}$
satisfies the continuity equation
$\unicode[STIX]{x2202}_{\unicode[STIX]{x1D707}}\boldsymbol{Q}^{\unicode[STIX]{x1D707}}=0$
and if its components vanish outside a finite spatial region, then the ‘charge’
${\mathcal{Q}}$
defined as the space integral of its time component
$\boldsymbol{Q}^{0}$
$$\begin{eqnarray}{\mathcal{Q}}=\int \text{d}^{3}x\boldsymbol{Q}^{0},\end{eqnarray}$$
is constant in time and is a Lorentz invariant. We will make use of this result in combination with (3.4) for different choices of the advected scalar
$S$
in order to rephrase in a Lorentz invariant framework the magnetic flux conservation (Alfvén theorem) that is usually presented in a fixed reference frame.
5.2 Flux conservation through a comoving 2-D surface
First we take
$$\begin{eqnarray}\boldsymbol{Q}_{{\mathcal{A}}}^{\unicode[STIX]{x1D707}}=\boldsymbol{u}^{\unicode[STIX]{x1D707}}\boldsymbol{b}^{\unicode[STIX]{x1D708}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D708}}S_{{\mathcal{A}}},\end{eqnarray}$$
where the function
$S_{{\mathcal{A}}}$
is defined in a given frame at
$t=0$
in terms of the characteristic function of a finite size, disk-like domain bounded by a 2-D spatial surface
${\mathcal{S}}(\boldsymbol{a})$
of the initial conditions (the base of the disk), by the corresponding surface displaced by the infinitesimal shift
$\unicode[STIX]{x0394}\boldsymbol{x}^{\unicode[STIX]{x1D707}}=\boldsymbol{u}^{\unicode[STIX]{x1D707}}\unicode[STIX]{x0394}\unicode[STIX]{x1D70F}$
(the top of the disk) and by the ‘ribbon’ connecting the rims of the two surfaces (the side of the disk). The function
$S_{{\mathcal{A}}}$
is advected by the flow 4-velocity
$\boldsymbol{u}^{\unicode[STIX]{x1D707}}$
, i.e. for all times
$t$
in the chosen frame
$S_{{\mathcal{A}}}=1$
inside the advected domain and
$S_{{\mathcal{A}}}=0$
outside.
Since
$\boldsymbol{Q}_{{\mathcal{A}}}^{\unicode[STIX]{x1D707}}$
is left invariantFootnote
3
by the gauge transformation (2.12) it can be rewritten as
$$\begin{eqnarray}\boldsymbol{Q}_{{\mathcal{A}}}^{\unicode[STIX]{x1D707}}=(\boldsymbol{u}^{\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6FE})\boldsymbol{B}^{i}\unicode[STIX]{x2202}_{i}S_{{\mathcal{A}}},\quad \text{so that}~\quad {\mathcal{Q}}_{{\mathcal{A}}}=\int \text{d}^{3}x\boldsymbol{B}^{i}\unicode[STIX]{x2202}_{i}S_{{\mathcal{A}}}=0.\end{eqnarray}$$
The last equality in (5.3) follows from the fact that
$\boldsymbol{B}$
(which is taken at
$t=0$
) is divergence free and that, because of the choice of the function
$S_{{\mathcal{A}}}$
, the space integral reduces to the surface integral over the closed surface
$\unicode[STIX]{x2202}\bar{{\mathcal{A}}}$
delimiting the spatial projection
$\bar{{\mathcal{A}}}$
of the domain
${\mathcal{A}}$
(only spatial derivatives are present in (5.3) because of the gauge used).
For the sake of clarity we will now refer to a
$3+1$
notation and use the fields
$\boldsymbol{v}$
,
$\boldsymbol{B}$
and
$\boldsymbol{E}$
so as to make the correspondence with the standard derivation of the Alfvén theorem explicit. There are three contributions to the vanishing flux in (5.3): the flux through the base surface, that through the ribbon and that through the top surface. From the space components of
$\unicode[STIX]{x2202}_{i}S_{{\mathcal{A}}}$
at the ribbon and using
$\boldsymbol{E}=\boldsymbol{B}\times (\boldsymbol{v}/c)$
we see that the flux of
$\boldsymbol{B}$
through the ribbon equals
$\unicode[STIX]{x0394}t=\unicode[STIX]{x1D6FE}\unicode[STIX]{x0394}\unicode[STIX]{x1D70F}$
times the circulation of
$\boldsymbol{E}$
along the rim of the domain
$\bar{{\mathcal{A}}}$
. By virtue of the induction equation this flux cancels the difference between the flux of
$\boldsymbol{B}(t)$
and that of
$\boldsymbol{B}(t+\unicode[STIX]{x0394}t)\sim \boldsymbol{B}(t)+[\unicode[STIX]{x2202}_{t}\boldsymbol{B}(t)]\unicode[STIX]{x0394}t$
through the top surface. Thus (5.3) implies that the flux of
$\boldsymbol{B}(t=0)$
through the base surface (having inverted the direction of the normal to this surface) and that of
$\boldsymbol{B}(t=\unicode[STIX]{x0394}t)$
through the same surface shifted by
$\unicode[STIX]{x0394}\boldsymbol{x}=\boldsymbol{v}\unicode[STIX]{x0394}t$
(the top surface) are the same. This equality, being valid for all times
$t$
, recovers the flux conservation theorem in an ideal plasma and makes it Lorentz invariant by virtue of the proposition in § 5.1.
Finally we note when the domain
${\mathcal{A}}$
is Lorentz boosted to a moving frame the points on the boosted base surface (and similarly for the points on the top surface) are no longer simultaneous because simultaneity between points that are spatially separated is not preserved by a Lorentz transformation. This lack of simultaneity in the boosted frame can be corrected, exploiting the flux conservation result derived above for each infinitesimal portion of the base and top surfaces, by a procedure of ‘time resetting’ along the trajectories of the 4-velocity
$\boldsymbol{u}^{\unicode[STIX]{x1D707}}$
that follows the method adopted in Pegoraro (Reference Pegoraro2012, Reference Pegoraro2016), see section below, equation (9) in Pegoraro (Reference Pegoraro2012) or Pegoraro (Reference Pegoraro2016, §5.1).
5.3 Invariant vanishing of the flux through an advected closed 2-D surface
Here take
$$\begin{eqnarray}\boldsymbol{Q}_{{\mathcal{D}}}^{\unicode[STIX]{x1D707}}=\boldsymbol{u}^{\unicode[STIX]{x1D707}}\boldsymbol{b}^{\unicode[STIX]{x1D708}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D708}}S_{{\mathcal{D}}},\end{eqnarray}$$
where the function
$S_{{\mathcal{D}}}$
is defined in terms of the characteristic function of a finite spatial domain
${\mathcal{D}}(\boldsymbol{a})$
of the initial conditions.
The derivation now follows with only minor changes the one in § 5.2 and we rewrite
$\boldsymbol{Q}_{{\mathcal{D}}}^{\unicode[STIX]{x1D707}}$
as
$$\begin{eqnarray}\boldsymbol{Q}_{{\mathcal{D}}}^{\unicode[STIX]{x1D707}}=(\boldsymbol{u}^{\unicode[STIX]{x1D707}}/\unicode[STIX]{x1D6FE})\boldsymbol{B}^{i}\unicode[STIX]{x2202}_{i}S_{{\mathcal{D}}},\quad \text{so that}\quad {\mathcal{Q}}_{{\mathcal{D}}}=\int \text{d}^{3}x\boldsymbol{B}^{i}\unicode[STIX]{x2202}_{i}S_{{\mathcal{D}}}=0.\end{eqnarray}$$
We take the simple case where
$\unicode[STIX]{x2202}{\mathcal{D}}$
is isomorphic to a sphere (i.e. it has no ‘holes’ and thus can be deformed into a sphere without changing its topological structure) and consider a closed curve
$\ell$
on
$\unicode[STIX]{x2202}{\mathcal{D}}$
that splits
$\unicode[STIX]{x2202}{\mathcal{D}}$
into two separate parts. Then (5.5) recovers the classical result according to which the flux through a 2-D surface bounded by a closed curve
$\ell$
is independent of the specific surface that is chosen and shows that this result is Lorentz invariant and that, obviously, it is conserved by the domain advection. Clearly, when the domain
${\mathcal{D}}$
is Lorentz boosted to a moving frame, it does not remain purely spatial (i.e. at constant time). This lack of simultaneity in the boosted frame can be corrected by the procedure of ‘time resetting’ mentioned above in § 5.2.
6 Conclusions
A first aim of this article was the completion of the Lorentz invariant formulation of the topological invariants begun in Pegoraro (Reference Pegoraro2012, Reference Pegoraro2016), where the concepts of covariant magnetic connections and of 2-D connection hypersurfaces were introduced, by proving that the time resetting procedure first introduced in Pegoraro (Reference Pegoraro2012) can be applied in order to obtain a Lorentz invariant formulation of the Alfvén theorem for the conservation of the magnetic flux through a surface comoving with an ideal plasma.
The construction of this invariant formulation moves from the definition of a Lorentz invariant ‘potential magnetic field’ that corresponds to the potential vorticity in the non-relativistic Ertel’s theorem. As an additional bonus this definition makes it possible to set in Lorentz invariant form the so called Cauchy conditions for the magnetic field components in the mapping between Eulerian and Lagrangian variables.
We remark that a Lorentz invariant definition of the magnetic flux conservation is important both from a theoretical and in particular from an experimental point of view. The latter is the case for example when observing a relativistically expanding plasma, because the distinction between electric and magnetic fields is frame dependent and because simultaneity between events at different spatial locations is not maintained in reference frames moving with different velocities with respect to the observer.
The results obtained in this article can be extended to non-ideal plasmas that obey a generalized ideal Ohm’s law, including e.g. electron inertia, as discussed explicitly in Pegoraro (Reference Pegoraro2015). A possible extension to more general fluid theories (see e.g. Kawazura, Miloshevich & Morrison (Reference Kawazura, Miloshevich and Morrison2017) in the non-relativistic limit) and in particular to those theories that involve an antisymmetric tensor that unifies the electromagnetic and the fluid fields, (Asenjo & Comisso Reference Asenjo and Comisso2015; Asenjo, Comisso & Mahajan Reference Asenjo, Comisso and Mahajan2015) should also be investigated. On the contrary a possible extension to an ideal plasma in curved space–time must include the fact that in general relativity covariant derivatives do not commute. This would result in a modification of (3.1) involving the curvature tensor.
Finally, we remark that the definition of potential magnetic field can be generalized following Hide (Reference Hide1996) by considering expressions of the form
$H(\boldsymbol{b}^{\unicode[STIX]{x1D708}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D708}}S)/N$
with
$H$
an arbitrary Lorentz scalar function. In this case a term proportional to
$\unicode[STIX]{x2202}_{\unicode[STIX]{x1D70F}}H$
is added to (3.5). As shown in Hide (Reference Hide1996), such an extension can provide a convenient tool in order to derive additional relationships involving the magnetic and the plasma velocity fields.
Acknowledgement
The author thanks Professor P. J. Morrison for useful information.
