Rotating clouds of charged Vlasov matter in general relativity

The existence of stationary solutions of the Einstein–Vlasov–Maxwell system which are axially symmetric but not spherically symmetric is proven by means of the implicit function theorem on Banach spaces. The proof relies on the methods of Andréasson et al (2014 Commun. Math. Phys. 329 787–808) where a similar result is obtained for uncharged particles. Among the solutions constructed in this article there are rotating and non-rotating ones. Static solutions exhibit an electric but no magnetic field. In the case of rotating solutions, in addition to the electric field, a purely poloidal magnetic field is induced by the particle current. The existence of toroidal components of the magnetic field turns out to be not possible in this setting.


Stationary solutions of the Einstein-Vlasov(-Maxwell) system
The Einstein-Vlasov-Maxwell system (EVM-system) describes an ensemble of charged particles whose motion is governed by gravity and an electro-magnetic field but which do not interact via collisions. Both the space-time curvature and the electro-magnetic field are generated collectively by the particles themselves. In contrast to the Einstein-Vlasov system, which only takes into account gravity, particles described by the EVM-system are not freely falling, i.e. their trajectories are not geodesics. Stationary solutions of the Einstein-Vlasov systems have been studied since the early 1990's. In [32] the existence of spherically symmetric, static solutions of the Einstein-Vlasov system with matter quantities of compact support is proven. In fact, in spherical symmetry, a large variety of different solutions is known, including balls, shells, highly relativistic and not highly relativistic solutions [9,29,30]. For massless particles, the existence of static, spherically symmetric solutions is known, too [6]. It is however conjectured that these solutions need to be highly relativistic. For charged particles the same variety of different static, spherically symmetric solutions can be constructed, at least for sufficiently small particle charges [5,37]. In contrast to the Vlasov-Poisson system, where rigorous existence of axisymmetric solutions which are not necessarily close to spherically symmetric are known [19], there is only little analytical understanding of the stationary solutions of the Einstein-Vlasov system beyond spherical symmetry. The only analytical results available are [7,8] where rotating and non-rotating axially symmetric, stationary solutions of the Einstein-Vlasov system are constructed as perturbations of spherically symmetric, static solutions. For the case of uncharged particles there exist numerical studies, see e.g. [1,2,35,36]. In these studies solutions have been constructed numerically that are far from spherical symmetry. Torus-shaped, disc-shaped and spindle-shaped solutions have been observed.
is proved, to the case where the particles are charged and hence induce an electro-magnetic field. Thereby, to the author's knowledge, the first existence result for non-spherically symmetric solutions of the EVM-system is presented. In the context of kinetic theory this method has already been used in [28] to show the existence of stationary, rotating solutions of the Vlasov-Poisson system. The idea of this method is to introduce a parameter λ to the system which can 'turn on' rotation and to perturb the system around a spherically symmetric, static solution without rotation. To this end one considers a functional F : X × [−δ, δ] → X, where X is a suitable function space which will contain the solution and [−δ, δ] is the interval in which the parameter λ will lie. The operator is constructed such that if F(ζ, λ) = 0 then ζ is a collection of functions which constitute a solution of the Vlasov-Poisson system with the parameter λ. The solution ζ 0 , corresponding to λ = 0, is known and we have F(ζ 0 , 0) = 0. The main part of the work consists in showing that the implicit function theorem can be applied. Then it follows that to each λ ∈ (−δ, δ) there exists ζ λ ∈ X such that F(ζ λ , λ) = 0. This collection ζ λ of functions consequently solves the Vlasov-Poisson system and this solution is axially symmetric but not spherically symmetric. It is in the nature of this method that the obtained rotating solutions have small overall angular momentum.
In [8] a similar method with a different set up has been used to show the existence of axially but not spherically symmetric, static solutions of the Einstein-Vlasov system. In this context it was used that the Vlasov-Poisson system is the non-relativistic limit of the Einstein-Vlasov system, in the sense that a solution of the Einstein-Vlasov system converges to a solution of the Vlasov-Poisson system if the speed of light c goes to infinity. So besides λ, the speed of light c has been introduced to the system as a second parameter. Perturbing off a spherically symmetric, static solution of the Vlasov-Poisson system in those two parameters λ and c yields an axially but not spherically symmetric, static solution of the Einstein-Vlasov system. The deviation from spherical symmetry is small but by a scaling argument the solution can be made fully relativistic, i.e. c = 1. In [7] further technical insights made it possible to include rotation into the picture.
Lichtenstein developed a method based on the implicit function theorem to construct rotating fluid bodies [23,24] in Newtonian gravity. This approach has later been reformulated in a modern mathematical language [22] and improved [21]. In [3,4] the authors use an implicit function argument to construct axially symmetric static and rotating elastic bodies in Einstein gravity. In a series of papers of which the last one is [15] the authors construct stationary solutions of the Einstein equations with negative cosmological constant without any symmetries. Many different matter models can be included, such as a scalar field, Maxwell, or Yang-Mills.

Rotating space-times with charged matter
Space-times with rotating, charged matter configurations have been studied in the literature by analytical and numerical means, see e.g. [13,14,18]. An important motivation for these studies is the modelling of rotating stars or neutron stars with a magnetic field. In these articles the matter is modelled as a perfect fluid and different shapes of the magnetic field can be observed depending on the assumptions on the fluid, like an equation of state or conductivity properties. For example rotating solutions with no poloidal magnetic field can be constructed, see [18]. These works can serve as a source of intuition for the study of rotating clouds of Vlasov matter. There is however an important difference. When studying a perfect fluid, the Einstein-Euler system (which describes a space-time containing matter of the type of a perfect fluid) has to be supplemented by an equation of state which captures the physical properties of the fluid under consideration. Depending on the choice of the equation of state, different matter configurations and different electro-magnetic fields can be constructed. For Vlasov matter however there is much less variety in the physical properties of the solutions that can be obtained. The basic assumptions on the particles' behaviour and how the energy and the angular momentum is distributed among the particles (this is sometimes referred to as a microscopic equation of state) already determines the macroscopic character of the solutions. It turns out that rotating solutions of the EVM-system must have a poloidal magnetic field but no toroidal magnetic field.
We briefly mention that in the non-relativistic setting a variety of different axially symmetric solutions can be constructed explicitly, see for example [12]. A well studied class of these solutions are disk solutions which serve as models for disk shaped galaxies and which are used to study some physical properties of these galaxies. The so called Morgan & Morgan disk solutions, introduced in [26], are important in this context. In [27] the authors construct comparable axially symmetric solutions in Newtonian gravity with general relativistic corrections. Surprisingly these general relativistic corrections account for changes of the solutions far from the galaxy core-a region where it was expected that Newtonian gravity describes the physics well and general relativistic effects do not play a significant role. This observation adds to the motivation of studying axially symmetric configurations of collisionless particles in the fully general relativistic picture.

The result and technical difficulties
The present article generalises [7] to the case of charged particles, i.e. solutions of the EVMsystem are constructed by perturbing off a non-trivial, spherically symmetric, static solution of the Vlasov-Poisson system. The following theorem is proved.
Theorem. There exist asymptotically flat, stationary solutions of the EVM-system with nonvanishing particle charge parameter, which are axially symmetric but not spherically symmetric. These solutions have no toroidal magnetic field and they have a non-trivial poloidal magnetic field if and only if they are not static, i.e. rotating.
It is assumed that the particles are charged with a particle charge q, i.e. an electro-magnetic field is included into the framework. A priori this can be done in two different ways. Either one considers q as a third (a priori small) parameter which 'turns on' charge. In this case one still perturbs off a spherically symmetric, static, uncharged solution of the Vlasov-Poisson system. The other way is to use the fact that in the non-relativistic limit the Maxwell equations reduce to the Poisson equation as well and one perturbs around a charged solution of the Vlasov-Poisson system. It turns out that the first approach is easier from a technical point of view since the operator F that the implicit function theorem will be applied to is changed only insignificantly by the included Maxwell equations. However, the result would be restricted to small particle charge parameters q. In the second approach arbitrary values 0 q < m p of the particle charge parameter can be treated, where m p denotes the mass of the particles. In this case the operator F has additional terms. In this article the second approach is presented.
In an axially symmetric, static setting the EVM-system reduces to a system of coupled, non-linear Poisson equations in different dimensions and a first order PDE. The solution of this system consists in a collection of functions which we denote ζ. A solution operator to these Poisson equations can be constructed via the Green's function G of the Laplace operator.
where ' * ' denotes the convolution and 'source[ζ]' schematically denotes the source terms of the Poisson equations which depends on the solution functions ζ. Consequently, a necessary condition for F to be well defined is that G * source [ζ] has at least the same regularity as ζ, i.e. the source terms of these Poisson equations need to be sufficiently regular. At this point a difficulty occurs which is new in the charged case. It can be illustrated as follows.
After the variable substitution A ϕ = 2 a one obtains for the ϕ-component of the electromagnetic four potential A the equation On the right hand side only some a priori problematic terms are written out explicitly. The functions ν and h are part of the collection ζ of solution functions of the EVM-system. These terms are a priori problematic because they are singular at the axis = 0. Looking a bit closer one notices that the right member of equation (1.1) is not singular if h and ν are axially symmetric functions of a certain regularity. However, by dividing by one 'looses derivatives'. For this reason the function space X has to be chosen such that the individual functions of the collection ζ have an appropriate hierarchy in regularity. For equation (1.1) for example one needs that h and ν are of higher regularity than a such that for example G * ∂ h/ has at least the regularity of the function space of a.
This article is a generalisation of [7] and the proof follows the same scheme. Including charge into the framework does not only increase the number of equations in the system but it also increases significantly the number of terms in each equation. Some of these terms require some care in the analysis but clearly not all of them. Still all required properties of the system have to be checked term by term. In order to make the presentation more concise this article resorts more to shorthands and schematic or symbolic notation than [7,8].

Discussion of the result
An important step in the construction of stationary solutions done in this article is exploiting that the particle distribution function f can be expressed in terms of two conserved quantities, the particle energy E and the z-component of the angular momentum L. We work with ansatz functions of the form where φ and ψ are regular functions specified in section 5.
If one aims for constructing solutions to the Einstein-Vlasov system, or the EVM-system, which are not spherically symmetric it is natural to include a dependency on the angular momentum L into the ansatz for the particle distribution function f . If the distribution function f is only a function of E any resulting static solution is spherically symmetric [20]. In a certain sense collisionless matter prefers spherically symmetric configurations if no physical effect, as for example rotation, accounts for more structure and thus non-spherically symmetric configurations.
For the Vlasov-Poisson system it is known that for all spherically symmetric, time-independent solutions the particle distribution function can be written as f = Φ(E, L), i.e. as a function which only depends on the particle energy and the angular momentum. This statement is referred to as Jeans' theorem. For the Einstein-Vlasov system this is not true [34]. Consequently, in the axially symmetric setting in particular, the ansatz (5.7) does not represent all possible solutions.
All results on the Einstein-Vlasov system in axial symmetry, in particular the numerical studies [1,2,35,36], use ansatz functions of the class (5.7). With this class of ansatz functions it turned out to be possible to model many of the observed shapes of galaxies which is an indication for that this class is not too restrictive.
Indirectly, the choice of ansatz function determines the properties of the resulting solution of the axially symmetric EVM-system. On the other hand, if the ansatz function is not consistent with the laws of Physics, it will not give rise to a non-trivial solution. Physically it is very intuitive that the solutions have a poloidal magnetic field if an overall rotation is present, since a current of charged particles induces a magnetic field. A non-trivial toroidal magnetic field could then only be present if the particles have an overall motion which induces such a magnetic field. Such a field would be induced by a particle cloud with a poloidal particle current. It is however unclear what mechanism could drive such a poloidal particle current. Exploring this possibility would constitute an interesting mathematical question and the study of corresponding ansatz functions for the particle distribution function would be a novelty.
The axially symmetric solutions of the EVM-system obtained in this article can be characterised as follows. Since the solutions are obtained essentially by a perturbation argument the overall rotation will be small. Furthermore, since the solutions are obtained by perturbing around a solution of the Vlasov-Poisson system which is the non-relativistic limit of the EVM-system (i.e. the limit where c → ∞), the parameter c is a priori very large. By a scaling argument one obtains solutions with c = 1. These solutions are then rather not very relativistic, i.e. no very strong local gravitational effects will occur. However, results like [27] suggest that studying the fully general relativistic system still entails phenomenological differences, as already mentioned above. Moreover, the particle charge parameter of the solutions does not need to be small.
The physical interpretation of collisionless, general relativistic, charged particles could be electrons or ions in a galactic nebula. Describing a galactic nebula by a slowly rotating solution of the EVM-system, as constructed in this article, would however be a strong idealisation neglecting a lot of structure. The most important contribution of this article is to get a more complete mathematical understanding of a whole class of physically relevant, non-linear partial differential equations.

Outline of the paper
In the next section the EVM-system will be introduced. Then, in section 3, the result of this article will be stated in detail and an outline of the proof will be given. The rest of the article is devoted to the introduction of the technical setup. In section 4, the EVM-system is formulated in axial symmetry and in section 5 the ansatz for the particle distribution function is discussed. The remaining sections contain the definition of the relevant objects, i.e. function spaces and solution operators, and the proofs of important properties of these operators.

The Einstein-Vlasov-Maxwell system
A solution of the EVM-system for particles with mass m p 0 and charge 0 q < 1 is a Lorentzian metric g ∈ T * M ⊗ T * M defined on a four dimensional manifold M , a particle distribution function f ∈ C 1 (TM ; R + ), defined on the tangent bundle of M , and an electromagnetic field tensor F ∈ Λ 2 (TM ) such that the EVM-system, 3) is satisfied. Here G µν is the Einstein tensor and we choose units such that G = 1 (G is the gravitational constant) but we leave c as parameter in the system. We give a brief explanation of the involved quantities, consult however e.g. [33] for a more detailed introduction to the EVM-system. The particle distribution function f = f (x, p) describes the particle number density at a certain point in x ∈ M with a certain four-momentum p ∈ T x M . The particle number can be obtained via integration. The quantity m p , defined by the relation is interpreted as the particles' rest mass. It can be shown that it stays constant along the characteristic curves of the Vlasov equation (2.4). Consequently the particle distribution function f describing an ensemble of particles where all particles have the same rest mass m p can be assumed to be supported on the mass shell P mp , a seven dimensional submanifold of TM which is defined to be In the remainder of this article we assume m p = 1 for all particles, and we denote the corresponding mass shell simply by P. The volume form dvol Px on the mass shell fibre P x over x ∈ M is given by dvol Px = | det(g µν (x))| −p 0 dp 1 ∧ dp 2 ∧ dp 3 , (2.9) and the transport operator T is given by It is tangent to any mass shell P [33]. Assume that we have a solution (g, f , F) of the EVM system and that on M we have coordinates t, x 1 , x 2 , x 3 , where t is the time coordinate. Assume further that ∂ t is a Killing field. Then the solution is asymptotically flat if the boundary conditions are satisfied, where η denotes the Minkowski metric.

The result
In this article we prove the following result.
of the EVM-system (2.1)-(2.6) with particle charge parameters q ∈ [0, 1), which are axially symmetric but not spherically symmetric. Such a solution has no toroidal magnetic field and it has a non-trivial poloidal magn etic field if and only if the solution is not static, i.e. rotating.
Proof. The proof which is given at this place is rather an outline of the proof, the technical details are given in the subsequent sections. The proof follows the same structure as in [7] where the existence of stationary, rotating, axially symmetric solutions is proved for uncharged particles. Each step is however a bit more involved and some arguments have to be formulated differently due to the additional Maxwell equations. We comment on the modifications in the respective sections.
Step 1: Elimination of the Vlasov equation. For the particle distribution function we use the ansatz f (x, p) = φ(E(x, p))ψ(λ, L(x, p)), see (5.8). So the particle distribution depends only on the particle energy E(x, p) and the z-component of the angular momentum L(x, p), see the definitions (5.2) and (5.1). Since the quantities E and L are conserved along its characteristics the Vlasov equation is automatically satisfied for such an ansatz, see section 5. Furthermore, we introduce a parameter λ which 'turns on' the dependency of f on L. This means that if λ = 0 then ψ ≡ 1, i.e. for each value of the z-component of the angular momentum there are equally many particles.
Step 2: Reduction of the remaining system. First we express the EVM-system (2.1)-(2.6) in cylindrical coordinates. The assumptions that the solution is asymptotically flat, axially symmetric, and time independent yield simplifications of the system of equations. We call this simplified system the reduced EVM-system, see definition 6.1 and it is stated in section 6, equations (6.19)-(6.28) where any value of c ∈ (0, ∞) is admitted. The solution of the reduced EVM-system is determined by the collection ζ = (ν, h, ξ, ω, A t , a) ∈ X of six functions, defined in a suitably chosen function space X (defined in section 7). Proposition 8.1 states that a solution of the reduced EVM-system with any parameter c can be converted into an axially symmetric, stationary solution of the EVM-system with the parameter c = 1.
Step 3: Introduction of the solution operator F. A solution of the reduced EVM-system with parameters γ := c −2 , λ ∈ [0, 1) × (−1, 1) is then obtained as perturbation of a spherically symmetric solution of the Vlasov-Poisson system. This spherically symmetric solution of the Vlasov-Poisson system we denote by ζ 0 ∈ X . To this end in section 9 an operator F : X × [0, 1) × (−1, 1) → X with the following properties is defined. Firstly, a collection of functions ζ ∈ X is a solution of the reduced EVM-system with parameters γ, λ if and In section 10 we show that this operator is well defined. The mentioned properties are shown in proposition 9.1 and lemma 9.3.
Step 4: Application of the implicit function theorem. The aim is to apply the implicit function theorem on Banach spaces, see for example [16, theorem 15.1]. This theorem implies the existence of δ > 0 such that there exists a mapping Z : i.e. Z(γ, λ) is a solution of the reduced EVM-system with parameters γ, λ. This solution Z(γ, λ) then gives rise to a solution of the EVM-system with the asserted properties, by proposition 8.1.
The implicit function theorem can be applied in this way if the operator F is con- exists and is continuous, and if this Fréchet derivative L is a bijection. These properties are established in section 11. Proposition 12.1 contains the details how it is made sure that the boundary conditions for an asymptotically flat solutions are satisfied.
Step 5: Characterisation of the electro-magnetic field. The assertion that the solution comprises a poloidal magnetic field if and only if the solution is rotating follows from the structure of the reduced EVM-system, see remark 6.2. For the assertion that there is no toroidal magnetic field, see lemma 6.3. □

Axial symmetry
Let By abuse of notation, we will use the same symbol for the original function on R 2 , f in this example, and the induced axially symmetric functions f on R n for different dimensions n.
Remark 4.1. At some places in the analysis presented in this article it will be useful to view an axially symmetric function f : R n → R as a function in and z defined on R 2 , by extending it as an even function to negative values of . The obtained function on R 2 then has the same regularity as the axially symmetric function on R n . We now introduce a coordinate gauge and the functions in terms of which we will formulate the reduced EVM-system. Consider the four dimensional manifold M which is assumed to be homeomorphic to R 4 and which is equipped with the cylindrical coordinates t, , z, ϕ. A stationary Lorentzian metric is characterised by the four time independent, axially symmetric functions ν, µ, ω : M → R and H : M → R + . It can be written in the form see [10] for details. The electro-magnetic field tensor F is given as the exterior derivative of the electro-magnetic four potential A ∈ Λ 1 (M ), i.e. F = dA. With respect to the coordinate co-basis of t, , z, ϕ the electro-magnetic potential A takes the form We assume that all components are time independent and axially symmetric.
In terms of the electro-magnetic field tensor F the electric field E ∈ Λ 1 (M ) and the magnetic field B ∈ Λ 1 (M ) are defined as follows. The electric field E is defined by the splitting F = E ∧ dt + B, where the two form B includes no term with dt. The magnetic field is defined by the splitting is the Hodge star operator and E is a two-form with no dt-term. See [17] for details. Define β := ∂ z A − ∂ A z . Then a calculation yields that the toroidal magnetic field component B ϕ takes the form (4.5) and the poloidal magnetic field components, B and B z , contain only the t-and the ϕ-component of A. In fact a calculation yields Next we introduce the parameter γ = 1 c 2 and the orthonormal frame e a = e a α ∂ α , α = t, , z, ϕ, where the non-trivial matrix elements are The corresponding co-frame reads α a = e a α dx α , where (e a α ) = (e a α ) −1 (the inverse matrix), and via the relation (4.9) In the remainder of this article we work with the coordinates on the tangent bundle TM . In these frame coordinates the mass shell relation (2.7) becomes and on P we consequently have (4.12)

Ansatz function for the particle distribution
The Vlasov equation (2.4) can be dealt with by the method of characteristics which is now described.

Lemma 5.1. The quantities E and L, defined on the tangent bundle TM , by
are conserved along the characteristic curves of the Vlasov equation, i.e.
Proof. The assertion of this lemma can be shown via a direct calculation and it is moved to the appendix. □ Remark 5.2. Unlike the uncharged case, in the charged case the characteristic curves of the Vlasov equations are not the lifts of the geodesics to TM . Consequently the conserved quantities cannot be obtained by g (X, p), where X is a Killing vector field and p is the canonical momentum. However, this structure can still be recognised in the present case. If we definẽ it turns out that the quantities E and L can be obtained from Ẽ and L by taking into account a suitable correction due to the electro-magnetic field. We have with some functions φ,ψ ∈ C 1 (R; R + ), solves the Vlasov equation (2.4) and is axially symmetric and time independent.
Proof. Since TE = TL = 0 we have by the chain rule Tf = 0. The remaining asserted properties of f are inherited from the metric functions ν , µ, H, and ω. □ A more general statement than corollary 5.1 is true, for ansatz functions that do not have the product structure (5.7). The corollary is however stated this way because in this article only ansatz functions of the form (5.7) are considered.
From now on we work with the ansatz where E and L are the conserved quantities, given in (5.2) and (5.1), respectively, and λ ∈ [0, 1] is the parameter which 'turns on' anisotropy in momentum of the particle distribution. The functions φ and ψ are assumed to fulfil the assumptions listed below. For an integrable func- We assume that the functions φ and ψ in (5.8) have the following properties.
leads to a compactly supported, spherically symmetric steady state ( f N , U N ) of the Vlasov-Poisson system for particles with mass 1 − q 2 , i.e. there exists a solution U N ∈ C 2 (R 3 ), of the equa- (1) is clearly satisfied. Condition (2) is also satisfied, see [11,31].
By the same proof as for [8, lemma 7.1] it can be shown that the third condition is satisfied for polytropes with exponent k sufficiently close to 7/2. To this end one uses the equa- Then merely the constant 4π has to be replaced by 4π(1 − q 2 ) in the proof of [8, lemma 7.1]. It is essential that 1 − q 2 > 0, the precise value is however irrelevant for the argument. □

The reduced system of equations
Before the reduced system of equations is presented some notation and shorthands shall be introduced. Partial derivatives ∂ ν , ∂ z A ϕ , etc will be denoted as ν , , A ϕ,z , etc. We define the functions ξ, h, a by the following changes of variables: Further, we call (ν, h, ξ, ω, A t , a) the solution functions and in the remainder of this article we will use the shorthand We do not include the components A and A z of the four-potential A into the solution functions ζ since it will turn out that in the current setting they must vanish everywhere, see lemma 6.3. In [7,8], where the existence of axially symmetric solutions of the Einstein-Vlasov system with uncharged particles is proven, a reduced system of equations is considered as well. The reduced EVM-system presented in the following coincides with the reduced system in [7] if the charge parameter q is set to zero. When the Maxwell equations are added to the framework not only the number of equations increases but also the number of terms in the Einstein equations increases by a multiple. For this reason, below, we are going to introduce source functions to collect these terms. This allows to present the reduced system in a compact way and also facilitates the presentation of the subsequent analysis. Moreover, we will introduce matter functions which basically consist in combinations of components T µν of the Vlasov part of the energy momentum tensor, as in [7].
In the subsequent analysis it will be necessary to show different properties of the matter functions and the source functions, like regularity with respect to the coordinates and z, decay properties, symmetries, or Fréchet differentiability with respect to the solution functions ζ. This means that at some occasions the source functions and the matter functions have to be seen as functions of and z which are parameterised by the solution functions. At other occasions they have to be seen as functions which take both the coordinate and the solution functions ζ (and their derivatives) as arguments. Moreover, for the analysis of the matter functions several different integral representations will be necessary. In order to give a clear presentation we deem it favourable to resort to symbolic notation in a larger extent than in [7].  is antisymmetric in v 3 .

Remark 6.2.
If the ansatz function f = φ(E)ψ(λ, L) for the matter distribution satisfies in addition to the conditions listed on page 15 that ψ is even in L, then the equations (6.19)-(6.28) possess solutions such that ω ≡ a ≡ 0, i.e. static solutions without rotation. Note that the corresponding matter functions vanish, see (6.11). So the equations exhibit the physical connection between rotation and the magnetic field. Intuitively one would think of this connection in the following way. If there is no overall rotation, i.e. ω ≡ 0, then there is consequently no electric current and no magnetic field is induced.
If there is rotation, however, the moving charges induce a poloidal magnetic field. Inspecting equations (6.22) and (6.24), we see that ω ≡ a ≡ 0 is a solution, whereas it is not possible that only one of these functions is zero everywhere because they appear mutually as source terms in the equation of each other. Lemma 6.3. For each continuous solution of (6.19)-(6.28) the combination β = A ,z − A z, vanishes everywhere, i.e. there is no toroidal magnetic field. (The toroidal component of the magnetic field is given in (4.5)).
The axially symmetric solutions of the EVM-system which are constructed in this article are obtained as perturbations around spherically symmetric solutions of the Vlasov-Poisson system. For this reason we discuss the non-relativistic limit of the EVM-system, i.e. the limit where γ → 0.
Define for the spherically symmetric steady state of the Vlasov Poisson system for particles of mass 1 − q 2 the potential at infinity U ∞ by Then by condition (2) on φ we clearly have U ∞ > E 0 and there exists R ∈ (R N , ∞) such that , for all r > R. (6.31) (Recall that R N is such that U N (R N ) = E 0 .) It turns out, that in the limit γ → 0, only the equations (6.19) and (6.23) of the reduced EVM-system remain non-trivial and they reduce to the Poisson equations ∆ν N = 4πρ νN +qAN , (6.32) where we use the notation ρ νN +qAN , introduced in (5.9), on the right hand side. See the proof of lemma 9.3 for details. The system (6.32) and (6.33) equipped with the boundary conditions ν N (0) = 0, A N (0) = 0, (6.34) and the equation are equivalent in the sense that a solution of (6.32) and (6.33) gives rise to a solution of (6.35) via U N = ν N + qA N and a solution of (6.35) gives rise to a solution of (6.32) and (6.33) via In lemma 10.7 we will furthermore see that the limits ν ∞ = lim |x|→∞ ν and A ∞ = lim |x|→∞ A t exist for any γ ∈ (0, ∞) and that in the limit γ → 0 there holds A ∞ = −qν ∞ , which is consistent.
We are going to linearise around a solution of the system in the limit (γ, λ) → (0, 0). We denote this solution by ζ 0 , i.e. Proof. The particle energy E converges to the Newtonian particle energy E N , given by in the non-relativistic limit where γ → 0. Using the expansions e x = 1 + x + . . . and and since ν → ν N , A t → A N , ω → 0, we see E → E N as γ → 0 which is the Newtonian particle energy with potential U N = ν N + qA N . Now, since ν + qA t − U N ∞ → 0, as γ → 0, there is γ 0 > 0 such that for all 0 γ γ 0 we have E > ν + qA t > E 0 for all |x| > R. □

The function space of the solution
In this paragraph the function spaces are defined in which a solution ζ = (ν, h, ξ, ω, A t , a) of the reduced EVM-system will be constructed. In [7,8] the considered function spaces contain axially symmetric functions on R 3 . Taking account for the fact that the reduced EVM-system is formulated as Poisson equations in different dimensions we define the function spaces for functions in the according dimensions. Furthermore, for the analysis of the source terms of these Poisson equations a hierarchy in regularity among the individual solution functions is needed, see lemma 10.5. For this reason the assumed regularity is a bit stronger than in [7]. Let α ∈ (0, 1/2) be a fixed parameter and We define the following spaces of axially symmetric functions, −z), and ν X1 < ∞}, (7.1) −z), and h X2 < ∞}, (7.2) −z), and ξ X3 < ∞}, (7.3) −z), and ω X4 < ∞}, (7.4) and Let β ∈ (0, 1) be another fixed parameter. Then the corresponding norms are defined to be and Finally we define

Solutions of the reduced system solve the full EVM-system
In this article we construct solutions to the reduced EVM-system (6.19)-(6.28). These solutions to the reduced EVM-system correspond to spherically symmetric, time independent solutions of the EVM-system (2.1)-(2.6). The relations between these systems is the subject of the following proposition. As already mentioned, this article generalises [7] to the case of charged particles and the reduced system treated here coincides with the reduced system considered in [7] if the charge parameter q is set to zero.

Proof.
We check the laws for A t and a. For the other functions, see [7]. For the Laplace operator we have the transformation law Then we use the Maxwell equations (6.23) and (6.24) for A t and a, respectively. Note that for example For the matter function corresponding to A t we obtain the expression and for a we have the matter function 1 + γ|v| 2 dv 1 dv 2 dv 3 . First we describe how the reduced EVM-system can be derived from the EVM-system. We start with the equations (6.19)-(6.22) which-without electromagnetic field terms of course-have been considered in [7]. Write down all Einstein equations in the coordinates t, , ϕ, z and take into account the symmetries by substituting the ansatz (4.3) for g. Suitable combinations of the Einstein equations yield the equations (6.19)-(6.22) for ν , h, ξ, and ω. For equation (6.19) take the combination For equation (6.20) take (1 + h)(G + G zz ), for equation (6.22) take 2e 2ξ 2 (1+h) 2 (G tϕ + ωG ϕϕ ), and for equation (6.21) take It is important to take the right combination of Einstein equations for the method to work and we follow [8].
The components G µν of the Einstein tensor and the components τ µν of the electro-magnetic part of the energy momentum tensor yield the left members and the source functions of equations (6.19)-(6.22). The matter functions M (γ,λ) i , i = 1, 2, 4 are obtained as explained now. First, using the ansatz (5.8) for the particle distribution function f and the orthonormal frame (4.9) one can write the components of the kinetic part T µν of the energy momentum tensor, defined in (2.2), as the integral expression.
For this formula the mass shell relation (4.12) needs to be used. Furthermore, the variables p µ , µ = 0, . . . , 3 can in terms of the frame components v 1 , v 2 , v 3 , be expressed as

(8.10)
Now taking the corresponding combinations of T µν and substituting the expressions (8.10) for the p -variables one obtains after simplification the matter functions. These matter functions coincide with the corresponding matter terms in [7], the only difference consists in the quantities E and L. The matter quantity M (γ,λ) 3 vanishes due to the symmetry T = T zz . The equations (6.23) and (6.24) for A t and a, respectively, are new with respect to [7] and they are obtained by suitable combinations of the Maxwell equation ∇ α F αβ = −4πqJ β for β = t and β = ϕ. These combinations are are obtained by taking the respective combinations of the components of the matter current J β , defined in (2.6). Using the orthonormal frame (4.9) it can be written as The variables p µ , µ = 0, . . . , 3, are given in terms of the frame coordinates as (8.14) So far it has been proved that a solution of the EVM-system implies a solution of the reduced EVM-system since the latter one is obtained by linear combinations of certain components of the former one. It remains to verify that the converse is also true, i.e. that a solution to the reduced EVM-system with parameter c ∈ [1, ∞) implies an axially symmetric, time independent solution of the EVM-system with c = 1. First we note that by the scaling laws (lemma 8.1) a solution to the reduced EVM-system with c = 1 can always be obtained. The Maxwell equations are already fulfilled since the number of equations has not been reduced. For the Einstein equations however the number of equations has been reduced, so situation is less clear. We define the quantity The non-trivial components are E tt , E , E zz , E ϕϕ , E tϕ , and E z . The other components are trivially zero since the Einstein tensor vanishes under the symmetry assumptions incorporated into the metric ansatz (4.3). It remains to show that the components E tt , E , E zz , E ϕϕ , E tϕ , and E z vanish, too. This can be done by using the same argument as given in [7, section 6] since the Einstein part of the reduced EVM-system that we are working with consists in the same linear combinations of Einstein equations which has been considered in [7]. A subtlety, which has to be dealt with, consists in the fact that ξ is only C 1,α , whereas Einstein's equations are of second order. Since in the present setup ξ has the same regularity as in the setup of [7] the arguments of [7] apply however. Finally, the boundary conditions (6.27) clearly imply the boundary conditions (2.11). □

Definition of the solution operator F
The equations (6.19), (6.20) and (6.22)-(6.24) of the reduced EVM-system are semi-linear Poisson equations. For this reason the solution operators corresponding to these equations are basically given in terms of the Greens function of the Laplace operator. If q is set to zero, the solution operator introduced here coincides with the solution operator defined in [7]. First, we recall some facts about the Poisson equation. Define for n 3 the n-dimensional Greens function G n y (x) of the Laplace operator ∆ n by  [25, theorem 6.21]. Now we give the definition of F. To this end we first define the operators G i : U → X i , i = 1, . . . , 6 (by X 5 and X 6 we understand X 1 and X 4 , respectively). We define Proof. Clearly g i [ζ; γ] and M i [ζ; γ, λ] are axially symmetric and even in z if ζ is. Consider the following prototype term. Let f : R n → R n be an axially symmetric function that is even in x n = z. One can check straight forwardly that G n [ f ] is axially symmetric and even in z by performing and appropriate change of variables in the integral, i.e. we have for A ∈ SO(n − 1) Remark 9.2. The operators G 1 and G 5 have been defined such that the Fréchet derivative of F with respect to ν , A t , at (ζ 0 ; 0, 0) is zero at ( , z) = 0. Observe the Ĝ in equation (9.4). This property is important in the proof that the Fréchet derivative at (ζ 0 ; 0, 0) is a bijection, see lemma 11.1. Proof. We adopt the notation ρ N := ρ UN , α N := α UN . The Einstein equations (6.20)-(6.22) for h, ξ, and ω are trivially satisfied for ζ = ζ 0 . So it remains to consider equation (6.19) for ν . The source function g 1 [ζ 0 ; 0, 0] is zero. For the matter function M 1 a calculation yields M 1 [ζ 0 ; 0, 0]( , z) = 4πρ N (r), where r = 2 + z 2 . This is the energy density induced by the ansatz (5.8) in the Newtonian case.

F is well defined
We have to verify that for all (ζ; γ, λ) ∈ U the functions G i [ζ; γ, λ] satisfy the regularity conditions and the decay behaviour stated in the definition of X , for i = 1, . . . , 6. Before we prove the regularity properties of G[ζ; γ, λ] we collect a few facts on axially symmetric functions, proven in [8] and [7].  Lemma 7.1 in [7]). Let u : R n → R be axially symmetric and u(x) =ũ( , z) where ũ : [0, ∞) × R → R. Let k ∈ {1, 2, 3} and α ∈ (0, 1). Then and all derivatives of ũ of order up to k which are of odd order in vanish for = 0,
Next we establish regularity of the matter functions. can be treated with the same ideas. We perform in the integrals of the formulas (6.5)-(6.9) for the matter functions M (γ,λ) i , i = 1, 2, 4, 5, 6, a change of variables, given by Then the domain of integration can be parameterised by η ∈ ((e γν − 1)/γ, ∞), s ∈ (−m, m). Further, for a function g = g(s, η, h, ν, ω), which will be chosen among the choices we define M (γ,λ) to be the operator which assigns to g the function Here we view ζ ∈ X as even functions in , see remark 4.1. By the observation on M (γ,λ) which is mentioned above the first term is a product of functions that are even in . For the second term we observe that the fraction is odd in since it contains as explicit factor. The second factor is also odd in by the upper observation. So in total the second term is even in . □

Lemma 10.4 (Regularity of the matter functions). Let
(10.12) We write this in a schematic form in order to make the analysis clearer. Let x = (x 1 , . . . , x 6 ). In the following this vector represents ( , ν, h, ω, A t , a). We write Note that ∈ C ∞ (R 3 ), since l ∈ C ∞ (R 3 ) already. To see the latter remind that h > −1/2 is assumed on the domain of M (γ,λ) . We have for i = 1, . . . , 6 Now we see that each additional derivative ∂ xj , j = 1, . . . , 6 leads to a derivative acting on φ, unless ĝ(s, (s, x), x) = 0. In this case, only if there are three or more derivatives, there act one or more derivatives on φ. Since ψ , ∈ C ∞ , and φ and ψ are compactly supported, the regularity of φ and g determines the regularity of M  We view A t , a, ν , and h now as functions in , z on R 2 that are even in , see remark 4.1. The functions A t, a, ν , a, and h , a are odd in and in C 2,α (R 2 ), so in particular in C 2 (R 2 ). So, by lemma 10.2, the functions (10.20) are in C 1 (R 2 ) and consequently also in C 0,α (R 2 ). This is sufficient to prove the asserted regularity. Finally we consider the operator G 3 [ζ; γ, λ]. The asserted regularity is easy to see since the source function g 3 [ζ; γ, λ] is obviously sufficiently regular, i.e. C 0,α . □ Next we check the decay properties of G[ζ; γ, λ]. First we recall a technical lemma.
Finally, by inspecting the formula (9.4) for the solution operators G 1 and G 5 corresponding to ν and A t , respectively, we see that |y| dy decay towards spatial infinity, also by lemma 10.6. □ Remark 10.8. Note that in lemma 10.6 we have seen that for the functions ν and A t the decay is improved, form (1 + |x|) −(1+β) to (1 + |x|) −2 , i.e. assuming the weaker decay of ν, A t ∈ X 1 we obtain the stronger decay of G 1 [ζ; γ, λ], G 5 [ζ; γ, λ]. This is important in the proof that the Fréchet derivative of these components at (ζ 0 ; 0, 0) is a compact operator, which in turn plays a role in the proof that this derivative is a bijection, see lemma 11.1 and [8, lemma 6.2].
All required properties of G[ζ; γ, λ] are now verified, thus the operator F is well defined.

The Fréchet derivative of F
We denote the functions ν , h, ξ, ω, A t , a constituting the collection ζ by ζ 1 , . . . , ζ 6 , if convenient. The Fréchet derivative of G i with respect to ζ j at (ζ; γ, λ) is a linear operator from X j to X i , i, j = 1, . . . , 6. Here and in the remainder of the article by X 5 and X 6 we mean X 1 and X 4 , respectively, since these are the function spaces corresponding to ζ 5 and ζ 6 , respectively. We denote the Fréchet derivative by Proof. The operators G i , i = 1, 2, 4, 5, 6 are of similar structure and we will start by analysing these operators. Schematically one can write these operators as sums of expressions of the form where the function Φ (γ,λ) : R 19 → R is a placeholder for either g . In order to write this in a compact and handy way we define the functional G n (which is slightly different from G n , see the definition (9.3) of G n ) bỹ G n Φ (γ,λ) , ζ ( , z) := R n G n y (| |, 0, . . . , 0, z) Φ (γ,λ) ( (y), ζ(y), ζ , (y), ζ ,z (y)) dy.

(11.3)
We will check now that the Fréchet derivative of G Φ with respect to ζ j is given by So we have to check that Here X Φ is the function space corresponding to Φ (γ,λ) . I.e. if Φ (γ,λ) is for example M (γ,λ) 1 then X Φ is X 1 . Define m as the number how often functions in X Φ are continuously differentiable, i.e. the largest number such that X Φ ⊂ C m,α . By the standard elliptic estimate [25, theorem 10.3] and the inclusion C m+1 ⊂ C m,α it suffices to check (11.6) It turns out that (11.6) holds if the functions Φ (γ,λ) are sufficiently regular, i.e. in C m to be precise. Now, Φ (γ,λ) is either a source function g For the matter functions M (γ,λ) i , i = 1, 2, 4, 5, 6, defined in equations (6.5)-(6.9), we note that they do not depend on derivatives of ζ and that the regularity is determined by the functions M (γ,λ) [g] which are all C 3 by lemma 10.4 and this is sufficient.

Application of the implicit function theorem
In the preceding sections we have established that the solution operator F fulfils the assumptions of the implicit function theorem for Banach spaces. Now we can prove the following proposition.
Proof. The solution ζ = (ν, h, ξ, ω, A t , a) exists by virtue of the implicit function theorem. The functions ω, ξ, h, and a fulfil the boundary condition lim |( ,z)→∞ (|ω| + |ξ| + |h| + |a|) = 0 by construction. For ω and a see the definition (7.9) of the norm of the space X 4 . Analogously, with lemma 10.6, it follows that h fulfils the boundary condition. By inspecting the structure (9.6) of the solution operator G 3 one easily sees that the boundary condition ξ(0, z) = ln(1 + h(0, z)) (12.1) is satisfied, too. For the boundary condition of ξ at infinity one infers first from (12.1) that lim |z|→∞ ξ(0, z) = 0, and then the decay as → ∞ can be deduced from the decay of the integrand of the solution operator G 3 , see formula (9.6) and [8, proposition 2.3]. The solution functions ν , A t obtained from the implicit function theorem do however a priori not satisfy the boundary condition