Stationary solutions of the axially symmetric Einstein-Vlasov system: present status and open problems

Abstract

The purpose of this work is to review the status about stationary solutions of the axially symmetric Einstein-Vlasov system with a focus on open problems of both analytical and numerical nature.
For the latter we emphasize that the code used to construct stationary solutions in \cite{Ames2016,Ames2019} is open source, see \cite{Ames2023joss}.
In the analytical setting the open problems include establishing methods for proving existence of axisymmetric stationary solutions which are far from spherically symmetric, both in the general case and for certain special classes of solutions pointed out in the text.
In the numerical setting there are intriguing properties of highly relativistic solutions that demand further attention, such as whether a sequence of such stationary solutions can approach a Kerr black hole, or if they necessarily approach the thin ring limit reminiscent of cosmic strings.
The question of whether stationary solutions include states with thin-disk like morphologies as seen in many galaxies is also open.
Finally, there are opportunities to extend this research to new settings such as the case of massless particles and coupled black hole-matter systems.
We believe that some of the open problems highlighted here are of central importance for the understanding of nature.

Full text

Classical and Quantum Gravity
Class. Quantum Grav. 41 (2024) 073001 (19pp) https://doi.org/10.1088/1361-6382/ad29e7
Topical Review
Stationary solutions of the axially
symmetric Einstein–Vlasov system:
present status and open problems
Ellery Ames1,∗and Håkan Andréasson2
1Department of Mathematics, California Polytechnic University, Humboldt,
Arcata, CA 95521, United States of America
2Mathematical Sciences, Chalmers University of Technology, University of
Gothenburg, SE-41296 Gothenburg, Sweden
E-mail: ellery.ames@humboldt.edu and hand@chalmers.se
Received 27 September 2023; revised 7 January 2024
Accepted for publication 15 February 2024
Published 1 March 2024
Abstract
The purpose of this work is to review the status about stationary solutions of the
axially symmetric Einstein–Vlasov system with a focus on open problems of
both analytical and numerical nature. For the latter we emphasize that the code
used to construct stationary solutions in Ames et al (2016 Class. Quantum
Grav. 33 155008; 2019 Phys. Rev. D99 024012) is open source, see Ames
and Logg (2023 J. Open Source Softw. 85979). In the analytical setting the
open problems include establishing methods for proving existence of axisym-
metric stationary solutions which are far from spherically symmetric, both in
the general case and for certain special classes of solutions pointed out in the
text. In the numerical setting there are intriguing properties of highly relativ-
istic solutions that demand further attention, such as whether a sequence of
such stationary solutions can approach a Kerr black hole, or if they necessar-
ily approach the thin ring limit reminiscent of cosmic strings. The question
of whether stationary solutions include states with thin-disk like morpholo-
gies as seen in many galaxies is also open. Finally, there are opportunities to
extend this research to new settings such as the case of massless particles and
∗Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution
4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the
title of the work, journal citation and DOI.
© 2024 The Author(s). Published by IOP Publishing Ltd
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Class. Quantum Grav. 41 (2024) 073001 Topical Review
coupled black hole-matter systems. We believe that some of the open problems
highlighted here are of central importance for the understanding of nature.
Keywords: relativity, stationary solutions, Einstein–Vlasov
1. Introduction
There are many reasons why studying the axially symmetric Einstein equations coupled to
matter is of great interest. In 1955 Wheeler wrote in his seminal work on geons [50]:
The simple toroidal geon forms the most elementary object of geon theory much
as a simple circular orbit constitutes the rst concept of planetary theory. But
the simplest physics does not go in the geon case with the simplest mathematics.
Wheeler here expresses the wish to study his ideas on geons in the axially symmetric
case but due to the complicated nature of the equations in that case he instead invented a
spherically symmetric toy model which he called ‘Idealized spherically symmetric geons’.
Although the research eld has made tremendous progress since this work was published,
analytic results about Einstein’s equations coupled to matter for isolated bodies are still very
limited in the axially symmetric case. The development of numerical solutions has perhaps
been even more dramatic since Wheeler’s work from 1955 (where Wheeler in fact solved the
Einstein equations numerically in the spherically symmetric case) but numerical studies about
Einstein’s equations coupled to matter often still concern toy models for the matter such as dust
or a scalar eld. There are of course exceptions, but at least for the Einstein–Vlasov system
only a few investigations have been carried out. Shapiro and Teukolsky were pioneers and ini-
tiated an important study in the early 90’s on the axisymmetric Einstein–Vlasov system. They
considered both the dynamical case [43–45] and the stationary case [46,47]. More recently
this line of research has been continued in [2–5]. Many questions have been answered in these
works but several central questions remain open.
The purpose of the present contribution is to review the status about stationary solutions of
the axially symmetric Einstein–Vlasov system, including both analytical and numerical work,
and in particular to discuss open problems. We hope this discussion will stimulate activity in
the eld so that progress can be made on central problems. The open source code GECo [6],
or further development of this code, may be useful in addressing some of the open problems
discussed below.
In order to put the discussion below into context, in the remainder of this section we briey
discuss the spherically symmetric setting from a numerical perspective in section 1.1, and in
section 1.2 include some equations and concepts from the axially symmetric literature. For a
general background on the Einstein–Vlasov system we refer to [1,10,39]. In section 2we
briey summarize the known results in the eld. Further discussion of results in the context
of open problems is presented in section 3. Finally in section 4we make a few remarks on the
evolutionary setting.
1.1. Comparison of numerical methods in the spherically symmetric setting
Spherically symmetric static solutions to the Einstein–Vlasov system are well-studied both
analytically and numerically, see [10]. There is a crucial difference in how solutions in the
spherically symmetric case and solutions in the axially symmetric case are obtained. In the
former case the equations can be solved as an initial value problem in the sense that given data
at r=0, a non-linear system of integro-differential equations can be solved straightforwardly,
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Class. Quantum Grav. 41 (2024) 073001 Topical Review
at least numerically. In this way a large variety of solutions can be constructed. An example is
the class of highly relativistic multi-peak solutions that was discovered in [17]. These solutions
are believed to be unstable, but still they are easy to obtain numerically.
In the axially symmetric case the situation is different. The numerical methods that have
been used in previous works [2,3,46,47] rely on an iteration procedure which must converge
in order to produce a solution. Such iterative methods have also been employed in the spher-
ically symmetric setting, where it is observed that convergence is only achieved for a certain
range of the parameters. In fact there are indications that this convergent set corresponds with
solutions which are dynamically stable. In particular, the highly relativistic multi-peak solu-
tions mentioned above are not accessible with the iterative method. An analysis of the xed
point map discussed above would provide useful insight on this observation—see discussion in
section 3.1. For now, these numerical studies provide some evidence that the solutions obtained
by the iteration procedure used in [2,3] are dynamically stable.
1.2. The axisymmetric Einstein–Vlasov system
In order to review the topic and to describe open problems in a meaningful way we nd it
useful to formulate the system of equations and to introduce some of the quantities that we use
to characterize the solutions.
1.2.1. Equations. The Einstein–Vlasov system consists of the coupled equations for the met-
ric tensor gand density function fon phase space, which in arbitrary coordinates and geometric
units (G=c=1) reads
Ric(g)ij −1
2R(g)gij =8πT(g,f)ij ,pi∂xif−Γk
ij (g)pipj∂pkf=0,
where the second equation is called the Vlasov equation. Here Ric(g),R(g)are the Ricci tensor
and scalar of the metric g,Γk
ij(g)are the Christoffel symbols of the metric g, and Tij(g,f)
is the energy momentum tensor associated with the Vlasov matter. In a coordinate frame
(p0,p1,p2,p3)on the tangent space at x∈Mthe energy momentum tensor takes the form
Tij (x) := ˆPx
pipjf(x,p)−detg(x)
−p0
dp1dp2dp3(1)
where Pxis the mass shell at point xand p0=g0ipi. For the stationary axisymmetric spacetimes
considered in the papers [2,3] the metric is written in axial coordinates (t,ρ, z, ϕ)as
g=−e2νdt2+e2µdρ2+e2µdz2+ρ2B2e−2ν(dφ−ωdt)2,(2)
where the metric elds ν,µ, B, ω depend only on the coordinates ρ,z. Note that ρ=0 is the
axis of symmetry, and that (ρ,z)are cylindrical coordinates at innity in the sense that in the
appropriate limit ρis the radius of the symmetry group orbits. The metric eld ωidentically
vanishes for solutions with no net rotation.
In order to construct stationary solutions it is useful to make an ansatz for the density func-
tion fon phase space, namely we assume that fdepends only on the particle energy Eand
angular momentum Lzabout the axis
f(x,p) = KΦ(E,Lz).(3)
Here Kis a positive constant, which we call the amplitude, and Φis a given function. In
principle the amplitude could be incorporated into Φbut since it plays an important role in the
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Class. Quantum Grav. 41 (2024) 073001 Topical Review
numerical xed point methods used in [2,3,46,47] we separate it out. The quantities Eand
Lzare given by
E=−g∂t,pi=e2νp0+ω(ρB)2e−2νp3−ωp0
Lz=g∂ϕ,pi= (ρB)2e−2νp3−ωp0,
which are constant along the geodesic ow. As a consequence, when fis given by the ansatz (3),
the Vlasov equation is satised.
In order to work with the integral expressions in the energy momentum tensor it is useful
to introduce new momentum variables as follows
v0=eνp0,v1=eµp1,v2=eµp2,v3=ρBe−νp3−ωp0.(4)
In terms of these coordinates the energy momentum tensor can be written as
Tij (x) = ˆPx
pi(v)pj(v)f(x,v)d3v
1+|v|2.(5)
where the mass shell condition gijpipj=−1, expressed in the new variables as (v0)2=1+|v|2,
has been used to eliminate p0. We take the positive root representing that all particles move
forward in time. The particle angular momentum and energy can be expressed respectively as
Lz=ρBe−νv3=: ρs
and
E=eν1+|v|2+ωLz=: h+ωρs.
By plugging in the ansatz for the density function in the expression for the energy
momentum tensor the components become integral expressions in the metric elds. The EV
system then takes the form a non-linear system of integro-partial differential equations which
we state for completeness:
∆ν=4πΦ00 + Φ11 +1+ (ρB)2e−4νω2Φ33 +2e−4νωΦ03(6)
−1
B∇B· ∇ν+1
2e−4ν(ρB)2∇ω· ∇ω,
∆B=8πBΦ11 −1
ρ∇ρ· ∇B,(7)
∆µ=−4πΦ00 + Φ11 +(ρB)2e−4νω2−1Φ33 +2e−4νωΦ03(8)
+1
B∇B· ∇ν− ∇ν· ∇ν+1
ρ∇ρ· ∇µ+1
ρ∇ρ· ∇ν
+1
4e−4ν(ρB)2∇ω· ∇ω, (9)
∆ω=16π
(ρB)2Φ03 + (ρB)2ωΦ33−3
B∇B· ∇ω+4∇ν· ∇ω(10)
−2
ρ∇ρ· ∇ω, (11)
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Class. Quantum Grav. 41 (2024) 073001 Topical Review
where ∆u:= ρ−1∂ρ(ρ∂ρu) + ∂z∂zuand ∇u= (∂ρu,∂zu). The matter components are given
by
Φ00 =e2µ−2νT00 (12)
=2π
Be2µ−2νˆ∞
eνˆsl
−sl
E(h,s)2Φ(E(h,s), ρs)dsdh,
Φ11 =Tρρ +Tzz
=2π
B3e2µ+2νˆ∞
eνˆsl
−sls2
l−s2Φ(E(h,s), ρs)dsdh,(13)
Φ33 = (ρB)−2e2µ+2νTφ φ (14)
=2π
B3e2µ+2νˆ∞
eνˆsl
−sl
s2Φ(E(h,s), ρs)dsdh,
Φ03 =e2µ+2νT0φ(15)
=−2π ρB−1e2µ+2νˆ∞
eνˆsl
−sl
sE(h,s)Φ(E(h,s), ρs)dsdh.
Here
sl:= Be−νe−2νh2−1.(16)
There are also auxiliary equations which we choose not to write out but refer to [2].
The system above must be complemented with boundary conditions. The following condi-
tions guarantee that the solutions are asymptotically at
ν,µ, ω →0,and B→1,as r=|(ρ, z)|→∞.
In addition we require that the metric is regular at the axis which implies that
ν(0,z) + µ(0,z) = lnB(0,z)(17)
for all zin the solution domain.
1.2.2. Solution characteristics. Our numerical solutions may be characterized by several
quantities. In this review we only use a subset of these quantities and we refer to [2,3] for a
more complete picture.
Two important properties of a solution are the total ADM mass Mand the total angular
momentum J. We use the Komar expression for the mass and obtain
M:= 2πˆ∞
−∞ ˆ∞
0
g(ρ,z)ρdρdz,(18)
where
g(ρ,z) = e2µ−2νρBT00 +ρB(Tρρ +Tzz ) + e2µ+2ν
ρBTφ φ −e2µ−2νρBω2Tφ φ .
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Class. Quantum Grav. 41 (2024) 073001 Topical Review
The mass plays an essential role in our iteration scheme. At each step of the iteration the
ansatz function is renormalized such that the total mass is unity. We also have a Komar integral
expression for the total angular momentum
J=−2πˆ∞
−∞ ˆ∞
0
e2µ−2νB(T0ϕ+ωTϕ ϕ)ρdρdz.(19)
The total angular momentum is particularly important when we construct highly compact
solutions.
An important measure of our solutions is the radius of support of the matter distribution. In
spherical symmetry the ratio 2M/R0, where R0is the radius of support in areal coordinates,
is a measure of how relativistic a solution is. The ratio 2M/R0is often referred to as the
compactness ratio.
If we express the metric (2) in spherical coordinates, the radial coordinate ris the isotropic
radius. This can be related to the areal radial coordinate Rthrough
R=r(1+M/(2r))2.
In this paper we denote the areal radius of support by R0and the isotropic radius of support
by r0. For a spherically symmetric solution the radius of support can be determined from the
cutoff energy E0(which is specied in the ansatz function) by the matching condition with a
Schwarzschild exterior. The expression in terms of both the areal and isotropic coordinates is
E0=1−2M/R0= (1− M/(2r0))/(1+M/(2r0)) .(20)
In spherical symmetry a black hole forms if the mass Mbecomes conned within a
Schwarzschild radius of R=2M. The compactness 2M/R0is thus a useful characteriza-
tion of the solution. There is no such well-dened criteria in axisymmetry. In our setting, a
natural measure of the radius is the length of the axisymmetric Killing vector eld which we
denote Rcirc := ρBe−ν. This quantity provides a natural length scale for the solution, in particu-
lar when restricted to the reection plane (z=0) and evaluated near the boundary of the matter.
For Vlasov matter, which typically has an extended atmosphere, it is useful to take the radius at
which the compactness inside a cylinder of radius ρis maximum. We dene the compactness
parameter Γ := maxρ∈(0,∞)2m(ρ)/¯
Rcirc(ρ), where ¯
Rcirc := (Rcirc)|z=0and where
m(ρ) := 2πˆ∞
−∞ ˆρ
0
g(˜ρ,z) ˜ρd˜ρdz.(21)
Note that m(ρ) = Mwhen ρexceeds the matter support. For the regular solutions we construct
Γ∈(0,1).
2. Brief survey of known results
In this section we briey summarize the current knowledge regarding stationary axisymmetric
solutions of the Einstein–Vlasov system. Relevant details are discussed below in section 3in
the context of open problems.
2.1. Numerical
Stationary solutions to the axially symmetric Einstein–Vlasov system were for the rst time
constructed numerically in 1993 by Shapiro and Teukolsky in [46,47]. In these works several
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Class. Quantum Grav. 41 (2024) 073001 Topical Review
sequences of stationary solutions are investigated with prolate and toroidal spatial density pro-
les, and include solutions with non-vanishing total angular momentum. An exciting problem
that these works left open was the question whether or not stationary solutions can be con-
structed which admit ergoregions. In numerical work together with Anders Logg, the authors
answered this question afrmatively in [2]. In this work stationary solutions with a variety of
morphologies are generated, including disk-like, spindle-like, toroidal, and solutions formed
from a composition of ansatz functions (multi-species solutions). See [6] for the numerical
code used to generate these solutions.
Further investigations of highly rotating and relativistic sequences of toroidal solutions were
carried out in [3]. The numerical results suggest two distinct possible limiting spacetimes
depending on whether the total angular momentum is larger or less than the mass squared.
Sequences for which Jbecomes less than M2eventually terminate, presumably becoming
unstable to gravitational collapse. Sequences for which Jstays larger than M2approach what
we dub the ‘thin-ring limit’, for which the solutions appear to have properties similar to those
of cosmic strings, in particular a locally conical geometry about the string characterized by
a decit angle. A bisection search is carried out that tunes between these two extremes, and
the results suggest a possible quasistationary transition to an extremal Kerr black hole for the
critical solution sequence.
In all of the numerical studies just cited the reduction scheme presented in section 1.2.1
and an ansatz of the form equation (3) is used. The ansatz function Φ(E,Lz)is taken to have a
product structure, i.e.
Φ(E,Lz) = ϕ(E)ψ(Lz).(22)
In many cases the ansatz function for the energy is taken to be the polytropic one ϕ(E)=(E0−
E)k
+, where (x)+=xif x⩾0 and (x)+=0 if x<0, and E0,kare parameters. The parameter E0
has a natural interpretation as the cutoff energy for the particles, and its presence is important
for obtaining solutions with compact support [38]. Different choices of ψresult in different
morphologies for the spatial density, among other properties. We note that if ψis an even
function the resulting solutions have zero net angular momentum. If instead the ansatz function
ψ(Lz)vanishes for Lz<0 then all particles have angular momentum of the same sign. Solutions
generated by such an ansatz have a net angular momentum and we call them rotating.
2.2. Analytic
In 2011 the existence of static axially symmetric solutions was shown for the rst time by the
second author together with Andréasson et al in [15]. In this case the total angular momentum
vanishes. This result was generalized in 2014 to the stationary, rotating case [16], and extended
to the Einstein–Vlasov–Maxwell case (in which the particles have charge) in 2020 [48]. These
analytic results rely on a perturbation argument with the consequence that the constructed
solutions can only be guaranteed to deviate slightly from spherically symmetric solutions and
to have small total angular momentum. Essentially the same system of equations as presented
in section 1.2 is used in the works [15,16], with the technical difference that perturbation
parameters corresponding to the angular momentum and relativistic nature of the solutions are
introduced. The authors also make use of an equation for ξ:= ν+µ.
In addition to the existence results discussed above, there is a recent result by Jabiri [29]
which uses a related method to construct stationary solutions. The solutions are obtained as
bifurcations from the Kerr spacetime and thus give a generalization to the case where a black
hole is surrounded by Vlasov matter.
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Class. Quantum Grav. 41 (2024) 073001 Topical Review
We mention also that in the static and cylindrically symmetric setting (in which the solution
has unbounded support in one-direction), the existence of solutions and compact support in
the radial direction is proved in [27].
3. Discussion and open problems
In this section we discuss properties of the solutions that were constructed analytically in [15,
16] and numerically in [2,3] with the aim of formulating open problems, both analytical and
numerical. The rst part concerns existence of analytic solutions and ideas on how to extend
previous methods to more general ones. The second part concerns highly relativistic compact
solutions where ergoregions, the quasistationary transition to black holes and the thin ring limit
are discussed. The third part concerns models of galaxies where the aim is to nd solutions
with the morphology of galaxies as observed in nature.
3.1. Existence of far from spherically symmetric axially symmetric solutions
A common feature of the existence results is that the solutions are obtained as perturbations
of known solutions. Hence solutions which are far from spherically-symmetric in the sense of
spatial density and net angular momentum—in particular, solutions containing ergoregions—
are not covered by these results. Accordingly, there is extensive room for analytic progress on
the existence of stationary solutions.
One way to attack this problem is to mimic the numerical approach, i.e. to analytically
investigate the iteration scheme of the numerical algorithm and show convergence in some
domain of parameter space. In fact there are several reasons why this may be of importance,
some of which are discussed below. To this end, let us describe the problem in the simplied
case of the spherically symmetric Vlasov–Possion system. This is the Newtonian analogue
of the Einstein–Vlasov system. Although existence of static solutions to the Vlasov–Poisson
system is well understood, it is nevertheless an interesting question if existence can be shown
via the method suggested by the numerical algorithm.
The Vlasov–Poisson system reads
∂tf+v·∂xf−∂xU·∂vf=0,
∆U=4πρ, lim
|x|→∞ U(t,x) = 0,
ρ(t,x) = ˆf(t,x,v)dv.
This system has the same general structure as the Einstein–Vlasov system but it is much less
involved. The aim is to prove the existence of static solutions by the following strategy. Let an
ansatz Φ(E,L)for the particle distribution be given, i.e.
f= Φ(E,L).(23)
Here the particle energy Eand the modulus of angular momentum Lare given by
E=1
2|v|2+U(x),L=|x×v|2.
For a given spatial density ρ=ρ(x)of nite mass M>0 and compact support we dene its
induced gravitational potential by
Uρ(x) = −ˆρ(y)
|x−y|dy.
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Class. Quantum Grav. 41 (2024) 073001 Topical Review
If Uρis substituted into the ansatz (23) we obtain a new spatial density
˜ρ(x) = ˆΦ1
2|v|2+Uρ(x),Ldv.
We now dene an amplitude
K(˜ρ) = Mˆ˜ρ(x)dx−1
so that the new spatial density K(˜ρ) ˜ρagain has mass M. Let us consider the map T:D→D
dened by T(ρ) = K(˜ρ)˜ρ, where
˜ρ=ˆΦ1
2|v|2+Uρ,Ldv.
Here Dis a suitable domain. If this map has a xed point ρ∗then
ρ∗=K(ρ∗)−1ˆΦ1
2|v|2+Uρ∗,Ldv,
so that (f∗,ρ∗,Uρ∗)is a static solution of the Vlasov–Poisson system where f∗is given by the
new ansatz
f∗=K(ρ∗)−1Φ(E,L).
Hence the exact problem that is solved is a priori not known, it is determined once K(ρ∗)is
known. It is an open problem to show that the map Thas a xed point.
Similar xed point problems, based on the same strategy, can be formulated for several
related systems; e.g. the spherically symmetric Einstein–Vlasov system and the axially sym-
metric Einstein–Vlasov system. It is an open problem in each case to show the existence of a
xed point.
3.1.1. Motivations. One reason why this problem is important is obvious, it would give strong
support that the numerical solutions obtained by this method are true solutions. Moreover,
progress on the xed point problem could give a new method to generate static solutions in
cases where previous strategies have failed. We have not only in mind the axially symmetric
Einstein–Vlasov system but also in the case of the Vlasov–Poisson system there is hope that
this method can be used to construct new solutions, namely at steady states. Presently there
are only limited results in the literature about such solutions, see [37], and they are of interest
as models of disk galaxies, see [18].
A further interesting aspect of this method is related to stability. Namely, as mentioned
above, there are indications that solutions obtained by this algorithm are dynamically stable.
This observation is particularly exciting in view of the open problem of non-linear stability
for the spherically symmetric Einstein–Vlasov system. Hence, if a link between non-linear
stability and the static solutions obtained by this algorithm can be established it would certainly
be of great interest.
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Class. Quantum Grav. 41 (2024) 073001 Topical Review
3.2. Highly relativistic solutions
In this section we discuss properties of highly relativistic solutions, by which we mean that
the compactness ratio Γis large. Such solutions contain ergoregions resembling black hole
solutions. A central open problem is whether or not a quasistationary transition to an extremal
black hole is possible. This is discussed below.
3.2.1. Ergoregions. An exciting result in [2,3] was the discovery that there exist regular
stationary toroidal solutions of the Einstein–Vlasov system which admit ergoregions (see for
example gures 3 and 4 in [3]). An ergoregion is typically associated with a Kerr black hole but
not with a regular stationary solution. The denition of an ergoregion is that the Killing eld ∂t
which corresponds to the time translation symmetry becomes spacelike. In our parametrization
of the metric it follows that an ergoregion is the set for which
e2ν−ρ2B2ω2e−2ν<0.(24)
In both [2,3] in which ergoregions were observed, the following ansatz function was used
Φ(E,Lz)=(E0−E)0
+(Lz−L0)0
+.
This takes the form of equation (22) with k=0, and ψ(Lz)taking a similar polytropic form.
Note however that the parameter L0represents a lower bound, resulting in a solution with
net angular momentum. In order to obtain sufciently relativistic solutions we construct a
sequence of solutions and ‘gently’ decrease the parameter E0, seeding the solver for each new
solution with the previous solution in the sequence.
The solutions admitting ergoregions that we obtain are all highly relativistic and highly
rotating, each satisfying the inequality
|J | >M2.(25)
We note that the Kerr metric for which (25) holds possesses a naked singularity. Hence, a
regular stationary solution satisfying (25) is likely stable (with respect to axially symmetric
perturbations) in view of the weak cosmic censorship conjecture. A highly relativistic solu-
tion for which (25) does not hold is most likely dynamically unstable; it will collapse to a
Kerr black hole if it is perturbed, even in axisymmetry. This is consistent with the observation
above in section 1.1 that there are indications that our numerical algorithm only converges to
dynamically stable solutions, and hence we are unable to obtain solutions with ergoregions
for which |J | <M2. The above discussion is illustrated in gure 1where convergence is lost
when E0<0.72 for the sequence which does not satisfy the inequality (25). Perhaps a differ-
ent numerical scheme will be more successful in nding solutions with ergoregions and with
|J | <M2.
As alluded to in section 2.1, Shapiro and Teukolsky also looked for solutions containing
ergoregions in [46]. They used a delta function based ansatz for both the particle energy and
angular momentum3, and obtained highly rotating and relativistic solutions. At the limits of
the resolution available at the time, Shapiro and Teukolsky were able to compute solutions
corresponding to an E0-value of about 0.745. In the step-function based ansatz studied in [2,3],
solutions with an ergoregion appear in the sequences of solutions studied at around E0≈0.66.
Given this, one might speculate that the solutions obtained in [46] were not relativistic enough
3Solutions obtained from distributions with imposed delta functions are typically not representative of general solu-
tions to the Einstein–Vlasov system, see for example [5].
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Class. Quantum Grav. 41 (2024) 073001 Topical Review
Figure 1. Ratio of total angular momentum to total mass squared. Reprinted (gure)
with permission from [3], Copyright (2019) by the American Physical Society.
to contain ergoregions. It would be interesting to look for ergoregions in families of solutions
obtained from other (non-step function) ansatzes.
3.2.2. Quasistationary transition to black holes. The presence of ergoregions suggests that
the sequence of stationary regular solutions may be approaching a family of rotating black
hole solutions. Such a quasistationary transition to black hole solutions does not occur for the
spherically symmetric Einstein–Vlasov system due to a Buchdahl type inequality, which for a
body of mass Mand radius Rreads 2M/R<8/9, see [8]. Hence there is a gap and 2M/R
cannot be arbitrary close to one. However, if one allows for charge a similar bound relating the
mass, radius, and total charge is known [9], and in this case there is no gap; a quasistationary
transition to a Reissner–Nordström black hole may thus be possible in spherical symmetry.
Indeed, such a transition to the extremal Reissner–Nordström black hole has been shown by
Meinel and Hütten [34] in the case of charged dust. Since charge is often considered as the
poor man’s angular momentum it is natural to ask if a similar transition is possible for rotating
solutions. We note that in the case of disk solutions for dust, Meinel [31–33] has answered
this question afrmatively by analytic arguments. More general cases have been investigated
numerically by Ansorg et al [19,26,33].
As discussed in section 2.1 this question is investigated in [3]. The angular momentum
parameter L0is tuned between highly rotating solutions, and more slowly rotating solutions.
For each value a sequence of solutions with gradually decreasing E0values is constructed, as
shown in gure 1. The high L0solution sequences approach the thin-ring limit, while the low
L0solution sequences eventually terminate, presumably becoming unstable to gravitational
collapse. Each such sequence, consisting of tens of solutions, is computationally expensive
to complete. The most relativistic solutions we compute are with L0=0.806 25. As shown in
gure 1, the ratio J/M2becomes very close to 1 as E0↘0.58. Indeed, the compactness Γ
also obtains its maximum value of roughly 0.8 at E0=0.58, before decreasing again as the
solution sequence bends towards the thin-ring limit (see [3] gure 2(b)). An extremal Kerr
black hole has a compactness of Γ = 1.
Given the presumed instability of solutions with J ∼ M2, and the conjectured relationship
between stability and the iterative solution method, it is not surprising that obtaining solutions
very near the extremal Kerr solution, even when Jis just greater than M2, is challenging. With
more computational effort and resolution can one continue this bisection search and reach all
the way to a black hole solution? It is not clear that the answer to this question is afrmative.
The numerical method used in the uid case, see [19,26,33], is different from the one used in
[3] and it is not straightforward to adapt that method to the Einstein–Vlasov system. Hence,
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Class. Quantum Grav. 41 (2024) 073001 Topical Review
from this discussion an essential question arises: is a transition of stationary solutions to black
hole solutions possible? Although such a transition has been established in the case of dust
there is no a priori requirement that a quasistationary transition to black hole solutions must
occur for solutions of the Einstein–Vlasov system.
Since solutions of the Einstein–Vlasov system share many properties of solutions of eld
theoretical models such as the Einstein–Dirac system, see [12], we nd it to be a question
of fundamental importance to understand whether or not such a transition is possible. Perhaps
there is a maximum value of the compactness parameter Γwhich is strictly below one as in the
spherically symmetric case. Perhaps solutions necessarily approach the thin ring limit when
E0is successively decreased independently of the value of L0. This would be surprising, since
as mentioned above, static spherically symmetric charged solutions exist which are arbitrary
close to black hole solutions, see [9,13]. It may be that a different numerical algorithm is
required to answer these questions or it may be that higher resolution is sufcient.
3.2.3. Existence of a thin-ring limit to sequences of rotating toroidal solutions. The thin-ring
limit discussed in the section above has interesting properties of its own. As shown in [3] the
limiting members of solution sequences approaching this limit appear to display a local conical
geometry around the matter, reminiscent of cosmic strings. Due to the near Dirac nature of the
spatial matter distribution in this limit (i.e. it is highly focused on a ring), proving existence
of such solutions may be analytically tractable. Indeed, in the spherically symmetric case it
was utilized in [7] that the highly compact solutions approached an innitely thin shell where
the matter sources become Dirac distributions. This feature was essential for deriving upper
and lower bounds on the compactness of the solutions, i.e. upper and lower bounds on 2m/r.
Perhaps a similar study is possible in the axially symmetric case by utilizing the Dirac nature
of the thin ring solutions.
3.3. Models of galaxies
One goal in [2] was to construct solutions which resemble galaxies as observed in nature.
While the solutions obtained in [2] make no attempt to be full-edged galaxy models, it is still
of interest to compare their properties with real galaxies in order to better understand which
properties of galaxies are captured by the fundamental axially symmetric gravitational physics.
We point out that models of galaxies are most often obtained within the Newtonian frame-
work, see [23] for models based on the Vlasov–Poisson system. One reason why relativistic
models could be of interest in this context is that net rotation has no effect on the gravitational
eld in Newtonian gravity, yet it noticeably alters the geometry in general relativity, even for
solutions which are not very relativistic.
3.3.1. Solutions with disk-like morphology. An important class of galaxies are disk galaxies
such as the Milky Way. Given the prevalence of disk galaxies in the observable Universe, it is
reasonable to ask whether such a morphology is represented in the space of stationary solutions
of the Einstein–Vlasov system. There exist models of disk galaxies which are conned to the
plane, see e.g. [18,42], but it would be desirable to obtain three-dimensional solutions which
are disk-like in the sense that the core of the density is as close to planar as possible.
An ansatz in which the z-component of the angular momentum is taken to have a Gaussian
prole generates oblate spheroidal solutions. The ansatz is given by
ψ(E,Lz) = 1
L0
expL2
z/L2
0.(26)
12

Class. Quantum Grav. 41 (2024) 073001 Topical Review
Figure 2. L0-parameterized sequence of disc-like solutions in the Einstein–Vlasov
model. Reproduced from [2]. © IOP Publishing Ltd. All rights reserved.
This ansatz is symmetric in Lzand therefore generates non-rotating solutions of the Einstein–
Vlasov system. A rotating version can be constructed by additionally imposing that all particles
have Lzof the same sign. In the limit L0→ ∞ the ansatz becomes independent of Lz, thus gen-
erating a spherically symmetric spatial density. As L0is decreased, particles with higher angu-
lar momentum are more heavily weighted compared to those with low angular momentum.
This yields a more attened shape.
In [2] the attening of such non-rotating oblate spheroids is studied. Parameters kand E0
for the particle energy are xed, while the parameter L0is gradually reduced from 10 to 1.5.
Figure 2shows the spatial density for a selection of solutions. Within the parameter range
shown the spatial density distribution stretches to its most attened form, while for parameters
L0<1.5 our numerical algorithm does not converge. We interpret this lack of convergence as a
failure of the conguration to remain gravitationally bound. The nal solution in the sequence
does not have a very attened form, though higher density contours appear far more disk-like
than lower density contours—see also gures 9 and 10 in [2].
In [2] models with net angular momentum are also studied. It is clear that rotation has an
effect on the geometry, and hence the morphology, but it does not give rise to the very attened
form that we look for (see for example gure 9 of [2]). It should be stressed that a systematic
investigation of this effect was not carried out in [2] but it would be surprising if merely an
increase of the angular momentum would result in sufciently attened solutions.
In conclusion, our solutions do not satisfyingly resemble the extremely attened galaxies
which are observed in nature. We nd it to be a central open problem to answer the question
whether or not it is possible to obtain solutions of the Einstein–Vlasov system which resemble
realistic disk galaxies. Perhaps a different choice of ansatz function will work out, or a different
numerical method, or that it is simply not possible. It would be a great contribution to nd out
the answer to this question.
A different aspect is that it is often claimed that halos of dark matter may be crucial to the
pronounced attened shape of disk galaxies. It is a question of great importance to investigate
if such halos surrounding disk-like solutions do improve the convergence of the numerical
13

Class. Quantum Grav. 41 (2024) 073001 Topical Review
algorithm. If this turns out to be the case, it would give signicant support to the established
claim that disk galaxies are embedded in dark matter halos. To our knowledge this has not been
investigated for the Einstein–Vlasov system and we nd it to be an essential open problem.
3.3.2. Spindle-like, toriodal and composite solutions. In [2] we also construct spindle-like
and toroidal solutions similar to what was done in [46]. The ansatz function we use for spindle
solutions is given by
ψspindle (Lz) = (1−Q|Lz|)l,|Lz|<1/Q
0,|Lz|⩾1/Q,(27)
and the ansatz function for the toroidal solutions takes the form
ψtorus (Lz) = (|Lz| − L0)l,|Lz|>L0
0,|Lz|⩽L0.(28)
Here Q,L0and lare parameters. As a remark we mention that the ansatz (27) was used for
the Vlasov–Poission system in [18] to investigate the rotation velocities of stars in galaxies. In
that setting, this ansatz gives rise to at rotation curves similar to those found in observations.
Interestingly, astrophysical objects with spindle-like and toroidal structures have recently
received attention by astrophysicists. In 2017 galaxy surveys revealed that prolate spindle-like
galaxies are much more common than previously thought [49]. In 2019 it was announced that
the VLA telescope had directly imaged a toroidal structure within an active galactic nucleus
[24]. In view of the observational evidence of these types of objects we nd that a more care-
ful study of spindle solutions and toroidal solutions is motivated, where the features of the
numerical solutions should be compared with the features of the observed objects.
In the context of galaxy morphology, let us also discuss composite objects which are
obtained by combining different ansatz functions. Examples of composite astrophysical
objects are numerous, and include disk galaxies with a central bulge, galaxies with dark matter
halos, as well as ring galaxies. In the case of the Vlasov–Poisson system there are several res-
ults in the literature about composite solutions, but for the Einstein–Vlasov system they were
rst obtained in [2].
It turns out to be sensitive to combine ansatz functions. Our numerical algorithm does not
converge for an arbitrary non-trivial linear combination of ansatz functions even if they indi-
vidually give rise to solutions. For instance, we are not able to obtain convergence by combin-
ing an ansatz for a polytropic central bulge with a toroidal ansatz. The spindle solutions turned
out to be useful in constructing composite objects. In [2] we used the following form of ansatz
function for a composite solution
Φ(E,Lz) = KsΦspindle (E,Lz) + KtΦtorus (E,Lz),
where Ksand Ktare amplitudes, or weights, of the two ansatz functions. An example of a
solution inspired by Hoag’s object [28] and obtained in this way is shown in gure 3. An open
issue is to understand which combinations of ansatz functions give rise to composite solutions
and to nd out if these solutions resemble the morphology of composite objects found in nature.
3.4. The massless Einstein–Vlasov system
A question related to the opening of this review and to the existence of compact solutions is
whether or not massless axially symmetric solutions exist. In the spherically symmetric case
14

Class. Quantum Grav. 41 (2024) 073001 Topical Review
Figure 3. Point cloud representation of a composite solution formed from spindle (27)
an torus (28) ansatzes, and inspired by Hoag’s object [28].
massless solutions exist if the solutions are sufciently compact, see [11,14]. Indeed, highly
compact spherically symmetric massless solutions have the property that the density function
vanishes at some nite radius Rand the solution can be glued to a vacuum Schwarzschild
solution at r=R. Without the gluing procedure the density function fwill eventually become
strictly positive at a sufciently large radius, see [14]. This particular feature is only present
in the massless case. Hence, the crucial question is whether or not massless axially symmetric
solutions can be constructed such that the density function vanishes outside a compact set.
If so, it would be possible to glue the solution to a vacuum solution—although no explicit
vacuum solution exists as in the spherically symmetric case.
We nd this question exciting in view of the quote by Wheeler given in the introduction.
Wheeler wanted to nd and investigate the properties of regular solutions of the Einstein–
Maxwell system. He named such solutions geons. In spherical symmetry the only solution to
the Einstein–Maxwell system is the Reissner–Nordström solution which is not regular. Hence,
spherically symmetric geons do not exist strictly speaking, although Wheeler did introduce the
concept of idealized spherically symmetric geons [50]. His ultimate wish was to study the axi-
ally symmetric case. Since an electromagnetic eld can be modeled as a photon gas [22,30],
a possible way to obtain understanding of the properties of solutions of the Einstein–Maxwell
system is to study solutions of the massless Einstein–Vlasov system. In the spherically sym-
metric case highly relativistic solutions of the two systems are indeed very similar, see [14].
Hence, the question whether or not axially symmetric solutions of the massless Einstein–
Vlasov system exist is closely related to the existence of Wheeler’s original conception of
geons.
3.5. Regular solutions about central black holes
An interesting direction in which to extend the study of regular axisymmetric stationary solu-
tions of the Einstein–Vlasov system, is to consider solutions with a central black hole. This
15

Class. Quantum Grav. 41 (2024) 073001 Topical Review
imposes inner boundary conditions at the black hole horizon. One motivation is to better under-
stand coupled matter-black hole systems and the effects that matter can have on central black
holes. Another reason such a line of investigation is of interest is to provide foundational res-
ults for current research on accretion disks. To date and to the authors’ knowledge, such studies
make use of a xed black hole background solution, as in for example [35], and focus on the
plasma physics in this extreme gravitational environment. In the purely gravitational setting,
Rioseco and Sarbach have several works studying the dynamics of Vlasov matter on black
hole backgrounds, for example [40,41].
The self-gravitating case is much more challenging. In the spherically symmetric setting,
Andréasson proves the existence of solutions with a central Schwarzschild black hole in the
massless case [11], and in [36] Rein establishes a such existence in the massive case. As men-
tioned in section 2.2, Jabiri has proved existence of self-gravitating solutions using a perturba-
tion argument about the Kerr spacetimes [29]. This result represents the rst proof of existence
for axially symmetric self-gravitating solutions about a central black hole. By the nature of the
proof however, the matter must be small (close to vacuum), and it remains an open problem to
nd general axially symmetric self-gravitating solutions of the Einstein–Vlasov system with
a central black hole.
A sensible starting place is to probe this problem numerically. One approach would be to
modify the code used in [2,3] and documented in [6]. Related studies exist in the case of
uniformly rotating uids about black holes, see for example [20,21].
3.6. Extension to the Einstein–Vlasov–Maxwell system
Another interesting direction of research, and possible extension of the code [6], is the addition
of charge. Allowing the particles to be charged (in addition to having mass) and interact addi-
tionally through the electromagnetic eld leads to the Einstein–Vlasov–Maxwell system. This
system was studied numerically in 2009 by Andréasson and coauthors [13] in the spherically
symmetric setting, and in 2020 Thaller used perturbation methods to prove existence of solu-
tions in the axisymmetric setting [48]. This result is similar to the work of [16] except that the
reference solution is a charged solution of the spherically symmetric Vlasov–Poisson system,
and as a result the particle charge is not restricted to be small. The total angular momentum
and the strength of the gravitational eld are still restricted, and an understanding of general
solutions to the axially symmetric Einstein–Vlasov–Maxwell system is lacking.
4. Remarks on the evolution problem
We have reviewed the status of stationary solutions of the axially symmetric Einstein–Vlasov
system. The question of their stability, and more generally, the fate of any axially symmetric
initial data requires an evolution code. Shapiro and Teukolsky were pioneers in developing
such a code using the particle in cell method. More recently, several results on the evolution
problem have been carried out, see [4,5,25,51]. We have in this work left out a discussion
about these results. One reason being that, to our knowledge, no evolution code is open source
and it is an extensive work to develop such a code. Needless to say there is immense room
for improvements and developments of this topic. We will not enter into this here but let us at
least bring up one open problem which is related to the discussion above. The highly compact
solutions that were constructed in [3] require very high resolution. It is an outstanding problem
to determine whether or not these solutions are stable. The high resolution needed to construct
the stationary solutions is clearly a severe obstacle. How can the high resolution needed for the
16

Class. Quantum Grav. 41 (2024) 073001 Topical Review
stationary solution be carried over to the evolution problem to make a simulation practically
feasible?
Data availability statement
No new data were created or analysed in this study.
Acknowledgments
The second author acknowledges support from the Erwin Schrödinger Institute where parts of
this work was carried out during the program ‘Spectral Theory and Mathematical Relativity’,
in June 2023.
ORCID iDs
Ellery Ames https://orcid.org/0000-0001-9444-585X
Håkan Andréasson https://orcid.org/0000-0003-4953-6001
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