Univerzita Karlova v Praze
Matematicko-fyzikálnı́ fakulta

DIPLOMOVÁ PRÁCE

Norbert Požár
Vybrané vlastnosti stacionárnı́ch axiálně symetrických polı́v obecné relativitě
Matematický ústav Univerzity Karlovy
Vedoucı́ diplomové práce: Doc. RNDr. Oldřich Semerák, Dr.
Studijnı́ program: Fyzika, matematické
a počı́tačové modelovánı́ ve fyzice a techniceDěkuji vedoucı́mu své diplomové práce, docentu Oldřichu Semerákovi, zazadánı́ zajı́mavého tématu a za pomoc při četných konzultacı́ch.

Prohlašuji, že jsem svou diplomovou práci napsal samostatně a výhradně
s použitı́m citovaných pramenů. Souhlası́m se zapůjčovánı́m práce.

V Praze dne 21. dubna 2006

Norbert Požár

Contents
1 Introduction

1

2 General relativity
2.1 Einstein equations . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Derivation from the action principle . . . . . . . . . . . . . . .
2.3 Stationary axially symmetric field . . . . . . . . . . . . . . . .
2.4 Ernst equation . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 Examples of Ernst potentials . . . . . . . . . . . . . . . . . . .
2.5.1 Minkowski solution . . . . . . . . . . . . . . . . . . . .
2.5.2 Kerr solution . . . . . . . . . . . . . . . . . . . . . . .
2.6 Einstein-Maxwell equations in stationary axisymmetric case .

4
4
5
5
6
8
8
8
8

3 Riemann surfaces
3.1 Topology of Riemann surfaces . . . . . . . . . . . . . . . . . .
3.2 Intersection theory on compact surfaces . . . . . . . . . . . . .
3.3 Differential forms . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Abelian differentials on compact surfaces . . . . . . . . . . . .
3.5 Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Abelian mapping . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Hyperelliptic surfaces . . . . . . . . . . . . . . . . . . . . . . .

11
13
14
15
17
17
19
20

4 Theta Functions
23
4.1 Riemann theta function . . . . . . . . . . . . . . . . . . . . . 23
4.2 Theta functions associated with a Riemann surface . . . . . . 24
5 Riemann-Hilbert problem
27
5.1 Scalar Riemann-Hilbert problem . . . . . . . . . . . . . . . . . 27
5.2 Matrix Riemann-Hilbert problem with quasi permutation monodromy matrices . . . . . . . . . . . . . . . . . . . . . . . . . 30
6 Solution of evolution nonlinear systems
31
6.1 Solutions of the nonlinear Schrödinger equation . . . . . . . . 31
7 Solutions of the Ernst equation
7.1 Linear system for the Ernst equation . . . . . . . . . . . . . .
7.2 Construction of the Ernst potential . . . . . . . . . . . . . . .
7.3 Symmetric linear system . . . . . . . . . . . . . . . . . . . . .

34
34
38
48

8 Numerical evaluation of the theta solutions
8.1 Numerics on a hyperelliptic Riemann surface . . . . . . . . . .
8.1.1 Square root . . . . . . . . . . . . . . . . . . . . . . . .
8.1.2 Numerical integration of the periods . . . . . . . . . .
8.1.3 Theta function . . . . . . . . . . . . . . . . . . . . . .
8.1.4 Mass and angular momentum . . . . . . . . . . . . . .
8.2 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . .

51
52
53
55
60
60
61

9 Conclusions

67

10 References68Název práce: Vybrané vlastnosti stacionárnı́ch axiálně symetrických polı́ v obecné relativitě
Autor: Norbert Požár
Katedra (ústav): Matematický ústav Univerzity Karlovy
Vedoucı́ diplomové práce: Doc. RNDr. Oldřich Semerák, Dr., Ústav teoretické fyziky
e-mail vedoucı́ho: semerak@mbox.troja.mff.cuni.cz
Abstrakt: Tato práce je věnována použitı́ metod algebraické geometrie při hledánı́ stacionárnı́ch axiálně symetrických prostoročasů jako přesných řešenı́ Einsteinových rovnic
obecné teorie relativity. Úloha se dá převést na Ernstovu rovnici – úplně integrabilnı́ nelineárnı́ parciálnı́ diferenciálnı́ rovnici druhého řádu. Ačkoli je známa řada konkrétnı́chřešenı́ této rovnice popisujı́cı́ch gravitačnı́ pole izolovaných objektů (černých děr, prstencových a diskových zdrojů) i široké třı́dy velmi obecných řešenı́ zı́skané tzv. generačnı́mitechnikami, nenı́ zatı́m plně zvládnuta okrajová úloha pro dostatečně obecnou situaci, např.
pro přı́pad vı́cenásobných zdrojů (zejména černých děr s disky či prstenci), který je nejen
teoreticky zajı́mavý, ale i astrofyzikálně významný. Jako slibné se ukazuje převedenı́ Ernstovy rovnice na odpovı́dajı́cı́ lineárnı́ úlohu, jejı́ formulace jako Riemannova-Hilbertova
problému a řešenı́ pomocı́ theta funkcı́ na Riemannových plochách. Po stručném přehledu
fyzikálnı́ motivace a nezbytných matematických partiı́ uvádı́me hlavnı́ výsledky nedávnýchpracı́ Kleina a Korotkina (et al.), přičemž upřesňujeme nebo vı́ce rozvádı́me některé bodyjejich postupu. Navrhujeme poněkud odlišný důkaz dvou klı́čových vět a také ’rychlý’ numerický kód pro vyčı́slenı́ řešenı́ Ernstovy rovnice ve tvaru theta-funkcı́ (podobný program
mezitı́m uveřejnili Klein a Frauendiener).
Klı́čová slova: přesná řešenı́ Einsteinových rovnic, Ernstova rovnice, metoda inverznı́horozptylu, algebraická geometrie, Riemannův-Hilbertův problém
Title: Selected properties of stationary axially symmetric fields in general relativity
Author: Norbert Požár
Department: Mathematical Institute of Charles University
Supervisor: Doc. RNDr. Oldřich Semerák, Dr., Institute of Theoretical Physics
Supervisor’s e-mail address: semerak@mbox.troja.mff.cuni.cz
Abstract: This thesis is devoted to the usage of the methods of algebraic geometry in finding
stationary axially symmetric spacetimes as exact solutions to Einstein equations of generalrelativity. The task can be transformed into Ernst equation – completely integrable secondorder non-linear partial differential equation. Although a number of particular solutions of
this equation are known describing gravitational field of isolated objects (black holes, ringor disc sources), as well as wide classes of very general solutions obtained by the so called
generation techniques, there remains not fully managed the boundary-value problem forsufficiently generic situation, e.g. for multiple sources (mainly black holes with discs orrings) which is both theoretically interesting and astrophysically important. It is promising
to translate the Ernst equation into the corresponding linear task, to formulate this as
Riemann-Hilbert problem and to solve the latter in terms of theta functions on Riemannsurfaces. After a brief survey of physical motivation and of necessary mathematical parts,
we give main results of the recent works by Klein and Korotkin (et al.), trying to makethem more precise and/or detailed in certain points. We suggest a somewhat different
proof of two key theorems and also a ’fast’ numerical code for evaluation of the Ernstequation solution in terms of theta-functions (in the meantime, a similar program has
been published by Klein and Frauendiener).
Keywords: exact solutions of Einstein equations, Ernst equation, inverse scattering method,
algebraic geometry, Riemann-Hilbert problem

1 INTRODUCTION

1

1Introduction

The first theory of gravitation was developed by Isaac Newton in 17th century. Since then it was successfully used to describe and explain the behaviorof the Solar system. But when the special theory of relativity appeared at
the beginning of the 20th century, problems arose, in particular with superluminal (infinite) velocity of the gravitational force. A brand new point of view
was brought by the general theory of relativity (general relativity) publishedin 1915 by Albert Einstein. It described gravitation as a manifestation of acurvature of the spacetime itself. It was in accordance with special relativity,
but it called for a completely new model of our universe. Since then manyother theories of gravitation have been developed but general relativity is
still assumed to be the correct one in a wide range of situations, although
not all of its predictions have been confirmed by experiments yet.
The fundation-stone of general relativity are Einstein equations. They
describe the interaction between physical fields and bodies and the spacetime. But they are much more involved then the simple equation for thegravitational force in the Newton theory. The Einstein equations are nonlinear because the gravitation interaction itself has energy. The equations
are so complex that even Einstein thought that no any exact solution would
be ever found. But in 1916 Karl Schwarzschild published his famous exact
solution in a special case. Since then many methods to solve the Einsteinequations were developed. One can find exact or numerical solutions, or solutions to linearized or perturbed systems. In this work, we are interested inexact solutions to Einstein equations in a stationary axisymmetric vacuum
spacetime. In this case, the Einstein equations can be reduced to one partial differential equation (Ernst equation) for one complex function known as
Ernst potential. The most interesting and useful classes of known solutions(Kerr) belong here. However, they were found “ad-hoc”. We are interested
in more systematic approach.
Exact solutions of physical equations are important for theoretical reasons as well as in understanding and modelling specific natural phenomena.
Although there are approximate (perturbative) and numerical techniques,
the outcomes of the former are typically valid within a narrow range of parameters only, while those of the latter go short of generality: not sure whether
a given numerical result represents a typical or just a marginal case, one canhardly recognise, analyse and interpret different classes of solutions. Also —from a purely mathematical view — the richness embodied in the equations
can only be fully revealed on their exact solutions.
During 1960s, new powerful methods (“inverse scattering” methods) weredeveloped to generate exact solutions of non-linear partial differential equa-1 INTRODUCTION

2

tions, based on their groups of symmetries. Such symmetries imply the existence of a certain linear system which is equivalent to the original non-linear
system in the sense that the latter is its integrability condition. The problem
is thus translated into the solution of a linear system for a matrix function
which is dependent on an additional (“spectral”) parameter. The methods
were first successfully applied to completely integrable non-linear evolution(hyperbolic) PDEs, and later to elliptic PDEs like the Ernst equation, too.
The Ernst equation is a non-linear second-order system of two equations
for the real and imaginary parts of the Ernst potential. It is completely
integrable in the Hamiltonian sense: the equations yield the same number ofconserved quantities as degrees of freedom [18]. The symmetry group is theinfinite dimensional Geroch group [2].
The study of the Ernst equation by means of Bäcklund transformations
and solitonic methods has brought a lot of new solutions [13,28,30,32–34,38].
They are given in terms of algebraic or exponential functions and can involvean infinite number of parameters [31]. However, with the increasing numberof parameters, the solutions become pretty cumbersome and only a very few
of them have been interpreted in detail up to now. It is probable that almost
all of them do not correspond to any physical situation. Namely, within theseapproaches it is very problematic to set certain physical requirements on thesolution in advance, so they yield rather “pot shot” results.
In the 1980s, a new rich class of solutions was found by the methods ofalgebraic geometry. Loosely speaking, it consists in posing the linear problem
as a Riemann-Hilbert problem which can be solved in terms of theta functions living on certain Riemann surfaces. The approach, already considered
for several other physical equations before, was employed for the Ernst equation by Korotkin [23, 24, 26, 27] and later worked out by Klein and Richter[10, 15, 16, 20, 21]. The papers from Klein and Richter were published as abook [22]. The solutions have many promising features and contain physically interesting subclasses which might hopefully describe fields of realistic
sources. However, they have not attracted much attention. One of the likely
reasons is the non-trivial theory needed for their formulation, which is still
being developed [19, 22]. Even a code for their numerical evaluation is difficult to construct except of rather special cases [10–12]. One of the purposesof the present work is to address this problem.
In section 2, we shortly introduce general relativity and the Einstein equations, we derive the Ernst equation in detail and we also briefly discuss theelectro-vacuum case based on the papers by Ernst [7, 8]. In sections 3 and 4,
we present the theory of Riemann surfaces and theta functions based on
the book Farkas & Kra [9] that is necessary for algebro-geometric methods
studied later. In section 5, we summarize the results on Riemann-Hilbert1 INTRODUCTION

3

problems that we will use later. Section 6 illustrates the algebro-geometricmethods for solving nonlinear evolution systems, it is based on the book by
Belokolos et al. [1].
We present our own results in sections 7 and 8. We focused on algebrogeometric methods for construction of the Ernst equation. We investigate two
versions of the linear system used in literature. We prove two theorems thatsummarize the properties of the matrix functions that solve these systems.
Our proofs are more detailed and somewhat more precise then those foundin papers [18, 21, 27]. We also give a very detailed proof that the function
constructed in [21] using a scalar Riemann-Hilbert problem is really a solution
to the Ernst equation. The relevance of hyperelliptic Riemann surfaces isdiscussed in a great detail.
The numerical evaluation is rather challenging. We present an implementation of a code that is powerful enough to evaluate quantities of interest on
hyperelliptic Riemann surfaces in section 8. Even though the code is far
from complete, we obtained some results that are comparable to the resultin [11, 12].

2 GENERAL RELATIVITY

2

4

General relativity

General relativity is the geometrical theory of gravitation published by Albert Einstein in 1915. It unifies special relativity and Isaac Newton’s law of
universal gravitation with the insight that gravitation is not viewed as being
due to a force (in the traditional sense) but rather a manifestation of curved
space and time, this curvature being produced by the mass-energy content
of the spacetime.
In this section, we briefly introduce Einstein equations as a complete setof equations governing the interaction between bodies or physical fields andgeometry of spacetime. We show how these equations can be simplified in
the case of stationary axisymmetric fields. At the end we discuss some simple
examples of solutions to these equations. For details, see some of the standard
book of general relativity, for example Wald [38]. For a more mathematicalpoint of view, see Hawking & Ellis [14].

2.1

Einstein equations

The main result of general relativity are Einstein equations, a system of 10partial differential equations describing interaction between mass and energy
and geometry of spacetime. The equations can be written in the standard
form
1
8πG
Rµν − Rgµν + Λgµν = 4 Tµν ,2cwhere Rµν is the Ricci tensor, R is the Ricci scalar, gµν is the metric tensor and Tµν is the stress-energy tensor. The constants G, c and Λ are thephysical constants in the equations. We will use metrized units, where thegravitational constant G and the speed of light c are set to be 1. We willassume that the cosmological constant Λ is zero.
The Ricci tensor and Ricci scalar are obtained from the Riemann curvature tensor that measures a local curvature of the spacetime. On the other
hand, the stress-energy tensor measures the amount and distribution of energy and mass in the spacetime.
Since the Bianchi identities hold for the Riemann curvature tensor, only
6 of the Einstein equations are equations for the metric functions. The other4 are the law of conservation of energy and momentum (which is a very interesting and important feature of these equations). Even though we have only
6 equations for 10 unknown metric functions, we have enough equations tofound the complete metric. The metric tensor has to be independent of the
choice of coordinates, i. e. up to a diffeomorphism given by 4 functions, andthus by fixing a coordinate system we fix 4 metric functions. This depen-2 GENERAL RELATIVITY

5

dence on a coordinate system causes problems when solving the equations
globally, because particular coordinates are good only locally and one needs
to continue the solution to other coordinate systems to obtain a solution on
the whole spacetime. It is also difficult to recognize positions of singularities of the spacetime or to see that two different solutions describe the same
spacetime.
The Einstein equations comprise a hyperbolic system of 6 nonlinear partial differential equations for 6 independent functions. The initial value problem is well-posed with initial conditions given on a 3-dimensional submanifold. The nonlinearity of the equations and the dependence on coordinates,
however, make analytic or numerical solution difficult. Some restrictions on
the metric functions and on the coordinates are usually imposed. In thiswork, we are interested in a stationary axially symmetric case in vacuum.

2.2

Derivation from the action principle

Einstein equations can be derived using an action principle (Carroll [3]). The
action (so called Hilbert action) is the integral of the simplest Lagrangiandensity over spacetimeZ
SH = LH d4 x.
(2.1)√
The Lagrange density is a tensor density, which can be expressed as −gtimes a scalar. Any nontrivial scalar must contain at least second derivativesof the metric. The only independent scalar the contains no higher then second
derivatives of the metric and it depends on the second derivatives linearly is
the Ricci scalar R. The simplest possible choice of the Lagrangian density
as proposed by Hilbert for a vacuum spacetime is therefore√(2.2)
LH = −gR.
Varying the action SH with respect to metric gµν gives the equation of motion,
the Einstein equations in vacuum. To obtain the full set of the Einsteinequation, the action has to be modified slightly.

2.3

Stationary axially symmetric field

The stationary axially symmetric solutions to the Einstein equations are
astrophysically important since they can describe the gravitational fields ofrotating bodies in equilibrium. It is generally believed that most of the stars
and galaxies can be described in a good approximation as stationary axiallysymmetric fluid bodies. In addition, they are usually reflectionally symmetric2 GENERAL RELATIVITY

6

with respect to some “equatorial” plane. A relativistic treatment is necessary
in particular for very compact objects such as neutron stars and black holes,
whose extremely strong fields are not described properly by Newton’s theoryof gravitation.
The stationary axially symmetric spacetime is characterized by two commuting Killing fields1 , the timelike2 Killing field ∂t and the spacelike Killingfield ∂φ with closed orbits. Every vacuum stationary axisymmetric metric
can always be cast into the Weyl-Lewis-Papapetrou form
ds2 = −e2U (dt + a dφ)2 + e−2U (e2k (dρ2 + dζ 2 ) + ρ2 dφ2 ),(2.3)

where t and φ are coordinates adapted to Killing fields and ρ, ζ span thepolar sections. The functions U , k and a are functions of Weyl coordinates ρand ζ only. The metric function U reduces to gravitational potential in the
Newtonian limit. a is the “gravitomagnetic” potential. In the following, we
will use a new function f = e2U .

2.4

Ernst equation

Ernst showed (see [7]) that the vacuum field equations in the stationary
axisymmetric case are equivalent to the Ernst equation for the complex potential E = E(ρ, ζ)(ℜE)∆E = ∇E.∇E,(2.4)
where ∆ is the Laplace operator in cylindrical coordinates (ρ, ζ, φ) ∆ =∂ρρ + ρ1 ∂ρ + ∂ζζ , and ∇ is a gradient in cylindrical coordinates ∇ = (∂ρ , ∂ζ , 0)(∂φ is omitted due to axisymmetry).
The Ernst equation can be derived using tensor methods from Einsteinequations, but the simpler way is to use the Lagrangian density3 (Ernst [7])11
L = − ρf −2 ∇f · ∇f + ρ−1 f 2 ∇a · ∇a.
224
Varying the functions f and a we obtain the field equations

1

f ∆f = ∇f · ∇f − ρ−2 f 4 ∇a · ∇a,∇ · (ρ−2 f 2 ∇a) = 0.

(2.5)
(2.6)

A Killing field is a vector field such that the Lie derivative of the metric along thisfield is zero.
2
Negative norm.
3
The terms with e2k are removed since they do not change the resulting formulas andthe metric function k can be expressed in terms
R of the other metric functions.
d4
L(f + th)ρdρdζ t=0 in f of the functional
We compute the Gâteaux derivative dt
Ω
R
L
in
any
direction
h
such
that
h
→
0
as
ρ
→ ∞ or ζ → ±∞. Then we can applyΩ
the Green’s theorem and we get (2.5). When we compute the Gâteaux derivative for ainstead, we get (2.6).

2 GENERAL RELATIVITY

7

If n is a unit vector in the azimuthal direction and ϕ is any reasonable
function independent of azimuth, we have the identity5∇ · (ρ−1 n × ∇ϕ) = 0.

(2.7)

ρ−1 f 2 ∇a = n × ∇b.

(2.8)

f −2 ∇b = −ρ−1 n × ∇a,(2.9)∇ · (f −2 ∇b) = 0(2.10)

Now equation (2.6) may be regarded6 as the integrability condition for the
existence of the function b = b(ρ, ζ) independent of azimuth defined byThis relation is equivalent to7

therefore the identity (2.7) implies8 the field equation
for the new potential b. When we express equation (2.5) in terms of thefunction ϕ and compare it with equation (2.10), we see that the complex
function
E = f + ib
(2.11)
satisfies the simple differential equation (2.4).
In the Weyl-Lewis-Papapetrou coordinate system (2.3), the Ernst equation has the form
1
(E + Ē)(Eζζ + Eρ + Eρρ ) = 2(Eζ2 + Eρ2 ),(2.12)
ρ
where the bar denotes complex conjugation in C. The metric functions U , k
and a can be expressed in terms of the Ernst potential E as ([18])(E − Ē)ξ
¯ Eξ Ēξ ,, kξ = (ξ − ξ)(2.13)
2(E + Ē)(E + Ē)2
where ξ is a new complex coordinate ξ = ζ − iρ.
Since ∂ξ = 12 (∂ζ + i∂ρ ) and ∂ξ̄ = 21 (∂ζ − i∂ρ ), the Ernst equation can be
rewritten using the coordinate ξ in an equivalent form
2
1
Eξ Eξ̄ .
(2.14)(Eξ − Eξ̄ ) =Eξξ̄ −¯
E + Ē
2(ξ − ξ)e2U = f = ℜE,aξ = 2ρ

The metric function U follows directly from a solution E by definition of
the Ernst potential. a and k can be obtained from (2.13) via quadratures.
The Ernst equation (2.14) is the integrability condition for (2.13).
5

2
Use the product rule and observe that ∇ × n = − ∇ρ
ρ × n + ρ ∇ζ and ∇ζ = ∇ρ × n.

This implies ∇ϕ · ( ∇ρ
ρ × n) = ∇ϕ · (∇ × n).
6(2.6) implies (2.7)
7
Apply n× to both sides.
8
Apply ∇· to both sides of (2.9) and use (2.7) for ϕ := a.

2 GENERAL RELATIVITY

2.5

8

Examples of Ernst potentials

Here we will present Ernst potentials for simple well-known solutions.
2.5.1

Minkowski solution

The Minkowski metric is the metric of flat spacetime. The corresponding
Ernst potential is constant, E = 1.
2.5.2

Kerr solution

The Kerr solution belongs to the class of stationary axisymmetric spacetimes
and thus must be a solution of the Ernst equation. Neugebauer and Meinel
showed how to derive this solution from the boundary data on the axis [34].
The Ernst potential can be expressed in the form [17]E=e−iϕ r+ + eiϕ r− − 2m cos ϕ,e−iϕ r+ + eiϕ r− + 2m cos ϕp(ζ ± m cos ϕ)2 + ρ2 . The solution is parametrized by two
where r± =parameters m and ϕ. They can be related to the Arnowitt-Deser-Misner(ADM) mass m and the angular momentum J = m2 sin ϕ. The black hole
has a horizon located on the axis ρ = 0 in the interval ζ ∈ [−m cos ϕ, m cos ϕ].
For ϕ = 0 we get the Schwarzschild solution and for ϕ = π/2 the solution
becomes the extreme Kerr solution.

2.6

Einstein-Maxwell equations in stationary axisymmetric caseIn this section, we release the condition that the spacetime is empty and allow
it to contain an electromagnetic field. We can simplify the Einstein equations(Einstein-Maxwell equations in this case) in a way similar to Section 2.4 as
was first noted by Ernst [8]. We can write the metric in the form (2.3) again.
In this case, the Lagrangian density is1
1
L = − ρf −2 ∇f · ∇f + ρ−1 f 2 ∇a · ∇a + 2ρf −1 At ∇At · ∇At22− 2ρ−1 f (∇Aφ + a∇At ) · (∇Aφ + a∇At ),where Aφ and At are the φ and t components of the electromagnetic 4potential, respectively. By variation of the 4 functions f, a, Aφ and At , we

9

2 GENERAL RELATIVITY

2

4
1

0.6
0.4
0.2
Re
0
-0.2

3
0
Ζ

2
-1

1
-2

Ρ

0
(a) ℜE

0.4
0.2
0
Im
-0.2
-0.4
2

4
1

3
0
Ζ

2
-1

1
-2

Ρ

0
(b) ℑE

Figure 1: The Ernst potential of the Kerr solution with parameters m = 0.5
and ϕ = 0.9.

10

2 GENERAL RELATIVITY
can derive the field equations in the form
(Re E + ΦΦ̄)∆E = (∇E + 2Φ̄∇Φ) · ∇E,(Re E + ΦΦ̄)∆Φ = (∇E + 2Φ̄∇Φ) · ∇Φ,(2.15)

where the new complex functions E and Φ are defined by
E = (e2U − ΦΦ̄) + ib′ ,Φ = At + iA′φ ,where b′ and A′φ are new potentials independent of azimuth, defined byn × ∇A′φ = ρ−1 f (∇Aφ + a∇At ),n × ∇b′ = −ρ−1 f 2 ∇a − 2n × Im(Φ∇Φ).

n is a unit vector in azimuthal direction. For details of the derivation, see[8] or [18]. The Ernst equation (2.4) can be obtained from (2.15) under theassumption Φ = 0.

11

3 RIEMANN SURFACES

3

Riemann surfaces

The theory of Riemann surfaces appear in many areas of mathematics. In
this thesis we will use it as a tool for the construction of exact solutions to theErnst equation. Riemann surfaces are studied in detail for example in Farkas& Kra [9], we give just a short account here. We will be especially interested
in compact hyperelliptic Riemann surfaces, because they will appear in thesolutions.
Definition 3.1. A Riemann surface is a connected one-dimensional complex
analytic manifold M with a maximal set of charts {Uα , zα }, α ∈ A, on M
such that the transition functions
fαβ = zα ◦ zβ−1 : zβ (Uα ∩ Uβ ) → zα (Uα ∩ Uβ )(3.1)

are holomorphic whenever (Uα ∩ Uβ ) 6= ∅.
The simplest Riemann surfaces are the complex plane C, the one pointcompactification, CP1 ≡ C ∪ {∞}, of C (known as the extended plane or
Riemann sphere) or any domain (connected open subset) on any Riemannsurface.
Definition 3.2. A compact Riemann surface is called closed, a non-compact
surface is called open.
We do not need to specify a maximal set of analytic coordinate charts,
merely a cover by any set of analytic coordinate charts (see [9], p. 10).
Definition 3.3 (Holomorphic mapping). A continuous mapping
f :M →N(3.2)

between Riemann surfaces is called holomorphic (or analytic) if for every
local coordinate {U, z} on M and every local coordinate {V, ζ} on N withU ∩ f −1 (V ) 6= ∅, the mappingζ ◦ f ◦ z −1 : z(U ∩ f −1 (V )) → ζ(V )(3.3)

is holomorphic (as a mapping from C to C).
Definition 3.4. A holomorphic mapping into C is called a holomorphic
function. A holomorphic mapping into C∪{∞}, other then mapping sending
all points of M to ∞, is called a meromorphic function. The vector space of
all meromorphic functions on a Riemann surface M is denoted K(M ). The
mapping f : M → N is called constant if f (M ) is a point.

12

3 RIEMANN SURFACES

A holomorphic mapping is open9 . It can be shown that if f : M → N
is a holomorphic mapping and M is compact, then f is either constant or
surjective. In the latter case, N is also compact. In particular, the only
holomorphic functions on a compact Riemann surface are constants, because
C is not compact. On the other hand, there are non-constant meromorphic
functions since the Riemann sphere C ∪ {∞} is compact.
We now want to define the multiplicity of a holomorphic mapping f .
WConsider a non-constant holomorphic mapping between Riemann surfaces
f : M → N . For a given point P ∈ M we choose local coordinates z̃ on M
vanishing at P and ζ vanishing at f (P ). In terms of these local coordinates,
we can writeXζ = f (z̃) =ak z̃ k , n > 0, an 6= 0.
z≥n

We also have

ζ = z̃ n h(z̃ n ) = (z̃h(z̃))n ,where h is holomorphic and h(0) 6= 0. z̃h(z̃) is another local coordinate
vanishing at P , and in terms of this new coordinate the mapping f is givenbyζ = zn.
(3.4)
Definition 3.5. The number n defined above is the ramification number off at P . We say that f takes on the value f (P ) n-times at P or f has
multiplicity n at P . The number (n − 1) is called the branch number of f at
P and denoted bf (P ).
The following theorem can be proved:
Theorem 3.1. Let f : M → N be a non-constant holomorphic mapping
between compact Riemann surfaces. There exists a positive integer m suchthat every Q ∈ N is assumed precisely m times on M by f , accounting for
multiplicities. That is, for all Q ∈ N ,X(bf (P ) + 1) = m.
P ∈f −1 (Q)Definition 3.6. The number m above is called the degree of f and denoted
deg f . We say that f is an m-sheeted cover of N by M.
9Open sets map to open sets.

3 RIEMANN SURFACES

3.1

13

Topology of Riemann surfaces

In this section, a curve on M is a continuous map c of the closed interval
I = [0, 1] into M . The point c(0) is called the initial point of the curve andc(1) is called the terminal or end point of the curve.
If P , Q are two points of M and c1 and c2 are two curves on M with initialpoint P and terminal point Q, we say that c1 is homotopic to c2 (c1 ∼ c2 )provided that there is a continuous map h : I × I → M with the properties
h(t, 0) = c1 (t), h(t, 1) = c2 (t), h(0, u) = P and h(1, u) = Q (for all t, u ∈ I).
In other words, one curve can be continuously deformed into the other.
We now consider all closed curves on M which pass through any point P ,of M i. e. all curves with initial and terminal point P . Curves c1 and c2 areequivalent whenever they are homotopic. The set of equivalence classes ofclosed curves through P forms an Abelian group. The sum of the equivalenceclass of the curve c1 with the equivalence class of the curve c2 is the equivalence class of the curve c1 followed by c2 . The inverse of the equivalence classof the curve c is the curve c parametrized in the reverse direction. The groupof equivalence classes constructed in this way is called the fundamental groupof M based at P .
The fundamental group based at P and the fundamental group based at
Q are almost canonically isomorphic as groups. The isomorphism between
them depends only on the homotopy class10 of the path from P to Q.
Definition 3.7. The fundamental group of M , π1 (M ), is defined to be thefundamental group of M based at P , for any P ∈ M .
For our applications, the dependence of π1 (M ) on the base point P willbe irrelevant.
The fundamental group is isomorphic the first simplicial homology group
H1 (M ) (see [9], p. 15).
For a compact Riemann surface M , it can be shown (e. g. [9]) that thefundamental group is either trivial (all closed curves are equivalent to a point)
or generated by 2g closed curves a1 , . . . , ag , b1 , . . . , bg .
Definition 3.8 (Genus). In the former case we say that the genus of M iszero and in the latter case we say that the genus is g.
Consider a non-constant holomorphic mapping f : M → N between
compact Riemann surfaces. Assume that M is a surface of genus g, N isa surface of genus γ. Assume that f is of degree n. We define the total
10

The equivalence class of homotopic curves.

14

3 RIEMANN SURFACES
branching number of f byB=X

bf (P ).

P ∈M

Theorem 3.2 (Riemann-Hurwitz Relation). We haveg = n(γ − 1) + 1 + B/2.
We will also need the following definition:
Definition 3.9 (Covering manifolds). The manifold M ∗ is said to be a covering manifold of the manifold M provided there is a continuous surjectivemap (called a covering map) f : M ∗ 7→ M with the following property: foreach P ∗ ∈ M ∗ there exist a local coordinate z ∗ on M ∗ vanishing at P ∗ , alocal coordinate z on M vanishing at f (P ), and an integer n > 0 such thatf is given by z = z ∗n in terms of these local coordinates. Here the integer ndepends only on the point P ∗ ∈ M ∗ . If n > 1, P ∗ is called a branch point oforder n − 1 or a ramification point of order n.

3.2Intersection theory on compact surfaces

A single non-negative integer, called genus, provides a complete topologicalclassification of compact Riemann surfaces. Every compact Riemann surface
is topologically equivalent to a sphere with g handles. The curves from thebasis of the fundamental group π1 (M ) are denoted {a1 , . . . , ag , b1 , . . . , bg }.
We want to construct a basis called the canonical basis of the fundamentalgroup by curves that intersect in a specific way.
Let a and b be two closed curves on the Riemann surface M . We define
the intersection number of a and b, a · b, as the number of intersections of thetype shown in Figure 2 minus the number of intersections of the type from
Figure 2 when the curves a and b are interchanged. The intersection number
is well defined and only depends on the homology classes of a and b.
Definition 3.10. Any basis {a1 , . . . , ag , b1 , . . . , bg } of H1 (M ) with the following intersection properties
aj · bk = δjk
aj · ak = 0 = b j · b k ,for all j, k = 1, . . . , g is called a canonical homology basis for M .

15

3 RIEMANN SURFACESba

Figure 2: Intersection of curves that contributes by 1 to the intersection
index a · bEvery curve (its equivalence class to be precise) can be decomposed into
the curves of a canonical homology basis byc=gX

m j aj + n j b j ,j=1

where mj , nj ∈ Z are given by
m j = c · bj ,3.3

nj = aj · c.

(3.5)

Differential forms

Let M be a Riemann surface. A 0-form on M is a function on M. A differential form (1-form or differential ) is an assignment of two continuousfunctions f and g to each local coordinate z = x + iy on M such thatf dx + g dy

(3.6)

is invariant under coordinate changes. A 2-form on M is an assignment of acontinuous function f to each local coordinate z such thatf dx ∧ dy
is invariant under coordinate changes. We shall consider differential forms ofthe typeu(z)dz + v(z)dz̄,
(3.7)
where
dz = dx + i dy,dz̄ = dx − i dy,

and comparing with (3.6) we get
f = u + v,

g = i(u − v).

16

3 RIEMANN SURFACES

A 1-form ω can be integrated over a finite union of paths. Thus, if
the piece-wise differentiable path c is contained in a single coordinate discz = x + iy, c : I → M , I = [0, 1], and if ω is given by (3.6), then

Z
Z 1
dx
dy
f (x(t), y(t)) + g(x(t), y(t))
ω=
dt.
dt
dt
c
0
The integral is independent of the choice of z.
For forms with C 1 coefficients, we introduce the differential operator d.
For C 1 functions f we define
df = fx dx + fy dy = fz dz + fz̄ dz̄,where fz = fx − ify and fz̄ = fx + ify . For a C 1 form ω given by (3.7) wedefinedω = (vz − uz̄ )dz ∧ dz̄.
Definition 3.11. A 1-form ω is called exact if ω = df for some C 2 function
f on M . ω is called closed if it is C 1 and dω = 0. A 1-form ω is called
holomorphic provided that locally ω = df with f holomorphic.
A differential ω of the form (3.7) is holomorphic if and only if v = 0 and
u is a holomorphic function of the local coordinate.
A meromorphic differential ( abelian differential) on a Riemann surface
is an assignment of a meromorphic function f to each local coordinate z suchthatf (z) dz
(3.8)
is invariant under coordinate changes.
For an abelian differential ω we define the residue of ω at P by
resP ω = a−1 ,where ω is given by (3.8) in terms of the local coordinate z that vanishes at
P , and the Laurent series of f is
f (z) =∞
X

an z n .

n=N

The smallest n such that an 6= 0 is called order of f at P and is denoted byordf P .
The abelian differentials which are holomorphic will be called of the firstkind, while the meromorphic abelian differentials with zero residues will becalled of the second kind. A general abelian differential (which may have

17

3 RIEMANN SURFACES

residues) will be called of the third kind. We define abelian integral Ω(P ) ona Riemann surface M as a meromorphic function that is given as an integralof an abelian differential dΩZ PΩ(P ) =dΩ, P ∈ M,
P0

where P0 ∈ M is some point. The abelian integral Ω(P ) will be called of thefirst, second or third kind according to the abelian differential.

3.4

Abelian differentials on compact surfaces

On a compact Riemann surface M of genus g, the vector space H1 (M ) ofholomorphic differentials has the dimension g. For a canonical homology
basis {a1 , . . . , ag , b1 , . . . , bg }, there exists a unique basis {ω1 , . . . , ωg } for the
space of holomorphic abelian differentials (space H1 (M )) with the propertyZωk = δjk ,j, k = 1, . . . , g.
(3.9)
aj

The matrix Π = πjk with
πjk =

Z

ωk

(3.10)

bjis symmetric with a positive definite imaginary part.
If {a′ , b′ } = {a1 , . . . , a′g , b′1 , . . . , b′g } is another canonical homology basis,and ω ′ = {ω1 , . . . , ωg } is the basis for H1 (M ) dual to this basis, the new basis
can be decomposed into the old basis by  ′ 
a
A B
a
=
′
b
C D
b
and the new matrix Π′ of b′ -periods is given byΠ′ = (C + DΠ)(A + BΠ)−1 .

3.5

Divisors

For this section, let M be a compact Riemann surface of genus g ≥ 0.
Definition 3.12 (Divisor). A divisor on M is a formal symbol
A = P1α1 P2α2 · · · Pkαk ,with Pj ∈ M , αj ∈ Z.

(3.11)

18

3 RIEMANN SURFACES
We can write the divisor A asA=Y

P α(P ) ,(3.12)P ∈M

where α(P ) ∈ Z, α(P ) 6= 0 for only finitely many P ∈ M .
We introduce the group of divisors on M denoted byQDiv(M ). It is acommutative group (written multiplicately, i. e. AB = P ∈M P α(P )+β(P ) )on the points in M . The unit element of the group Div(M ) is denoted by 1
(divisor with no points, α(P ) = 0 for P ∈ M ).
For A ∈ Div(M ) given by (3.12), we defineX
deg A =
α(P ).
P ∈M

If f ∈ K(M ) \ {0}, then f determines a divisor (f ) ∈ Div(M ) byY(f ) =
P ordP f .
P ∈M

A divisor in the image of (·) : K(M ) → Div(M ) is called principal. The
group of divisors modulo principal divisors is known as the divisors classgroup. Divisors A, B areQcalled equivalent (A ∼ B) provided that AB−1 isprincipal, where B−1 = P ∈M P −β(P ) .
The divisor A of (3.12) is integral (A ≥ 1) provided that α(P ) ≥ 0 for
all P . This introduces a partial ordering on divisors: A ≥ B if and only if
AB−1 ≥ 1.
For every integer n ≥ 1, Mn stands for the set of the integral divisors ofdegree n on M .
A function f ∈ K(M ), f 6= 0, is said to be a multiple of a divisor A
provided that (f ) ≥ A. For a simplification, we assume convention that(0) ≥ A for all divisors A ∈ Div(M ).
For a divisor A on M , we set
L(A) = {f ∈ K(M ) | (f ) ≥ A},
the vector space of functions that are multiples of the divisor A. Its dimension
is denoted r(A). We also setΩ(A) = {τ | τ is an abelian differential on M with (τ ) ≥ A},
the vector space of abelian differentials that are multiples of the divisor A.
Its dimension is denoted i(A). The most important result on divisors is the
Riemann-Roch theorem

19

3 RIEMANN SURFACES

Theorem 3.3 (Riemann-Roch). Let M be a compact Riemann surface ofgenus g and A be any divisor on M , Thenr(A−1 ) = deg A − g + 1 + i(A).
Proof. See [9], III.4.
Definition 3.13. Divisor D is called special provided that r(D−1 ) > 1.

3.6

Abelian mapping

Let M be a compact Riemann surface of genus g > 0.
We define the lattice L(M ) generated by the 2g-columns of the g × 2g
matrix (I, Π), where I is the g × g identity matrix. Denote these columnsby e(1) , . . . , e(g) , π (1) , . . . , π (g) . These vectors are clearly linearly independent
over the reals11 . The lattice L(M ) is the set
)
( g
g
X
X
nj π (j) | m, n ∈ Zg .
mj e(j) +L(M ) =j=1

j=1

The lattice L(M ) introduces in Cg the equivalence relationx ∼ y ⇔ x − y ∈ L(M ).
The quotient of Cg by this equivalence relation, J(M ) = Cg /L(M ), is called
the Jacobian variety of M. We define a map
ϕ : M → J(M )by choosing a point P0 ∈ M and setting
Z P
Z P 
ϕP0 (P ) =ω1 , . . . ,
ωg .
P0

P0

This map is called the Abelian mapping. We shall drop the subscript P0 if it
is clear what base point we are using. We can extend the map ϕ to integral
divisors on M ,ϕ : Mn → J(M )11 (k)e ∈ Rg are vectors of the standard basis of Rg and since ℑΠ > 0, the vectors π (k) are
independent over R. Furthermore, ℑπ (k) 6= 0 for all k, thus e(k) and π (k) are all linearly
independent over the reals.

20

3 RIEMANN SURFACES
by setting
ϕ(D) =nX

ϕ(Pj )

j=1

for D ∈ Mn , D = P1 · · · Pn .
For general divisor D ∈ Div(M ) of the form (3.12), we set
X
ϕ(D) =
α(P )ϕ(P ).
P ∈MA point x ∈ J(M ) is called a point of order 2 when x 6= 0 and 2x = 0 in
J(M ).

3.7

Hyperelliptic surfaces

We will be interested in the simplest Riemann surfaces, so called hyperelliptic Riemann surfaces. In this section we will briefly discuss some of theirimportant properties.
Definition 3.14. A compact Riemann surface M is called hyperelliptic provided that there exists an integral divisor D on M withdeg D = 2,

r(D−1 ) ≥ 2.

Equivalently, M is hyperelliptic if and only if M admits a non-constantmeromorphic function with precisely 2 poles.
If a Riemann surface M of genus g admits such a function f with exactly
two poles, the degree of f is 2 and f is a 2-sheeted cover of the Riemannsphere C ∪ {∞} by M . Since the only possible branch numbers of f are
0 or 1 at each point (by virtue of Theorem 3.1), all branch points havethe branching number 1. By the Riemann-Hurwitz relation (Theorem 3.2),
the total branching number of f is 2g + 2 and we can therefore describea hyperelliptic surface of genus g as a two-sheeted covering of the spherebranched at 2g + 2 points.
Every surface of genus ≤ 2 is hyperelliptic. Surfaces of genus 1 (tori) areoften called elliptic.
Let λ be a meromorphic function of degree 2 on a hyperelliptic surface M
and let P1 , . . . , P2g+2 be the branch points of λ, without the loss of generality
such thatλ(Pj ) 6= ∞,j = 1, . . . , 2g + 2.
The function

v
u2g+2
uY
µ=t
(λ − λ(Pj ))j=1(3.13)

21

3 RIEMANN SURFACES

is a meromorphic function on M . This function can be used for constructionof holomorphic differentials on M . One can verify that the g differentials
νj =

λj dλ
,
µ

j = 0, . . . , g − 1,

(3.14)

form a basis for the Abelian differentials of the first kind on M .
We can also proceed in the other direction. We can construct a hyperelliptic Riemann surface using the two functions λ and µ. In this case, thehyperelliptic surface of genus g is defined as the set of points of C2 definedby equation
2g+2Y2(λ − Ej )(3.15)
µ =
j=1

with

Ej ∈ C, Ej 6= Ek ,j, k = 1, . . . , 2g + 2, j 6= k.

The local parametrization is defined by the homeomorphism (µ, λ) 7→ λ in
the neighborhoods ofpthe points (µ0 , λ0 ) with λ0 6= Ej , ∀j and by the homeomorphism (µ, λ) 7→ λ − Ej in the neighborhood of each point (0, Ej ). For
this hyperelliptic curve there are two points at infinity and they will be denoted by the symbols ∞+ and ∞− . They are distinguished by the conditions(µ, λ) → ∞± ⇔ λ → ∞, µ ∼ ±λg+1 ,(3.16)and the local parameter in the neighborhoods of both points is given by the
homeomorphism (µ, λ) 7→ λ−1 .
To include all general surfaces, we can allow one of the points Ej (without
the loss of generality j = 2g + 2) to be the complex infinity12 . In this casethe equation (3.15) has the form
2g+1
2

µ =

Y
j=1

(λ − Ej )with the same assumptions on Ej except E2g+2 = ∞ now. In this case there
is only one point at infinity denoted by ∞, it is distinguished by the condition1(µ, λ) → ∞ ⇔ λ → ∞, µ ∼ λg+ 2

and the local parameter in its neighborhood is given by the homeomorphism(µ, λ) 7→ λ−1/2 .
12

Points Ej correspond to λ(Pj ) in (3.13). Since λ is a meromorphic function of degree2, one of its branch points can be in general at ∞.

22

3 RIEMANN SURFACES
b2
b1
a2

a1
E1

E2

E3

E4

E5

E6

Figure 3: Construction of canonical homology basis for a hyperelliptic surface
using the two-sheeted covering of the Riemann sphere
A construction of the canonical homology basis for a hyperelliptic surface
M is a simple task when we know the two-sheeted covering of the Riemannsphere by M . In this case, we can construct the basis on the Riemann sphereC ∪ {∞} as seen in figure 3.

4 THETA FUNCTIONS

4

23

Theta Functions

The most important part of the theory of Riemann surfaces is the construction of meromorphic functions. The theory of theta functions is a very goodtool for it. In this section we will review definitions and some basic properties of theta functions and especially of theta functions associated with a
Riemann surface. All definitions are based on the book by Farkas & Kra [9].
More properties of theta functions can be found in Rodin [37].

4.1

Riemann theta function

For fixed integer g ≥ 1, the symbol Hg denotes the Siegel upper half space ofgenus g. It is the space of complex symmetric g × g matrices with positive
definite imaginary part
Hg = τ ∈ Cg×g | τ = τ T , ℑτ > 0 ,where T denotes the matrix transposition. It is an open subset of the manifold of complex symmetric g × g matrices.
We define Riemann’s theta function by


X
1 T
T
N τN + N z ,(4.1)
θ(z, τ ) =exp 2πi2g
N ∈Zwhere z ∈ Cg is a complex vector and τ ∈ Hg is a complex symmetric matrix
with positive definite imaginary part. This function converges on compact
subsets of Cg × Hg and it is a holomorphic function [36], §7.
The periodicity of the theta function is the most interesting property. In
the following text, I will denote the g × g identity matrix and e(k) will be its
k-th column, and τ ∈ Hg and τ (k) denote the k-th column of τ .
Theorem 4.1. Let µ, ν ∈ Zg . Then



1 T
T
θ(z + Iν + τ µ, τ ) = exp 2πi −µ z − µ τ µ θ(z, τ )
2

(4.2)

for all z ∈ Cg and τ ∈ Hg .
In addition to this periodicity formula, θ(z, τ ) is an even function of z,
that isθ(−z, τ ) = θ(z, τ )for all z ∈ Cg , all τ ∈ Hg .

We can consider in place of θ(z, τ ) the translated function θ(z + w, τ ) forsome w ∈ Cg . We have immediately
θ(z + w + e(k) , τ ) = θ(z + w)

24

4 THETA FUNCTIONS
and

h
τkk i
θ(z + e + τ (k) , τ ) = exp 2πi −zk − ek −θ(z + w, τ ).
2
Because the columns of the g × 2g period matrix (I, τ ) are linearly independent over R by the same reasons as in section 3.6, every w ∈ Cg can be
expressed as
 
ε
ε′
ε
w =I +τ ≡ ′ε2
2for some ε′ , ε ∈ Rg . Thusεε′θ(z + w, τ ) = θ z + I + τ , τ
2
2




.

(4.3)

This suggests13 that we should define a theta function with characteristics,
a function on Cg × Hg , by εθ ′ (z, τ ) =
ε
 


X
1
ε′
ε T 
ε Tε exp 2πi
.
z+
N+
+ N+
τ N+
2
2
2
2
2
g
N ∈Z
The relation with the theta function (4.1) is
 


 
ε′
1 T
1 T ′
ε
1 T
ε
θ ′ (z, τ ) = exp 2πi ε τ ε + ε z + ε ε θ z + I + τ , τ , (4.4)
ε
8
2
4
2
2
in particular 
0(z, τ ) = θ(z, τ ).
θ0

For a construction of solutions to PDEs, the most important case is when
ε and ε′ are integer vectors (ε, ε′ ∈ Zg ). In this case, θ [εε′ ] (z, τ ) is called thefirst order theta function with integer characteristic [εε′ ]. The characteristic[εε′ ] is called non-singular (resp. singular ), if gradz θ [εε′ ] (z, τ )|z=0 6= 0 (resp.
= 0).

4.2

Theta functions associated with a Riemann surface

In the previous section we defined the Riemann’s theta function as a holomorphic function defined on Cg × Hg . In this section we want to associate
the theta function with a compact Riemann surface M .
13

The theta function with characteristic has the same divisor of zeros as the translated
function (4.3).

25

4 THETA FUNCTIONS

Let M be a surface of genus g ≥ 1. Let {a1 , . . . , ag , b1 , . . . , bg } be acanonical basis on M , a = {a1 , . . . , ag }T , b = {a1 , . . . , bg }T , and let ω T ={ω1 , . . . , ωg } be a basis for holomorphic differentials H1 (M ) dual to thecanonical homology basis.
Since the matrix Π of b-periods defined in section 3.4 is a complex symmetric matrix with positive definite imaginary part, i. e. Π ∈ Hg , we candefine first order theta functions with characteristics θ [εε′ ] (z, Π) using thismatrix.
We will study the theta functions associated with a surface M as a function on M . We will consider the function θ [εε′ ] (·, Π)◦ϕ (for the sake of brevityθ ◦ ϕ), where ϕ is the Abelian mapping ϕ : M → J(M ) introduced in section
3.6 and [εε′ ] is an integer characteristic. Function θ ◦ ϕ is not single valued
on M because ϕ is not single valued as a function M → Cg and depends on
the path of integration. The function θ ◦ ϕ has a very simple multiplicative
behavior in the sense that the continuation of θ [εε′ ] (·, Π) ◦ ϕ along the closed
curve aj and bj beginning at a point P ∈ M multiplies it byhε ijexp 2πi
2
and


ε′j
πjj
exp 2πi − −− ϕj (P ) ,22respectively.
Even though θ ◦ ϕ has multiple values, the set of zeros is a well defined set
on M . There are two possibilities only. Either θ ◦ ϕ vanishes identically on
M or it has only a finite number of zeros on M . By evaluating the integralZ1d log θ ◦ ϕ
2πialong a curve constructed from all the curves in the canonical homology basis(in both directions, so called fundamental polygon of M , for details see VI.2.4
in [9]) and by taking care of the analytic continuation of the integrand, wecome to exactly g zeros of the function θ ◦ ϕ (counting multiplicities).
The set of zeros of a theta function is described in a quite detailed way by
the Riemann theorem on the divisor of zeros of the theta function. The formulation presented here is from [1], 2.7. More general formulations together
with proofs can be found in [9], VI.3.
Theorem 4.2 (Riemann Theorem). Let M be a compact Riemann surface
with a canonical basis {a1 , . . . , ag , b1 , . . . , bg } and let K ∈ Cg be a vector of

26

4 THETA FUNCTIONS
Riemann constants

g Z
πjj X
ϕj ωk
−
Kj =
2
k=1 akfor j = 1, . . . , g. Let w ∈ J(M ) be a vector such that the Riemann theta
function F (P ) = θ(ϕ(P ) − w − K, Π) does not vanish identically on M .
Then(i). the function F (P ) has on M exactly g zeros P1 , . . . , Pg that give asolution to the Jacobi inversion problem
ϕ(P1 · · · Pg ) = w(4.5)(ii). the divisor D = P1 · · · Pg is nonspecial (see section 3.5),(iii). the points P1 , . . . , Pg are defined from (4.5) uniquely up to a permutation.
This theorem implies that for the nonspecial divisor D = P1 · · · Pg , thefunction F (P ) = θ(ϕ(P )−w−K, Π) has on M exactly g zeros P = P1 , . . . , Pg .
Remark. The definition of the Riemann’s theta function (4.1) is different
in some publications (for example [1]). The difference is in the conventionof definition of the b-period matrix Π. A canonical homology basis is often
normalized, in contrast to (3.9), by the conditionsZω˜k = 2πiδjkaj

and this changes the properties of the b-period matrix Π̃, namely in thiscase it is a complex symmetric matrix with negative definite real part. By
definition (3.10), the relation of b-period matrices is simply Π̃ = 2πiΠ then.
The Riemann’s theta function has therefore to be defined as


X
1 T
T
exp
θ̃(z̃, τ̃ ) =N τ̃ N + N z ,2
N ∈Zg
where z̃ ∈ Cg is again a complex vector and τ̃ ∈ Cg×g is a complex symmetricmatrix with negative definite real part. The arguments of the theta function
defined in this way are related to the argument of our theta function by the
relations z̃ = 2πiz and τ̃ = 2πiτ .

27

5 RIEMANN-HILBERT PROBLEM

5

Riemann-Hilbert problem

The Riemann-Hilbert problem or the Riemann boundary problem is the problem to determine analytic matrices or functions on a Riemann surface thathave a certain jump across a contour on the surface. This problem appears
in different areas of mathematics and physics. It was formulated by B. Riemann and was studied by D. Hilbert, C. Hazeman, J. Plemelj and others. The
Riemann-Hilbert problem has a wide range of physical applications, such as
in contact problems of elasticity, dispersion relations in quantum mechanics,
flow problems in hydrodynamics, diffraction theory, and so on [36].
Although solution of the general Riemann-Hilbert problem is not known,
there are two cases when explicit solution can be given in terms of theta
functions on Riemann surfaces and that are useful for generating solutions
to the Ernst equation – the scalar Riemann-Hilbert problem (section 5.1) and
the Riemann-Hilbert problem with quasi-permutation monodromy matrices[25] (section 5.2).

5.1

Scalar Riemann-Hilbert problem

The solution of this problem was given by S. I. Zverovich (1971) [36]. We
present here the version of the problem on a hyperelliptic surface LH .
Let LH be a hyperelliptic Riemann surface of genus g with branch points¯ E1 , F1 , . . . , Eg , Fg on CP1 . Let θ(z) be the theta funcwith projections ξ, ξ,tion on LH . Let ϕ be the Abelian mapping LH → J(LH ) with a base point
p0 with the projection ξ on CP1 . Let p denote a point on LH .
Let Γ be a piecewise smooth contour without self-intersections on LH .
Let Λ = p1 · · · pr be a divisor on Γ consisting of a finite number of mutuallydifferent points subject to the condition that Γ\Λ decomposes into a finite setof connected components {Γj }, j = 1, . . . , N , each of which is homeomorphic
to the interval (0, 1). Each Γj has a starting and an end point, given by two
points of Γ. We define the functions α(p, Γj ) for p ∈ Γ by(
1 if p ∈ Γj,α(p, Γj ) =0 otherwise
for j = 1, . . . , N . On each curve Γj let there be defined a Hölder continuousfunction Gj (p), which is finite and nonzero. We will denote
G(p) =NX
j=1α(p, Γj )Gj (p),p ∈ Γ \ Λ.

(5.1)

28

5 RIEMANN-HILBERT PROBLEM

Let there be a divisor A of degree m, consisting of points of the divisor Λ,
taken at arbitrary degree. Let on LH \ Γ be another divisor B of degree n.
Definition 5.1. The homogeneous scalar Riemann-Hilbert problem is formulated as follows: Determine a function ψ in LH \ Γ holomorphic and continuous up to the boundary satisfying the boundary condition on Γ,
ψ + (p) = G(p)ψ − (p),p ∈ Γ,(5.2)

with ψ ∈ L(A−1 B−1 ).
The key point in the construction of the solution to the problem 5.1 is1
an analogue to the usual Cauchy kernel t−z. This analogue is given by anormalized Abelian differential of the third kind with poles at p and p0 andis denoted by ωpp0 (see sect. 3.4). The solution to the problem 5.1 is thengiven by [21, 36]
ψ̃(p) = eγ(p) ,
(5.3)
with

1
γ(p) =2πi

Z

ln G ωpp0 ,

(5.4)

Γ

where p0 ∈/ Γ. The function ψ̃(p), however, is not single-valued on LH but it
has multiplicative periods along the cycles bj given by [36], p. 39,
Zψ̃j = exp
ln G ωj .
(5.5)
Γ

The function ψ̃(p) changes its value when continued along the cycle bj asψ̃(p) → ψ̃j ψ̃(p). One can check that (5.3) solves problem 5.1 by applying the
Plemelj-Sokhotsky formulae [36], p. 26,Z11±γ(p) = ± ln G(p) +(5.6)
ln G ωpp0 , p ∈ Γ \ Λ,22πi Γ
where γ ± (p) denotes the limiting values of γ(p) at Γ. Therefore


Z
1
1
+
ψ̃ (p) = exp
ln G(p) +ln G ωpp0 = exp(ln G(p))ψ̃ − (p),22πi Γfor p ∈ Γ \ Λ.
Since the function ψ̃(p) is not single-valued on LH , we need to modify
it to be useful as a solution of the Ernst equation. Let p0 ∈ LH be a fixed
point not coinciding with the singularities of ψ̃ or the branch points Ei and5 RIEMANN-HILBERT PROBLEM

29

Fi . Let Ω(p) be a linear combination of normalized14 Abelian integrals of
the second kind and of the third kind such that the singularities of Ω do not¯
coincide with the branch points Ei and Fi and are independent of ξ and ξ.
Let D = p1 + · · · + pg be a fixed non-special divisor on LH . Let 2πiv be thevector of b-periods of the Abelian differential dΩ given byZdΩ(p).
vj =bj

Let 2πiu be the vector of additive b-periods of γ(p) (5.4) given using (5.5) asZ1uj =ln G ωj .
2πi ΓUsing all these assumptions, the following theorem was formulated and provedin [21], for example.
Theorem 5.1. The generalized solution to the scalar Riemann-Hilbert problem 5.1 with the assumptions above is given by the function
ψ(p) = ψ0

θ(ϕ(p) − ϕ(D) + u + v − K)θ(ϕ(p) − ϕ(D) − K)Z1
× exp Ω(p) +ln G ωpp0 , (5.7)
2πi Γwhere ψ0 is a normalization constant and K is the vector of Riemann constants from theorem 4.2. The paths of integration have to be the same for all
integrals in ϕ and Ω.
Proof. Since the function ψ̃(p) is already a solution to the scalar RiemannHilbert problem 5.1, it remains to prove that ψ(p) is a single-valued function
on LH . If we continue ψ(p) along a closed loop c that can be decomposed in
the canonical homology basis of π1 (LH ) as
c=

g
X
j=1

m j aj +gXnj bj = M T a + N T b,

j=1

with mj , nj ∈ Z and M = (m1 , . . . , mg ), N = (n1 , . . . , ng ), the integralschange values according to the formulasϕ(p) → ϕ(p) + M + ΠN,Ω(p) → Ω(p) + 2πiN T v,
γ(p) → γ(p) + 2πiN T u.

14

R

aj

dΩ(p) = 0.

5 RIEMANN-HILBERT PROBLEM

30

Under this transformation, the theta function transforms according to theperiodicity property (4.2) and we observe that ψ(p) transforms as
ψ(p) → ψ(p) exp(−2πiN T (u + v)) exp(2πiN T (u + v)) = ψ(p).
Thus the function (5.7) is a single valued function on LH .
The function ψ(p) has g simple poles on LH at the points p1 , . . . , pg fromthe Riemann theorem 4.2 and g simple zeros. Additional poles, zeros, andessential singularities can be obtained by a suitable choice of Abelian integralsin Ω. Without loss of generality, we can choose D to consist only of branch
points of LH since D gives the poles of ψ due to the zeros of the theta
function in the denominator and this can always be compensated for by asuitable choice of zeros and poles of ψ which arise from the integrals in Ω.
All P ∈ D shall have multiplicity 1.

5.2

Matrix Riemann-Hilbert problem with quasi permutation monodromy matrices

An explicit solution to a general Riemann-Hilbert problem is not known. A
quite general class of Riemann-Hilbert problems was, however, solved in apaper by Korotkin in 2004 [25]. It is a Riemann-Hilbert problem to find asquare N × N matrix Ψ on CP1 that has singularities at some points γj , andhas jumps across some contours connecting γj to some fixed point γ0 thatare given by
Ψ+ (γ) = Ψ− (γ)M (γ).
Matrices M are assumed to be quasi-permutation N × N matrices, i. e. matrices with exactly one nonzero element in each column and row. A solution
to this Riemann-Hilbert problem can be given on a Riemann surface that is a
N -sheeted cover of CP1 with branch points projected on the points γj using
a generalization of the Cauchy kernel, the so-called Szegö kernel. This topic
will not be pursued further here, for details of the construction of a solution,see [25].

31

6 SOLUTION OF EVOLUTION NONLINEAR SYSTEMS

6

Solution of evolution nonlinear systems

In the theory of completely integrable partial differential equations, various
types of generating techniques were successfully used to find general classesof solutions that lead to a better understanding of those systems and evento understanding of important physical phenomena such as solitons. Especially algebro-geometric methods are a systematic approach to systems witha known linear system, so called Lax represantation.

6.1

Solutions of the nonlinear Schrödinger equation

The nonlinear Schrödinger (NS) system is one of important systems in theoretical physics. It defines the evolution of complex valued functions y(x, t)
and y ∗ (x, t) by two equations
iyt + yxx − 2y ∗ y 2 = 0,−iyt∗ + yxx − 2yy ∗2 = 0.

(6.1)
(6.2)

The Lax representation
Ut (λ) − Vx (λ) = [V (λ), U (λ)],λ ∈ C,(6.3)

for this system was first found by Zakharov and Shabat (see [1], p. 87). In
this case U and V are 2 × 2 matrices of the form


0
iy
,
(6.4)
U (λ) ≡ −iλσ3 +−iy ∗ 0−iyy ∗ yx
V (λ) ≡ 2λU (λ) +,(6.5)−yx∗ iyy ∗
1 0is the third Pauli’s matrix. The Lax representation
where σ3 =0 −1(6.3) is equivalent to the compatibility condition of the associated linear
systemΨx = U Ψ,Ψt = V Ψ

(6.6)
(6.7)

for the 2 × 2 matrix function Ψ = Ψ(λ, x, t). This function plays an important role in the construction of solutions of the NS system. The standard
procedure is an investigation of the most general properties of Ψ that satisfies(6.6) with matrices U and V having the form (6.4). The following theorem
summarizes the results from [1]:

6 SOLUTION OF EVOLUTION NONLINEAR SYSTEMS

32

Theorem 6.1. Let Ψ(λ, x, t) be a 2 × 2 matrix function holomorphic in λin some neighborhood of infinity without the infinity point on the Riemannsphere CP1 and depending smoothly on x and t, with the asymptotic expansion"#∞XΨk (x, t)λ−k exp(−iλxσ3 − 2iλ2 tσ3 )C(λ)Ψ(λ, x, t) = I +k1

at λ = ∞ that can be differentiated by terms with respect to x and t, whereC(λ) ∈ GL(2, C) is constant in x and t. Then the logarithmic derivatives ofΨ have the asymptotic behavior
Ψx Ψ−1 = U (λ) + O(λ−1 ),Ψt Ψ−1 = V (λ) + O(λ−1 )as λ → ∞ and the matrices U and V are of the form (6.4). Functions y andy ∗ that are the solution of (6.1) are proportional to non-diagonal elements ofthe matrix Ψ1 (x, t),y(x, t) = 2(Ψ1 )12 ,y ∗ (x, t) = 2(Ψ1 )21 .
To construct a solution Ψ of the linear system, we fix an arbitrary hyperelliptic Riemann surface X given by the relation
2g+2

µ2 =

Y
j=1

(λ − Ej ),where Ej ∈ C are branch points of X, Ej 6= Ek whenever j 6= k. We alsoneed the Abelian integrals Ω1 (P ), Ω2 (P ), Ω3 (P ), P ∈ X which are fixed by
the conditionsZdΩk = 0, ∀j, k,ajΩ1 (P ) = ±(λ + O(1)),Ω2 (P ) = ±(2λ2 + O(1)),Ω3 (P ) = ±(ln λ + O(1))

as P → ∞± , λ = λ(P ) is the projection of P on CP1 , and Ωj (P ) have no
singularities at the points different from ∞± . Let
Q D be an arbitrary divisor
with deg D = g of general position, i. e. D = gi=j Pj , λ(Pj ) 6= Ek , ∀j, k,
and j 6= k ⇒ λ(Pj ) 6= λ(Pk ).
The function Ψ can begiven in terms of the vector-valued Baker-Akhiezer
ψ1that is uniquely determined by two conditions:function ψ(P, x, t) =ψ2

6 SOLUTION OF EVOLUTION NONLINEAR SYSTEMS

33

1. ψ(P ) is meromorphic on X \ {∞± } and its divisor of poles coincides
with D.
2. condition for the asymptotic behavior of ψ at ∞± with essential singularities at ∞±

 
1
−1
+ O(λ ) exp(−iλx − 2iλ2 t), P → ∞−ψ(P ) =
0

 
0
−1
+ O(λ ) exp(iλx + 2iλ2 t), P → ∞+ , α ∈ C,
ψ(P ) = αλ1where λ is the projection λ = λ(P ).
The Baker-Akhiezer function is given by the formulaψ0 (P ) = cθ(ϕ(P )) + v − ϕ(D) − K)
exp Ω(P ),θ(ϕ(P )) − ϕ(D) − K)

where ϕ is the Abelian mapping on X with the base point P0 ∈ X, D is adivisor of degree g, Ω(P ) is an Abelian integral of the second kind with poles
at some points Q1 , . . . , Qn , the principal parts of Ω(P ) coincide with somepolynomials qj (zj ), j = 1, . . . , n, where zj is a local coordinate at Qj , and v
a vector of b-periods of the integrals Ω(P ),Z
vj =dΩ(P ), j = 1, . . . , g.
bj

The path of integration in Ω(P ) and ϕ(P ) is chosen to be the same. The
Baker-Akhiezer function has poles only at points of the divisor D and inneighborhood of every pont Qj the estimate
ψ0 exp(−qj (zj (P ))) = O(1)holds. Theorem 2.24 in [1] summarizes the properties of Baker-Akhiezer
functions. It also states that a Baker-Akhiezer function with some divisor D
and polynomials qj is unique up to a constant c ∈ C.
A Baker-Akhiezer function ψ(P, x, t) used to solve the nonlinear Schrödinger system is constructed using the Abelian integrals Ω1 (P ), Ω2 (P ), Ω3 (P )and the asymptotic behavior of ψ. The matrix Ψ(P ) is then given byΨ(λ) = (ψ(P + ), ψ(P − )),where P ± ∈ X are the two preimages of λ so that P ± → ∞± when λ → ∞.
The resulting formulas are quite long and the topic is not pursued further
here. Details of the construction are given in [1], chapt. 4. The construction
procedure is, however, very similar to that we will use in section 7.

7 SOLUTIONS OF THE ERNST EQUATION

7

34

Solutions of the Ernst equation

First, let us show that the solutions of the Ernst equation can be given
as solutions to certain Riemann-Hilbert problems on hyperelliptic Riemannsurfaces. We will review this approach here.

7.1

Linear system for the Ernst equation

The Ernst equation is completely integrable (see, for example, Maison [29]).
This means that it can be considered as the integrability condition of an
overdetermined linear differential system for a matrix-valued function Φ that
contains an additional variable, the so-called spectral parameter. Several
forms of the linear system are known in the literature which are related
through gauge transformations (e. g. [4]). The choice of a specific form of
the linear system is equivalent to a gauge fixing.
In the literature, the most common form of the associated linear problem¯ ,is the system for 2 × 2 matrix function Ψ(K, ξ, ξ)K − ξ¯ 0 M
M 0
−1
+
Ψξ Ψ
=
≡ V,
(7.1)
0 N
N 0
µ0




K −ξ 0 N
N 0
−1
Ψξ̄ Ψ
=
+
≡ W.
(7.2)
0 M
M 0
µ0
¯ but not on K, and have
The functions M and N depend only on ξ and ξ,
the form
Eξ
Eξ
, N=
,
M=
E +E
E +E
where E is the Ernst potential. In [34] it is shown that the compatibility
condition Ψξξ̄ = Ψξ̄ξ of the linear system is the Ernst equation. The spectral
parameter K lives on the Riemann surface L of genus 0 given by the relation¯µ20 (K) = (K − ξ)(K − ξ).

(7.3)

Points P ∈ L are given in the formP ≡ (K, µ0 (K)),

K ∈ C.

The sign15 of µ0 (K) distinguishes the sheet of the cover of CP1 by L. Thisis the first indication of the relevance of Riemann surfaces in the context of
the Ernst equation. The branch points ξ, ξ¯ of the surface L depend on the15

The branch of the square rootp

µ20 (K).

7 SOLUTIONS OF THE ERNST EQUATION

35

spacetime coordinates. This is the special feature of the Ernst equation notpresent in the theory of evolution equations.
The surface admits two important automorphisms. The holomorphic involution16 σ interchanging sheets acting on P ∈ L byP ≡ (K, µ0 (K)) → σ(P ) ≡ P σ ≡ (K, −µ0 (K))

and the antiholomorphic involutionP ≡ (K, µ0 (K)) → τ (P ) ≡ P ≡ (K, µ0 (K)).

¯ 0) on L simply as ξ
We denote the points P0 ≡ (ξ, 0) and P0 ≡ (ξ,+−¯and ξ, respectively. ∞ and ∞ are the points with projection ∞ on CP1 ,distinguished by (3.16). In the following, Ψ, Ψξ Ψ−1 and Ψξ̄ Ψ−1 stand for therespective functions on L, i. e. Ψ(P ), Ψξ (P )Ψ−1 (P ) (≡ Ψξ Ψ−1 (P )), etc. The
point P ∈ L denotes (K, µ0 (K)). The following theorem is the basic tool in
the construction of solutions to (7.1) [20–22].
Theorem 7.1 (Analytic properties of Ψ(P )). Let Ψ(P ) be a 2 × 2 matrix
function on L with the following properties:(i ). Ψ(P ) is holomorphic and invertible at the branch points
ξ, ξ¯ such√−1
17
that the logarithmic derivativeΨξ Ψ diverges as 1/ K − ξ at ξ andp−1¯¯
Ψξ̄ Ψ diverges as 1/ K − ξ at ξ.
(ii ). All singularities of Ψ(P ) on L (poles, essential singularities, zeros ofdet Ψ, branch-cuts and branch points) are regular, which means thatthe logarithmic derivatives Ψξ Ψ−1 and Ψξ̄ Ψ−1 are holomorphic in theneighborhood of the singular point (thus they have to be independent of¯ In particular, Ψ(P ) should have
ξ and ξ).
(a) regular singularities at the points Ai ∈ L, i = 1, . . . , n, which do¯
not depend on ξ, ξ,(b) regular essential singularities at the points Si ∈ L, i = 1, . . . , m,which do not depend on ξ,(c) boundary values at a set of orientable, piecewise smooth contours
Γi ⊂ L, i = 1, . . . , l, independent of ξ, ξ¯ which are related on bothsides of the contours via
Ψ− (P ) = Ψ+ (P )Gi (P )|P ∈Γi ,(7.4)

where Gi (P ) are invertible Hölder continuous matrices independent¯of ξ, ξ.
16 2
17

σ = Id.
¯ respectively.
Ψξ Ψ−1 and Ψξ̄ Ψ−1 have poles of order 1 at ξ, ξ,

7 SOLUTIONS OF THE ERNST EQUATION

36(iii ). Ψ(P ) satisfies the reduction condition
Ψ(P σ ) = σ3 Ψ(P )γ(P ),

(7.5)


1 0
is the third Pauli matrix and γ(P ) is a 2 × 2
where σ3 =0 −1¯
invertible matrix independent of ξ, ξ.
(iv ). Ψ(P ) satisfies the normalization and reality condition


Ē 1
+
.
Ψ(∞ ) =
E −1

(7.6)

Then the function E in (7.6) is a solution of the Ernst equation (2.4).
Let us remark that the independence of the singular points on ξ, ξ¯ means¯
that K is constant in P = (K, µ0 (K)) although µ0 (K) itself depends on ξ, ξ.
The theorem was proved in [21] or [18]. Here we present a different proof
which is more detailed and precise. In particular, the first part of the proofs
in these papers do not give clear arguments why Ψξ Ψ−1 and Ψξ̄ Ψ−1 have to
be in the form of (7.1). We exploit the Riemann-Roch theorem to prove it
here.
Proof. By virtue of (i) and (ii), the function Ψξ Ψ−1 is holomorphic on L\{ξ}and its local behavior at ξ is given by
p

α−1
−1
√
Ψξ Ψ (P ) =+ α0 + O
K −ξ .
K −ξ

Since ξ is the branch
point of the Riemann surface L, the local parameter√at ξ is given by K − ξ (sect. 3.7). Therefore, Ψξ Ψ−1 (P ) is a meromorphic
function on L with a pole of order 1 at ξ.
Let D denote the divisor of the point ξ. Because the genus g of L is 0 andi(D) ≤ i(1) = 0, the Riemann-Roch theorem 3.3 implies that r(D−1 ) = 2.
In other words, all meromorphic functions on L with a pole of order 1 at ξ
are given by a linear combination of a constant function and one particularmeromorphic function with a pole of order 1 at ξ. One such function iss
K − ξ¯h(P ) =
, P ≡ (K, µ0 (K)).
K −ξ

Indeed, the function h(P ) is a holomorphic function L → CP1 . In local
coordinates at ξ and ξ¯ it is clear that it has a pole of order 1 at ξ and a zero

37

7 SOLUTIONS OF THE ERNST EQUATION¯ Function Ψξ Ψ−1 (P ) must be thus given byof order 1 at ξ.
s
K − ξ¯Ψξ Ψ−1 (P ) = β0
+ β1 ,
K −ξ

where β0 , β1 ∈ C2×2 are constant matrices (independent of P , but depending¯on ξ, ξ).
The structure of matrices β0 and β1 follows from (iii). The involution σ
acts on Ψξ Ψ−1 asΨξ (P σ )Ψ−1 (P σ ) = σ3 Ψξ (P )Ψ−1 (P )σ3(7.7)

and on h(P ) as h(P σ ) = −h(P ). Only those components of matrix β0 are
nonzero that correspond to the components of Ψξ Ψ−1 that change sign under
the involution σ in (7.7). Similarly, β1 has nonzero components where Ψξ Ψ−1
has components that do not change sign under σ. Hence, β0 and β1 are of
the form




× 0
0 ×
.
,
β1 =
β0 =
0 ×
× 0
At P = ∞+ , the value of Ψξ Ψ−1 follows from (iv),
1
Ēξ Ēξ+−1+Ψξ (∞ )Ψ (∞ ) =,E + Ē Eξ Eξ
and h(∞+ ) = 1. Thus1
β0 + β1 =E + Ē



Ēξ ĒξEξ Eξ

and we can conclude that Ψξ Ψ−1 has the structure given by the first equationof the linear system (7.1). The structure of Ψξ̄ Ψ−1 is obtained in the sameway.
The solution to the linear system (7.1) is not unique for a given Ernst
potential E. It is easy to show that if Ψ(P ) is a 2 × 2 matrix function subject
to the conditions of theorem 7.1 and C(K) is a 2 × 2 matrix function that
only depends on K, with properties
C(K) = α1 (K)I + α2 (K)σ1 ,α1 (∞) = 1,
α2 (∞) = 0,
0 1
is the first Pauli matrix, then the matrix Ψ′ (P ) =where σ1 =
1 0
Ψ(P )C(K) also satisfies the conditions of theorem 7.1 and Ψ′ (∞+ ) = Ψ(∞+ ).
This matrix leads to the same Ernst potential E as Ψ(P ) [18].
7 SOLUTIONS OF THE ERNST EQUATION

7.2

38

Construction of the Ernst potential

Theorem 7.1 can be used to construct solutions to the Ernst equation by
constructing the matrix function Ψ(P ) in accordance with the conditions (i)– (iv). The relevance of the hyperelliptic Riemann surface for constructionof Ψ(P ) is motivated by a monodromy matrix L, i. e. a matrix function¯ such that LΨ is a solution of (7.1) as well. It is a solution to theL(K, ξ, ξ)systemLξ = [W, L], Lξ̄ = [V, L].
It can be chosen to be traceless and then it is a 2 × 2 holomorphic matrix
function with the structure [21]A(K) B(K)L=
C(K) −A(K)and its eigenvalues µ̂ are given as the solution to the equationµ̂2 (K) = A(K)2 + B(K)C(K),which is an equation of a Riemann surface. In general, it has infinite genus
but we can restrict our analysis to the case of a regular curve with finite
genus. Then the Riemann surface is given by an equation of the formgY(K − Ej )(K − Fj ),µ̂ (K) =
2

(7.8)

j=1

¯ Thiswhere Ej , Fj ∈ C are independent of the physical coordinates ξ, ξ.
equation represents a two-sheeted covering of the Riemann surface L andthus a four-sheeted covering of the Riemann sphere CP1 . A point P̂ ∈ L̂ canbe given by P̂ ≡ (K, µ0 (K), µ̂(K)). The structure of the surface is showedat the top in figure 4.
It is useful to compute the genus ĝ of the surface L̂ using the RiemannHurwitz relation, theorem 3.2. L̂ is a 2-sheeted18 cover of L which is a surface
of genus γ = 0. Thus the projection π : L̂ → L is a holomorphic mappingof degree n = 2. π is branched at points where µ̂(K) = 0, i. e. at the
points (Ej , ±µ0 (Ej ), 0) and (Fj , ±µ0 (Fj ), 0) in (7.8). The branch number bπ
at these points is 1. Hence the total branching number B is given by the
number of these points, B = 4g. The Riemann-Hurwitz relation then yieldsĝ = n(γ − 1) + 1 + B/2 = 2g − 1.
18

Another way to compute the genus ĝ is to think of L̂ as of the 4-sheeted cover of the
Riemann sphere branched at 4g + 4 points, but it yields the same result for ĝ.

39

7 SOLUTIONS OF THE ERNST EQUATION
^
0

+ +

+ −
^

− +

− −
E1

F1

E2

F2
H

+
H

−
E1

F1

E2

F2

Figure 4: Structure of the Riemann surface L̂ and its quotient LH
However, the surface L̂ can be factored using an involution on it into a
two-sheeted cover of a sphere – a hyperelliptic surface LH . Then a powerful calculus of hyperelliptic surfaces can be used to study the solution Ψ.
This factorization is not as straightforward as is written in [20, 21]19 . The
involution we have to use acts on a point of L̂ asP̂ ≡ (K, µ0 (K), µ̂(K)) → σ̂(P̂ ) = (K, −µ0 (K), −µ̂(K)).
Let H = {Id, σ̂} be a subgroup of the group of automorphisms of L̂. We
define the orbit space L̂/σ̂ (the quotient of the action of the group H) as thespace of the orbits of points P̂ ∈ L̂,
H(P̂ ) = {h(P̂ ) | h ∈ H}.
We can topologize the orbit space L̂/σ̂ such that the natural projection
πH : L̂ → L̂/σ̂,19

πH (P̂ ) 7→ H(P̂ )(7.9)

In [20] it is said that the fixed points of the involution become branch points on thefactorized surface. This is not correct because the local coordinate at the image of abranch point is z k , where z is a local coordinate at the branch point and k = 2 is theorder of the stabilizer subgroup of H [9], III.7.8. That is, the fixed points of the involution
remove the branch points of the surface. Another way how to get an insight is to use the
Riemann-Hurwitz relation, thorem 3.2, which shows that factorization using an involution
with fixed points remove branch points of the surface.

7 SOLUTIONS OF THE ERNST EQUATION

40

is continuous. Than we introduce a complex structure on L̂/σ̂. Since theinvolution σ̂ fixes no points, then any local coordinate at P̂ serves as a local
coordinate at πH (P̂ ) [9], III.7.8. In particular, the branch points of L̂ map
onto branch points of L̂/σ̂. We conclude that LH = L̂/σ̂ is a hyperelliptic
Riemann surface LH given by¯µ2H = (K − ξ)(K − ξ)

g
Y
j=1

(K − Ej )(K − Fj ).

(7.10)

To verify that this factorization really leads to LH we compute the genus γof this surface using the Riemann-Hurwitz relation. The projection (7.9) isa holomorphic mapping of degree 2 with no branch points and we concludethatĝ + 1γ== g,
2which is the genus of the hyperelliptic surface given by (7.10). The factorization of L̂ is diagrammatized in figure 4.
Now we can use the results of section 5.1 to construct a matrix function
Ψ on LH and project it to L so it obeys the conditions of theorem 7.1.
Although we know only a solution of the scalar Riemann-Hilbert problem,
definition 5.1, we construct a matrix Ψ using this solution by exploiting its
symmetry.
To fulfill the reality condition (iv) of theorem 7.1 we must impose more
restrictions on the branch points of LH . Namely, we request that each pairof the branch points satisfies either Ej , Fj ∈ R or Ej = F̄j . Then LH admits
the antiholomorphic involution τH ,PH ≡ (K, µH (K)) → τH (PH ) ≡ PH = (K, µH (K)),
as well as the holomorphic involution σHPH ≡ (K, µH (K)) → σH (PH ) = (K, −µH (K)).
On LH we use the cut system and the canonical homology basis indicatedin figure 5. In the following, ϕ will be the Abelian mapping LH → J(LH )with the base point P0 ≡ (ξ, 0) and θ will be the theta function on LH .
Let Ω be the abelian differential defined in section 5.1, satisfying the reality
conditionΩ̄(PH ) = Ω(PH ),(7.11)
D be the divisor of degree g that consists only of the branch points of LH
and every point of D has multiplicity 1. Let Γ be a contour on LH subject
to the reality condition
PH ∈ Γ ⇒ PH ∈ Γ41

7 SOLUTIONS OF THE ERNST EQUATION
b3

b2

b1

E3

E1

a1

E2

a2

F2
...
a3

F1

F3

Figure 5: Cut system and canonical homology basis on LH viewed as a 2sheeted cover of CP1 . The dashed part of a cycle goes on the −-sheet of thecover.
and G be a Hölder continuous function of Γ satisfying
G(PH ) = G(PH ),PH ∈ Γ.

Let u and v be the vectors introduced in section 5.1. The function ψ(PH )(5.7) solves the scalar Riemann-Hilbert problem 5.1 on LH . We define anotherfunction on LH by
χ(PH ) = χ0

θ(ϕ(PH ) − ϕ(P¯0 ) − ϕ(D) + u + v − K)θ(ϕ(PH ) − ϕ(D) − K)Z1
ln G ωPH P0 , (7.12)
× exp Ω(PH ) +2πi Γ

where χ0 is a normalization constant. We will denote the argument of exponential by I(PH , P0 ),Z
1
I(PH , P0 ) = Ω(PH ) +ln G ωPH P0 .
2πi ΓThe analytic behavior of χ(PH ) is identical to that of ψ(PH ), except that itchanges sign when it is continued along any of the cycles bj . χ(PH ) is thus
not a single-valued function on LH but it can be regarded as a function that

7 SOLUTIONS OF THE ERNST EQUATION

42

changes sign at every cycle20 aj . However, it is single valued on L̂. L̂ canbe viewed as two copies of LH glued in the following way: Two copies of LH
cover the Riemann sphere CP1 by four sheets in total, with branch points
projecting on the same points on CP1 . We number the sheets of the first LH
as 1 and 2 and the sheets of the second one as 3 and 4. The branch cuts
interchange sheets 1, 2 and sheets 3, 4. We cut both surfaces along the loop
¯ and glue the sheets in such a way that this branchabove the branch cut [ξ, ξ]
cut interchanges the sheets 1, 3 and the sheets 2, 4. Then the lifts21 of thecycles aj from LH to L̂ divides L̂ into two disconnected components22 . We
will denote those two components L̂+ and L̂− and they shall be fixed by(ξ, 0, ±µ̂(ξ)) ∈ L̂± .
We normalize (if possible, see theorem 7.2) ψ and χ on LH such that
ψ(∞−H ) = 1,

χ(∞−H ) = −1.

(7.13)

Now we will define functions ψ̂ and χ̂ on L̂ by the relationsψ̂(P̂ ) = ψ(πH (P̂ )),χ̂(P̂ ) = ±χ(πH (P̂ )),P̂ ∈ L̂± .

(7.14)

Functions ψ̂ and χ̂ defined this way are single-valued and holomorphic on L̂.
We construct the matrix Ψ on L asψ̂(P ⊕ ) ψ̂(P ⊖ ).
(7.15)
Ψ(P ) =χ̂(P ⊕ ) χ̂(P ⊖ )where P ⊕ and P ⊖ are the possible lifts of P ≡ (K, µ0 (K)) ∈ L to L̂P ⊕ = (K, µ0 (K), +µ̂(K)),

P ⊖ = (K, µ0 (K), −µ̂(K)).

The main result of this section is the following theorem:
Theorem 7.2. Assume that θ(ϕ(∞−H )−ϕ(D)−K) 6= 0. Then the 2×2 matrix
function Ψ(P ) on L defined in (7.15) satisfies the conditions of theorem 7.1.
Proof. Verification of the reduction condition (iii) is straightforward. We
have to check that the action of σ on Ψ is in accordance with (7.5). The
20

Because the intersection number of any loop with aj cycles counts the number of bj
cycles in this loop, see 3.2.
21
The two components of the preimage of each cycle aj under the projection πH .
22
The components of L̂ \ ∪gj=1 πH −1 (aj ).

7 SOLUTIONS OF THE ERNST EQUATION

43

involution σ on L has a natural lift to L̂ that changes the sign in front ofµ0 (K) only in (K, µ0 (K), µ̂(K)). For a fixed point P ∈ L, let PH = πH (P ⊕ ).
Then σH (PH ) = πH (P ⊖ ) and the matrix (7.15) has the formψ(PH )
ψ(σH (PH )),Ψ(P ) =s1 χ(PH ) s2 χ(σH (PH ))with the signs s1 , s2 fixed according to (7.14). If P ⊕ and P ⊖ lie in different
components L̂± then s1 = −s2 , if they lie in the same component thens = s1 = s2 . Since πH (σ(P )⊕ ) = σH (πH (P ⊕ )) and πH (σ(P )⊖ ) = σH (πH (P ⊖ ))and because the involution σ changes the signs s1 , s2 only when P ⊕ and P ⊖lie in the same component, we get eitherψ(σH (PH ))ψ(PH )σ,Ψ(P ) =s1 χ(σH (PH )) s2 χ(PH )when the signs are different orψ(σH (PH ))ψ(PH )σ,Ψ(P ) =−sχ(σH (PH )) −sχ(PH )when the signs are the same and therefore we conclude thatΨ(P σ ) = σ3 Ψ(P )σ1 ,which is exactly the form of of the condition (iii).
Let B = {E1 , F1 , . . . , Eg , Fg } ⊂ C denote the set of projections of theconstant branch points of LH . By virtue of theorem 4.2 the functions ψ andχ have zeros on LH only at the points of the divisor D. These points lie onlyat the branch points E1 , F1 , . . . , Eg , Fg of LH . At all other points the matrix
function Ψ on L is therefore holomorphic and invertible. At a branch point,
the situation is different. Let P = (E, µ0 (E)) ∈ L and E ∈ C its projection
on CP1 , such that E ∈ B. In a neighborhood of this point, z = K − E is alocal coordinate. The point PH =√(E, 0)√is a branch point of LH then. In aneighborhood of this point, zH = z = K − E is a local coordinate. The
local coordinate at the point σH (PH ) is simply zH (σH (PH )) = −zH (PH ). If
PH ∈ D, functions χ and ψ have poles of order 1 at PH and they can be
written in the form
a−1
+ a0 + O(zH ),zH
b−1
χ(zH ) =+ b0 + O(zH ).
zHψ(zH ) =7 SOLUTIONS OF THE ERNST EQUATION44

The components of Ψ thus diverges as (K − E)−1/2 at P and the determinantdet Ψ has the formdet Ψ(z) =2a−1 b0 − 2a0 b−1+ O(1)zH

and diverges as (K −E)−1/2 as well. If PH ∈/ D, functions χ and ψ are regularat PH and they can be written in the form
ψ(zH ) = a0 + a1 zH + O(zH2 ),χ(zH ) = b0 + b1 zH + O(zH2 ).
The components of Ψ are thus regular at P and the determinant det Ψ,
det Ψ(z) = 2(a1 b0 − a0 b1 )zH + O(zH2 ),vanishes as (K − E)1/2 . Because Ψ is a function on LH it is not single
valued on L. It has a jump across every lift of branch cuts [Ej , Fj ] to L. If
Ej , Fj ∈ R, we get Ψ− = −Ψ+ σ2 on the cut between [Ej , Fj ], whereas we getΨ− = −Ψ+ σ1 when Ej = Fj [21]. The logarithmic derivatives Ψξ Ψ−1 andΨξ̄ Ψ−1 are, however, holomorphic at all of these points [21]. The condition (ii)¯ the functions Ψξ Ψ−1 and Ψξ̄ Ψ−1 haveis thus satisfied. At the points ξ and ξ,poles of order 1, respectively, in accordance with (i) [20]. These properties ofΨξ Ψ−1 and Ψξ̄ Ψ−1 follow from the modular properties23 of the theta function,
but it is not pursued in our work.
To verify the condition (iv), i. e. that the matrix Ψ(∞+ ) (7.15) has the
form


E 1
+
Ψ(∞ ) =,E −1
we rewrite the functions ψ and χ first. We know that K, ϕ(D) and ϕ(P0 )are points of order 2 in J(LH ) (see [9], VII.1). Therefore we can define thefollowing integer characteristics. Let p, q, α and β ∈ Zg be integer vectors
such that
 
 
p
α
= ϕ(D) + ϕ(P0 ) + K,
= ϕ(D) + K.
qβ
Since ϕ(P0 ) ∈ R, it follows that p = α. Then ψ(PH ) and χ(PH ) can be23

Dependence of the theta function on the shape of a Riemann surface it resides on.

7 SOLUTIONS OF THE ERNST EQUATION

45

written using (4.4) in the form p(ϕ(PH ) + u + v + ϕ(P0 ))θq ψ(PH ) = ψ0α(ϕ(PH ))
θ
β


1 T
1 T
T
exp πi p (u + v + ϕ(P0 )) + p q − α β exp I(PH , P0 )22
and similarly p(ϕ(PH ) + u + v)θq
 χ(PH ) = χ0α(ϕ(PH ))
θ
β


1 T
1 T
T
exp πi p (u + v) + p q − α β exp I(PH , P0 ).
2
2
Since θ [pq] (ϕ(∞−H ) + u + v) 6= 0, we can normalize ψ and χ according
to (7.13). That yields for ψ at the point ∞+
H
 
 
p
α
+
(ϕ(∞H ) + u + v + ϕ(P0 )) θ
(ϕ(∞−
θ
H ))
q
β
−
+
 
ψ(∞H ) =  
exp I(∞+
H , ∞H )
p
α
+
(ϕ(∞−
θ
H ) + u + v + ϕ(P0 )) θ β (ϕ(∞H ))q+−and similarly for χ(∞+H ). Because the lifts of ∞H and ∞H lie in different
components L̂± , the sign of χ(∞+H ) changes and thus the expressions for++ψ(∞H ) and χ(∞H ) differ only in the argument of the theta functions in thefirst fraction. The second fraction is 1, because [αβ] is an even characteristic−(see [9], VII.1) and ϕ(∞+H ) = −ϕ(∞H ) since the integration path in ϕ has
the same projection on CP1 .
We have to show that for the normalized functions ψ and χ the relation+ψ(∞+H ) = χ(∞H )holds. We will need the following lemma.
Lemma 7.3 (Reality properties). Let LH be a hyperelliptic suface of genus¯ E1 , F1 , . . . , Eg , Fg on CP1 thatg with branch points that have projections ξ, ξ,satisfies the following reality condition: For each pair of branch points Ej , Fj ,

46

7 SOLUTIONS OF THE ERNST EQUATION

j = 1, . . . , g, it is either Ej , Fj ∈ R or Ej = Fj . Let the canonical homology
basis be given as shown in figure 5. Then


1
1 1
ϕ(P0 ) =, ,...,
2 2
2
as a point in the Jacobian variety J(LH ). Furthermore, the matrix Π ofb-periods has the reality propertyΠ + Π = C,
where

d1 1
 1 d2C =  .. ..
. .
1 1


1
1

..  ,
.

···
···
...

· · · dg

(
1
dj =
0

if Ej , Fj ∈ R,
if Ej = Fj

j = 1, . . . , g.

Proof. We know that σH acts on the differentials of the normalized basis ωby σH (ω) = −ω. Let a0 be a cycle running around the branch points ξ, ξ¯ on
the +-sheet of LH in the counter-clockwise direction.PIt can be decomposedin the canonical homology basis using (3.5), a0 = − gj=1 aj . Then
Z ξ̄

1
ω=
2
ξ

1
1
ω = (−1, . . . , −1) = (1, . . . , 1).
2
2
a0

Z

The reality property of the matrix Π can be established by employing theantiholomorphic involution τH . We express τH (aj ) and τH (bj ), which is justanother canonical homology basis24 , in the canonical homology basis aj , bj
using (3.5) and we obtain(gX
aj if Ej , Fj ∈ Rak −τH (aj ) = −aj , τH (bj ) = bj −
.
0
if
E
=
F
j
j
k=1
k6=j

We see that



 
 
a
−I 0
τH (a)=b
I −C
τH (b)

and we immediately obtain

Π+Π=C
from (3.11).
24

Even though τH (aj ) · τH (bj ) = −1, we can call it a canonical homology basis for our
purposes

47

7 SOLUTIONS OF THE ERNST EQUATION

Corollary 7.4. Let θ(z) be the theta function on LH with reality properties
given by the above lemma. Then1
θ(z) = θ(z̄ + d).
2(7.16)

Proof. By expressing θ(z, Π) into the defining series


X
1 T
1 T
T
N ΠN − N z̄ − N CN
θ(z, Π) =exp 2πi22gN ∈Z
we observe that

πiN T CN = πi

g
X
j=1

Ni2 di + 2πigX

Nj Nk .

j=1
k=1
j6=k

The first term gives in the exponential the same result as N T d, while thesecond term gives 1, and we immediately obtain (7.16).
Now we have
1
+
−ϕ(∞+
H ) − ϕ(P0 ) − u − v − d = ϕ(∞H ) + u + v
2
and we conclude that the matrix Ψ(P ) obeys the normalization condition(iv) as well.
At this moment we have everything we need to write the formula for an
Ernst potential from the constructed matrix Ψ(P ).
Corollary 7.5 (Ernst potential). Let θ [pq] (ϕ(∞−H ) + u + v) 6= 0. Then thefunction
 
p
(ϕ(∞+
θ
H ) + u + v)q¯
E(ξ, ξ) =  
p
(ϕ(∞−
θ
H ) + u + v)
q


Z
1
+
−
× exp Ω(∞H ) − Ω(∞H ) +ln G ω∞+ ∞− .
2πi Γis a solution to the Ernst equation.
Proof. We know that the matrix Ψ(P ) is subject to the conditions of theorem 7.1 and therefore we obtain the Ernst potential from Ψ(∞+ ) directly.
The solutions studied here are a subclass of solutions found by Korotkin[23]. pq has to be an integer characteristic.

48

7 SOLUTIONS OF THE ERNST EQUATION

7.3

Symmetric linear system

There is another useful (more symmetric) form of the linear system (7.1)(see [27]). Introducing the real symmetric matrix2i(E − Ē)¯,
(7.17)g(ξ, ξ) =i(E − Ē)2E Ēthe Ernst equation can be rewritten as¯ ξ̄ g −1 )ξ = 0.
¯ ξ g −1 )ξ̄ + ((ξ − ξ)g((ξ − ξ)g

(7.18)

This equation is the compatibility condition of the linear systemgξ g −1
Ψξ =
Ψ,
1−γ

gξ̄ g −1
Ψξ̄ =
Ψ,
1+γ

(7.19)¯ is a 2×2 matrix function with continuous second derivatives.
where Ψ(γ, ξ, ξ)¯ is a spectral parameter subject to the compatible first
The function γ(ξ, ξ)order equations
γξ =

γ 1+γ
,
ξ − ξ¯ 1 − γ

γ 1−γγξ̄ = ¯.
ξ−ξ1+γThey can be solved by


¯ q
ξ
+
ξ
2
1
¯ =
¯ =
w
−
±
γ± (w, ξ, ξ)
(w
−
ξ)(w
−
ξ)
¯
2
ξ − ξ¯γ∓ (w, ξ, ξ)with w ∈ C a constant of integration, which can be regarded as a hidden
spectral parameter. In the sequel we will suppress the index ± and simply¯ ≡ γ± (w, ξ, ξ).
¯ It can be regarded as a meromorphic function
write γ(w, ξ, ξ)of degree 1 on the Riemann surface L of genus 0, given by the relation¯ This function is thus a holomorphic bijection L →µ2w = (w − ξ)(w − ξ).
1CP . It has a pole of order 1 at ∞+ and zero of order 1 at ∞− . We candescribe all quantities either on L as functions of P = (w, µw (w)) ∈ L, or
on CP1 as functions of γ ∈ CP1 . We will adhere to the latter approach here.
Furthermore, using the chain rule we getd∂γ 1+γ ∂≡+,dξ∂ξ ξ − ξ¯ 1 − γ ∂γ∂
γ 1−γ ∂d≡ ¯+ ¯.
¯dξ∂ ξ ξ − ξ 1 + γ ∂γ(7.20)As before, we investigate the behavior of Ψξ Ψ−1 and Ψξ̄ Ψ−1 as that offunctions of γ on the Riemann sphere CP1 . The following theorem summarizes the analytic properties of Ψ as a function of γ [27].

49

7 SOLUTIONS OF THE ERNST EQUATION¯
Theorem 7.6 (Analytic properties of Ψ(γ)). Let the 2 × 2 matrix Ψ(γ, ξ, ξ)be subject to the following conditions:(i ). As a function of γ the matrix Ψ is holomorphic and invertible everywhere on some cover of the Riemann γ-sphere CP1 with the exceptionof the points mentioned below.
¯ = γ(wj , ξ, ξ)¯ for(ii ). Ψ has regular singularities at the branch points γj (ξ, ξ)
j = 1, . . . , N with constants wj ∈ C, in the vicinity of which it behavesas¯ = Gj (ξ, ξ)Ψ
¯ j (γ, ξ, ξ)(γ¯ − γj )Tj Cj as γ ∼ γj ,Ψ(γ, ξ, ξ)¯ = I + O(γ − γj ) is holomorphic andwhere, for γ ∼ γj , Ψj (γ, ξ, ξ)invertible . The matrices Cj and Tj are constant invertible and constantdiagonal, respectively, while ξ, ξ¯ dependent matrices Gj are assumed to
be invertible.
¯ which connect(iii ). Across certain contours {Lj }, Lj ⊂ CP1 , Lj = Lj (ξ, ξ),the singular points γj to some arbitrarily choosen fixed and non-singular¯ the boundary values of Ψ− (γ) and Ψ+ (γ)
base point γ0 = γ(w0 , ξ, ξ),
are related by¯ = Ψ− (γ, ξ, ξ)M¯ j (w),
Ψ+ (γ, ξ, ξ)γ ∈ Lj ,where the invertible matrices Mj depend only on the constant spectral
parameter w.
(iv ). The normalization conditions¯ = g∞ ,Ψ(∞, ξ, ξ)¯ = g(ξ, ξ)¯
Ψ(0, ξ, ξ)(7.21)(7.22)¯
hold, where the matrix g∞ is constant invertible and the matrix g(ξ, ξ)
is invertible.
¯ solves (7.18).
Then Ψ obeys the linear system (7.19) and g(ξ, ξ)We present a more detailed proof here, exploiting the Riemann-Rochhtheorem again.
Proof. Conditions (i) – (iii) imply that Ψξ Ψ−1 is holomorphic in γ everywhere on CP1 away from γ = 1. Indeed, it has removable singularities atsingularities of Ψ. If γ ∼ γj , then from (7.20)
d(γ − γj )(1 + γ + γj − γγj )(γ − γj )Tj = Tj (γ − γj )Tj −I= Sj (γ − γj )Tjdξ(1 − γ)(1 − γj )50

7 SOLUTIONS OF THE ERNST EQUATION

with Sj holomorphic, and thus the singular terms (γ − γj )Tj and (γ − γj )−Tjin Ψξ Ψ−1 cancel each other at γ ∼ γj . At the contours Lj the situation iseven simpler becausedΨ+ −1 dΨ−dΨ− −1Ψ+ =
Mj (w)Mj −1 (w)Ψ− −1 =
Ψ− .
dξ
dξ
dξ
By virtue of (7.20), it has a pole of order 1 at γ = 1, with behaviorΨξ Ψ−1 = Ψγ Ψ−112+ O(1),¯
ξ−ξ1−γγ ∼ 1.

Now we use the same argument as in theorem 7.1. Ψξ Ψ−1 is a meromorphic
function of degree 1 with a pole at γ = 1. Hence, it must be of the formΨξ Ψ−1 = β01
+ β1 ,1−γ¯
where β0 and β1 are constant (independent of γ, but depending on ξ and ξ)
2 × 2 matrices. Both are fixed by the normalization (iv), β1 = 0 is fixed by(7.21) and β0 = gξ g −1 is fixed by (7.22). We conclude that Ψξ Ψ−1 has theform of the first equation in (7.19). The form of the equation for Ψξ̄ Ψ−1 isobtained in the same way, just with the pole at γ = −1.

¯ can be constructed using a solution to a RiemannA function Ψ(γ, ξ, ξ)Hilbert problem with quasi-permutation monodromy matrices, see section¯ obeys not only the equation 7.18, but also
5.2. To make sure that g(ξ, ξ)
is real and symmetric and an Ernst potential can be reconstucted from it,¯ see [27]. The
one has to impose further conditions on the function Ψ(γ, ξ, ξ),
Ernst potential obtained in this way can be rewritten in the form of the Ernst
potential found in section 7.2, see for example [18, 26]. The advantage of thismore general approach will be probably a possibility to construct a solution
to the Einstein-Maxwell equations in stationary axisymmetric case, see [18].

8 NUMERICAL EVALUATION OF THE THETA SOLUTIONS

8

51

Numerical evaluation of the theta solutions

The solutions of the Ernst equation in terms of theta functions on hyperelliptic Riemann surfaces have been known since the end of 1980s. Up to now,
only a few papers have addressed their numerical evaluation. However, a fastand reliable method for obtaining numerical values of metric coefficient is a
key condition of applicability of the solutions in astrophysics.
There are general packages from Deconinck et al. for computing values of general theta functions (see [6]) and parameters of general algebraic
curves (see [5]) that are distributed with Maple 8. Other examples are approaches via Schottky uniformization [1]. These methods work well for the
algebro-geometric solutions of evolution equations like Korteweg-de Vries,
Sine-Gordon, or nonlinear Schrödinger equation. Unfortunately, their direct
use for the Ernst-equation solutions is misleading. Namely, for each spacetime point the solution is implicitly parametrized by the underlying structureof the hyperelliptic Riemann surface it resides on. A change of spacetime coordinates moves two branch points of the surface and thus deforms the latter.
It is therefore necessary to compute all the essential quantities characterizing the surface at every point of the ρ-ζ half-plane, especially the b-period
matrix Π. To obtain those values, one has to integrate differentials on thesurface, which leads to hyperelliptic integrals where the analytical continuation of functions is imperative. Analyzing the latter is time expensive forgeneral Riemann surfaces. However, in the case of hyperelliptic surfaces it
can be done more quickly, because one can exploit the symmetry of such a
surface which can be regarded as a 2-sheeted cover of CP1 . This was noticed
by Frauendiener and Klein [10–12] who were able to handle numerically a
special case of solutions of genus 2 with a thin disk source. Here we present
and discuss an implementation of spectral methods used in the latter articles
to hyperelliptic surfaces of general genus in Mathematica 5.0 and then we
study the computed Ernst potential and metric functions.
We start from the property of a hyperelliptic surface LH of genus g that it
can be given by simple formula (3.15) that in the case of the Ernst equation
is parametrized by the physical coordinates ξ, ξ¯ as¯µ2 = (K − ξ)(K − ξ)gY(K − Ei )(K − Fi ),

(8.1)

i=1

where K is the spectral parameter. It is therefore a 2-sheeted cover of the
Riemann sphere CP1 . We denote by ∞+ , ∞− the points that project on
infinity of the sphere and which are distinguished by the condition (3.16).

8 NUMERICAL EVALUATION OF THE THETA SOLUTIONS

52

The basis of holomorphic differentials is simply given by
νj =

K j−1
dK,
µ

j = 1, . . . , g.

(8.2)

The normalized differentials ωj are their linear combinations. We define thenormalized Abelian differential of the third kind ω∞+ ∞− with poles at ∞+and ∞− and with residues +1 and −1, respectively.
It is normalized by theRcondition that all a-periods vanish, i. e. aj ω∞+ ∞− = 0 for j = 1, . . . , g. It
is given by the formula νg+1 = −K g dK/µ up to a holomorphic differential.
By virtue of the bilinear relations (see [9], III.3.6), the useful formulaZ1
ω∞+ ∞− = ϕ(∞+ ) − ϕ(∞− )2πi bkholds, where ϕ is the Abelian mapping (sect. 3.6). The solutions of the Ernst
equation can be written in the form [19]E=

θpq (ϕ(∞+ )).
θpq (ϕ(∞− )))(8.3)

Paths of integration in ϕ(∞+ ) and ϕ(∞− ) have the same projection on the
Riemann sphere. The constant vectors p and q must satisfy the reality condition Bp + q ∈ Rg .
For large values of |ξ| the Ernst potential expands as [11]
2m 2m2 2iJζ
1
E =1−
(8.4)
−
−
+O
|ξ|
|ξ|2
|ξ|3
|ξ|3
the constants (with respect to ξ) m and J being the mass and the angularmomentum of the spacetime, respectively. The mass m can in principle beimaginary. The real part of m is the ADM-mass (Arnowitt-Deser-Misner,[17]) and the imaginary part of m is called magnetic mass25 .

8.1

Numerics on a hyperelliptic Riemann surface

The underlying hyperelliptic Riemann surface LH parametrizes the solution (8.3) by the matrix of b-periods Π and by the values of ϕ(∞± ). To evaluate those quantities, we need to integrate holomorphic differentials (8.2) andthe Abelian differential of the third kind ω∞+ ∞− along the cycles a1 , . . . , ag ,25

It is connected to the NUT (Newman-Unti-Tamburino, [17]) parameter β0 by the
relation arctan β0 = Im m/ Re m

8 NUMERICAL EVALUATION OF THE THETA SOLUTIONS

53

E1
b1

a1

E2

F2

a2
b2
F1

Figure 6: Cut system and the basis of cycles for the numerical treatment ofa surface of genus 2
b1 , . . . , bg of the fundamental group π1 (LH ). Since LH is the 2-sheeted coverof the sphere, we can evaluate those integrals on LH as integrals on the Riemann sphere with careful tracing of the sheet. We will denote the 2 sheets
as + and − according to the point ∞± that they contain. Since the integralof a differential along the boundary of a simply connected domain vanishes
provided that the differential is holomorphic inside the domain (Cauchy theorem), we can always continuously deform the cycle without changing thevalue of the integral. By deforming the cycle we can always express theintegral along the cycle as a sum of line integrals of the type
Z F
Kj
p dK,
j = 0, . . . , g
(8.5)
µ2
E

between some branch points E and F . Evaluation of such integrals is thus
the main challenge of numerical treatment of the Ernst potential.
8.1.1

Square root

We need to track the sheet of thepcover as effectively as possible. The crucial
factor is the value of µ(K) = µ2 (K) defined by (8.1).
p The differentials
also depend on µ so the evaluation of the square root µ2 (K) for arbitrarycomplex number K is the essential part of the computation. In order tomake this a well defined problem, we introduce the cut-system as shown infigure 6 for a surface of genus 2, or as shown in figure 5. On the cut surface

8 NUMERICAL EVALUATION OF THE THETA SOLUTIONS

54

the square root µ(K) is defined as the product of square roots of monomials
µ=

p

K −ξ

q

K − ξ¯gYpp
K − Ei K − Fi .

(8.6)

i=1

We cannot simply use the square root routines such as the one available
in Mathematica. Namely, the expression (8.6) is holomorphic on the cut
surface, but the built-in square root function usually has its branch-cut along
the negative real axis. We need to adapt the square roots to out cut system.
Let E, F be two branch points connected by a branch-cut. Let α =arg(F − E), where arg(z) is the argument of√a complex number z with values(−π, π). Now we define the square root (α) · with branch-cut along√the raywith argument α by computing, for each z ∈ C, the square root s = z withthe help of the built-in routine and then putting(√s
α/2 < arg(s) < α/2 + π(α).
z=−s otherwise√
With the new square root (α) ·, we compute two factors in (8.6) as√√(α)
K − E (α) K − F .

The expression changes sign exactly on the branch-cut between E and F .
The whole expression√ (8.6) is computed by multiplying the pairs of factors(α)· corresponding to the branch-cuts.
with square roots
In the case of a nonlinear substitution to√the integrals we will use later,
however, the definition of the square root (α) · is useless. We have to continue the square root analytically along the path of integration. We use thefollowing simple method. We assume that the argument of the square root
is a complex-valued continuous function f of one real parameter t ∈ [a, b].
In our case it will be f (t) = µ2 (K(t)). Therefore, the image f ([t1 , t2 ]) of
the function is a connected curve in the complex plane. Furthermore, we assume that the square root is evaluated for a set of consecutive values f (tn ),a = t0 ≤ t2 ≤ · · · ≤ tN = b. Since the built-in square root has a branch-cut
on the negative real axis, we test if the line between consecutive argumentsf (ti−1 ) and f (ti ) of the square root in the complex plane crosses the negative
real axis. If yes, we multiply the new value of the square root by −1 andthen we alter all subsequent values of the square root in this way
p until thenegative real axis is crossed again. The branch of the first value f (t0 ) has
to be chosen appropriately and we use the value given by (8.6).

8 NUMERICAL EVALUATION OF THE THETA SOLUTIONS
8.1.2

55

Numerical integration of the periods

The task of following the analytical continuation of the functions K j /µ(K)in (8.5) is much simplified by the proper definition of square roots in (8.6).
However, the numerical integration of the line integrals (8.5) is still tricky.
1
The integrands have singularities of the type √K−E
at the end points of theline of integration. On the other hand, the integrand is analytic in between.
Using the linear parametrization K = F −E
(t + 1) + E for a real parameter2t ∈ (−1, 1), we get integrals of the form
Z 1
P (t)H(t)
√
dt,
(8.7)
I=
1 − t2−1
where P (t) = K j (t) is a polynomial of degree j and H(t) is an analytic
function of the formpp(α)K(t) − E (β) K(t) − F
F −E√√
H(t) = (α),µ(K(t))F − E (β) E − F
α and β being arguments of the branch-cuts corresponding to the branch
points E and F . This form of the integral suggests to use spectral methods. To evaluate the integral, we approximate the function H(t) by a linear
combination of Chebyshev polynomials (see below)
H(t) =NX

hn Tn (t).

n=0Then we express the polynomial P (t) in the basis of Chebyshev polynomials
up to order j,jX
pn Tn (t).
P (t) =
n=0

The integral I is then calculated with the help of the orthogonality propertiesof Chebyshev polynomials asjπX
pn hn .
I = πp0 h0 +
2 n=1

(8.8)

The Chebyshev polynomials Tn (x) are defined on the interval [−1, 1] by
the formula
Tn (cos(t)) = cos(nt),where x = cos(t),t ∈ [0, π].

8 NUMERICAL EVALUATION OF THE THETA SOLUTIONS

56

They form an ortogonal system on [−1, 1] with respect to the inner product
Z 1
f (x)ḡ(x)√hf, gi ≡dx.
1 − x2−1
The normalization isπhTm , Tn i = cm δmn ,2
where c0 = 2 and cm = 1 otherwise. In order to approximatePa continuousfunction f ∈ C([−1, 1]) by a series of Chebyshev polynomials N
n=0 an Tn forgiven N , we require thatfl ≡ f (xl ) =NXan Tn (xl ) at points xl = cos

n=0π(l + 12 )
N +1



,

(8.9)

where l = 0, . . . , N . By putting c0 = 2 and cn = 1 for n = 2, . . . , N anddefining the numbers Fn = cn an , we get
N
X

N
X


π(l + 12 )fl =an Tn (xl ) =an Tn cos
N +1
n=0
n=0
 X



N
N
X
πn(l + 21 )πn(l + 21 )
Fn
=
.
cos
=
an cos
N
+
1
c
N
+
1
n
n=0
n=0




This is exactly the discrete cosine series up to a constant. The coefficients Fn
are therefore related to the values fl of the function by the inverse discretecosine transform (DCT) [35]Nπn(l + 12 )2 X.
fl cosFn =N + 1 l=0
N +1
The inverse discrete cosine transform is related to the discrete Fourier transform of the extended data fl , l = 0, . . . , 2N + 1, extended symmetricallyaround n = N + 1/2, by [35]f2N +1−j = fj ,j = 0, . . . , N.

The discrete cosine transform consists then of the first N + 1 values of thediscrete Fourier transform divided by the constants cn and multiplied by thefactor eπin/(2N +2) ,
2N
+1
X
2
πin/(2N +2)e
fl eπinl/(N +1) ,cn Fn =
N +1
l=0

n = 0, . . . , N.

8 NUMERICAL EVALUATION OF THE THETA SOLUTIONS

57

The discrete Fourier transform can be computed very effectively by the fast
Fourier transform (FFT) routine26 . This relationship between Chebyshev
polynomials and the FFT is the basis for efficient computations.
A different set of test points xl was used in [11,12], namely xl = cos(πl/N ).
The advantage of our approach is that the test points (8.9) are the roots ofthe polynomial Tn+1 which helps to estimate an error of the approximation(8.10). According to our checks, both methods perform equally well.
The approximation by Chebyshev
R 1 polynomials of a function H(t) canbe used to compute the integral −1 H(t)dt. The formula known as the
Clenshaw-Curtis quadrature [35], chapt. 5.9, reads
Z 1

−1

[N/2]

H(t)dt = −2

h2k
,
(2k
−
1)(2k
+
1)
k=0

X

where [N/2] is the integer part of N/2. The Clenshaw-Curtis quadrature withthe test points xl = cos(πl/N ), used in [11, 12], is called the “trapezoidal” orGauss-Lobato variant.
Let us remark that a polynomial K j (t) can be quickly expressed in thebasis of Chebyshev polynomials up to order j by DCT with j + 1 sample
points x′l = cos πl/j.
The Chebyshev polynomial approximation is almost as good as the ‘holy
grail’ of approximating polynomials, the minimax polynomial [35], but it has
one serious drawback in our application. The estimation of the residue of
the Chebyshev polynomial approximation ismax |H(t) −−1≤t≤1NX
n=0

hn Tn (t)| ≤1

max |H (n+1) (t)|.

2n (n + 1)! −1≤t≤1(8.10)

So the approximation does a great job when approximating smooth function
with bounded derivations.
In our case, however, the function H(t) contains√
terms of the form 1/ t − t0 . The n-th derivation of this term is
2n+1
d1(2n − 1)!!
√(t − t0 )− 2 ,= (−1)nnndt
2t − t0where (2n − 1)!! = n(n − 2)(n − 4) · · · is the double factorial (product of
odd numbers not greater that 2n − 1). When |t − t0 | is small, the error
estimate (8.10) is even increasing with increasing N . Thus we do not get a
26

There also exists a fast cosine transform, but special packages for its implementation
are usually needed.

8 NUMERICAL EVALUATION OF THE THETA SOLUTIONS

58

better approximation with larger N and the applicability of the Chebyshev
approximation for the integration of (8.7) is very limited.
In our case problems occur when some other branch points are “too close”
to the path of integration in (8.5), i. e. to the line connecting branch points
E and F . “Too close” means that the numerical error of the Chebyshev
approximation is too high and cannot be reduced by increasing the number
of approximating polynomials. In some cases this problem can be avoided
by a different choice of integration paths (8.5), especially when the branch
points are distributed uniformly. But, unfortunately, there are situations
when some branch points are very close to each other; in particular this¯ to radial infinity whenapplies to the spacetime symmetry axis when ξ → ξ,27ξ, ξ¯ → ∞ and to the other branch points when ξ → Fi while ξ¯ → Ei .
To solve this problem, we will use the substitution of the form
K(t) = a sin(ct + d) + b,

a, b, c, d ∈ C,t ∈ (−1, 1) ⊂ R(8.11)in (8.5). We can regularize then the integrandKjp(K − R)(K − S) · · ·at exactly two singular points R, S ∈ C by a particular choice of constantsa = (S − R)/2 and b = (S + R)/2. That means that the integral (8.5) is put
into the form
Z t2
K j (t)(S − R) cos(ct + d)dt
cp(8.12)(S − R)(1 + sin(ct + d))(R − S)(1 − sin(ct + d)) · · ·t1p
and the terms cos(ct + d) and 1 − sin2 (ct + d) cancel out with the result±1. The sign has to be fixed so that we integrate the correct branch of thesquare root. With the particular choice of constants
c=

x−y
,
2

d=

x+y
,
2

where

2Q − R − S
2P − R − S, y = arcsin
,
S−R
S−R
we obtain an integral from −1 to 1 and we can use the Clenshaw-Curtis
quadrature. The advantage of the above substitution is that when we integrate between points R and S, i. e. when removing singularities at the ends
x = arcsin

27

In the theory of evolution equations, the situation when pairs of branch points arecollapsing and the surface is degenerating, the so-called solitonic limit, is paricularly important.

8 NUMERICAL EVALUATION OF THE THETA SOLUTIONS

E

59

F

F
E

(a) Connected points(b) Not connected points

Figure 7: Line integration between points connected or not connected by abranch-cut. Solid line represents a path of integration on the +-sheet.
of the line of integration, the constants c = π/2 and d = 0 are real in contrast
to the substitution used in [11, 12].
Hence, if we integrate (8.5) between two branch points E and F in thesituation when there are 2 other branch points E ′ near E and F ′ near F , wedivide the integration line in two and we integrate two integrals. In the first
integral between P = E and Q = (E + F )/2, we set R = E and S = E ′and we use the substitution (8.11) and the Clenshaw-Curtis quadrature. In
the second integral between P = (E + F )/2 and Q = F , we set R = F
and S = F ′ and we integrate using the Clenshaw-Curtis quadrature again.
The only problem with the substitution (8.11) is that the integration patht → a sin(ct+d)+b is not a straight line between P and Q anymore28 and canin principle approach another critical point of the integrand. Fortunately, thepath stays close enough to the original straight line.
To compute the integral over a whole loop, we have to decompose theloop into a sequence of lines between the branch points of the surface. The
integral along a line between two branch points connected by a branch-cut isequal to one half of the integral along a loop containing only the two branch
points in the counter-clockwise direction on the +-sheet (or in the clockwise
direction on the −-sheet), see Figure 7(a). On the other hand, the integral
along a line between two branch points E, F not connected by a branch-cut isequal to one half of the integral along a loop containing only the two branch
points that goes from the branch-cut corresponding to E to the branch-cut
corresponding to F on the +-sheet and back to the branch-cut at E on the−-sheet, Figure 7(b).
In the code, we use the spectral method (8.8) when the line of integration√
That’s why we cannot use the square root (α) · anymore and we have to continue thesquare root analytically.
28

8 NUMERICAL EVALUATION OF THE THETA SOLUTIONS

60

between points E and F is far enough from other branch points, because
it is the fastest methods. In other cases we use the latter approach with
Clenshaw-Curtis quadrature, but it is slightly slower since g + 1 DCTs ofdata of length N + 1 are needed.
Once we have computed the a- and b-periods of the differentials ν1 , . . . ,νg+1 , we compute the matrix Π of the b-periodsof the normalized basis ofRholomorphic differentials and the b-periods bj ω∞+ ∞− of the Abelian differR
ential of the third kind ω∞+ ∞− . Let Ãjk = aj νk be the matrix of a-periodsof the holomorphic differentials ν1 , . . . , νg . Then the matrix Π of b-periods is
given asZZ−1νl(8.13)
ωk = Ãlk
πjk =
bj

bj

and the b-periods of ω∞+ ∞− are given asZZZω∞+ ∞− =νg+1 − πjkνg+1 .
bj

8.1.3

bj

ak

Theta function

Although there are implementations for the evaluation of theta functions ([6])
in Maple 8, for example, there are no such implementations in Mathematica
5.029 . We implemented a simple method for evaluation of theta functionsof general genus using the defining series (4.1). In most cases, the numberof terms necessary to reach the machine precision is at most 15 in eachdirection which allows evaluation of theta function up to genus 4 with timethat is small compared to the time needed to compute the parameters of
the Riemann surface. The number of terms in the series (4.1) is determinedby the condition that they are strictly smaller than some threshold value ε
which is required to be smaller then the machine precision. The condition
for the maximal norm of N necessary in (4.1) to obtain the desired precisonreadsqkzk + kzk2 − π1 λ0 ln εkN k >,
λ0
with λ0 being the smallest30 eigenvalue of Im Π.
8.1.4

Mass and angular momentum

The mass and the angular momentum of the spacetime described by a solution of the Ernst equation can be inferred from the behavior of the Ernst
29
30

We are not aware of any of them.
Recall that Im Π > 0.

8 NUMERICAL EVALUATION OF THE THETA SOLUTIONS

61

potential E at infinity, i. e. in the limit |ξ| → ∞. The asymptotic expansionof the Ernst potential E is given by the formula (8.4). With the substitutionξ = ζ − iρ = k/t with k ∈ C, Im k < 0, it suffices to analyze the behavior of¯E(ξ(t), ξ(t))
at t → 0+.
¯
Numerically, we sample the function E(ξ(t), ξ(t))
atPsome points tl > 0
close enough to zero and we fit the polynomial p(t) = dn=0 pn tn of degreed ≥ 3 to the samples by the least-square method. Then we deduce the constants m (mass) and J (angular momentum) from the coefficients of the fitted
polynomial. This procedure yields a very quick and reliable determinationof the physical parameters of spacetime.

8.2

Numerical results

The implemented code was used to obtain Ernst potentials on hyperelliptic
surfaces of genus 1 and genus 2. The branch points of the surfaces were
¯ i, −i} and {ξ, ξ,¯ i − 1, −i − 1, i + 1, −i + 1}. Characteristic p, q was{ξ, ξ,
chosen to yield a ‘realistic’ Ernst potential. Namely, p = {0.2}, q = {0.} forgenus 1 and p = {0.2, 0.}, q = {0., 0.} for genus 2.
The first step is a computation of the matrix of b-periods Π and the valueof ϕ(∞+ ) = −ϕ(∞− ). For the integration, we constantly used Chebyshev
approximation by 32 polynomials. The speed of our code was 70 (resp. 30)
different positions of ξ, ξ¯ per second for genus 1 (resp. genus 2) solution on alow-end computer. We tested the performance of the numerical routines byintroducing two quantities – the norm of the asymmetric part of Π, denotedby b∆ ,
Π − ΠTb∆ =,2and the norm of the sum of a-periods of all differentials νj , j = 1, . . . , g + 1
together with the periods of all differentials around a curve a0 with a0 · bj =
−1,
!2
g Z
g+1
X
X
νk − 2πiδk(g+1) .
a2∆ =
k=1

j=0

aj

The term −2πiδk(g+1) is there because νg+1 has residue 1 at ∞+ . Both quantities should yield zero. Their magnitude is thus a measure of the error of
estimation of Π and ϕ(∞+ ). With a good choice of integration paths andthe integration method – spectral method or Clenshaw-Curtis quadrature –we got the value a∆ as low as the machine precision, i. e. a∆ ∼ 10−15 . The
magnitude of b∆ was higher, i. e. b∆ ∼ 10−8 . As an example of the dependence of numerical precision on a correct choice of integration paths, we8 NUMERICAL EVALUATION OF THE THETA SOLUTIONS

0-5log aD
-10
-15
-20

62

-0.5
-1
0.5
1
Ρ

Ζ

-1.5
1.5
2

-2

Figure 8: The error of numerical integration measured by a∆present a plot of a∆ for the hyperelliptic Riemann surface with branch points¯ −0.00001i, 0.00001i} (solitonic limit) versus the value of ξ(= ζ −iρ), see{ξ, ξ,
figure 8. It is clear that when some other branch points are too close to theintegration path (in this case the branch points −0.00001i and 0.00001i to¯ the numerical integration fails. The dependence ofthe line between ξ and ξ),
the minimal eigenvalue λ0 of ℑΠ is interesting as well. Figures 9(a) and 9(b)show this dependence for hyperelliptic surfaces of genus 1 and 2, respectively.
One can see that λ0 diverges at the branch points in the plot range.
Problems appeared in the evaluation of Ernst potential from the quantities on a surface. Although the obtained potentials have the right qualitative
behavior (asymptotic flatness, E → 1 when ξ → ∞, differentiability everywhere except points where conditions of theorem 7.1 do not hold), they do
not obey the Ernst equation exactly. These problems are probably causedby an incorrect implementation of function (8.3) and we have not been able
to solve them yet. As an illustration of our results, we present plots of theErnst potentials for genus 1 and genus 2 solutions on surfaces with branch
points mentioned above, see figure 10 and 11.
The genus 1 solution possess reflection symmetry because it obeys [30]E(ρ, ζ) = Ē(ρ, −ζ).

8 NUMERICAL EVALUATION OF THE THETA SOLUTIONS

2.5
Λ0 2
1.5
1
0.5

1
0.5
0
0.5
1
Ρ

Ζ

-0.5
1.5
2

-1

(a) Genus 1

1.25
Λ0 1
0.75
0.5
0.25

0
-0.5
-1
0.5
1
Ρ

Ζ

-1.5
1.5
2

-2

(b) Genus 2

Figure 9: Minimal eigenvalue of the matrix of b-periods Π63

8 NUMERICAL EVALUATION OF THE THETA SOLUTIONS

64

Imaginary part of E has a jump across the line ζ = 0, ρ ∈ [0, 1], whereas
the real part is continuous there. There is a possibility that this jump canbe explained by a thin disk of matter in this region. This conjecture isnot, however, verified. A study of the asymptotic behavior of E for ξ → ∞suggests, that the mass of the object is purely imaginary, therefore very
exotic.
The genus 2 solution is presented here only in the range ζ < 0 because
the code is not able to change the integration paths automatically yet. In
this case, a jump across the line ζ ∈ [−1, 1], ρ = 1 is present in both real andimaginary parts of E.

8 NUMERICAL EVALUATION OF THE THETA SOLUTIONS

0.6
1

0.4

0.5

0.2
0
0.5
1
Ρ

Ζ

-0.5
1.5
2

-1

(a) ℜE

0.1
1
0
0.5

-0.1
0
0.5
1
Ρ

Ζ

-0.5
1.5
2

-1

(b) ℑE

Figure 10: The Ernst potential of a genus 1 solution

65

8 NUMERICAL EVALUATION OF THE THETA SOLUTIONS

0.8
0.7
0.6
0.5
0.4

0
-0.5
-1
0.5
1
Ρ

Ζ

-1.5
1.5
2

-2

(a) ℜE

0

0

-0.2
-0.5
-0.4
-1
0.5
1
Ρ

Ζ

-1.5
1.5
2

-2

(b) ℑE

Figure 11: The Ernst potential of a genus 2 solution

66

9 CONCLUSIONS

967Conclusions

We investigated algebro-geometric methods for construction of solutions to
the Ernst equation, in particular using the scalar Riemann-Hilbert problem
on hyperelliptic Riemann surfaces, following the approach in [21]. We gave
more detailed and perhaps more precise proofs of two key theorems that summarize the analytic properties of matrix functions that satisfy linear systems
for the Ernst equation. We also discussed relevance of hyperelliptic Riemannsurfaces.
We implemented “fast” and accurate code for evaluation of quantities ofinterest on hyperelliptic surfaces. Although the code is far from complete,
we obtained some results and plots comparable to a similar program byFrauendiener & Klein [11, 12].
There are still many open problems. In this work, we did not address
physical properties of the theta solutions although more detailed investigation of solutions of higher genus has not been published yet. There is ahigh chance to identify subclasses of these solutions describing more complex
physical systems, in particular a black hole–disk system. There is no direct
way to infer the jump data from the boundary value problem one wants tosolve. The explicit form of the theta solutions, however, possibly offers a
different approach to boundary value problems: one can try to identify thefree parameters in the solutions from the boundary problem.
An interesting problem is also a construction of theta solutions to the
Einstein-Maxwell equations. In this case, the linear system is similar to thelinear systems for the Ernst equation, but this time it is a system for a 3 × 3
matrix . The solutions can be constructed using the recent solution to the
Riemann-Hilbert problem with quasi-permutation monodromies [25]. The
solution lives on a 3-sheeted cover of CP1 and therefore the structure of thissurface will not be as simple as in the case of hyperelliptic surfaces. Moreinvolved theory will have to be used and the numerical code will have to beextended to manage numerics on such surfaces.
The code can also be improved considerably so it will be able to handle
general hyperelliptic surfaces without any assumptions on reality properties
or genus. A black-box program for a general use, working with general hyperelliptic surfaces without any need of instigation of the surface, would bevery helpful for astrophysical and other applications.

10 REFERENCES

10

68

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[2]

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[3]

Carroll S. M., Lecture Notes on General Relativity, arXiv:gr-gc/9712019 (1997).

[4]

Cosgrove C. M., Relationship between the group-theoretic and soliton-theoretic techniques for generating stationary axisymmetric gravitational solutions, J. Math. Phys.
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