BACHELOR THESIS

Robert Jurcık

Centre of the Kerr and Appellspace-times

Institute of Theoretical Physics

Supervisor of the bachelor thesis: doc. RNDr. Oldrich Semerak, DScStudy programme: Physics (B1701)

Study branch: FOF (1701R026)

Prague 2021

I declare that I carried out this bachelor thesis independently, and only with thecited sources, literature and other professional sources. It has not been used toobtain another or the same degree.I understand that my work relates to the rights and obligations under the ActNo. 121/2000 Sb., the Copyright Act, as amended, in particular the fact that theCharles University has the right to conclude a license agreement on the use of thiswork as a school work pursuant to Section 60 subsection 1 of the Copyright Act.

In . . . . . . . . . . . . . date . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Author’s signature

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Firstly, I would like to thank doc. RNDr. Oldrich Semerak, DSc for his sugges-tions, patient guidance and time.

Secondly, my thanks go to my dearest partner Bc. Katerina Mlada for sup-porting me during studies and to Bc. Radek Zajıcek for his attempt to understandthis thesis.

Lastly, I thank to Mgr. Deniska Jurcıkova for her help with the language.

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Title: Centre of the Kerr and Appell space-times

Author: Robert Jurcık

Institute: Institute of Theoretical Physics

Supervisor: doc. RNDr. Oldrich Semerak, DSc, Institute of Theoretical Physics

Abstract: One of the most important solutions of Einstein equations is the Kerrmetric. At the very centre of this space-time, there lies a ring curvature singular-ity. The singularity encircles a surface which joins together two asymptoticallyflat sheets of the manifold. The surface is intrinsically flat and is standardlyinterpreted as a planar disc. However, an article has been recently publishedwhich claims that the central surface is actually a dicone, with vertex (vertices)on the symmetry axis. In this thesis we analyse various geometric characteristicsof the surface, in order to check which of the pictures is more adequate. We alsoexamine the same surface of the Appell space-time which has the same spatialstructure as the Kerr one.

Keywords: general relativity, geometry of space-time, Kerr metric

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Contents

Introduction 2

1 Space-times considered 31.1 Kerr space-time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Boyer-Lindquist coordinates . . . . . . . . . . . . . . . . . 31.1.2 Kerr-Schild coordinates . . . . . . . . . . . . . . . . . . . . 4

1.2 Appell space-time . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Weyl cylindrical coordinates . . . . . . . . . . . . . . . . . 51.2.2 Oblate spheroidal coordinates . . . . . . . . . . . . . . . . 6

2 Curvature invariants 72.1 Intrinsic curvature . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Extrinsic curvature . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Geometry of the central r = 0 disc 123.1 Regularity of the metric coefficients . . . . . . . . . . . . . . . . . 123.2 Embedding of the central disc into the Euclidean space . . . . . . 143.3 The radius-circumference relation . . . . . . . . . . . . . . . . . . 153.4 Calculating the curvature invariants for the Kerr space-time . . . 163.5 Calculating the curvature invariants for the Appell space-time . . 19

Conclusion 23

Bibliography 24

List of Figures 25

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IntroductionThe general theory of relativity stands as the greatest and most comprehensivetheory of gravity for more than a century. Along with the theory came interestingtheoretical predictions, which have been gradually confirmed since then. One ofthese predictions is the existence of black holes. In brevity, the black hole isan object with such a strong gravitational field that nothing can escape fromit – neither particles nor electromagnetic radiation. In general relativity, thereare four basic types of exact, isolated, stationary, and asymptotically flat blackhole solutions – the Schwarzschild, Reissner-Nordstrom, Kerr and Kerr-Newmansolution. These solutions differ in parameters, which characterize the space-time.The Schwarzschild solution is characterized only by its mass. The Reissner-Nordstrom solution has, in addition to mass, a charge. The Kerr solution ischaracterized by mass and spin. Lastly, the Kerr-Newman solution has all threeparameters – mass, spin, and a charge.

From astrophysical observations of the exteriors of black holes, it seems thatthe Kerr solution is the most relevant. Apart from the exterior, we can studyits interior mathematically as well, which is performed through the so-calledmaximal extensions. Unfortunately, we currently do not have mechanisms forthe experimental study of this interior but this does not mean the mathematicalanalysis is meaningless. We may be able to extract some information from theinside of a black hole in the future and even if not, this analysis can lead to deeperunderstanding of general relativity as such. Inside the aforementioned Kerr blackhole, there is a ring curvature singularity. We focus on the surface enclosed bythis singular ring.

We base our thesis on the conflict between two articles. The first article isfrom W. Israel and has been published in 1970 [1]. The other has been writtenby H. Garcıa-Compean and V. S. Manko and has been published in 2015 [2]. In[1], Israel establishes the general view on the geometric nature of the central disc.He states that the singular ring encloses an intrinsic plane and calls this area adisc. On the other hand, the authors claim in [2] that the long acknowledgedinterpretation is wrong and that they have found the true geometric nature ofthe surface – it is a dicone. This claim implies certain behaviour not discussedin [2], which we focus on in Chapter 3. In Chapter 1 and 2 we introduce the keyconcepts.

Simultaneously, we calculate the same properties for the Appell space-timeas well, because the Appell space-time is know to be Newtonian analogue forthe Kerr space-time. In [3], this analogy is shown for the Euler field whichis fundamentally the same as the Appell space-time. Additionally, we refer toAppendix A in [4] where some similarities and differences are mentioned as well.

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1. Space-times consideredPhysics, through mathematics, creates models, which we set side by side withthe real world. In this thesis, we use general relativity as a model and its centralconcept is the metric tensor. The metric tensor (or shortly the metric) is asolution of Einstein equations and it contains the whole information about thestudied system. In other words, it generates a space-time which we analyse andidentify with nature.

Throughout this chapter, we introduce two metric tensors, both of which areexpressed in two different types of coordinates.

1.1 Kerr space-timeOne of the simplest and the most basic solutions of general relativity is, without adoubt, the Schwarzschild solution. It describes the space-time outside of sphericalmass. The solution has only one parameter - the mass. However, objects in theuniverse have other properties we need to take into consideration, as was said inthe introduction. One of them is their rotation.

The solution which takes both mass and spin into account, has been found byR. P. Kerr, after whom it is named. It is worth mentioning that this solution hasbeen published in 1963, which is almost 50 years after the Schwarzschild solution.

The Kerr solution can be expressed in several types of coordinates and eachof them might reveal different properties of the space-time. For studying the be-haviour of the central disc, the Boyer-Lindquist coordinates are the most suitablesince the central disc is described as r = 0.

1.1.1 Boyer-Lindquist coordinatesThe Kerr metric represented in the Boyer-Lindquist (B-L) coordinates (t, r, θ, ϕ)has the form [5]

ds2 = −(

1 − 2Mr

Σ

)dt2 − 4Mr

Σ a sin2 θ dtdϕ+ AΣ sin2 θ dϕ2 + Σ

∆dr2 +Σdθ2, (1.1)

where Σ, ∆, and A read

Σ = r2 + a2 cos2 θ, (1.2)∆ = r2 − 2Mr + a2, (1.3)A = (r2 + a2)2 − ∆a2 sin2 θ. (1.4)

Parameter M represents mass of the source and a is the rotational angular mo-mentum divided by mass. For brevity, a is called spin.

The substantiation for naming parameter M mass can be seen when onechooses a = 0 in (1.1). The outcome is Schwarzschild metric, where we know themeaning of that sole parameter. Naming parameter a spin is based on invarianceof the metric under the transformations a → −a ⇒ t → −t , a → −a ⇒φ → −φ and on its behaviour at large radii.

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In order to acquaint ourselves with the presented metric, we can study itsproperties. For example, what happens, if we choose M = 0? The metric reducesto the Minkowski space-time in the oblate spheroidal coordinates

ds2 = −dt2 + Σr2 + a2 dr2 + Σdθ2 + (r2 + a2) sin2 θdϕ2. (1.5)

Indeed, if we express the Minkowski metric in the following coordinates:

x =√

r2 + a2 sin θ sin ϕ, (1.6)y =

√r2 + a2 sin θ cos ϕ, (1.7)

z = r cos θ, (1.8)

we obtain (1.5). However, recently it has been pointed out [6] that the limit Mapproaches zero is not so trivial. The article [6] claims that the result of thislimit is the locally flat wormhole and not flat Minkowski space-time as is usuallyassumed. Nevertheless, this claim does not take into account the work [2] and hismaximal extension as has been commented in [7]. This supports the necessity ofcorrect interpretation of the central surface and consequently the construction ofcorrect maximal extension.

It is evident that the metric (1.1) has a singularity at Σ = 0. For a = 0, thiscondition can be reformulated using (1.2). This gives us Σ = 0 ⇔ (r = 0 ∧ θ = π

2 ).The nature of this singularity is more visible in the Kerr-Schild coordinates.

The metric tensor also diverges for ∆ = 0 ⇔ r1,2 = M ±√

M2 − a2. However,these are not curvature singularities but merely the singularities of coordinates(horizons). In the third section, we see that no invariant diverges on horizons.

There are other properties of the Kerr metric: the metric is stationary (exceptbetween the horizons), axially symmetric and reflection symmetric with respectto the plane θ = π

2 (called the equatorial plane).

1.1.2 Kerr-Schild coordinatesAs we mentioned before, there are other useful coordinates, namely the Kerr-Schild (K-S) ones (T, ϱ, φ, z). In this coordinate system, the Kerr metric has theform [8]

ds2 = −dT 2 + dϱ2 + ϱ2dφ2 + dz2 + 2Mr3

r4 + a2z2

(dT + rϱdϱ − aϱ2dφ

r2 + a2 + zdz

r

)2

,

(1.9)where r satisfies the following equation:

r4 − (ϱ2 − a2 + z2)r2 − a2z2 = 0, (1.10)

which has the following four solutions:

r1,2 = ±

ϱ2 − a2 + z2 +√

(ϱ2 − a2 + z2)2 + 4a2z2

2 , (1.11)

r3,4 = ±

ϱ2 − a2 + z2 −√

(ϱ2 − a2 + z2)2 + 4a2z2

2 . (1.12)

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Depending on our position, one of the roots is selected.Once again, with the choice M = 0, the metric assumes the form of Minkowski

metric, but this time it is expressed in the cylindrical coordinates. As we men-tioned in the previous section, this standard interpretation does not have to bethe right one [6].

In the B-L coordinates, we have noticed that the Kerr metric has a singularityat (r = 0 ∧ θ = π

2 ). Therefore, the singularity in the Kerr-Schild coordinates liesat z = 0, ρ = a. As we have foretold in the previous section, the true nature ofthis singularity is revealed in the K-S coordinates – it is a ring in the equatorialplane with the radius a.

1.2 Appell space-timeAs well as in the Kerr case, the Appell solution is the exact and vacuum solutionwithout cosmological constant. The solution has been found by solving a cylin-drical Laplace equation in the variable φ, because solving Einstein equations fora static axisymmetric space-time is principally the same as solving a problem inNewtonian gravity [9].

In this thesis, the main motivation for using the Appell solution is in a topo-logical similarity with the Kerr solution. In addition, they have almost the samestructure of momenta. However, the Appell metric is simpler in the sense that,unlike the Kerr metric, it is static (Weyl type), so there exists a set of coordinatesin which it is diagonal and additionally, there are no horizons.

1.2.1 Weyl cylindrical coordinatesThe Appell solution is once more a two parametric solution, but now it is givenby the Weyl metric. The Weyl metrics are a class of axially symmetric and staticsolutions of Einstein equations. They can be expressed in the Weyl cylindricalcoordinates (t, ρ, φ, z) [8],

ds2 = −e2νdt2 + e−2ν [ρ2dφ2 + e2λ(dρ2 + dz2)], (1.13)

where ν and λ stand for parameters (also called potentials).The Appell solution is given by the following choice of parameters:

ν = ∓ M√2Σ

√Σ + ρ2 + z2 − a2, (1.14)

λ = M2

8a2

[1 − ρ2 + z2 + a2

Σ − 2a2ρ2(Σ2 − 8z2a2)Σ4

], (1.15)

where a is the Weyl radius, M is mass, and Σ stands for

Σ =√

(ρ2 − a2 + z2)2 + 4a2z2. (1.16)

The singularity occurs for Σ = 0, which is equivalent to (ρ = a ∧ z = 0). Thisleads us to the interpretation of the singularity as a ring in the equatorial planez = 0 with the radius ρ = a, which is the same result as we have obtained in theK-S coordinates for the Kerr space-time.

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1.2.2 Oblate spheroidal coordinatesAnother way to describe the Appell space-time is through the oblate spheroidalcoordinates (t, R, ϕ, θ). The metric has the following form [8]:

ds2 = −e2νdt2 + e2ν(R2 + a2) sin2 θdϕ2 + Σe2(λ−ν)(

dR2

R2 + a2 + dθ2)

, (1.17)

where ν, λ, and Σ are given by

ν = −MR

Σ , (1.18)

λ = −M2 sin2 θ

4Σ

[1 + (R2 + a2)(Σ2 − 8R2a2 cos2 θ)

Σ3

], (1.19)

Σ = R2 + a2 cos2 θ. (1.20)

For a = 0, the aforementioned singularity transforms to (R = 0 ∧ θ = π2 ).

In addition to the already mentioned resemblances, there are other which canbe easily seen through a comparison of the metrics expressed either the Kerrmetric in the B-L coordinates and the Appell metric in the oblate spheroidalcoordinates or the Kerr metric in the K-S coordinates and the Appell metric inthe Weyl cylindrical coordinates. One example can be the same placement of thecurvature singularity.

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2. Curvature invariantsInvariants, by definition, provide information independent of the choice of coor-dinate system. We have chosen two invariants which together contain the entireinformation about the curvature of a surface. They are the intrinsic and extrin-sic curvature. For the sake of simplicity, we use the Gaussian curvature as arepresentative of the intrinsic curvature.

In this chapter, we intuitively introduce how to characterize curvature of asurface. We start with the intrinsic curvature of a curve and its generalisationto a surface. Afterwards, we introduce the extrinsic curvature of a surface. Itis possible to state all the equations that we use for calculations throughout thethesis. However, the beauty of physics lies in the process of discovery. Addi-tionally, we gain intuition by reasoning and examining the geometry. For thatreason, we introduce several definitions of curvature and show their equivalence.This approach is inspired by Box 14.1 Perspectives on curvature from [10].

2.1 Intrinsic curvatureLet us assume a curved line, e.g. Fig. 2.1. How to characterize the curvature ofsuch a curve? The most intuitive way is to set a referential direction and studythe change of an angle θ between the referential direction and tangent vector tothe curve. Naturally, the change of the angle along the curve characterizes thecurvature. It is thus possible to define the curvature of a curve as follows [10]

κ = ds(θ)dθ

, (2.1)

where s is an affine parameter of the curve.

Figure 2.1: The example of a curve parametrized by s. The angle between thetangent line and reference is denoted by θ, ρ stands for the curvature radius [10].

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In literature, we usually encounter definition by means of a radius of curvatureR (in Fig. 2.1 it is denoted by ρ). In such a case, the curvature is defined as [10]

κ = 1R

. (2.2)

We have two definitions and therefore, we are obliged to show the equivalencebetween them. Let us suppose we have an arbitrary continues curve. This means,that for every point of the curve, there exists a neighbourhood. We can work onsuch neighbourhood1 that the curve coincides with circle of radius R (Fig. 2.2).Next, we make tangent lines at points A and B. We denote the angles betweenreferential and tangent lines by θ1, θ2, respectively. Geometrical analysis leads usto the equation

dθ = θ2 − θ1 = dφ. (2.3)The last piece of information is the relationship between the small angle dφ andthe arc of a circle with the radius R, denoted by ds,

ds = Rdφ. (2.4)

Finally, we can insert the aforementioned expressions for dθ and ds into theequation (2.1), which is the desired conclusion.

Figure 2.2: Approximation of a curve (grey line) by a circle with the radius Rwith two tangent lines (red lines) and angles between tangent lines and referentialdirection (green line). The angle dφ is an angle between radii.

The equation (2.2) can be generalized for 2D-surfaces as follows [10]

κ = 1R1R2

, (2.5)

where R1 and R2 are the radii of curvature.This outcome is easily grasped and gives us an insight into the concept of the

Gaussian curvature. However, it is of no use in general relativity. We need todefine the intrinsic curvature in the form of the metric tensor and his derivatives.Hopefully, it is a part of every introductory course of general relativity thatthe Riemann tensor contains the whole information about the curvature. It ispossible to find that the Riemann tensor of 2D-surfaces has only one independentcomponent (for example from symmetries). Clearly, this component needs to be

1To be precise, we are working on an infinitesimally small neighbourhood, because one ofthe definitions is using a derivative.

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related to the Gaussian curvature. However, which component should we choose?We need to choose either one of the 24 components or we can use an invariant –the scalar curvature! Let us use Occam’s razor and assume the relation betweenthe Gaussian curvature and the scalar curvature is linear, that is

κ = α · R + β, (2.6)

where α, β are arbitrary constants. The value of β needs to be zero because theGaussian curvature and the Riemann tensor of a plane are zero which leads tozero scalar curvature. We find α from the example of a sphere with the radius ofa. The metric of the sphere is

ds2 = a2dθ2 + a2 sin2 θdφ2. (2.7)

From this metric, it is necessary to determine the Riemann tensor. Afterwards,the scalar curvature is calculated as the double contraction of the Riemann tensor.The outcome is R = 2

a2 . The Gaussian curvature of the sphere is κ = 1a2 [10].

Thus from equation (2.6), we obtain α = 12 .

In the rest of the thesis, we use the following relation for calculating theGaussian curvature

κ = R

2 . (2.8)

Is the whole information about curvature of a plane in the Gaussian curvature?From the relation to the metric, we see it contains the whole intrinsic informationbut from the definition (2.5), we see that in the extreme cases (i.e. one radius iszero or infinity), we lose information about the second radius.

There are only three types of spatial geometric bodies with zero Gauss cur-vature - a plane, cone, and cylinder. One can easily verify that these bodies havezero Gauss curvature by imagining their curvature radii (immediately, we see thatone radius is infinite). The proof that there are only these three bodies is beyondthe scope of this thesis.

2.2 Extrinsic curvatureThe whole intrinsic information about curvature is in the Gaussian curvature, aswe have seen in the previous section. However, it does not contain the wholeinformation about the bending into higher dimension, especially when one of thecurvature radii is infinite. How to modify the definition (2.5) so that there isno loss of information when one radius is infinite? The easiest way is to definecurvature as [10]

Kext = 1R1

+ 1R2

. (2.9)

In that way, when one radius is infinite it does not suppress information about thesecond radius. The curvature defined by equation (2.9) is in [10] called extrinsiccurvature. 2

Once again, we need to find a way how to characterize the extrinsic curvaturein a language of differential geometry. Let as suppose we have a cylinder (Fig.2.3).

2To be precise naming Kext the extrinsic curvature is little-bit misleading because the ex-trinsic curvature is a tensor of second rank and Kext corresponds only to its trace.

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Figure 2.3: The examples of two 2D-surfaces with zero intrinsic curvature. Theone on the left has zero extrinsic curvature and the other has non zero extrinsiccurvature [10].

Let us first focus on the left part of Fig. 2.3. When we visualize a paralleltransport of a vector along the triangle, we realize that neither the tangent nornormal vector changes. It is a consequence of zero intrinsic and extrinsic curva-ture. On the other hand, when we focus on the right part, there is a difference:after the parallel transport, the normal vector changes and we see that the extrin-sic curvature is not zero – new information has appeared. Therefore, we expectthe extrinsic curvature can be characterized by the change of parallel transportedvector along an enclosed trajectory.

The space-time is given by a four dimensional metric tensor and we explainhow to extract a normal vector to a (hyper-)surface and a metric tensor of a(hyper-)surface from it.

First, we need to do so-called ”3+1 splitting” of space-time which is splittingof the space-time (to space and time). In the case of adaptable coordinates (usedin this thesis), the splitting is done by fixing the value of some coordinate inwhich the 4D metric tensor is given (in the case of the ”3+1 splitting”, we fixtime).

We can always rewrite the metric in the following way:ds2 = −Ndt2 + hij(N idt + dxi)(N jdt + dxj), (2.10)

where N is the lapse function and N i is the shift. We define the normal vectorto the hyper-surface Σt (the hyper-surface at a constant time t) as [11]

nα = −Nδ0α, nα = 1

N(1, −N i). (2.11)

We separate the metric into two parts. The first part is given by the normalvector to Σt and the second part is the rest (which is written as hµν) [11]

gµν = −nµnν + hµν . (2.12)The same can be rewritten as

g00 = −N2 + h00 = −N2 + NjNj, (2.13)

g0i = h0i = Ni, (2.14)gi0 = hi0 = Ni, (2.15)gij = hij, (2.16)

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where hij is the pure spatial metric tensor of the Σt.Similarly, one can decompose the contravariant metric [11]

gµν = −nµnν + hµν , (2.17)

which can be, once again, rewritten as

g00 = − 1N2 + 0 = − 1

N2 , (2.18)

g0i = N i

N2 + 0 = N i

N2 , (2.19)

gi0 = N i

N2 + 0 = N i

N2 , (2.20)

gij = −N iNk

N2 + hij, (2.21)

where hij is the inverse to the spatial metric hij.Next, we need to perform 2+1 splitting within the hyper-surface Σt. The

decomposition is almost the same. We start with separating the metric of thehyper-surface Σt into the normal part to a surface Σts (surface of constant t andone spatial coordinate) and the rest (denoted by ˜hµν):

hµν = ˜nµ˜nν + ˜hµν . (2.22)

The extrinsic curvature ˜Kµν of the surface Σts is defined as [11]

˜Kµν = ˜nα|β˜hα

µ˜hβ

ν , (2.23)

where ˜nµ is the normal vector to Σts, mixed components ˜hαµ are obtained by

˜hαµ = ˜hµνgνα, (2.24)

and by an index-posed vertical line, we denote the covariant derivative corre-sponding to the Levi-Civita connection of the metric hµν . The invariant informa-tion is extracted as the following contraction

˜Kext = ˜hµν˜Kµν . (2.25)

The trace of the extrinsic curvature ˜Kext can be rewritten into a simpler form[11] ˜Kext = ˜nα

|α. (2.26)

The trace ˜Kext has a clear geometrical interpretation: it gives expansion ofa vector field nµ, which means Σts is convex for ˜Kext < 0 (i.e. normal vectorfield is converging) and ˜Kext > 0 implies that Σts is concave (i.e. normal field isdiverging).

Note that the fundamental difference between the intrinsic and extrinsic cur-vature is that the inhabitant of a surface can detect and measure the intrinsiccurvature but not the extrinsic one. In other words, inhabitant of a surface withzero intrinsic curvature can not decide if he is living on the plane, cone or cylinder.

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3. Geometry of the central r = 0discFirstly, let us take a look at the regularity of the metric components. The mainmotivation for this is the following statement: the geometrical nature of thesurface enclosed by the ring singularity is a dicone [2]. If this statement holds,one expects problems in regularity at the apexes. Then we embed the centraldisc in the Euclidean space. As a follow-up, we examine the behaviour of propercircumference and proper radius within the central disc. At the end, we calculatethe curvature invariants.

3.1 Regularity of the metric coefficientsAs we mentioned above, the interpretation of the central disc in [2] is a dicone.For that reason, we expect some problems in regularity at the apexes. We studythe enclosed disc for the Appell case as well. In this section, we choose to expressthe Kerr metric only in the B-L coordinates and the Appell space-time in theoblate coordinates. Let us start with the Kerr case.

We have calculated the derivatives of the metric coefficients up to the sec-ond order. We have found that they are regular everywhere except at the ringr = 0, θ = π

2 . This is a reasonable result because the Kerr space-time has thewell-known curvature singularity there. The metric coefficients for r = 0 reduceto

gtt = −1, gtϕ = 0, grr = cos2 θ,

gθθ = a2 cos2 θ, gϕϕ = a2 sin2 θ.

The derivatives with respect to t and ϕ are always zero because the metric isstationary and axially symmetric. The rest of the first order derivatives are

gtt,r = 2M sec2 θa2 , gtϕ,r = −2M tan2 θ

a, grr,r = 2M cos2 θ

a2 ,

gθθ,r = 0, gϕϕ,r = 2M sin2 θ tan2 θ,

gtt,θ = 0, gtϕ,θ = 0, grr,θ = −2 sin(2θ),

gθθ,θ = −2a2 sin(2θ), gϕϕ,θ = a2 sin(2θ).

The cross derivatives are

gtt,rθ = 4M tan θ sec2 θa2 , gtϕ,rθ = −4M tan θ sec2 θ

a, grr,rθ = −2M sin(2θ)

a2 ,

gθθ,rθ = 0, gϕϕ,rθ = 4M(1 + cos2 θ) tan3 θ,

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and the second order derivatives have the form

gtt,rr = −8M2 tan2 θ sec2 θa4 , gtϕ,rr = 0, grr,rr = 2a2−(a2−4M2) cos2 θ

a4 ,

gθθ,rr = 2, gϕϕ,rr = 2 sin2 θ,

gtt,θθ = 0, gtϕ,θθ = 0, grr,θθ = −2 cos(2θ),

gθθ,θθ = −2a2 cos(2θ), gϕϕ,θθ = 2a2 cos(2θ).

There is nothing problematic except for the expected curvature ring singularityat r = 0, θ = π

2 . In addition, we refer to [8], where different curvature invariantis used. This invariant expressed at the central disc r = 0 gives

|K − i∗K| = 48M2

a6 cos6 θ. (3.1)

Once again, the expression is problematic only at the point θ = π2 . In equation

(3.1), K stands for the Kretschmann invariant and ∗K is the Chern-Pontryaginscalar. As a reminder, the Kretchmann invariant is defined from the Riemanntensor Rµναβ as

K = RµναβRµναβ, (3.2)and the Chern-Pontryagin invariant is obtained as

∗K = ∗RµναβRµναβ, (3.3)

where ∗Rµναβ is the Riemann left dual.Let us move on to the Appell case. Again, the metric coefficients and their

first and second order derivatives are regular everywhere, with the exception ofR = 0, θ = π

2 . Unlike in the Kerr case, we show the coefficients and theirderivatives only at the point R = 0, θ = 0, because the equations for the wholecentral disc are too complicated. Same as for the Kerr case, the Appell space-timeis stationary and axially symmetric and therefore, we show only the derivativeswith respect to R and θ. The metric coefficients are

gtt = −1, gRR = 1, gθθ = a2, gϕϕ = 0,

and their derivatives

gtt,R = 2Ma2 , gRR,R = 2M

a2 , gθθ,R = 2M, gϕϕ,R = 0,

gtt,θ = 0, gRR,θ = 0, gθθ,θ = 0, gϕϕ,θ = 0,

and the second order derivatives have the form (the cross partial derivatives arezero at the aforementioned point)

gtt,RR = −4M2

a4 , gRR,RR = 4M2

a4 , gθθ,RR = 2 + 4M2

a2 , gϕϕ,RR = 0,

gtt,θθ = 0, gRR,θθ = −2 − 2M2

a2 , gθθ,θθ = −2(a2 + M2), gϕϕ,θθ = 2a2.

Once again, there is nothing problematic except for the expected curvature ringsingularity at R = 0, θ = π

2 , which is the same result as in the Kerr case.

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3.2 Embedding of the central disc into the Eu-clidean space

One of the methods for visualization of a 2D surface is its isometric embeddinginto the Euclidean space E3. This method is presented for example in [11], whereit is used for embedding the equatorial plane in the Schwarzschild space-time.Firstly, we embed the equatorial plane in the Kerr (the B-L coordinates) andthen in the Appell case (the oblate spheroidal coordinates).

For the embedding of the central disc t =const., r = 0 in the Kerr space-time(the B-L coordinates), we need the metric tensor of the corresponding surface

ds2 = a2 sin2 θdϕ2 + a2 cos2 θdθ2. (3.4)

Now, we want to embed it into E3 which is defined by the metric tensor

dσ2 = dr2 + r2dϕ2 + dz2. (3.5)

In our case, the Kerr space-time is axially symmetric, which means we can requirez = z(r). We can calculate its total differential

dz(r) = dz(r)dr

dr, (3.6)

and substitute it in equation (3.5). Thus, we end up with

dσ2 =⎛⎝1 +

(dz(r)

dr

)2⎞⎠ dr2 + r2dϕ

2. (3.7)

By comparing (3.4) and (3.7), we obtain

r = ±a sin θ, (3.8)z(r) = const., (3.9)

ϕ = ϕ. (3.10)

From z(r), we see that the central disc might be a plane in E3.The result above can be easily seen from the following substitution in (3.4):

r = ±a sin θ, (3.11)dr = ±a cos θdθ. (3.12)

After the substitution, the metric (3.4) takes the form

ds2 = r2dϕ2 + dr2, (3.13)

which is a flat plane in polar coordinates.

14

Let us concentrate on the Appell case. Once again, we need to obtain themetric tensor of the equatorial slice t = const., R = 0:

ds2 = a2 sin2 θdϕ2 + a2 cos2 θe2λdθ2, (3.14)

where λ readsλ = −M2 sin2 θ

4a2 cos4 θ

(1 + cos2 θ

). (3.15)

We want to embed the surface defined by (3.14) in E3, which is given by (3.5).We work, once again, with axially symmetric space-time, which means we canrequire z = z(r) with total differential (3.6). From the comparison of (3.14) and(3.7), one obtains

r = ±a sin θ, (3.16)ϕ = ϕ, (3.17)

z(r) =∫ √

e2λ − 1 dr, (3.18)

where λ as a function of r is

λ = −(

Mr

2a (a2 − r2)

)2 (2a2 − r2

). (3.19)

The definition domain for r is [0, a]. Therefore, the integrand in (3.18) is a pureimaginary number with the exception of r = 0 for which it is zero.

The result for the Kerr case is equivalent with the zero Gaussian curvaturebecause we calculated it only from the intrinsic metric tensor. Of course, theplane can still be embedded in E3 in various ways, with the correspoding extrinsiccurvature not necessarily vanishing. On the other hand, we are not able tointerpret the result for the Appell case. The outcome (3.18) in the Appell case isnot physically relevant because it is imaginary. This can be caused by a wrongmethod of embedding but, having said that, we are not able to say the reasonwhy our method does not work.

3.3 The radius-circumference relationAnother possible test of whether a given surface is flat, follows from the relationbetween the proper circumference ξ and proper radius l of any circle. In theEuclidean case, the ratio of these quantities is 2π at every point. As mentionedbefore, the results differ for the Kerr and Appell case because the metrics aredifferent for the central disc. Let us first focus on the Kerr solution. The ratiois calculated ∀Ω ∈ [0, π

2 ]. The meaning of Ω can be seen clearly in the followingcomputation

ξ

l=∫ 2π

0√

gϕϕ|θ=Ω dϕ∫ Ω0

√gθθ dθ

=√

a2 sin2 Ω∫ 2π

0 dϕ∫ Ω0

√a2 cos2 θ dθ

= a sin Ω · 2π

a sin Ω = 2π.

In the first equality, the definition of the proper radius and proper circumferenceis used. In the second equality, we utilize equation (3.4) and the fact that we

15

work with axially symmetric space-times. Afterwards, we calculate integrals andsimplify the result.

We approach the Appell case in the same manner.

ξ

l=∫ 2π

0√

gϕϕ|θ=Ω dϕ∫ Ω0

√gθθ dθ

=√

a2 sin2 Ω∫ 2π

0 dϕ∫ Ω0

√a2 cos2 θe2λ dθ

= sin Ω · 2π∫ Ω0 cos θeλ dθ

.

We are not able to find analytical solution for the proper radius. However, we cantell that for M = 0, the ratio is not equal to 2π, ∀Ω ∈ (0, π

2 ], which is importantinformation about curvature in the central disc. As Ω approaches zero the ratiotends to 2π. This limit can by calculated using L’Hospital’s rule and fundamentallemma of calculus.

The result for the Kerr case shows that the surface is a flat plane and inparticular no conical singularity is present at θ = 0. This outcome carries theintrinsic information as was the case in Section 3.2. It is also equivalent to zeroGaussian curvature everywhere. In contrast, the Appell case is more complicated.Even though, we were not able to calculate the integral in the case of the properradius, the result shows that the enclosed surface has non zero Gaussian curvatureas was calculated in [8].

3.4 Calculating the curvature invariants for theKerr space-time

Let us now calculate the Gaussian curvature in the whole equatorial plane, bothin the B-L and K-S coordinates. The behaviour of the Gaussian curvature isdepicted in contours figures and afterwards, we calculate the extrinsic curvatureof the central disc. Both the figures and the curvature invariants themselves arecalculated using our code in Mathematica.

By restricting the Kerr metric (1.1) to the equatorial plane t =const., θ = π2

we obtainds2 = r2

r2 − 2Mr + a2 dr2 + r3 + a2(2M + r)r

dϕ2. (3.20)

By inserting the metric into Mathematica, we acquire the Gaussian curvature

κ = M (a4r (8M2 + 2Mr − 7r2) + a2r4(11M − 5r) − a6(5M + 3r) − r7)r4 (a2(2M + r) + r3)2 .

(3.21)We can compare the result above with [8], their result matches ours. We use thisoutcome as verification of our code in Mathematica.

In order to visualize the outcome in the equatorial plane, we illustrate theequation (3.21) in Fig. 3.1 and 3.2. The colours in the figures denote ranges ofthe Gaussian curvature. At each point (r, a), the Gaussian curvature is calculatedfrom corresponding equation (e.g. for Fig. 3.1, we use equation 3.21) using thevalues of the point in the figure. We assume M = a for all contour figures.Note that both the Kerr and Appell space-times are axially symmetric. For thatreason, we use a slice of an arbitrary φ =const. (or ϕ =const.) in all contourfigures.

16

Figure 3.1: Contours of theGaussian curvature plottedwithin the equatorial planefor the Kerr solution with theassumption M = a. There isspin a on the horizontal axisand the radius r on the verticalaxis. Both axes have the rangeof 0-0.5.

Figure 3.2: Contours of theGaussian curvature plottedwithin the equatorial planefor the Kerr solution with theassumption M = a. There isspin a on the horizontal axisand the radius r on the verti-cal axis. Both axes have therange of 0-0.02.

We see that the Gaussian curvature diverges at r = 0. The divergence, onsmall neighbourhood of r = 0, leads to the fact that Mathematica can no longercalculate the Gaussian curvature due to memory overflow. This leaves the partof the figure white, which causes the serrated edge of the figure. This phenomenaoccurs in several other figures as well.

We can take a second look at the Gaussian curvature outside of ring singularitybut this time in the K-S coordinates. By restricting the metric tensor to thesurface T =const., z = 0 and choosing ϱ > a, we obtain

ds2 =(

1 + 2M√

ϱ2 − a2

ϱ2

)dϱ2 − 4Ma

ϱdϱdφ +

(ϱ2 − 2Ma2

√ϱ2 + a2

)dφ2. (3.22)

The Gaussian curvature corresponding to the metric tensor (3.22) has the form

κ = −M(a2(9M

√ϱ2 − a2 + 10M2 + ϱ2

)+ 2Mϱ2√ϱ2 − a2 − 2a4 + ϱ4

)(a2 − ϱ2)2

(√ϱ2 − a2 + 2M

)3 . (3.23)

The Gaussian curvature is illustrated in Fig. 3.3. In equation (3.23), we see thatthe Gaussian curvature diverges for ϱ = a. The same can be seen in Fig. 3.3.

17

Figure 3.3: Contours of the Gaussian curvature plotted within the equatorialplane for the Kerr solution in the K-S coordinates with the assumption M = a.There is spin a on the horizontal axis with the range (0,0.005) and the radius ϱon the vertical axis with the range (0,0.010).

Next, we take a look at the geometry of the central disc. The metric of thecentral disc in the B-L coordinates is obtained by choosing t =const., r = 0.We have already made this calculation in Section 3.2., the outcome is equation(3.4). Therefore, we can calculate the Gaussian curvature of the disc

κ = 0. (3.24)

The same result has been calculated for example in [2]. Moreover, we have seenthe intrinsic geometry in Section 3.2 and 3.3, and it is intrinsically flat as well.

Finally, we calculate the extrinsic curvature of the central disc. The metrictensor of the hyper-surface Σt (hyper-surface of the constant time) is

ds2 = AΣ sin2 θdϕ2 + Σ

∆dr2 + Σdθ2, (3.25)

where Σ, ∆, and A are given by (1.2-1.4).From (2.11), we find the normal vectors to the surfaces living within the

hypersurface t =const.

˜nµ =⎛⎝0,

√Σ∆ , 0, 0

⎞⎠ , ˜nµ =⎛⎝0,

√∆Σ , 0, 0

⎞⎠ . (3.26)

Using Mathematica, we calculate the trace of the extrinsic curvature fromboth definitions (2.25) and (2.26), and obtain the result

˜Kext = M sin2 θ

a2| cos3 θ|. (3.27)

We see that the result is non-trivial and equals zero only at θ = 0. Note that thesign of the trace of the extrinsic curvature is based on conventions – firstly in thedefinition (2.26) and secondly in the normal vector.

18

Even thought the same outcome has been already calculated in [12]1, an inter-pretation is still needed. However, it does not seem to be describing a (di-)coneas suggested in [2] because the value at the apex (θ = 0) is zero.

We calculate the trace of the extrinsic curvature of r = 0 in the K-S coordi-nates, too. For brevity, we transform the metric tensor from the B-L coordinatesonly in t −→ T and ϕ −→ φ, which is done by the following formulae [8]:

dT = dt − 2Mr

∆ dr, dφ = dϕ − 2Mar

(r2 + a2)∆dr, dr = dR. (3.28)

After transforming the metric tensor (1.1) and restricting it onto the hyper-surfaceT =const., the metric takes the form

ds2 = Σ(r2 + 2Mr + a2)(r2 + a2)2 dR2 + 4Mra sin2 θ

r2 + a2 dRdφ + AΣ sin2 θdφ2 + Σdθ2. (3.29)

The (co-)normals to the surfaces T =const., R =const. which are living in thehypersurface T =const., are

nµ =√

AΣ(Σ + 2Mr)

(0, 1, 0, − 2MraΣ

A(r2 + a2)

), (3.30)

nµ =√

Σ(Σ + 2Mr)A

(0, 1, 0, 0) . (3.31)

The obtained trace of the extrinsic curvature for r = 0 is the same as (3.27).In order to interpret the result of the extrinsic curvature, we need further

examination. The examination can be done through the embedding diagramswhich are based on the extrinsic curvature instead of the intrinsic one. We havebeen able to find an article which develops such a method [13]. However, thismethod is beyond the scope of this thesis.

3.5 Calculating the curvature invariants for theAppell space-time

In Section 3.2, we have seen that the embedding in the case of the Appell space-time was cumbersome and, consequently, we expect complicated results for theinvariants in this area. We take the same approach as in the Kerr case.

The metric tensor of the equatorial plane t =const., θ = π2 in the oblate

spheroidal coordiantes is

ds2 = R2 + a2

N2 dϕ2 + R2e2λ

N2dR2

R2 + a2 , (3.32)

where λ and N read

λ = −M2(2R2 + a2)4R4 ,

N = e− MR .

1Actually, in [12] the mean curvature is calculated. This mean curvature is defined as a ratioof the trace of extrinsic curvature to the dimension of sub-manifold.

19

The Gaussian curvature reads

κ = −M (a2R2 (2M2 − MR + 3R2) + a4M2 + R4 (M2 − MR + R2))

e−M(a2M+2R2(M−2R))

2R4 R9. (3.33)

There is a strong singularity at R = 0. Using Mathematica we visualize equation(3.33) in Fig. 3.4 . This figure shows the singularity at R = 0.

Figure 3.4: Contours of the Gaussian curvature plotted within the equatorialplane for the Appell solution in the oblate spheroidal coordinates with the as-sumption M = a. There is spin a on the horizontal axis with the range (0,0.005)and the radius R on the vertical axis with the range (0,0.055).

Let us now focus our attention on the equatorial plane in the Weyl coordinates.The metric tensor for the surface t =const., z = 0 with the restriction ρ ≥ a 2

changes into

ds2 = e−2ν[ρ2dϕ2 + e2λdρ2

], (3.34)

where ν and λ are given by

ν = ∓ M√ρ2 − a2 ,

λ = −M2(2ρ2 − a2)4(ρ2 − a2)2 .

The Gaussian curvature is obtained as

κ = −M(ρ4(−M

√ρ2 − a2 + M2 + ρ2

)+ a2Mρ2√ρ2 − a2 − 3a4ρ2 + 2a6

)Ξ

(ρ2 − a2)9/2 ,

(3.35)2This allows us to use simplifications such as

√(ρ2 − a2)2 = ρ2 − a2.

20

where Ξ isΞ = exp

(−1

2M

(M (a2 − 2ρ2)

(a2 − ρ2)2 + 4√ρ2 − a2

)). (3.36)

As in the previous section, we display it in Fig. 3.5. Furthermore, we expect thatequations (3.33) and (3.35) describe the same curvature outside of the ring sin-gularity. In order to support this claim, one can transform from one coordinatesto other or take a look at asymptotic behaviour of the equations.

Figure 3.5: Contours of the Gaussian curvature plotted within the equatorialplane for the Appell solution in the Weyl coordinates with the assumption M = a.There is spin a on the horizontal axis with the range (0,0.005) and the radius ρon the vertical axis with the range (0,0.010).

We can see the singularity at ρ = a. We decided to do the calculation onlyoutside of the singular ring by the restriction above (ρ > a). This causes thewhite area above ρ = a. The area inside is analysed next in oblate spheroidalcoordinates.

The Gaussian curvature of the surface R = 0 reads, in oblate spheroidalcoordinates,

κ = −M2 sec6 θ

a4 eη, (3.37)

where η reads

η = M2 tan2 θ (sec2 θ + 1)2a2 . (3.38)

The Gaussian curvature of the surface enclosed by the singular ring is not zeroas it is in the Kerr case. This has been foreseen in Section 3.2 and 3.3.

The extrinsic curvature of the central disc for the Appell case was calculatedfrom the metric of hyper-surface Σt

ds2 = Σe2λ

N2

(dR2

R2 + a2 + dθ2)

+ (R2 + a2) sin2 θ

N2 dϕ2, (3.39)

21

where λ is given by (1.19). The normal (co-)vector to the surface ΣT is given as

nµ =

Σe2λ

N2(R2 + a2)δRµ , (3.40)

nµ =√

N2(R2 + a2)Σe2λ

δµR, (3.41)

and from equation (2.26) follows

˜Kext = − 2M

a2 | cos3 θ |eΥ, (3.42)

where Υ readsΥ = M2 tan2 θ (sec2 θ + 1)

4a2 . (3.43)

The result is quite complicated. The interpretation of this result needs to be donewith the respect to non zero Gaussian curvature, which should lead to complicatedgeometry of the central surface. Unfortunately, we are unable to carry out suchanalysis properly within this thesis.

22

ConclusionIn our thesis, we confront the conflict between two geometric interpretations ofthe central disc enclosed by ring-shaped singularity in the Kerr space-time. Weexamine this issue for the Appell case as well.

In the first chapter, we introduce both space-times in two types of coordinatesand sum up their basic properties.

In the second chapter, we intuitively introduce two curvature invariants andexplain their role for characterizing the curvature of a surface in the Euclideanspace.

In the last chapter, we confront the aforementioned issue. We calculate be-haviour of the metric coefficients and their derivatives up to second order insideof the central surface (in the B-L coordinates it is t =const., r = 0). We seethat, with the exception of the ring singularity at r = 0, θ = π

2 (B-L coordi-nates), there is nothing problematic. The same result is calculated for the Appellcase (oblate spheroidal coordinates) as well. Next, we embed the central disc inthe Euclidean space. The outcome signifies the intrinsic flatness of the centraldisc in the Kerr case. However, the result for the Appell case is complicated andit shows that the central disc has non-trivial geometry and yet, is still regularat axis of symmetry. The same results are calculated in Section 3.3. Section 3.4focuses solely on the Kerr case. We calculate the Gaussian curvature in the wholeequatorial plane outside of the singularity in both B-L and K-S coordinates. Af-terwards, we focus our attention on the central disc. We determine the Gaussiancurvature as zero for the whole central disc. The extrinsic curvature is calculatedand it is not zero anywhere except at the central point. We calculate the extrinsiccurvature in the K-S coordinates as well. The same invariants are calculated forthe Appell case, where the Gaussian curvature and extrinsic curvature are non-zero. The outcome for the Appell case (Section 3.5) shows that the geometry ofthe central disc is non trivial because of non zero Gaussian curvature.

The results for the Kerr case (unlike for the Appell case) show that the centraldisc is intrinsically flat, as has been stated by W. Israel [1]. The claim presentedby H. Garcıa-Compean and V.S. Manko that the well-known interpretation ofthe central surface as a disc is wrong [2], is misleading because the interpretationfrom Israel [1] refers only to the intrinsic properties which are crucial for physics.However, the fact that the trace of extrinsic curvature is non zero needs furtherexaminations. In this thesis, we have proposed one potential method for furtherexamination from article [13] which can unquestionably decide the geometricnature of the central disc.

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Bibliography[1] W. Israel. Source of the Kerr Metric. Phys. Rev. D, 2:641–646, Aug 1970.

[2] V.S. Manko H. Garcıa-Compean. Are known maximal extensions of the Kerrand Kerr–Newman spacetimes physically meaningful and analytic? Progressof Theoretical and Experimental Physics, 2015(4), Apr 2015.

[3] K. Glampedakis and T. A Apostolatos. The separable analogue of Kerr inNewtonian gravity. Classical and Quantum Gravity, 30(5):055006, feb 2013.

[4] O. Semerak, T. Zellerin, and M. Zacek. The structure of superposed weylfields. Monthly Notices of the Royal Astronomical Society, 308(3):691–704,1999.

[5] R. H. Boyer and R. W. Lindquist. Maximal analytic extension of the Kerrmetric. J. Math. Phys., 8:265, 1967.

[6] G. W. Gibbons and M. S. Volkov. Zero mass limit of Kerr spacetime is awormhole. Phys. Rev. D, 96(2):024053, 2017.

[7] V. S. Manko. Comment on ”zero mass limit of Kerr spacetime is a wormhole”,2017.

[8] O. Semerak. Static axisymmetric rings in general relativity: How diversethey are. Physical Review D, 94(10), Nov 2016.

[9] R. J. Gleiser and J. A. Pullin. Appell rings in general relativity. Classicaland Quantum Gravity, 6(7):977–985, jul 1989.

[10] K. S. Thorne Ch. W. Misner and J. A. Wheeler. Gravitation. W. H. Freeman,San Francisco, 1973.

[11] J. Bicak and O. Semerak. Relativistic physics. Lecture notes for a coursetaught at Prague math-phys, May 2021.

[12] W. Krivan and H. Herold. 2-surfaces of constant mean curvature in physicalspacetimes. Classical and Quantum Gravity, 12(9):2297–2308, sep 1995.

[13] J. L. Lu and W. M. Suen. Extrinsic Curvature Embedding Diagrams. GeneralRelativity and Gravitation, 35(7):1175–1189, Jul 2003.

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List of Figures

2.1 The example of a curve parametrized by s. The angle betweenthe tangent line and reference is denoted by θ, ρ stands for thecurvature radius [10]. . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Approximation of a curve (grey line) by a circle with the radius Rwith two tangent lines (red lines) and angles between tangent linesand referential direction (green line). The angle dφ is an anglebetween radii. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 The examples of two 2D-surfaces with zero intrinsic curvature. Theone on the left has zero extrinsic curvature and the other has nonzero extrinsic curvature [10]. . . . . . . . . . . . . . . . . . . . . . 10

3.1 Contours of the Gaussian curvature plotted within the equatorialplane for the Kerr solution with the assumption M = a. There isspin a on the horizontal axis and the radius r on the vertical axis.Both axes have the range of 0-0.5. . . . . . . . . . . . . . . . . . . 17

3.2 Contours of the Gaussian curvature plotted within the equatorialplane for the Kerr solution with the assumption M = a. There isspin a on the horizontal axis and the radius r on the vertical axis.Both axes have the range of 0-0.02. . . . . . . . . . . . . . . . . . 17

3.3 Contours of the Gaussian curvature plotted within the equatorialplane for the Kerr solution in the K-S coordinates with the as-sumption M = a. There is spin a on the horizontal axis with therange (0,0.005) and the radius ϱ on the vertical axis with the range(0,0.010). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 Contours of the Gaussian curvature plotted within the equatorialplane for the Appell solution in the oblate spheroidal coordinateswith the assumption M = a. There is spin a on the horizontal axiswith the range (0,0.005) and the radius R on the vertical axis withthe range (0,0.055). . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.5 Contours of the Gaussian curvature plotted within the equatorialplane for the Appell solution in the Weyl coordinates with theassumption M = a. There is spin a on the horizontal axis withthe range (0,0.005) and the radius ρ on the vertical axis with therange (0,0.010). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

25