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Catalogue of Spacetimes

Catalogue of Spacetimes

Catalogue of Spacetimes

e2 e1 x2 = 2 x = 2 q ∂x2 1 x2 = 1 ∂x1 x1 = 1

x1 = 0 x2 = 0

M

Authors: Thomas Müller Visualisierungsinstitut der Universität Stuttgart (VISUS) Allmandring 19, 70569 Stuttgart, Germany [email protected]

Frank Grave formerly, Universität Stuttgart, Institut für Theoretische Physik 1 (ITP1) Pfaffenwaldring 57 //IV, 70550 Stuttgart, Germany [email protected]

URL: http://go.visus.uni-stuttgart.de/CoS

Date: 21. Mai 2014 Co-authors

Andreas Lemmer, formerly, Institut für Theoretische Physik 1 (ITP1), Universität Stuttgart Alcubierre Warp Sebastian Boblest, Institut für Theoretische Physik 1 (ITP1), Universität Stuttgart deSitter, Friedmann-Robertson-Walker Felix Beslmeisl, Institut für Theoretische Physik 1 (ITP1), Universität Stuttgart Petrov-Type D Heiko Munz, Institut für Theoretische Physik 1 (ITP1), Universität Stuttgart Bessel and plane wave Andreas Wünsch, Institut für Theoretische Physik 1 (ITP1), Universität Stuttgart Majumdar-Papapetrou, extreme Reissner-Nordstrøm dihole, energy momentum tensor

Many thanks to all that have reported bug ﬁxes or added metric descriptions. Contents

1 Introduction and Notation1 1.1 Notation...... 1 1.2 General remarks...... 1 1.3 Basic objects of a metric...... 2 1.4 Natural local tetrad and initial conditions for geodesics...... 3 1.4.1 Orthonormality condition...... 3 1.4.2 Tetrad transformations...... 4 1.4.3 Ricci rotation-, connection-, and structure coefﬁcients...... 4 1.4.4 Riemann-, Ricci-, and Weyl-tensor with respect to a local tetrad...... 4 1.4.5 Null or timelike directions...... 5 1.4.6 Local tetrad for diagonal metrics...... 5 1.4.7 Local tetrad for stationary axisymmetric spacetimes...... 5 1.5 Newman-Penrose tetrad and spin-coefﬁcients...... 6 1.6 Coordinate relations...... 7 1.6.1 Spherical and Cartesian coordinates...... 7 1.6.2 Cylindrical and Cartesian coordinates...... 7 1.7 Embedding diagram...... 8 1.8 Equations of motion and transport equations...... 9 1.8.1 Geodesic equation...... 9 1.8.2 Fermi-Walker transport...... 9 1.8.3 Parallel transport...... 9 1.8.4 Euler-Lagrange formalism...... 9 1.8.5 Hamilton formalism...... 10 1.9 Special topics...... 10 1.9.1 Timelike circular geodesics...... 10 1.10 Units...... 10 1.11 Energy momentum tensor...... 11 1.11.1 Energy conditions...... 11 1.11.2 Examples for energy momentum tensors...... 11 1.12 Tools...... 13 1.12.1 Maple/GRTensorII...... 13 1.12.2 Mathematica...... 13 1.12.3 Maxima...... 15 1.12.4 Sympy...... 16

2 Spacetimes 19 2.1 Minkowski...... 19 2.1.1 Cartesian coordinates...... 19 2.1.2 Cylindrical coordinates...... 19 2.1.3 Spherical coordinates...... 20 2.1.4 Conform-compactiﬁed coordinates...... 20 2.1.5 Rotating coordinates...... 21 2.1.6 Rindler coordinates...... 22 2.2 Schwarzschild spacetime...... 23

i ii CONTENTS

2.2.1 Schwarzschild coordinates...... 23 2.2.2 Schwarzschild in pseudo-Cartesian coordinates...... 25 2.2.3 Isotropic coordinates...... 25 2.2.4 Eddington-Finkelstein...... 27 2.2.5 Kruskal-Szekeres...... 28 2.2.6 Tortoise coordinates...... 29 2.2.7 Painlevé-Gullstrand...... 30 2.2.8 Israel coordinates...... 32 2.3 Alcubierre Warp...... 33 2.4 Barriola-Vilenkin monopol...... 34 2.5 Bertotti-Kasner...... 36 2.6 Bessel gravitational wave...... 38 2.6.1 Cylindrical coordinates...... 38 2.6.2 Cartesian coordinates...... 38 2.7 Cosmic string in Schwarzschild spacetime...... 39 2.8 Einstein-Rosen wave with Weber-Wheeler-Bonnor pulse...... 41 2.9 Ernst spacetime...... 42 2.10 Extreme Reissner-Nordstrøm dihole...... 44 2.11 Friedman-Robertson-Walker...... 47 2.11.1 Form 1...... 47 2.11.2 Form 2...... 48 2.11.3 Form 3...... 49 2.12 Gödel Universe...... 53 2.12.1 Cylindrical coordinates...... 53 2.12.2 Scaled cylindrical coordinates...... 54 2.13 Halilsoy standing wave...... 56 2.14 Janis-Newman-Winicour...... 57 2.15 Kasner...... 59 2.16 Kastor-Traschen...... 60 2.17 Kerr...... 61 2.17.1 Boyer-Lindquist coordinates...... 61 2.18 Kottler spacetime...... 64 2.19 Majumdar-Papapetrou spacetimes...... 66 2.20 Melvin universe...... 70 2.21 Morris-Thorne...... 71 2.22 Oppenheimer-Snyder collapse...... 73 2.22.1 Outer metric...... 73 2.22.2 Inner metric...... 74 2.23 Petrov-Type D – Levi-Civita spacetimes...... 76 2.23.1 Case AI...... 76 2.23.2 Case AII...... 76 2.23.3 Case AIII...... 77 2.23.4 Case BI...... 77 2.23.5 Case BII...... 78 2.23.6 Case BIII...... 78 2.23.7 Case C...... 78 2.24 Plane gravitational wave...... 81 2.25 Reissner-Nordstrøm...... 82 2.26 de Sitter spacetime...... 84 2.26.1 Standard coordinates...... 84 2.26.2 Conformally Einstein coordinates...... 84 2.26.3 Conformally ﬂat coordinates...... 85 2.26.4 Static coordinates...... 85 2.26.5 Lemaître-Robertson form...... 87 2.26.6 Cartesian coordinates...... 88 CONTENTS iii

2.27 Stationary axisymmetric spacetimes in Weyl Coordinates...... 89 2.28 Straight spinning string...... 90 2.29 Sultana-Dyer spacetime...... 92 2.30 TaubNUT...... 94

Bibliography 95

Chapter 1

Introduction and Notation

The Catalogue of Spacetimes is a collection of four-dimensional Lorentzian spacetimes in the context of the General Theory of Relativity (GR). The aim of the catalogue is to give a quick reference for students who need some basic facts of the most well-known spacetimes in GR. For a detailed discussion of a metric, the reader is referred to the standard literature or the original articles. Important resources for exact solutions are the book by Stephani et al[SKM+03] and the book by Grifﬁths and Podolský[GP09]. Most of the metrics in this catalogue are implemented in the Motion4D-library[MG09] and can be visu- alized using the GeodesicViewer[MG10]. Except for the Minkowski and Schwarzschild spacetimes, the metrics are sorted by their names.

1.1 Notation

The notation we use in this catalogue is as follows: Indices: Coordinate indices are represented either by Greek letters or by coordinate names. Tetrad indices are indicated by Latin letters or coordinate names in brackets. Einstein sum convention: When an index appears twice in a single term, once as lower index and once as upper index, we build the sum over all indices:

3 µ µ ζµ ζ ∑ ζµ ζ . (1.1.1) ≡ µ=0

µ Vectors: A coordinate vector in x direction is represented as ∂xµ ∂µ . For arbitrary vectors, we use ≡ µ boldface symbols. Hence, a vector a in coordinate representation reads a = a ∂µ . µ Derivatives: Partial derivatives are indicated by a comma, ∂ψ/∂x ∂µ ψ ψ µ , whereas covariant ≡ ≡ , derivatives are indicated by a semicolon, ∇ψ = ψ;µ . Symmetrization and Antisymmetrization brackets:

1 1 a b = aµ bν + aν bµ , a b = aµ bν aν bµ (1.1.2) ( µ ν ) 2 [ µ ν ] 2 − 1.2 General remarks

The Einstein ﬁeld equation in the most general form reads[MTW73] 8πG Gµν = κTµν Λgµν , κ = , (1.2.1) − c4 1 with the symmetric and divergence-free Einstein tensor Gµν = Rµν Rgµν , the Ricci tensor Rµν , the − 2 Ricci scalar R, the metric tensor gµν , the energy-momentum tensor Tµν , the cosmological constant Λ, Newton’s gravitational constant G, and the speed of light c. Because the Einstein tensor is divergence- µν free, the conservation equation T ;ν = 0 is automatically fulﬁlled.

1 2 CHAPTER 1. INTRODUCTION AND NOTATION

A solution to the ﬁeld equation is given by the line element

2 µ ν ds = gµν dx dx (1.2.2)

µν with the symmetric, covariant metric tensor gµν . The contravariant metric tensor g is related to the νλ λ covariant tensor via gµν g = δµ with the Kronecker-δ. Even though gµν is only a component of the µ ν metric tensor g = gµν dx dx , we will also call gµν the metric tensor. ⊗ Note that, in this catalogue, we mostly use the convention that the signature of the metric is +2. In general, we will also keep the physical constants c and G within the metrics.

1.3 Basic objects of a metric

The basic objects of a metric are the Christoffel symbols, the Riemann and Ricci tensors as well as the Ricci and Kretschmann scalars which are deﬁned as follows: Christoffel symbols of the ﬁrst kind:1

1 Γ = g + g g (1.3.1) νλ µ 2 µν,λ µλ,ν − νλ,µ with the relation

gνλ,µ = Γµνλ +Γµλν (1.3.2)

Christoffel symbols of the second kind:

1 Γ µ = gµρ g + g g (1.3.3) νλ 2 ρν,λ ρλ,ν − νλ,ρ which are related to the Christoffel symbols of the ﬁrst kind via

µ µρ Γνλ = g Γνλρ (1.3.4)

Riemann tensor:

Rµ = Γ µ Γ µ +Γ µ Γ λ Γ µ Γ λ (1.3.5) νρσ νσ,ρ − νρ,σ ρλ νσ − σλ νρ or

λ λ λ Rµνρσ = g R = Γνσ µ ρ Γνρµ σ +Γ Γ Γ Γ (1.3.6) µλ νρσ , − , νρ µσλ − νσ µσλ with symmetries

Rµνρσ = Rµνσρ , Rµνρσ = Rνµρσ , Rµνρσ = Rρσ µν (1.3.7) − − and

Rµνρσ + Rµρσν + Rµσνρ = 0 (1.3.8)

Ricci tensor:

ρσ ρ Rµν = g Rρµσν = R µρν (1.3.9)

Ricci and Kretschmann scalar:

µν µ αβγδ γδ αβ R = g Rµν = R µ , K = Rαβγδ R = R αβ R γδ (1.3.10)

1 Rindler CoS The notation of the Christoffel symbols of the ﬁrst kind differs from the one used by Rindler[Rin01], Γµνλ = Γνλ µ . 1.4. NATURAL LOCAL TETRAD AND INITIAL CONDITIONS FOR GEODESICS 3

Weyl tensor: 1 Cµνρσ = Rµνρσ g R g R + Rg g (1.3.11) − µ[ ρ σ ]ν − ν[ ρ σ ]µ 3 µ[ ρ σ ]ν If we change the signature of a metric, these basic objects transform as follows: µ µ Γ Γ , Rµνρσ Rµνρσ , Cµνρσ Cµνρσ , (1.3.12a) νλ 7→ νλ 7→ − 7→ − Rµν Rµν , R R, K K . (1.3.12b) 7→ 7→ − 7→ Covariant derivative

∇λ gµν = gµν;λ = 0. (1.3.13) Covariant derivative of the vector ﬁeld ψ µ : µ µ µ µ λ ∇ν ψ = ψ;ν = ∂ν ψ +Γνλ ψ (1.3.14) Covariant derivative of a r-s-tensor ﬁeld: a1...ar a1...ar a1 d...ar ar a1...ar 1d ∇cT b ...b = ∂cT b ...b +Γdc T b ...b + ... +Γdc T − b ...b 1 s 1 s 1 s 1 s (1.3.15) Γ d T a1...ar Γ d T a1...ar b1c d...bs ... bsc b1...bs 1d − − − − Killing equation:

ξµ;ν + ξν;µ = 0. (1.3.16)

1.4 Natural local tetrad and initial conditions for geodesics

We will call a local tetrad natural if it is adapted to the symmetries or the coordinates of the spacetime. µ µ The four base vectors e(i) = e(i)∂µ are given with respect to coordinate directions ∂/∂x = ∂µ , compare Nakahara[Nak90] or Chandrasekhar[Cha06] for an introduction to the tetrad formalism. The inverse or (i) (i) µ dual tetrad is given by θ = θµ dx with (i) µ (i) (i) ν ν θµ e( j) = δ( j) and θµ e(i) = δµ . (1.4.1) Note that we us Latin indices in brackets for tetrads and Greek indices for coordinates.

1.4.1 Orthonormality condition

To be applicable as a local reference frame (Minkowski frame), a local tetrad e(i) has to fulfill the orthonormality condition e e ge e g eµ eν ! (i), ( j) g = (i), ( j) = µν (i) ( j) = η(i)( j), (1.4.2) where η = diag( 1, 1, 1, 1) depending on the signature sign(g) = 2 of the metric. Thus, the (i)( j) ∓ ± ± ± ± line element of a metric can be written as 2 (i) ( j) (i) ( j) µ ν ds = η(i)( j)θ θ = η(i)( j)θµ θν dx dx . (1.4.3) (i) To obtain a local tetrad e(i), we could first determine the dual tetrad θ via Eq. (1.4.3). If we combine all 1 four dual tetrad vectors into one matrix Θ, we only have to determine its inverse Θ − to find the tetrad vectors,  (0) (0) (0) (0) e0 e0 e0 e0  θ0 θ1 θ2 θ3 (0) (1) (2) (3) θ (1) θ (1) θ (1) θ (1) e1 e1 e1 e1  Θ =  0 1 2 3  Θ 1 =  (0) (1) (2) (3). (1.4.4)  (2) (2) (2) (2) − e2 e2 e2 e2  θ0 θ1 θ2 θ3  ⇒  (0) (1) (2) (3) (3) (3) (3) (3) e3 e3 e3 e3 θ0 θ1 θ2 θ3 (0) (1) (2) (3) There are also several useful relations: ν µ (a) e(a)µ = gµν e(a), η(a)(b) = e(a)e(b)µ , e(b)µ = η(a)(b)θµ , (1.4.5a) (b) (a)(b) (a) (a)(b) (a) (b) µν θµ = η e(a)µ , gµν = e(a)µ θν , η = θµ θν g . (1.4.5b) 4 CHAPTER 1. INTRODUCTION AND NOTATION

1.4.2 Tetrad transformations Instead of the above found local tetrad that was directly constructed from the spacetime metric, we can also use any other local tetrad k eˆ(i) = Ai e(k), (1.4.6) where A is an element of the Lorentz group O(1,3). Hence AT ηA = η and (detA)2 = 1. a T p 2 Lorentz-transformation in the direction n = (sin χ cosξ,sin χ sinξ,cosξ) = na with γ = 1/ 1 β , − 0 0 a a a a a Λ = γ, Λ = βγna, Λ = βγn , Λ = (γ 1)n nb + δ . (1.4.7) 0 a − 0 − b − b 1.4.3 Ricci rotation-, connection-, and structure coefﬁcients

The Ricci rotation coefficients γ(i)( j)(k) with respect to the local tetrad e(i) are defined by : g eµ eλ g eµ eν eλ g eµ eν eλ λ eβ γ(i)( j)(k) = µλ (i)∇e(k) ( j) = µλ (i) (k)∇ν ( j) = µλ (i) (k) ∂ν ( j) +Γνβ ( j) . (1.4.8) They are antisymmetric in the first two indices, γ = γ , which follows from the definition, (i)( j)(k) − ( j)(i)(k) Eq. (1.4.8), and the relation

β ν 0 = ∂µ η(i)( j) = ∇µ gβν e(i)e( j) , (1.4.9)

where ∇µ gβν = 0, compare [Cha06]. Otherwise, we have

(i) (i) ν λ λ ν (i) γ = θ e ∇ν e = e e ∇ν θ . (1.4.10) ( j)(k) λ (k) ( j) − ( j) (k) λ The contraction of the ﬁrst and the last index is given by

(k) (k)(i) ν γ = γ = η γ = γ + γ + γ + γ = ∇ν e . (1.4.11) ( j) ( j)(k) (i)( j)(k) − (0)( j)(0) (1)( j)(1) (2)( j)(2) (3)( j)(3) ( j) (m) The connection coefficients ω( j)(n) with respect to the local tetrad e(i) are defined by ω(m) : (m) eµ (m)eα eµ (m)eα eµ µ eβ ( j)(n) = θµ ∇e( j) (n) = θµ ( j)∇α (n) = θµ ( j) ∂α (n) +Γαβ (n) , (1.4.12) compare Nakahara[Nak90]. They are related to the Ricci rotation coefficients via

(m) γ(i)( j)(k) = η(i)(m)ω(k)( j). (1.4.13) ν µ ν µ ν Furthermore, the local tetrad has a non-vanishing Lie-bracket [X,Y] = X ∂µY Y ∂µ X . Thus, − (k) (k) (k) e(i),e( j) = c(i)( j)e(k) or c(i)( j) = θ e(i),e( j) . (1.4.14)

(k) The structure coefficients c(i)( j) are related to the connection coefficients or the Ricci rotation coefficients via k k k c( ) = ω( ) ω( ) = η(k)(m) γ γ = γ(k) γ(k) . (1.4.15) (i)( j) (i)( j) − ( j)(i) (m)( j)(i) − (m)(i)( j) ( j)(i) − (i)( j) 1.4.4 Riemann-, Ricci-, and Weyl-tensor with respect to a local tetrad The transformations between the coordinate representations of the Riemann-, Ricci-, and Weyl-tensors and their representation with respect to a local tetrad e(i) are given by µ ν ρ σ R(a)(b)(c)(d) = Rµνρσ e(a)e(b)e(c)e(d), (1.4.16a) µ ν R(a)(b) = Rµν e(a)e(b), (1.4.16b) µ ν ρ σ C(a)(b)(c)(d) = Cµνρσ e(a)e(b)e(c)e(d) 1 R = R η R η R + η η . (1.4.16c) (a)(b)(c)(d) − 2 (a)[ (c) (d) ](b) − (b)[ (c) (d) ](a) 3 (a)[ (c) (d) ](b) 1.4. NATURAL LOCAL TETRAD AND INITIAL CONDITIONS FOR GEODESICS 5

1.4.5 Null or timelike directions (i) A null or timelike direction υ = υ e(i) with respect to a local tetrad e(i) can be written as

(0) (0) υ = υ e(0) + ψ sin χ cosξ e(1) + sin χ sinξ e(2) + cos χ e(3) = υ e(0) + ψn. (1.4.17)

In the case of a null direction we have ψ = 1 and υ(0) = 1. A timelike direction can be identiﬁed with ± an initial four-velocity u = cγ (e0 + βn), where

u2 = u,u = c2γ2 e + βn,e + βn = c2γ2 1 + β 2 = c2, sign(g) = 2. (1.4.18) h ig (0) (0) − ∓ ± Thus, ψ = cβγ and υ0 = cγ. The sign of υ(0) determines the time direction. ±

e(3)

υ

χ ψ

ξ e(2)

Figure 1.1: Null or timelike direction υ e(1) with respect to the local tetrad e(i).

The transformations between a local direction υ(i) and its coordinate representation υ µ read

µ (i) µ (i) (i) µ υ = υ e(i) and υ = θµ υ . (1.4.19)

1.4.6 Local tetrad for diagonal metrics If a spacetime is represented by a diagonal metric

2 0 2 1 2 2 2 3 2 ds = g00(dx ) + g11(dx ) + g22(dx ) + g33(dx ) , sign(g) = 2, (1.4.20) ± the natural local tetrad reads 1 1 1 1 e(0) = ∂0, e(1) = ∂1, e(2) = ∂2, e(3) = ∂3, (1.4.21) √ g00 √ g11 √ g22 √ g33 ∓ ± ± ± given that the metric coefﬁcients are well behaved. Analogously, the dual tetrad reads

(0) 0 (1) 1 (2) 2 (3) 3 θ = √ g00 dx , θ = √ g11 dx , θ = √ g22 dx , θ = √ g33 dx . (1.4.22) ∓ ± ± ± 1.4.7 Local tetrad for stationary axisymmetric spacetimes The line element of a stationary axisymmetric spacetime is given by

2 2 2 2 2 ds = gtt dt + 2gtϕ dt dϕ + gϕϕ dϕ + grrdr + gϑϑ dϑ , (1.4.23) where the metric components are functions of r and ϑ only. The local tetrad for an observer on a stationary circular orbit, (r = const,ϑ = const), with four velocity u = cΓ ∂t + ζ∂ϕ can be deﬁned as, compare Bini[BJ00],

1 1 e(0) = Γ ∂t + ζ∂ϕ , e(1) = ∂r, e(2) = ∂ϑ , (1.4.24a) √grr √gϑϑ e = ∆Γ (gtϕ + ζgϕϕ )∂t (gtt + ζgtϕ )∂ϕ , (1.4.24b) (3) ± ∓ 6 CHAPTER 1. INTRODUCTION AND NOTATION

where 1 1 Γ = q and ∆ = q . (1.4.25) 2 2 gtt + 2ζgtϕ + ζ gϕϕ g gtt gϕϕ − tϕ − 2 The angular velocity ζ is limited due to gtt + 2ζgtϕ + ζ gϕϕ < 0 r r 2 gtt 2 gtt ζmin = ω ω and ζmax = ω + ω (1.4.26) − − gϕϕ − gϕϕ

with ω = gtϕ /gϕϕ . − For ζ = 0, the observer is static with respect to spatial inﬁnity. The locally non-rotating frame (LNRF) has angular velocity ζ = ω, see also MTW[MTW73], exercise 33.3. Static limit: ζ = 0 gtt = 0. min ⇒ The transformation between the local direction υ(i) and the coordinate direction υ µ reads (1) (2) 0 (0) (3) 1 υ 2 υ 3 (0) (3) υ = Γ υ υ ∆w1 , υ = , υ = , υ = Γ υ ζ υ ∆w2 , (1.4.27) ± √grr √gϑϑ ∓ with

w1 = gtϕ + ζgϕϕ and w2 = gtt + ζgtϕ . (1.4.28) The back transformation reads 0 3 0 3 (0) 1 υ w2 + υ w1 (1) 1 (2) 2 (3) 1 ζυ υ υ = , υ = √grr υ , υ = √gϑϑ υ , υ = − . (1.4.29) Γ ζw1 + w2 ±∆Γ ζw1 + w2

µ Note, to obtain a right-handed local tetrad, det e(i) > 0, the upper sign has to be used.

1.5 Newman-Penrose tetrad and spin-coefﬁcients

The Newman-Penrose tetrad consists of four null vectors e? = l,n,m,m¯ , where l and n are real and m (i) { } and m¯ are complex conjugates; see Penrose and Rindler[PR84] or Chandrasekhar[Cha06] for a thorough discussion. The Newman-Penrose (NP) tetrad has to fulﬁll the orthonormality relation  0 1 0 0  D E  1 0 0 0  e? ,e? = η? with η? =  . (1.5.1) (i) ( j) (i)( j) (i)( j)  0 0 0 1  0 0 1− 0 − A straightforward relation between the NP tetrad and the natural local tetrad, as discussed in Sec. 1.4, is given by

1 1 1 l = e 0 + e 1 , n = e 0 e 1 , m = e 2 + ie 3 , (1.5.2) ∓√2 ( ) ( ) ∓√2 ( ) − ( ) ∓√2 ( ) ( ) where the upper/lower sign has to be used for metrics with positive/negative signature. The Ricci rotation-coefficients of a NP tetrad are now called spin coefficients and are designated by specific symbols:

1 κ = γ , ρ = γ , ε = γ + γ , (1.5.3a) (2)(1)(1) (2)(0)(3) 2 (1)(0)(0) (2)(3)(0) 1 σ = γ , µ = γ , γ = γ + γ , (1.5.3b) (2)(0)(2) (1)(3)(2) 2 (1)(0)(1) (2)(3)(1) 1 λ = γ , τ = γ , α = γ + γ , (1.5.3c) (1)(3)(3) (2)(0)(1) 2 (1)(0)(3) (2)(3)(3) 1 ν = γ , π = γ , β = γ + γ . (1.5.3d) (1)(3)(1) (1)(3)(0) 2 (1)(0)(2) (2)(3)(2) 1.6. COORDINATE RELATIONS 7

1.6 Coordinate relations

1.6.1 Spherical and Cartesian coordinates The well-known relation between the spherical coordinates (r,ϑ,ϕ) and the Cartesian coordinates (x,y,z), compare Fig. 1.2, are

x = r sinϑ cosϕ, y = r sinϑ sinϕ, z = r cosϑ, (1.6.1) and p p r = x2 + y2 + z2, ϑ = arctan2( x2 + y2,z), ϕ = arctan2(y,x), (1.6.2) where arctan2() ensures that ϕ [0,2π) and ϑ (0,π). ∈ ∈

z

ϑ r

ϕ y

Figure 1.2: Relation between spherical x and Cartesian coordinates.

The total differentials of the spherical coordinates read

xdx + ydy + zdz xzdx + yzdy (x2 + y2)dz ydx + xdy dr = , dϑ = p − , dϕ = − 2 2 , (1.6.3) r r2 x2 + y2 x + y whereas the coordinate derivatives read ∂x ∂y ∂z ∂r = ∂x + ∂y + ∂z = sinϑ cosϕ ∂x + sinϑ sinϕ ∂y + cosϑ ∂z, (1.6.4a) ∂r ∂r ∂r ∂x ∂y ∂z ∂ = ∂x + ∂y + ∂z = r cosϑ cosϕ ∂x + r cosϑ sinϕ ∂y r sinϑ ∂z, (1.6.4b) ϑ ∂ϑ ∂ϑ ∂ϑ − ∂x ∂y ∂z ∂ϕ = ∂x + ∂y + ∂z = r sinϑ sinϕ ∂x + r sinϑ cosϕ ∂y, (1.6.4c) ∂ϕ ∂ϕ ∂ϕ − and ∂r ∂ϑ ∂ϕ cosϑ cosϕ sinϕ ∂x = ∂r + ∂ + ∂ϕ = sinϑ cosϕ ∂r + ∂ ∂ϕ , (1.6.5a) ∂x ∂x ϑ ∂x r ϑ − r sinϑ ∂r ∂ϑ ∂ϕ cosϑ sinϕ cosϕ ∂y = ∂r + ∂ + ∂ = sinϑ sinϕ ∂r + ∂ + ∂ , (1.6.5b) ∂y ∂y ϑ ∂y ϕ r ϑ r sinϑ ϕ ∂r ∂ϑ ∂ϕ sinϑ ∂z = ∂r + ∂ + ∂ϕ = cosϑ ∂r ∂ . (1.6.5c) ∂z ∂z ϑ ∂z − r ϑ

1.6.2 Cylindrical and Cartesian coordinates The relation between cylindrical coordinates (r,ϕ,z) and Cartesian coordinates (x,y,z) is given by p x = r cosϕ, y = r sinϕ, and r = x2 + y2, ϕ = arctan2(y,x), (1.6.6) 8 CHAPTER 1. INTRODUCTION AND NOTATION

z

z ϕ y

r Figure 1.3: Relation between cylindrical x and Cartesian coordinates. where arctan2() again ensures that the angle ϕ [0,2π). ∈ The total differentials of the spherical coordinates are given by

xdx + ydy ydx + xdy dr = , dϕ = − , (1.6.7) r r2

and

dx = cosϕ dr r sinϕ dϕ, dy = sinϕ dr + r cosϕ dϕ. (1.6.8) − The coordinate derivatives are ∂x ∂y ∂r = ∂x + ∂y = cosϕ ∂x + sinϕ ∂y, (1.6.9a) ∂r ∂r ∂x ∂y ∂ϕ = ∂x + ∂y = r sinϕ ∂x + r cosϕ ∂ym (1.6.9b) ∂ϕ ∂ϕ −

and ∂r ∂ϕ sinϕ ∂x = ∂r + ∂ϕ = cosϕ ∂r ∂y, (1.6.10a) ∂x ∂x − r ∂r ∂ϕ cosϕ ∂y = ∂r + ∂ = sinϕ ∂r + ∂y. (1.6.10b) ∂y ∂y ϕ r

1.7 Embedding diagram

A two-dimensional hypersurface with line segment

2 2 2 dσ = grr(r)dr + gϕϕ (r)dϕ (1.7.1)

can be embedded in a three-dimensional Euclidean space with cylindrical coordinates, " # dz 2 dσ 2 = 1 + dρ2 + ρ2dϕ2. (1.7.2) dρ

2 With ρ(r) = gϕϕ (r) and dr = (dr/dρ)dρ, we obtain for the embedding function z = z(r), s 2 dz d√gϕϕ = grr . (1.7.3) dr ± − dr

2 If gϕϕ (r) = r , then d√gϕϕ /dr = 1. 1.8. EQUATIONS OF MOTION AND TRANSPORT EQUATIONS 9

1.8 Equations of motion and transport equations

1.8.1 Geodesic equation The geodesic equation reads

D2xµ d2xµ dxρ dxσ = +Γ µ = 0 (1.8.1) dλ 2 dλ 2 ρσ dλ dλ with the affine parameter λ. For timelike geodesics, however, we replace the affine parameter by the proper time τ. The geodesic equation (1.8.1) is a system of ordinary differential equations of second order. Hence, to solve these differential equations, we need an initial position xµ (λ = 0) as well as an initial direction (dxµ /dλ)(λ = 0). This initial direction has to fulfill the constraint equation

dxµ dxν g = κc2, (1.8.2) µν dλ dλ where κ = 0 for lightlike and κ = 1, (sign(g) = 2), for timelike geodesics. ∓ ± The initial direction can also be determined by means of a local reference frame, compare sec. 1.4.5, that automatically fulﬁlls the constraint equation (1.8.2). If we use the natural local tetrad as local reference frame, we have

µ dx µ (i) µ = υ = υ e(i). (1.8.3) dλ λ=0

1.8.2 Fermi-Walker transport µ µ The Fermi-Walker transport, see e.g. Stephani[SS90], of a vector X = X ∂µ along the worldline x (τ) µ µ with four-velocity u = u (τ)∂µ is given by FuX = 0 with

µ µ dX µ ρ σ 1 σ µ σ µ ρ FuX := +Γρσ u X + (u a a u )gρσ X . (1.8.4) dτ c2 − The four-acceleration follows from the four-velocity via

D2xµ Duµ duµ aµ = = = +Γ µ uρ uσ . (1.8.5) dτ2 dτ dτ ρσ

1.8.3 Parallel transport µ If the four-acceleration vanishes, the Fermi-Walker transport simpliﬁes to the parallel transport PuX = 0 with µ µ µ DX dX µ ρ σ PuX := = +Γρσ u X . (1.8.6) dτ dτ

1.8.4 Euler-Lagrange formalism A detailed discussion of the Euler-Lagrange formalism can be found, e.g., in Rindler[Rin01]. The La- grangian L is deﬁned as

µ ν ! 2 L := gµν x˙ x˙ , L = κc , (1.8.7) where xµ are the coordinates of the metric, and the dot means differentiation with respect to the afﬁne parameter λ. For timelike geodesics, κ = 1 depending on the signature of the metric, sign(g) = 2. For ∓ ± lightlike geodesics, κ = 0. 10 CHAPTER 1. INTRODUCTION AND NOTATION

The Euler-Lagrange equations read

d ∂L ∂L = 0. (1.8.8) dλ ∂x˙µ − ∂xµ If L is independent of xρ , then xρ is a cyclic variable and

ν pρ = gρν x˙ = const. (1.8.9)

2 Note that [L ] = length for timelike and [L ] = 1 for lightlike geodesics, see Sec. 1.10. U time2 U

1.8.5 Hamilton formalism The super-Hamiltonian H is deﬁned as

1 ! 1 H := gµν p p , H = κc2, (1.8.10) 2 µ ν 2 ν where pµ = gµν x˙ are the canonical momenta, see e.g. MTW[MTW73], para. 21.1. As in classical me- chanics, we have µ dx ∂H dpµ ∂H = and = µ . (1.8.11) dλ ∂ pµ dλ − ∂x

1.9 Special topics

1.9.1 Timelike circular geodesics Given a spacetime in spherical or polar coordinates, timelike circular geodesics with respect to the radial coordinate can be found by means of the equation for acceleration

duµ aµ = +Γ µ uν uλ (1.9.12) dτ νλ

t ϕ p 2 and the ansatz for the four-velocity u = cγ e + βe = u ∂t +u ∂ϕ with γ = 1/ 1 β . To be geodetic, (t) (ϕ) − all components of the four-acceleration (1.9.12) must vanish. Because u = const, duµ /dτ = 0. In spherical coordinates, the remaining equations read

t t t t t t ϕ t ϕ ϕ 2 2 h t t t t t t 2 t t t i ! a = Γtt u u + 2Γtϕ u u +Γϕϕ u u = c γ Γtt e(t)e(t) + 2βΓtϕ e(t)e(ϕ) + β Γϕϕ e(ϕ)e(ϕ) = 0, (1.9.13a) r r t t r t ϕ r ϕ ϕ 2 2 h r t t r t t 2 r t t i ! a = Γtt u u + 2Γtϕ u u +Γϕϕ u u = c γ Γtt e(t)e(t) + 2βΓtϕ e(t)e(ϕ) + β Γϕϕ e(ϕ)e(ϕ) = 0, (1.9.13b) ϑ ϑ t t ϑ t ϕ ϑ ϕ ϕ 2 2 h ϑ t t ϑ t t 2 ϑ t t i ! a = Γtt u u + 2Γtϕ u u +Γϕϕ u u = c γ Γtt e(t)e(t) + 2βΓtϕ e(t)e(ϕ) + β Γϕϕ e(ϕ)e(ϕ) = 0, (1.9.13c) ϕ ϕ t t ϕ t ϕ ϕ ϕ ϕ 2 2 h ϕ t t ϕ t t 2 ϕ t t i ! a = Γtt u u + 2Γtϕ u u +Γϕϕ u u = c γ Γtt e(t)e(t) + 2βΓtϕ e(t)e(ϕ) + β Γϕϕ e(ϕ)e(ϕ) = 0. (1.9.13d)

1.10 Units

A ﬁrst test in analyzing whether an equation is correct is to check the units. Newton’s gravitational constant G, for example, has the following units

length3 [G] = , (1.10.1) U mass time2 · where [ ] indicates that we evaluate the units of the enclosed expression. Further examples are · U length 1 h i ds u RSchwarzschild RSchwarzschild 2 [ ]U = length, [ ]U = , [ trtr ]U = 2 , ϑϕϑϕ = length . (1.10.2) time time U 1.11. ENERGY MOMENTUM TENSOR 11 1.11 Energy momentum tensor

The Einstein ﬁeld equations (1.2.1) connect the geometry of the spacetime with the density of the energy and the momenta. For a given energy momentum tensor Tµν , they are a differential system for the spacetime components gµν . On the other hand, they give us the energy momentum tensor for a given spacetime geometry. In this case, Tµν has to satisfy the so called energy conditions to guarantee that the metric is physically reasonable. These conditions go back to Hawking and Ellis [HE99].

1.11.1 Energy conditions

Weak energy condition:

µ µ ν An observer moving with the four-velocity u measures the local energy density ρ := Tµν u u . It has to be non-negative for all causal uµ , that means for all timelike and lightlike uµ :

µ ν ρ = Tµν u u 0. (1.11.1) ≥ For lightlike uµ this is also called null energy condition.

Dominant energy condition:

µ µ ν µ An observer moving with the four-velocity u with Tµν u u 0 measures the local energy ﬂux w := µν µ ≥ T uν which has to be also a causal four-vector for all u . For the metric signature +2, this condition reads

µ ν gµν w w 0. (1.11.2) ≤ Strong energy condition:

ˆ 1 The tidal energy momentum tensor is deﬁned by Tµν := Tµν 2 Tgµν . The corresponding energy density µ ν − µ ρˆ := Tˆµν u u has to be non-negative for all causal four-velocities u : 1 µ ν ρˆ = Tµν Tgµν u u 0. (1.11.3) − 2 ≥

1.11.2 Examples for energy momentum tensors

Hawking-Ellis type I:

The energy momentum tensor for an observer whose world-line has the unit tangent vector e0 is

 2  ρ0c 0 0 0 a b  0 p1 0 0  T ( )( ) =   (1.11.4)  0 0 p2 0  0 0 0 p3 with the local energy density ρ0 and the pressures pi in the three spacelike directions (see [HE99]). We will consider a four-velocity with respect to the same local tetrad as T (a)(b). The observer moves without loss of generality in e -direction (k 1,2,3 ). Thus, it is (k) ∈ { } µ (0) (k) u = (u ,u e(k)) (1.11.5) with u(0) = cγ and u(k) = cβγ in the timelike and u(0) = u(k) = 1 in the lightlike case. The weak energy condition (1.11.1) yields

µ ν 4 2 2 2 2 ρ = Tµν u u = ρ0c γ + pkc β γ 0 (1.11.6) ≥ 12 CHAPTER 1. INTRODUCTION AND NOTATION

and thus, especially for the cases β = 0 and β = 1 the conditions 2 ρ0 0 and ρ0c + pk 0. (1.11.7) ≥ ≥ For the dominant energy-condition (1.11.2) we obtain

µ ν 6 2 2 2 2 2 2 gµν w w = c γ ρ + c β γ p 0. (1.11.8) − 0 k ≤ and especially for β = 0 and β = 1 2 2 ρ 0 and ρ0c pk . (1.11.9) − 0 ≤ ≥ | | The strong energy-condition (1.11.3) then yields

1 2 2 2 1 2 2 2 2 2 ρˆ = ρ0c + p1 + p2 + p3 c γ + ρ c + pk pk 1 pk 2 c β γ 0. (1.11.10) 2 2 0 − + − + ≥ and for β = 0 and β = 1 2 2 ρ0c + p1 + p2 + p3 0 and ρ0c + pk 0. (1.11.11) ≥ ≥ Perfect ﬂuid:

The energy momentum tensor of a perfect ﬂuid is given by p sign(g) T µν = ρ + vµ vν + pgµν (1.11.12) 0 c2 2 µ with the four-velocity v of the particles, the energy density in the rest frame ρ0, the isotropic pressure p and the signature of the metric sign(g) (sign(g) = +2 or sign(g) = 2). In a local rest frame with v(a) = − (c,0,0,0) it is  2  ρ0c 0 0 0 a b  0 p 0 0 T ( )( ) =   (1.11.13)  0 0 p 0 0 0 0 p which is a special case of the Hawking-Ellis type I energy momentum tensor (1.11.4). The resulting conditions from the weak energy condition are 2 ρ0 0 and ρ0c + p 0, (1.11.14) ≥ ≥ from the dominant energy condition 2 2 ρ 0 and ρ0c p , (1.11.15) − 0 ≤ ≥ | | and from the strong energy condition 2 2 ρ0c + 3p 0 and ρ0c + p 0. (1.11.16) ≤ ≥ Electromagnetic ﬁeld:

The energy momentum tensor of the electromagnetic ﬁeld reads (see [Wal84]) 1 ρ 1 ρσ Tµν = Fµρ Fν gµν Fρσ F (1.11.17) µ0 − 4

with the constant of the magnetic ﬁeld µ0, the electromagnetic ﬁeld strength tensor

Fµν = ∇µ Aν ∇ν Aµ (1.11.18) − and the four-potential Φ Aµ = ,A . (1.11.19) c 1.12. TOOLS 13

1.12 Tools

1.12.1 Maple/GRTensorII The Christoffel symbols, the Riemann- and Ricci-tensors as well as the Ricci and Kretschmann scalars in this catalogue were determined by means of the software Maple together with the GRTensorII package by Musgrave, Pollney, and Lake.2

A typical worksheet to enter a new metric may look like this:

> grtw(); > makeg(Schwarzschild);

Makeg 2.0: GRTensor metric/basis entry utility To quit makeg, type ’exit’ at any prompt. Do you wish to enter a 1) metric [g(dn,dn)], 2) line element [ds], 3) non-holonomic basis [e(1)...e(n)], or 4) NP tetrad [l,n,m,mbar]? > 2:

Enter coordinates as a LIST (eg. [t,r,theta,phi]): > [t,r,theta,phi]:

Enter the line element using d[coord] to indicate differentials. (for example, r^2*(d[theta]^2 + sin(theta)^2*d[phi]^2) [Type ’exit’ to quit makeg] ds^2 =

If there are any complex valued coordinates, constants or functions for this spacetime, please enter them as a SET ( eg. { z, psi } ).

Complex quantities [default={}]: > {}:

You may choose to 0) Use the metric WITHOUT saving it, 1) Save the metric as it is, 2) Correct an element of the metric, 3) Re-enter the metric, 4) Add/change constraint equations, 5) Add a text description, or 6) Abandon this metric and return to Maple. > 0:

The worksheets for some of the metrics in this catalogue can be found on the authors homepage. To determine the objects that are deﬁned with respect to a local tetrad, the metric must be given as non- holonomic basis. The various basic objects can be determined via µ Christoffel symbols Γνρ grcalc(Chr2); grcalc(Chr(dn,dn,up)); µ partial derivatives Γνρ,σ grcalc(Chr(dn,dn,up,pdn)); Riemann tensor Rµνρσ grcalc(Riemman); grcalc(R(dn,dn,dn,dn)); Ricci tensor Rµν grcalc(Ricci); grcalc(R(dn,dn)); Ricci scalar R grcalc(Ricciscalar); Kretschmann scalar K grcalc(RiemSq);

1.12.2 Mathematica The calculation of the Christoffel symbols, the Riemann- or Ricci-tensor within Mathematica could read like this:

Clearing the values of symbols: In[1]:= Clear[coord, metric, inversemetric, affine,

2The commercial software Maple can be found here: http://www.maplesoft.com. The GRTensorII-package is free: http://grtensor.phy.queensu.ca. 14 CHAPTER 1. INTRODUCTION AND NOTATION

t, r, Theta, Phi]

Setting the dimension: In[2]:= n := 4

Defining a list of coordinates: In[3]:= coord := {t, r, Theta, Phi}

Defining the metric: In[4]:= metric := {{-(1 - rs/r) c^2, 0, 0, 0}, {0, 1/(1 - rs/r), 0, 0}, {0, 0, r^2, 0}, {0, 0, 0, r ^2 Sin[Theta]^2}} In[5]:= metric // MatrixForm

Calculating the inverse metric: In[6]:= inversemetric := Simplify[Inverse[metric]]

In[7]:= inversemetric // MatrixForm

Calculating the Christoffel symbols of the second kind: In[8]:= affine := affine = Simplify[ Table[(1/2) Sum[inversemetric[[Mu, Rho]] ( D[metric[[Rho, Nu]], coord[[Lambda]]] + D[metric[[Rho, Lambda]], coord[[Nu]]] - D[metric[[Nu, Lambda]], coord[[Rho]]]), {Rho, 1, n}], {Nu, 1, n}, {Lambda, 1, n}, {Mu, 1, n}]]

Displaying the Christoffel symbols of the second kind: In[9]:= listaffine := Table[If[UnsameQ[affine[[Nu, Lambda, Mu]], 0], {Style[ Subsuperscript[\[CapitalGamma], Row[{coord[[Nu]], coord[[Lambda]]}], coord[[Mu]]], 18], "=", Style[affine[[Nu, Lambda, Mu]], 14]}], {Lambda, 1, n}, {Nu, 1, Lambda}, {Mu, 1, n}]

In[10]:= TableForm[Partition[DeleteCases[Flatten[listaffine], Null], 3], TableSpacing -> {1, 2}]

Defining the Riemann tensor: In[11]:= riemann := riemann = Table[D[affine[[Nu, Sigma, Mu]], coord[[Rho]]] - D[affine[[Nu, Rho, Mu]], coord[[Sigma]]] + Sum[affine[[Rho, Lambda, Mu]] affine[[Nu, Sigma, Lambda]] - affine[[Sigma, Lambda, Mu]] affine[[Nu, Rho, Lambda]], {Lambda, 1, n}], {Mu, 1, n}, {Nu, 1, n}, {Rho, 1, n}, {Sigma, 1, n}]

Defining the Riemann tensor with lower indices: In[12]:= riemannDn := riemannDn = Table[Simplify[ Sum[metric[[Mu, Kappa]] riemann[[Kappa, Nu, Rho, Sigma]], {Kappa, 1, n}]], {Mu, 1, n}, {Nu, 1, n}, {Rho, 1, n}, {Sigma, 1, n}]

In[13]:= listRiemann := Table[If[UnsameQ[riemannDn[[Mu, Nu, Rho, Sigma]], 0], {Style[Subscript[R, Row[{coord[[Mu]], coord[[Nu]], coord[[Rho]], coord[[Sigma]]}]], 16], "=", riemannDn[[Mu, Nu, Rho, Sigma]]}], {Nu, 1, n}, {Mu, 1, Nu}, {Sigma, 1, n}, {Rho, 1, Sigma}]

In[14]:= TableForm[Partition[DeleteCases[Flatten[listRiemann], Null], 3], TableSpacing -> {2, 2}]

Defining the Ricci tensor: In[15]:= ricci := ricci = Table[Simplify[ Sum[riemann[[Rho, Mu, Rho, Nu]], {Rho, 1, n}]], 1.12. TOOLS 15

{Mu, 1, n}, {Nu, 1, n}]

In[16]:= listRicci := Table[If[UnsameQ[ricci[[Mu, Nu]], 0], {Style[Subscript[R, Row[{coord[[Mu]], coord[[Nu]]}]], 16], "=", Style[ricci[[Mu, Nu]], 16]}], {Nu, 1, 4}, {Mu, 1, Nu}]

In[17]:= TableForm[Partition[DeleteCases[Flatten[listRicci], Null], 3], TableSpacing -> {1, 2}]

Defining the Ricci scalar: In[18]:= ricciscalar := ricciscalar = Simplify[Sum[ Sum[inversemetric[[Mu, Nu]] ricci[[Nu, Mu]], {Mu, 1, n}], {Nu, 1, n}]]

Defining the Kretschmann scalar: In[19]:= riemannUp := riemannUp = Table[Simplify[ Sum[inversemetric[[Nu, Kappa]] riemann[[Mu, Kappa, Rho, Sigma]], {Kappa, 1, n}]], {Mu, 1, n}, {Nu, 1, n}, {Rho, 1, n}, {Sigma, 1, n}]

In[20]:= kretschmann := kretschmann = Simplify[Sum[ Sum[Sum[Sum[ riemannUp[[Mu, Nu, Rho, Sigma]] riemannUp[[Rho, Sigma, Mu, Nu]], {Mu, 1, n}], {Nu, 1, n}], {Rho, 1, n}], {Sigma, 1, n}]]

Some example notebooks can be found on the authors homepage.

1.12.3 Maxima Instead of using commercial software like Maple or Mathematica, Maxima also offers a tensor package that helps to calculate the Christoffel symbols etc. The above example for the Schwarzschild metric can be written as a maxima worksheet as follows:

/* load ctensor package */ load(ctensor);

/* define coordinates to use */ ct_coords:[t,r,theta,phi];

/* start with the identity metric */ lg:ident(4); lg[1,1]:-c^2*(1-rs/r); lg[2,2]:1/(1-rs/r); lg[3,3]:r^2; lg[4,4]:r^2*sin(theta)^2;

/* computes the metric inverse and sets up the package for further calculations. */ cmetric();

/* calculate the christoffel symbols of the second kind */ christof(mcs);

/* calculate the riemann tensor Note the different ordering of the indices: R[mu,nu,rho,sigma]=lriem[nu,sigma,rho,mu] */ lriemann(true); RM(mu,nu,rho,sigma):=lriem[nu,sigma,rho,mu];

/* calculate the ricci tensor */ ricci(true);

/* simplify the ricci tensor */ ratsimp(ric[1,1]); ratsimp(ric[2,2]); 16 CHAPTER 1. INTRODUCTION AND NOTATION

/* calculate the ricci scalar */ scurvature();

/* calculate the Kretschmann scalar */ uriemann(false); rinvariant(); ratsimp(%);

Here, we have used maxima version 5.20.1.

1.12.4 Sympy Another alternative to commercial software is the SymPy package for python. The m4d module is par- tially based on...

import sys from sympy import * class Metric(object): """ Turn matrix into upper and lower metric """ def __init__(self,m): self.gdd = m self.guu = m.inv()

def __str__(self): return "g_dd = \n" + str(self.gdd)

def dd(self,i,j): return self.gdd[i,j]

def uu(self,i,j): return self.guu[i,j] class Gamma(object): """ Calculate Christoffel Gamma_ij^k symbols of metric g """ def __init__(self,g,x): self.g = g self.x = x

def ddu(self,i,j,k): g = self.g x = self.x chr = 0 for m in [0,1,2,3]: chr += g.uu(k,m)/2 * (g.dd(m,i).diff(x[j]) \ + g.dd(m,j).diff(x[i]) - g.dd(i,j).diff(x[m])) #return chr.simplify() return chr class Riemann(object): """ Calculate Riemann tensor R^mu_nu,rho,sigma """ def __init__(self,g,G,x): self.g = g self.G = G self.x = x

def uddd(self,mu,nu,rho,sigma): G = self.G x = self.x R = G.ddu(nu,sigma,mu).diff(x[rho]) - G.ddu(nu,rho,mu).diff(x[sigma]) for lam in [0,1,2,3]: R += G.ddu(rho,lam,mu)*G.ddu(nu,sigma,lam) - \ G.ddu(sigma,lam,mu)*G.ddu(nu,rho,lam) return R.simplify()

def dddd(self,mu,nu,rho,sigma): 1.12. TOOLS 17

g = self.g R = 0 for lam in [0,1,2,3]: R += g.dd(mu,lam)*self.uddd(lam,nu,rho,sigma) return R.simplify() class Ricci(object): """ Calculate Ricci tensor from Riemann tensor """ def __init__(self,R): self.R = R

def dd(self,mu,nu): R = self.R Ric = 0 for rho in [0,1,2,3]: Ric += R.uddd(rho,mu,rho,nu) return Ric.simplify() class RicciScalar(object): """ Calculate Ricci scalar from Ricci tensor """ def __init__(self,Ric,g): self.Ric = Ric self.g = g

def value(self): Ric = self.Ric g = self.g RS = 0 for mu in [0,1,2,3]: for nu in [0,1,2,3]: RS += g.uu(mu,nu)*Ric.dd(mu,nu) return RS.simplify() def pprint_christoffel_ddu(Gamma,i,j,k): pprint(Eq(Symbol(’Chr_%i%i^%i’ % (i,j,k)), Gamma.ddu(i,j,k))) def pprint_christoffels(Gamma): for i in [0,1,2,3]: for j in [0,1,2,3]: for k in [0,1,2,3]: if (Gamma.ddu(i,j,k)!=0): pprint_christoffel_ddu(Gamma,i,j,k) def pprint_riemann(Riemann): for i in [0,1,2,3]: for j in [0,1,2,3]: for k in [0,1,2,3]: for m in [0,1,2,3]: if (Riemann.uddd(i,j,k,m)!=0): pprint(Eq(Symbol(’R^%i_%i%i%i’ % (i,j,k,m)), Riemann.uddd(i,j,k,m))) def pprint_riemann_down(Riemann): for i in [0,1,2,3]: for j in [0,1,2,3]: for k in [0,1,2,3]: for m in [0,1,2,3]: if (Riemann.uddd(i,j,k,m)!=0): pprint(Eq(Symbol(’R_%i%i%i%i’ % (i,j,k,m)), Riemann.dddd(i,j,k,m))) def codeprint_metric(g,f=sys.stdout): for i in [0,1,2,3]: for j in [0,1,2,3]: print >>f, "g_compts[{0}][{1}] = {2};".format(i,j,ccode(g.dd(i,j))) def codeprint_christoffels(Gamma,f=sys.stdout): for i in [0,1,2,3]: for j in [0,1,2,3]: for k in [0,1,2,3]: print >>f, "christoffel[{0}][{1}][{2}] = {3};".format(i,j,k,ccode(Gamma.ddu(i,j,k))) 18 CHAPTER 1. INTRODUCTION AND NOTATION

def codeprint_chrisD(Gamma,X,f=sys.stdout): for i in [0,1,2,3]: for j in [0,1,2,3]: for k in [0,1,2,3]: for m in [0,1,2,3]: print >>f, "chrisD[{0}][{1}][{2}][{3}] = {4};" .format(i,j,k,m,ccode(Gamma.ddu(i,j,k).diff(X[m]).simplify())) def codeprint_riem(Riemann,f=sys.stdout): for i in [0,1,2,3]: for j in [0,1,2,3]: for k in [0,1,2,3]: for m in [0,1,2,3]: print >>f, "riem[{0}][{1}][{2}][{3}] = {4};".format(i,j,k,m,ccode(Riemann.uddd(i,j,k,m))) def codeprint_ricci(Ricci,f=sys.stdout): for i in [0,1,2,3]: for j in [0,1,2,3]: print >>f, "ricci[{0}][{1}] = {2}".format(i,j,ccode(Ricci.dd(i,j))) Chapter 2

Spacetimes

2.1 Minkowski

2.1.1 Cartesian coordinates The Minkowski metric in Cartesian coordinates t,x,y,z R reads { ∈ }

ds2 = c2dt2 + dx2 + dy2 + dz2. (2.1.1) −

All Christoffel symbols as well as the Riemann- and Ricci-tensor vanish identically. The natural local tetrad is trivial, 1 e = ∂ , e = ∂ , e = ∂ , e = ∂ , (2.1.2) (t) c t (x) x (y) y (z) z with dual

θ(t) = cdt, θ(x) = dx, θ(y) = dy, θ(z) = dz. (2.1.3)

2.1.2 Cylindrical coordinates The Minkowski metric in cylindrical coordinates t R,r R+,ϕ [0,2π),z R , { ∈ ∈ ∈ ∈ }

ds2 = c2dt2 + dr2 + r2dϕ2 + dz2, (2.1.4) − has the natural local tetrad 1 1 e = ∂ , e = ∂ , e = ∂ , e = ∂ . (2.1.5) (t) c t (r) r (ϕ) r ϕ (z) z Christoffel symbols:

r ϕ 1 Γ = r, Γr = . (2.1.6) ϕϕ − ϕ r Partial derivatives

ϕ 1 r Γr r = , Γ = 1. (2.1.7) ϕ, −r2 ϕϕ,r −

Ricci rotation coefﬁcients: 1 1 γ = and γ = . (2.1.8) (ϕ)(r)(ϕ) r (r) r

19 20 CHAPTER 2. SPACETIMES

2.1.3 Spherical coordinates In spherical coordinates t R,r R+,ϑ (0,π),ϕ [0,2π) , the Minkowski metric reads { ∈ ∈ ∈ ∈ }

ds2 = c2dt2 + dr2 + r2 dϑ 2 + sin2 ϑdϕ2. (2.1.9) −

Christoffel symbols: 1 Γ r = r, Γ r = r sin2 ϑ, Γ ϑ = , (2.1.10a) ϑϑ − ϕϕ − rϑ r ϑ ϕ 1 ϕ Γ = sinϑ cosϑ, Γr = , Γ = cotϑ. (2.1.10b) ϕϕ − ϕ r ϑϕ Partial derivatives

ϑ 1 ϕ 1 r Γ = , Γr r = , Γ = 1, (2.1.11a) rϑ,r −r2 ϕ, −r2 ϑϑ,r − 1 Γ ϕ = , Γ r = sin2 ϑ, Γ ϑ = cos(2ϑ), (2.1.11b) ϑϕ,ϑ −sin2 ϑ ϕϕ,r − ϕϕ,ϑ − Γ r = sin(2ϑ). (2.1.11c) ϕϕ,ϑ − Local tetrad: 1 1 1 e = ∂t , e = ∂r, e = ∂ , e = ∂ . (2.1.12) (t) c (r) (ϑ) r ϑ (ϕ) r sinϑ ϕ Ricci rotation coefﬁcients: 1 cotϑ γ = γ = , γ = . (2.1.13) (ϑ)(r)(ϑ) (ϕ)(r)(ϕ) r (ϕ)(ϑ)(ϕ) r The contractions of the Ricci rotation coefﬁcients read 2 cotϑ γ = , γ = . (2.1.14) (r) r (ϑ) r

2.1.4 Conform-compactiﬁed coordinates The Minkowski metric in conform-compactiﬁed coordinates ψ [ π,π],ξ (0,π),ϑ (0,π),ϕ [0,2π) { ∈ − ∈ ∈ ∈ } reads[HE99]

ds2 = dψ2 + dξ 2 + sin2 ξ dϑ 2 + sin2 ϑdϕ2. (2.1.15) − This form follows from the spherical Minkowski metric (2.1.9) by means of the coordinate transformation ψ + ξ ψ ξ ct + r = tan , ct r = tan − , (2.1.16) 2 − 2 resulting in the metric

dψ2 + dξ 2 sin2 ξ ds˜2 = − + dϑ 2 + sin2 ϑdϕ2, (2.1.17) 2 ψ+ξ 2 ψ ξ 2 ψ+ξ 2 ψ ξ 4cos 2 cos −2 4cos 2 cos −2

2 2 2 2 2 ψ+ξ 2 ψ ξ and by the conformal transformation ds = Ω ds˜ with Ω = 4cos 2 cos −2 . Christoffel symbols:

Γ ϑ = cotξ, Γ ϕ = cotξ, Γ ξ = sinξ cosξ, (2.1.18a) ξϑ ξϕ ϑϑ − Γ ϕ = cotϑ, Γ ξ = sinξ cosξ sin2 ϑ, Γ ϑ = sinϑ cosϑ. (2.1.18b) ϑϕ ϕϕ − ϕϕ − 2.1. MINKOWSKI 21

Partial derivatives 1 1 Γ ϑ = , Γ ϕ = , Γ ξ = cos(2ξ), (2.1.19a) ξϑ,ξ −sin2 ξ ξϕ,ξ −sin2 ξ ϑϑ,ξ − 1 Γ ϕ = , Γ ξ = cos(2ξ)sin2 ϑ, Γ ϑ = cos(2ϑ), (2.1.19b) ϑϕ,ϑ −sin2 ϑ ϕϕ,ξ − ϕϕ,ϑ − 1 Γ ξ = sin(2ξ)sin(2ϑ). (2.1.19c) ϕϕ,ϑ −2

Riemann-Tensor:

2 2 2 4 2 Rξϑξϑ = sin ξ, Rξϕξϕ = sin ξ sin ϑ, Rϑϕϑϕ = sin ξ sin ϑ. (2.1.20)

Ricci-Tensor:

2 2 2 Rξξ = 2, Rϑϑ = 2sin ξ, Rϕϕ = 2sin ξ sin ϑ. (2.1.21)

Ricci and Kretschmann scalars:

R = 6, K = 12. (2.1.22)

The Weyl tensor vanishs identically. Local tetrad: 1 1 e = ∂ , e = ∂ , e = ∂ , e = ∂ . (2.1.23) (ψ) ψ (ξ) ξ (ϑ) sinξ ϑ (ϕ) sinξ sinϑ ϕ

Ricci rotation coefﬁcients: cotϑ γ = γ = cotξ, γ = . (2.1.24) (ϑ)(ξ)(ϑ) (ϕ)(ξ)(ϕ) (ϕ)(ϑ)(ϕ) sinξ

The contractions of the Ricci rotation coefﬁcients read cotϑ γ = 2cotξ, γ = . (2.1.25) (ξ) (ϑ) sinξ

Riemann-Tensor with respect to local tetrad:

R(ξ)(ϑ)(ξ)(ϑ) = R(ξ)(ϕ)(ξ)(ϕ) = R(ϑ)(ϕ)(ϑ)(ϕ) = 1. (2.1.26)

Ricci-Tensor with respect to local tetrad:

R(ξ)(ξ) = R(ϑ)(ϑ) = R(ϕ)(ϕ) = 2. (2.1.27)

2.1.5 Rotating coordinates The transformation dϕ dϕ + ω dt brings the Minkowski metric (2.1.4) into the rotating form[Rin01] 7→ with coordinates t R,r R+,ϕ [0,2π),z R , { ∈ ∈ ∈ ∈ }

ω2r2 r2 ds2 = 1 [cdt Ω(r)dϕ]2 + dr2 + dϕ2 + dz2 (2.1.28) − − c2 − 1 ω2r2/c2 − with Ω(r) = (r2ω/c)/(1 ω2r2/c2). − Metric-Tensor:

2 2 2 2 2 gtt = c + ω r , gtϕ = ωr , grr = gzz = 1, gϕϕ = r . (2.1.29) − 22 CHAPTER 2. SPACETIMES

Christoffel symbols:

r 2 ϕ ω r ϕ 1 r Γ = ω r, Γ = , Γ = ωr, Γr = , Γ = r. (2.1.30) tt − tr r tϕ − ϕ r ϕϕ − Partial derivatives

r 2 ϕ ω r ϕ 1 r Γ = ω , Γ = , Γ = ω, Γr r = , Γ = 1. (2.1.31) tt,r − tr,r −r2 tϕ,r − ϕ, −r2 ϕϕ,r − The local tetrad of the comoving observer is 1 ω 1 e = ∂t ∂ϕ , e = ∂r, e = ∂ϕ , e = ∂z, (2.1.32) (t) c − c (r) (ϕ) r (z) whereas the static observer has the local tetrad 1 e(t) = p ∂t , e(r) = ∂r, e(z) = ∂z, (2.1.33a) c 1 ω2r2/c2 − p ωr 1 ω2r2/c2 e(ϕ) = p ∂t + − ∂ϕ . (2.1.33b) c2 1 ω2r2/c2 r − 2.1.6 Rindler coordinates The worldline of an observer in the Minkowski spacetime who moves with constant proper acceleration α along the x direction reads

c2 αt c2 αt x = cosh 0 , ct = sinh 0 , (2.1.34) α c α c where t0 is the observer’s proper time. The observer starts at x = 1 with zero velocity. However, such an observer could also be described with Rindler coordinates. With the coordinate transformation 1 1 (ct,x) (τ,ρ) : ct = sinhτ, x = coshτ, (2.1.35) 7→ ρ ρ where ρ = α/c2, the Rindler metric reads

1 1 ds2 = dτ2 + dρ2 + dy2 + dz2. (2.1.36) −ρ2 ρ4

Christoffel symbols: 1 2 Γ ρ = ρ, Γ τ = , Γ ρ = . (2.1.37) ττ − τρ −ρ ρρ −ρ Partial derivatives 1 2 Γ ρ = 1, Γ τ = , Γ ρ = . (2.1.38) ττ,ρ − τρ,ρ ρ2 ρρ,ρ ρ2 The Riemann and Ricci tensors as well as the Ricci and Kretschmann scalar vanish identically. Local tetrad:

2 e(τ) = ρ∂τ , e(ρ) = ρ ∂ρ , e(y) = ∂y, e(z) = ∂z. (2.1.39)

Ricci rotation coefﬁcients:

γ = ρ, and γ = ρ. (2.1.40) (τ)(ρ)(τ) (ρ) − 2.2. SCHWARZSCHILD SPACETIME 23

2.2 Schwarzschild spacetime

2.2.1 Schwarzschild coordinates In Schwarzschild coordinates t R,r R+,ϑ (0,π),ϕ [0,2π) , the Schwarzschild metric reads { ∈ ∈ ∈ ∈ }

rs 1 ds2 = 1 c2dt2 + dr2 + r2 dϑ 2 + sin2 ϑdϕ2, (2.2.1) − − r 1 rs/r − 2 where rs = 2GM/c is the Schwarzschild radius, G is Newton’s constant, c is the speed of light, and M is the mass of the black hole. The critical point r = 0 is a real curvature singularity while the event horizon, r = rs, is only a coordinate singularity, see e.g. the Kretschmann scalar. Christoffel symbols: 2 r c rs(r rs) t rs r rs Γtt = 3− , Γtr = , Γrr = , (2.2.2a) 2r 2r(r rs) −2r(r rs) − − ϑ 1 ϕ 1 r Γ = , Γr = , Γ = (r rs), (2.2.2b) rϑ r ϕ r ϑϑ − − ϕ r 2 ϑ Γ = cotϑ, Γ = (r rs)sin ϑ, Γ = sinϑ cosϑ. (2.2.2c) ϑϕ ϕϕ − − ϕϕ − Partial derivatives 2 r (2r 3rs)c rs t (2r rs)rs r (2r rs)rs Γtt,r = − 4 , Γtr,r = 2 − 2 , Γrr,r = 2 − 2 , (2.2.3a) − 2r −2r (r rs) 2r (r rs) − − ϑ 1 ϕ 1 r Γ = , Γr r = , Γ = 1, (2.2.3b) rϑ,r −r2 ϕ, −r2 ϑϑ,r − 1 Γ ϕ = , Γ r = sin2 ϑ, Γ ϑ = cos(2ϑ), (2.2.3c) ϑϕ,ϑ −sin2 ϑ ϕϕ,r − ϕϕ,ϑ − r Γ = (r rs)sin(2ϑ). (2.2.3d) ϕϕ,ϑ − − Riemann-Tensor: 2 2 2 2 c rs 1 c (r rs)rs 1 c (r rs)rs sin ϑ Rtrtr = , Rt t = − , Rtϕtϕ = − , (2.2.4a) − r3 ϑ ϑ 2 r2 2 r2 2 1 rs 1 rs sin ϑ 2 Rrϑrϑ = , Rrϕrϕ = , Rϑϕϑϕ = rrs sin ϑ. (2.2.4b) −2 r rs −2 r rs − − As aspected, the Ricci tensor as well as the Ricci scalar vanish identically because the Schwarzschild spacetime is a vacuum solution of the ﬁeld equations. Hence, the Weyl tensor is identical to the Riemann tensor. The Kretschmann scalar reads r2 K = 12 s . (2.2.5) r6

Here, it becomes clear that at r = rs there is no real singularity. Local tetrad: r 1 rs 1 1 e(t) = p ∂t , e(r) = 1 ∂r, e(ϑ) = ∂ϑ , e(ϕ) = ∂ϕ . (2.2.6) c 1 rs/r − r r r sinϑ − Dual tetrad: r (t) rs (r) dr (ϑ) (ϕ) θ = c 1 dt, θ = p , θ = r dϑ, θ = r sinϑ dϕ. (2.2.7) − r 1 rs/r − Ricci rotation coefﬁcients: r rs 1 rs cotϑ γ = , γ = γ = 1 , γ = . (2.2.8) (r)(t)(t) 2p (ϑ)(r)(ϑ) (ϕ)(r)(ϕ) (ϕ)(ϑ)(ϕ) 2r 1 rs/r r − r r − 24 CHAPTER 2. SPACETIMES

The contractions of the Ricci rotation coefﬁcients read

4r 3rs cotϑ γ = − , γ = . (2.2.9) (r) 2p (ϑ) 2r 1 rs/r r − Structure coefﬁcients: r t rs ϑ ϕ 1 rs ϕ cotϑ c( ) = , c( ) = c( ) = 1 , c( ) = . (2.2.10) (t)(r) 2p (r)(ϑ) (r)(ϕ) (ϑ)(ϕ) 2r 1 rs/r − r − r r − Riemann-Tensor with respect to local tetrad:

rs R = R = , (2.2.11a) (t)(r)(t)(r) − (ϑ)(ϕ)(ϑ)(ϕ) −r3 rs R = R = R = R = . (2.2.11b) (t)(ϑ)(t)(ϑ) (t)(ϕ)(t)(ϕ) − (r)(ϑ)(r)(ϑ) − (r)(ϕ)(r)(ϕ) 2r3 The covariant derivatives of the Riemann tensor read

3rs p R = R = r(r rs), (2.2.12a) (t)(r)(t)(r);(r) − (ϑ)(ϕ)(ϑ)(ϕ);(r) r5 − R(t)(r)(r)(ϑ);(ϑ) = R(t)(r)(t)(ϕ);(ϕ) = R(t)(ϑ)(t)(ϑ);(r) = R(t)(ϕ)(t)(ϕ);(r) = 3rs p = R = r(r rs), (2.2.12b) (r)(ϕ)(ϑ)(ϕ);(ϑ) −2r5 − 3rs p R = R = R = r(r rs). (2.2.12c) (r)(ϑ)(r)(ϑ);(r) (r)(ϑ)(ϑ)(ϕ);(ϕ) (r)(ϕ)(r)(ϕ);(r) 2r5 − Newman-Penrose tetrad: 1 1 1 l = e t + e r , n = e t e r , m = e ϑ + ie ϕ . (2.2.13) √2 ( ) ( ) √2 ( ) − ( ) √2 ( ) ( ) Non-vanishing spin coefﬁcients: r 1 rs rs cotϑ ρ = µ = 1 , γ = ε = , α = β = . (2.2.14) 2p −√2r − r 4√2r 1 rs/r − −2√2r − Embedding: The embedding function reads

z = 2√rs√r rs. (2.2.15) − Euler-Lagrange: The Euler-Lagrangian formalism, Sec. 1.8.4, for geodesics in the ϑ = π/2 hyperplane yields

2 2 1 1 k 1 rs h r˙2 +V = , V = 1 κc2 (2.2.16) 2 eff 2 c2 eff 2 − r r2 −

2 2 with the constants of motion k = (1 rs/r)c t˙, h = r ϕ˙, and κ as in Eq. (1.8.2). For timelike geodesics, the − effective potential has the extremal points

2 p 2 2 2 h h h 3c rs r = ± 2 − , (2.2.17) ± c rs where r+ is a maximum and r is a minimum. The innermost timelike circular geodesic follows from 2 2 2 − 3 h = 3c rs and reads ritcg = 3rs. Null geodesics, however, have only a maximum at rpo = 2 rs. The corresponding circular orbit is called photon orbit. Further reading: Schwarzschild[Sch16, Sch03], MTW[MTW73], Rindler[Rin01], Wald[Wal84], Chandrasekhar[Cha06], Müller[Mül08b, Mül09]. 2.2. SCHWARZSCHILD SPACETIME 25

2.2.2 Schwarzschild in pseudo-Cartesian coordinates The Schwarzschild spacetime in pseudo-Cartesian coordinates (t,x,y,z) reads

2 2 2 2 2 rs 2 2 x 2 2 dx 2 y 2 dy ds = 1 c dt + + y + z 2 + x + + z 2 − − r 1 rs/r r 1 rs/r r 2 − 2 − (2.2.18) 2 2 z dz 2rs + x + y + 2 + 2 (xydxdy + xzdxdz + yzdydz), 1 rs/r r r (r rs) − − where r2 = x2 + y2 + z2. For a natural local tetrad that is adapted to the x-axis, we make the following ansatz: 1 e(0) = p ∂t , e(1) = A∂x, e(2) = B∂x +C∂y, e(3) = D∂x + E∂y + F∂z. (2.2.19) c 1 rs/r −

1 gxy 1 A = , B = q − , C = q , (2.2.20a) gxx 2 2 √ gxx g /gxx + gyy g /gxx + gyy − xy − xy gxygyz gxzgyy gxzgxy gxxgyz √N D = − , E = − , F = , (2.2.20b) √NW √NW √W with

2 N = gxxgyy g , (2.2.21a) − xy 2 2 2 W = gxxgyygzz g gyy + 2gxzgxygyz g gzz gxxg . (2.2.21b) − xz − xy − yz 2.2.3 Isotropic coordinates Spherical isotropic coordinates The Schwarzschild metric (2.2.1) in spherical isotropic coordinates (t,ρ,ϑ,ϕ) reads

2 4 1 ρs/ρ ρs ds2 = − c2dt2 + 1 + dρ2 + ρ2 dϑ 2 + sin2 ϑdϕ2, (2.2.22) − 1 + ρs/ρ ρ where 2 ρs 1 p r = ρ 1 + or ρ = 2r rs 2 r(r rs) (2.2.23) ρ 4 − ± − is the coordinate transformation between the Schwarzschild radial coordinate r and the isotropic radial coordinate ρ, see e.g. MTW[MTW73] page 840. The event horizon is given by ρs = rs/4. The photon orbit and the innermost timelike circular geodesic read ρpo = 2 + √3 ρs and ρitcg = 5 + 2√6 ρs. (2.2.24)

Christoffel symbols:

4 2 ρ 2(ρ ρs)ρ ρsc t 2ρs ρ 2ρs Γtt = − 7 , Γtρ = 2 2 , Γρρ = , (2.2.25a) (ρ + ρs) ρ ρ −(ρ + ρs)ρ − s ϑ ρ ρs ϕ ρ ρs ρ ρ ρs Γρϑ = − , Γρϕ = − , Γϑϑ = ρ − , (2.2.25b) (ρ + ρs)ρ (ρ + ρs)ρ − ρ + ρs 2 ϕ ρ (ρ ρs)ρ sin ϑ ϑ Γϑϕ = cotϑ, Γϕϕ = − , Γϕϕ = sinϑ cosϑ. (2.2.25c) − ρ + ρs − 26 CHAPTER 2. SPACETIMES

Riemann-Tensor:

2 2 2 2 (ρ ρs) ρsc (ρ ρs) ρρsc Rtρtρ = 4 − 4 , Rtϑtϑ = 2 − 4 , (2.2.26a) − (ρ + ρs) ρ (ρ + ρs) 2 2 2 2 (ρ ρs) ρc ρs sin ϑ (ρ + ρs) ρs Rtϕtϕ = 2 − 4 , Rρϑρϑ = 2 3 , (2.2.26b) (ρ + ρs) − ρ 2 2 2 2 (ρ + ρs) ρs sin ϑ 4(ρ + ρs) ρs sin ϑ Rρϕρϕ = 2 , R = . (2.2.26c) − ρ3 ϑϕϑϕ ρ

The Ricci tensor and the Ricci scalar vanish identically. Kretschmann scalar: r2 r2 K = 192 s = 12 s . (2.2.27) 6 12 6 ρ (1 + ρs/ρ) r(ρ)

Local tetrad:

1 + ρs/ρ ∂t 1 e(t) = , e(r) = 2 ∂ρ , (2.2.28a) 1 ρs/ρ c [1 + ρ /ρ] − s 1 1 e ∂ e ∂ (ϑ) = 2 ϑ , (ϕ) = 2 2 ϕ . (2.2.28b) ρ [1 + ρs/ρ] ρ [1 + ρs/ρ] sin ϑ

Ricci rotation coefﬁcients:

2 2ρsρ ρ(ρ ρs) γ(ρ)(t)(t) = 3 , γ(ϑ)(ρ)(ϑ) = γ(ϕ)(ρ)(ϕ) = − 3 , (2.2.29a) (ρ + ρs) (ρ ρs) (ρ + ρs) − ρ cotϑ γ(ϕ)(ϑ)(ϕ) = 2 . (2.2.29b) (ρ + ρs)

The contractions of the Ricci rotation coefﬁcients read 2 2 2ρ(ρ ρρs + ρs ) ρ cotϑ γ(ρ) = − 3 , γ(ϑ) = 2 . (2.2.30) (ρ + ρs) (ρ ρs) (ρ + ρs) − Riemann-Tensor with respect to local tetrad:

rs R = R = , (2.2.31a) (t)(ρ)(t)(ρ) − (ϑ)(ϕ)(ϑ)(ϕ) −r(ρ)3 rs R = R = R = R = . (2.2.31b) (t)(ϑ)(t)(ϑ) (t)(ϕ)(t)(ϕ) − (ρ)(ϑ)(ρ)(ϑ) − (ρ)(ϕ)(ρ)(ϕ) 2r(ρ)3

Further reading: Buchdahl[Buc85].

Cartesian isotropic coordinates The Schwarzschild metric (2.2.1) in Cartesian isotropic coordinates (t,x,y,z) reads,

2 4 1 ρs/ρ ρs ds2 = − c2dt2 + 1 + dx2 + dy2 + dz2, (2.2.32) − 1 + ρs/ρ ρ

where ρ2 = x2 + y2 + z2 and, as before,

2 ρs r = ρ 1 + . (2.2.33) ρ 2.2. SCHWARZSCHILD SPACETIME 27

Christoffel symbols:

2 3 2 3 2 3 x 2c ρ ρs (ρ ρs)x y 2c ρ ρs (ρ ρs)y z 2c ρ ρs (ρ ρs)z Γtt = −7 , Γtt = −7 , Γtt = −7 , (2.2.34a) (ρ + ρs) (ρ + ρs) (ρ + ρs) 2ρsx 2ρsy 2ρsz Γ t = , Γ t = , Γ t = , (2.2.34b) tx ρ3 [1 ρ2/ρ2] ty ρ3 [1 ρ2/ρ2] tz ρ3 [1 ρ2/ρ2] − s − s − s x y z x x 2ρs x Γxx = Γxy = Γxz = Γyy = Γzz = 3 , (2.2.34c) − − − ρ 1 + ρs/ρ y x y z y 2ρs y Γxx = Γxy = Γyy = Γyz = Γzz = 3 , (2.2.34d) − − − ρ 1 + ρs/ρ z x z y z 2ρs z Γxx = Γxz = Γyy = Γyz = Γzz = 3 . (2.2.34e) − − − ρ 1 + ρs/ρ

2.2.4 Eddington-Finkelstein The transformation of the Schwarzschild metric (2.2.1) from the usual Schwarzschild time coordinate t to the advanced null coordinate v with

cv = ct + r + rs ln(r rs) (2.2.35) − leads to the ingoing Eddington-Finkelstein[Edd24, Fin58] metric with coordinates (v,r,ϑ,ϕ),

rs ds2 = 1 c2dv2 + 2cdvdr + r2 dϑ 2 + sin2 ϑdϕ2. (2.2.36) − − r

Metric-Tensor: 2 rs 2 2 2 gvv = c 1 , gvr = c, g = r , gϕϕ = r sin ϑ. (2.2.37) − − r ϑϑ Christoffel symbols:

2 crs c rs(r rs) crs 1 Γ v = , Γ r = − , Γ r = , Γ ϑ = , (2.2.38a) vv 2r2 vv 2r3 vr −2r2 rϑ r ϕ 1 v r r ϕ Γr = , Γ = , Γ = (r rs), Γ = cotϑ, (2.2.38b) ϕ r ϑϑ −c ϑϑ − − ϑϕ 2 v r sin ϑ r 2 ϑ Γ = , Γ = (r rs)sin ϑ, Γ = sinϑ cosϑ. (2.2.38c) ϕϕ − c ϕϕ − − ϕϕ − Partial derivatives 2 crs (2r 3rs)c rs crs Γ v = , Γ r = − , Γ r = , (2.2.39a) vv,r − r3 vv,r − 2r4 vr,r r3 ϑ 1 ϕ 1 v 1 Γ = , Γr r = , Γ = , (2.2.39b) rϑ,r −r2 ϕ, −r2 ϑϑ,r − c 1 sin2 ϑ Γ r = 1, Γ ϕ = , Γ v = , (2.2.39c) ϑϑ,r − ϑϕ,ϑ −sin2 ϑ ϕϕ,r − c r sin(2ϑ) Γ v = , Γ r = sin2 ϑ, Γ ϑ = cos(2ϑ), (2.2.39d) ϕϕ,ϑ − c ϕϕ,r − ϕϕ,ϑ − r Γ = (r rs)sin(2ϑ). (2.2.39e) ϕϕ,ϑ − − Riemann-Tensor: 2 2 c rs c rs(r rs) crs Rvrvr = , Rv v = − , Rv r = , (2.2.40a) − r3 ϑ ϑ 2r2 ϑ ϑ − 2r 2 2 2 c rs(r rs)sin ϑ crs sin ϑ 2 Rvϕvϕ = − , Rvϕrϕ = , R = rrs sin ϑ. (2.2.40b) 2r2 − 2r ϑϕϑϕ 28 CHAPTER 2. SPACETIMES

2 6 While the Ricci tensor and the Ricci scalar vanish identically, the Kretschmann scalar is K = 12rs /r . Static local tetrad: r 1 1 rs 1 1 e(v) = p ∂v, e(r) = p ∂v + 1 ∂r, e(ϑ) = ∂ϑ , e(ϕ) = ∂ϕ . (2.2.41) c 1 rs/r c 1 rs/r − r r r sinϑ − − Dual tetrad: r (v) rs dr (r) dr (ϑ) (ϕ) θ = c 1 dv p , θ = p , θ = r dϑ, θ = r sinϑdϕ. (2.2.42) − r − 1 rs/r 1 rs/r − − Ricci rotation coefﬁcients: r rs 1 rs cotϑ γ = , γ = γ = 1 , γ = . (2.2.43) (r)(v)(v) 2p (ϑ)(r)(ϑ) (ϕ)(r)(ϕ) (ϕ)(ϑ)(ϕ) 2r 1 rs/r r − r r − The contractions of the Ricci rotation coefﬁcients read

4r 3rs cotϑ γ = − , γ = . (2.2.44) (r) 2p (ϑ) 2r 1 rs/r r − Riemann-Tensor with respect to local tetrad:

rs R = R = , (2.2.45a) (v)(r)(v)(r) − (ϑ)(ϕ)(ϑ)(ϕ) −r3 rs R = R = R = R = . (2.2.45b) (v)(ϑ)(v)(ϑ) (v)(ϕ)(v)(ϕ) − (r)(ϑ)(r)(ϑ) − (r)(ϕ)(r)(ϕ) 2r3

2.2.5 Kruskal-Szekeres The Schwarzschild metric in Kruskal-Szekeres[Kru60, Wal84] coordinates (T,X,ϑ,ϕ) reads

3 2 4rs r/rs 2 2 2 2 ds = e− dT + dX + r dΩ , (2.2.46) r −

where r R 0 is given by means of the LambertW-function W , ∈ + \ { } 2 2 r r/rs 2 2 X T 1 e = X T or r = rs W − + 1 . (2.2.47) rs − − e The derivatives of the radial function r with respect to T and X read

2 2 ∂r 2rs(1 rs/r)T 2Trs r/rs ∂r 2rs(1 rs/r)X 2Xrs r/rs = − = e− and = − = e− . (2.2.48) ∂T − X2 T 2 − r ∂X X2 T 2 r − − The Schwarzschild coordinate time t in terms of the Kruskal coordinates T and X reads T t = 2r arctanh , r > r , (2.2.49a) s X s X t = 2r arctanh , r < r , (2.2.49b) s T s t = ∞, r = rs. (2.2.49c)

The transformations between Kruskal- and Schwarzschild coordinates read r r r r/(2rs) ct r r/(2rs) ct X = 1 e sinh , T = 1 e cosh , 0 < r < rs, (2.2.50a) − rs 2rs − rs 2rs r r r r/(2rs) ct r r/(2rs) ct X = 1e cosh , T = 1e sinh , r rs. (2.2.50b) rs − 2rs rs − 2rs ≥ 2.2. SCHWARZSCHILD SPACETIME 29

Christoffel symbols:

Trs(r + rs) Γ T = Γ X = Γ T = e r/rs , (2.2.51a) TT TX XX r2 −

X T X Xrs(r + rs) r/rs Γ = Γ = Γ = e− , (2.2.51b) TT TX XX − r2 2 2 ϑ ϕ 2rs T r/rs ϑ ϕ 2rs X r/rs Γ = Γ = e− , Γ = Γ = e− , (2.2.51c) Tϑ Tϕ − r2 Xϑ Xϕ r2 T r X r Γϑϑ = T, Γϑϑ = X, (2.2.51d) −2rs −2rs T r 2 X r 2 Γϕϕ = T sin ϑ, Γϕϕ = X sin ϑ, (2.2.51e) −2rs −2rs Γ ϕ = cotϑ, Γ ϑ = sinϑ cosϑ. (2.2.51f) ϑϕ ϕϕ − Riemann-Tensor:

7 4 rs 2r/rs 2rs r/rs RTXTX = 16 e− , RT T = e− , (2.2.52a) − r5 ϑ ϑ r2 4 4 2rs r/rs 2 2rs r/rs RTϕTϕ = e− sin ϑ, RX X = e− , (2.2.52b) r2 ϑ ϑ − r2 4 2rs r/rs 2 2 RXϕXϕ = e− sin ϑ, R = rrs sin ϑ. (2.2.52c) − r2 ϑϕϑϕ

The Ricci-Tensor as well as the Ricci-scalar vanish identically. Kretschmann scalar:

12r2 K = s . (2.2.53) r6

Local tetrad:

√r r/(2rs) √r r/(2rs) 1 1 e(T) = e ∂T , e(X) = e ∂X , e(ϑ) = ∂ϑ , e(ϕ) = ∂ϕ (2.2.54) 2rs√rs 2rs√rs r r sinϑ

Riemann-Tensor with respect to local tetrad:

rs R = R = R = R = , (2.2.55a) (T)(X)(T)(X) (X)(ϑ)(X)(ϑ) (X)(ϕ)(X)(ϕ) − (ϑ)(ϕ)(ϑ)(ϕ) −r3 rs R = R = . (2.2.55b) (T)(ϑ)(T)(ϑ) (T)(ϕ)(T)(ϕ) 2r3

2.2.6 Tortoise coordinates The Schwarzschild metric represented by tortoise coordinates (t,ρ,ϑ,ϕ) reads

rs rs ds2 = 1 c2dt2 + 1 dρ2 + r(ρ)2 dϑ 2 + sin2 ϑdϕ2, (2.2.56) − − r(ρ) − r(ρ)

2 where rs = 2GM/c is the Schwarzschild radius, G is Newton’s constant, c is the speed of light, and M is the mass of the black hole. The tortoise radial coordinate ρ and the Schwarzschild radial coordinate r are related by

r ρ ρ = r + rs ln 1 or r = rs 1 + W exp 1 . (2.2.57) rs − rs − 30 CHAPTER 2. SPACETIMES

Christoffel symbols:

2 c rs rs rs Γ ρ = , Γ t = , Γ ρ = , (2.2.58a) tt 2r(ρ)2 tρ 2r(ρ)2 ρρ 2r(ρ)2

ϑ 1 1 ϕ 1 1 ρ Γρϑ = , Γρϕ = , Γϑϑ = r(ρ), (2.2.58b) r(ρ) − rs r(ρ) − rs − Γ ϕ = cotϑ, Γ ρ = r(ρ)sin2 ϑ, Γ ϑ = sinϑ cosϑ. (2.2.58c) ϑϕ ϕϕ − ϕϕ − Riemann-Tensor:

2 2 2 c rs rs c rs rs Rtρtρ = 1 , Rt t = 1 , (2.2.59a) −r(ρ)3 − r(ρ) ϑ ϑ 2 − r(ρ) r(ρ) 2 2 c sin ϑ rs rs 1 rs rs Rtϕtϕ = 1 , R = 1 (2.2.59b) 2 − r(ρ) r(ρ) ρϑρϑ −2 − r(ρ) r(ρ) 2 sin ϑ rs rs 2 Rρϕρϕ = 1 , R = r(ρ)rs sin ϑ. (2.2.59c) − 2 − r(ρ) r(ρ) ϑϕϑϕ

The Ricci tensor as well as the Ricci scalar vanish identically because the Schwarzschild spacetime is a vacuum solution of the ﬁeld equations. Hence, the Weyl tensor is identical to the Riemann tensor. The Kretschmann scalar reads r2 K = 12 s . (2.2.60) r(ρ)6

Local tetrad: 1 1 1 1 e(t) = p ∂t , e(ρ) = p ∂ρ , e(ϑ) = ∂ϑ , e(ϕ) = ∂ϕ . (2.2.61) c 1 rs/r(ρ) 1 rs/r(ρ) r(ρ) r(ρ)sinϑ − − Dual tetrad:

r rs r rs θ(t) = c 1 dt, θ(ρ) = 1 dρ, θ(ϑ) = r(ρ)dϑ, θ(ϕ) = r(ρ)sinϑ dϕ. (2.2.62) − r(ρ) − r(ρ)

Riemann-Tensor with respect to local tetrad:

rs R = R = , (2.2.63a) (t)(ρ)(t)(ρ) − (ϑ)(ϕ)(ϑ)(ϕ) −r(ρ)3 rs R = R = R = R = . (2.2.63b) (t)(ϑ)(t)(ϑ) (t)(ϕ)(t)(ϕ) − (ρ)(ϑ)(ρ)(ϑ) − (ρ)(ϕ)(ρ)(ϕ) 2r(ρ)3

Further reading: MTW[MTW73]

2.2.7 Painlevé-Gullstrand The Schwarzschild metric expressed in Painlevé-Gullstrand coordinates[MP01] reads

r 2 rs ds2 = c2dT 2 + dr + cdT + r2 dϑ 2 + sin2 ϑdϕ2, (2.2.64) − r

where the new time coordinate T follows from the Schwarzschild time t in the following way:

r p ! r 1 r/rs 1 cT = ct + 2rs + ln p − . (2.2.65) rs 2 r/rs + 1 2.2. SCHWARZSCHILD SPACETIME 31

Metric-Tensor: r 2 rs rs 2 2 2 gTT = c 1 , gTr = c , grr = 1, g = r , gϕϕ = r sin ϑ. (2.2.66) − − r r ϑϑ

Christoffel symbols:

r 2 crs rs c rs(r rs) rs Γ T = , Γ r = − , Γ T = , (2.2.67a) TT 2r2 r TT 2r3 Tr 2r2 r r r crs rs T rs r r rs ΓTr = 2 , Γrr = 2 , Γrr = 2 , (2.2.67b) −2r r 2cr rs −2r r ϑ 1 ϕ 1 T r rs Γ = , Γr = , Γ = , (2.2.67c) rϑ r ϕ r ϑϑ −c r r r ϕ T r rs 2 Γ = (r rs), Γ = cotϑ, Γ = sin ϑ, (2.2.67d) ϑϑ − − ϑϕ ϕϕ −c r r 2 ϑ Γ = (r rs)sin ϑ, Γ = sinϑ cosϑ. (2.2.67e) ϕϕ − − ϕϕ − Riemann-Tensor: 2 2 r c rs c rs(r rs) crs rs RTrTr = , RT T = − , RT r = , (2.2.68a) − r3 ϑ ϑ 2r2 ϑ ϑ − 2r r 2 2 r c rs(r rs)sin ϑ crs rs 2 rs RTϕTϕ = − , RTϕrϕ = sin ϑ, Rr r = , (2.2.68b) 2r2 − 2r r ϑ ϑ −2r 2 rs sin ϑ 2 Rrϕrϕ = , R = rrs sin ϑ. (2.2.68c) − 2r ϑϕϑϕ The Ricci tensor and the Ricci scalar vanish identically. Kretschmann scalar:

2 6 K = 12rs /r . (2.2.69) For the Painlevé-Gullstrand coordinates, we can deﬁne two natural local tetrads. Static local tetrad: r 1 √rs rs 1 1 eˆ(T) = p ∂T , eˆ(r) = ∂T + 1 ∂r, eˆ(ϑ) = ∂ϑ , eˆ(ϕ) = ∂ϕ , (2.2.70) c 1 rs/r c√r rs − r r r sinϑ − − Dual tetrad: r ˆ(T) rs dr ˆ(r) dr ˆ(ϑ) ˆ(ϕ) θ = c 1 dT p , θ = p , θ = r dϑ, θ = r sinϑ dϕ. (2.2.71) − r − r/rs 1 1 rs/r − − Freely falling local tetrad: r 1 rs 1 1 e = ∂T ∂r, e = ∂r, e = ∂ , e = ∂ϕ . (2.2.72) (T) c − r (r) (ϑ) r ϑ (ϕ) r sinϑ Dual tetrad: r rs θ(T) = cdT, θ(r) = c dT + dr, θ(ϑ) = r dϑ, θ(ϕ) = r sinϑdϕ. (2.2.73) r

Riemann-Tensor with respect to local tetrad:

rs R = R = , (2.2.74a) (T)(r)(T)(r) − (ϑ)(ϕ)(ϑ)(ϕ) −r3 rs R = R = R = R = . (2.2.74b) (T)(ϑ)(T)(ϑ) (T)(ϕ)(T)(ϕ) − (r)(ϑ)(r)(ϑ) − (r)(ϕ)(r)(ϕ) 2r3 32 CHAPTER 2. SPACETIMES

2.2.8 Israel coordinates The Schwarzschild metric in Israel coordinates (x,y,ϑ,ϕ) reads[SKM+03]

y2dx ds2 = r2 4dx dy + + (1 + xy)2 dϑ 2 + sin2 ϑdϕ2 , (2.2.75) s 1 + xy

where the coordinates x and y follow from the Schwarzschild coordinates via y t = r 1 + xy + ln and r = r (1 + xy). (2.2.76) s x s

Christoffel symbols:

y(2 + xy) y3(3 + xy) y(2 + xy) Γ x = , Γ y = , Γ y = , (2.2.77a) xx − (1 + xy)2 xx (1 + xy)3 xy (1 + xy)2 y y x Γ ϑ = , Γ ϕ = , Γ ϑ = , (2.2.77b) xϑ 1 + xy xϕ 1 + xy yϑ 1 + xy ϕ x x x y y Γx = , Γ = (1 + xy), Γ = (1 xy), (2.2.77c) ϕ 1 + xy ϑϑ −2 ϑϑ −2 − x y y Γ ϕ = cotϑ, Γ x = (1 + xy)sin2 ϑ, Γ = (1 xy)sin2 ϑ, (2.2.77d) ϑϕ ϕϕ −2 ϕϕ −2 − Γ ϑ = sinϑ cosϑ. (2.2.77e) ϕϕ − Riemann-Tensor:

2 2 2 2 rs y rs rs Rxyxy = 4 , Rx x = 2 , Rx y = , (2.2.78a) − (1 + xy)3 ϑ ϑ − (1 + xy)2 ϑ ϑ −1 + xy 2 2 2 2 2 rs y sin ϑ rs sin ϑ 2 2 Rxϕxϕ = 2 , Rxϕyϕ = , R = (1 + xy)r sin ϑ. (2.2.78b) − (1 + xy)2 − 1 + xy ϑϕϑϕ s

The Ricci tensor as well as the Ricci scalar vanish identically. Hence, the Weyl tensor is identical to the Riemann tensor. The Kretschmann scalar reads 12 K = 4 6 . (2.2.79) rs (1 + xy)

Local tetrad: √1 + xy y √1 + xy e(0) = ∂x + ∂y, e(1) = ∂x, (2.2.80a) − 2rsy rs√1 + xy 2rsy 1 1 e(2) = ∂ϑ , e(3) = ∂ϕ . (2.2.80b) rs(1 + xy) rs(1 + xy)sinϑ

Dual tetrad:

rs√1 + xy 2rsy rs√1 + xy θ(0) = dy, θ(1) = dx + dy, (2.2.81a) y √1 + xy y (2) (3) θ = rs(1 + xy)dϑ, θ = rs(1 + xy)sinϑ dϕ. (2.2.81b) 2.3. ALCUBIERRE WARP 33

2.3 Alcubierre Warp

The Warp metric given by Miguel Alcubierre[Alc94] reads

2 2 2 2 2 2 ds = c dt + (dx vs f (rs)dt) + dy + dz (2.3.1) − − where dxs(t) v = , (2.3.2a) s dt q 2 2 2 rs(t) = (x xs(t)) + y + z , (2.3.2b) − tanh(σ(rs + R)) tanh(σ(rs R)) f (rs) = − − . (2.3.2c) 2tanh(σR) The parameter R > 0 deﬁnes the radius of the warp bubble and the parameter σ > 0 its thickness. Metric-Tensor: 2 2 2 gtt = c + v f (rs) , gtx = vs f (rs), gxx = gyy = gzz = 1. (2.3.3) − s − Christoffel symbols: 2 3 t f fxvs z 2 y 2 Γ = , Γ = f fzv , Γ = f fyv , (2.3.4a) tt c2 tt − s tt − s 3 4 2 2 2 2 2 3 f fxv c f fxv c ft vs f fxv f fxv Γ x = s − s − , Γ t = s , Γ x = s , (2.3.4b) tt c2 tx − c2 tx − c2 2 y fyvs fzvs f fyv Γ = , Γ z = , Γ t = s , (2.3.4c) tx 2 tx 2 ty − 2c2 2 3 2 2 2 3 2 f fyv + c fyvs f fzv f fzv + c fzvs Γ x = s , Γ t = s , Γ x = s , (2.3.4d) ty − 2c2 tz − 2c2 tz − 2c2 2 fxvs f fxv fyvs Γ t = , Γ x = s , Γ t = , (2.3.4e) xx c2 xx c2 xy 2c2 2 2 f fyv fzvs f fzv Γ x = s , Γ t = , Γ x = s , (2.3.4f) xy 2c2 xz 2c2 xz 2c2 with derivatives

d f (rs) vsσ (x xs(t)) h 2 2 i ft = = − − sech (σ(rs + R)) sech (σ(rs R)) (2.3.5a) dt 2rs tanh(σR) − −

d f (rs) σ (x xs(t)) h 2 2 i fx = = − sech (σ(rs + R)) sech (σ(rs R)) (2.3.5b) dx 2rs tanh(σR) − −

d f (rs) σy h 2 2 i fy = = sech (σ(rs + R)) sech (σ(rs R)) (2.3.5c) dy 2rs tanh(σR) − −

d f (rs) σz h 2 2 i fz = = sech (σ(rs + R)) sech (σ(rs R)) (2.3.5d) dz 2rs tanh(σR) − − Riemann- and Ricci-tensor as well as Ricci- and Kretschman-scalar are shown only in the Maple worksheet. Comoving local tetrad: 1 e = (∂ + v f ∂ ), e = ∂ , e = ∂ , e = ∂ . (2.3.6) (0) c t s x (1) x (2) y (3) z Static local tetrad: p 2 2 2 1 vs f c vs f e(0) = ∂t , e(1) = ∂t − ∂x, e(2) = ∂y, e(3) = ∂z. (2.3.7) pc2 v2 f 2 cpc2 v2 f 2 − c − s − s Further reading: Pfenning[PF97], Clark[CHL99], Van Den Broeck[Bro99] 34 CHAPTER 2. SPACETIMES

2.4 Barriola-Vilenkin monopol

The Barriola-Vilenkin metric describes the gravitational ﬁeld of a global monopole[BV89]. In spherical coordinates (t,r,ϑ,ϕ), the metric reads

ds2 = c2dt2 + dr2 + k2r2 dϑ 2 + sin2ϑ dϕ2, (2.4.1) −

where k is the scaling factor responsible for the deﬁcit/surplus angle. Christoffel symbols:

1 Γ r = k2r, Γ r = k2r sin2 ϑ, Γ ϑ = , (2.4.2a) ϑϑ − ϕϕ − rϑ r ϑ ϕ 1 ϕ Γ = sinϑ cosϑ, Γr = , Γ = cotϑ. (2.4.2b) ϕϕ − ϕ r ϑϕ Partial derivatives

ϑ 1 ϕ 1 r 2 Γ = , Γr r = , Γ = k , (2.4.3a) rϑ,r −r2 ϕ, −r2 ϑϑ,r − 1 Γ ϕ = , Γ r = k2 sin2 ϑ, Γ ϑ = cos(2ϑ), (2.4.3b) ϑϕ,ϑ −sin2 ϑ ϕϕ,r − ϕϕ,ϑ − Γ r = k2r sin(2ϑ). (2.4.3c) ϕϕ,ϑ − Riemann-Tensor:

R = (1 k2)k2r2 sin2 ϑ. (2.4.4) ϑϕϑϕ − Ricci tensor, Ricci and Kretschmann scalar:

2 2 2 2 2 2 1 k (1 k ) R = (1 k ), Rϕϕ = (1 k )sin ϑ, R = 2 − , K = 4 − . (2.4.5) ϑϑ − − k2r2 k4r4

Weyl-Tensor:

2 2 2 2 c (1 k ) c 2 c 2 2 Ctrtr = − , Ct t = (1 k ), Ctϕtϕ = (1 k )sin ϑ, (2.4.6a) − 3k2r2 ϑ ϑ 6 − 6 − 2 2 1 2 1 2 2 k r 2 2 Cr r = (1 k ), Crϕrϕ = (1 k )sin ϑ, C = (1 k )sin ϑ. (2.4.6b) ϑ ϑ −6 − −6 − ϑϕϑϕ 3 −

Local tetrad: 1 1 1 e = ∂t , e = ∂r, e = ∂ , e = ∂ . (2.4.7) (t) c (r) (ϑ) kr ϑ (ϕ) kr sinϑ ϕ Dual tetrad:

θ(t) = cdt, θ(r) = dr, θ(ϑ) = kr dϑ, θ(ϕ) = kr sinϑ dϕ. (2.4.8)

Ricci rotation coefﬁcients: 1 cotϑ γ = γ = , γ = . (2.4.9) (ϑ)(r)(ϑ) (ϕ)(r)(ϕ) r (ϕ)(ϑ)(ϕ) kr The contractions of the Ricci rotation coefﬁcients read 2 cotϑ γ = , γ = . (2.4.10) (r) r (ϑ) kr 2.4. BARRIOLA-VILENKIN MONOPOL 35

Riemann-Tensor with respect to local tetrad:

1 k2 R = − . (2.4.11) (ϑ)(ϕ)(ϑ)(ϕ) k2r2

Ricci-Tensor with respect to local tetrad:

1 k2 R = R = − . (2.4.12) (ϑ)(ϑ) (ϕ)(ϕ) k2r2

Weyl-Tensor with respect to local tetrad:

1 k2 C = C = − , (2.4.13a) (t)(r)(t)(r) − (ϑ)(ϕ)(ϑ)(ϕ) − 3k2r2 1 k2 C = C = C = C = − . (2.4.13b) (t)(ϑ)(t)(ϑ) (t)(ϕ)(t)(ϕ) − (r)(ϑ)(r)(ϑ) − (r)(ϕ)(r)(ϕ) 6k2r2

Embedding: The embedding function, see Sec. 1.7, for k < 1 reads p z = 1 k2 r. (2.4.14) − Euler-Lagrange: The Euler-Lagrangian formalism, Sec. 1.8.4, for geodesics in the ϑ = π/2 hyperplane yields

1 1 h2 1 h2 r˙2 +V = 1 , V = 2 κc2 , (2.4.15) 2 eff 2 c2 eff 2 k2r2 −

2 2 2 with the constants of motion h1 = c t˙ and h2 = k r ϕ˙.

The point of closest approach r for a null geodesic that starts at r = ri with y = e +cosξe +sinξe pca ± (t) (r) (ϕ) is given by r = ri sinξ. Hence, the rpca is independent of k. The same is also true for timelike geodesics.

Further reading: Barriola and Vilenkin[BV89], Perlick[Per04]. 36 CHAPTER 2. SPACETIMES

2.5 Bertotti-Kasner

The Bertotti-Kasner spacetime in spherical coordinates (t,r,ϑ,ϕ) reads[Rin98]

1 ds2 = c2dt2 + e2√Λct dr2 + dϑ 2 + sin2 ϑdϕ2, (2.5.1) − Λ

where the cosmological constant Λ must be positive. Christoffel symbols:

√Λ Γ r = c√Λ, Γ t = e2√Λct , Γ ϕ = cotϑ, Γ ϑ = sinϑ cosϑ. (2.5.2) tr rr c ϑϕ ϕϕ − Partial derivatives 1 Γ t = 2Λe2√Λct , Γ ϕ = , Γ ϑ = cos(2ϑ). (2.5.3) rr,t ϑϕ,ϑ −sin2 ϑ ϕϕ,ϑ −

Riemann-Tensor: 2 2 2√Λct sin ϑ Rtrtr = Λc e , R = . (2.5.4) − ϑϕϑϕ Λ

Ricci-Tensor:

2 2√Λct 2 Rtt = Λc , Rrr = Λe , R = 1, Rϕϕ = sin ϑ. (2.5.5) − ϑϑ The Ricci and Kretschmann scalars read

R = 4Λ, K = 8Λ 2. (2.5.6)

Weyl-Tensor:

2 2 2 2 2√Λct c c 2 Ctrtr = Λc e , Ct t = , Ctϕtϕ = sin ϑ, (2.5.7a) −3 ϑ ϑ 3 3 2 1 2√Λct 1 2√Λct 2 2 sin ϑ Cr r = e , Crϕrϕ = e sin ϑ, C = . (2.5.7b) ϑ ϑ −3 −3 ϑϕϑϕ 3 Λ

Local tetrad:

1 √Λct √Λ e = ∂t , e = e ∂r, e = √Λ∂ , e = ∂ . (2.5.8) (t) c (r) − (ϑ) ϑ (ϕ) sinϑ ϕ Dual tetrad: 1 sinϑ θ(t) = cdt, θ(r) = e√Λct dr, θ(ϑ) = dϑ, θ(ϕ) = dϕ. (2.5.9) √Λ √Λ

Ricci rotation coefﬁcients:

γ = √Λ, γ = √Λ cotϑ. (2.5.10) (t)(r)(r) − (ϑ)(ϕ)(ϕ) − The contractions of the Ricci rotation coefﬁcients read

γ(t) = √Λ, γ(ϑ) = √Λ cotϑ. (2.5.11)

Riemann-Tensor with respect to local tetrad:

R = R = Λ. (2.5.12) (t)(r)(t)(r) − (ϑ)(ϕ)(ϑ)(ϕ) − 2.5. BERTOTTI-KASNER 37

Ricci-Tensor with respect to local tetrad:

R = R = R = R = Λ. (2.5.13) (t)(t) − (r)(r) − (ϑ)(ϑ) − (ϕ)(ϕ) − Weyl-Tensor with respect to local tetrad:

2Λ C = C = , (2.5.14a) (t)(r)(t)(r) − (ϑ)(ϕ)(ϑ)(ϕ) − 3 Λ C = C = C = C = . (2.5.14b) (t)(ϑ)(t)(ϑ) (t)(ϕ)(t)(ϕ) − (r)(ϑ)(r)(ϑ) − (r)(ϕ)(r)(ϕ) 3

Euler-Lagrange: The Euler-Lagrangian formalism, Sec. 1.8.4, for geodesics in the ϑ = π/2 hyperplane yields

2 2 2 2√Λ ct 2 c t˙ = h e− +Λh κ (2.5.15) 1 2 − 2√Λ ct with the constants of motion h1 = re˙ and h2 = ϕ˙/Λ. Thus,

2 2√Λ ct 1 1 + q(t) 1 q(ti) h e λ = ln − , q(t) = 1 − + 1, (2.5.16) q 1 q t 1 q t 2 c√Λ Λh2 κ ( ) + ( i) Λh2 κ 2 − − − where ti is the initial time. We can also solve the orbital equation: q 2 2√Λ ct 2 h1e− +Λh2 κ r(t) = w(t) w(ti) + ri, w(t) = − , (2.5.17) − − h1√Λ where ri is the initial radial position.

Further reading: Rindler[Rin98]: “Every spherically symmetric solution of the generalized vacuum ﬁeld equations Ri j = Λgi j is either equivalent to Kottler’s generalization of Schwarzschild space or to the [...] Bertotti-Kasner space (for which Λ must be necessarily be positive).” 38 CHAPTER 2. SPACETIMES

2.6 Bessel gravitational wave

D. Kramer introduced in [Kra99] an exact gravitational wave solution of Einstein’s vacuum ﬁeld equations. According to [Ste03] we execute the substitution x t and y z. → → 2.6.1 Cylindrical coordinates The metric of the Bessel wave in cylindrical coordinates reads

2 2U 2K 2 2 2 2 2U 2 ds = e− e dρ dt + ρ dϕ + e dz . (2.6.1) −

The functions U and K are given by

U := CJ0 (ρ)cos(t), (2.6.2)

1 2 n h 2 2i 2 o K := C ρ ρ J0 (ρ) + J1 (ρ) 2J0 (ρ)J1 (ρ)cos (t) , (2.6.3) 2 − where Jn (ρ) are the Bessel functions of the ﬁrst kind. Christoffel symbols:

t ρ t ∂U ∂K ϕ z ∂U t 2K 2 ∂U Γ = Γ = Γ = + , Γ = Γ = , Γ = e− ρ , (2.6.4a) tt tρ ρρ − ∂t ∂t tϕ tz − ∂t ϕϕ − ∂t ρ t ρ ∂U ∂K ϕ 1 ∂U ρ 4U 2K ∂U Γ = Γ = Γ = + , Γ = , Γ = e − , (2.6.4b) tt tρ ρρ − ∂ρ ∂ρ ρϕ ρ − ∂ρ zz − ∂ρ ρ 2K ∂U z ∂U t 4U 2K ∂U Γ = ρe− ρ 1 , Γ = , Γ = e − . (2.6.4c) ϕϕ ∂ρ − ρz ∂ρ zz ∂t

Local tetrad:

U K U K 1 U U e = e ∂t , e = e ∂ , e = e ∂ , e = e ∂z. (2.6.5) (t) − (ρ) − ρ (ϕ) ρ ϕ (z) −

Dual tetrad:

(t) K U (ρ) K U (ϕ) U (z) U θ = e − dt, θ = e − dρ, θ = ρe− dϕ, θ = e dz. (2.6.6)

2.6.2 Cartesian coordinates p In Cartesian coordinates with ρ = x2 + y2 the metric (2.6.1) reads

2U 2 2(K U) 2 e− 2K 2 2 2 2K ds = e − dt + e x + y dx + 2xy e 1 dxdy − x2 + y2 − (2.6.7) + x2 + e2Ky2dy2 + e2U dz2.

Local tetrad: s 2 2 U K U x + y e = e − ∂t , e = e ∂x, (t) (x) e2Kx2 + y2 s (2.6.8) 2K 2 2 U K 2K U K e x + y e − e 1 U e(y) = e − 2 2 ∂y + xyp − ∂x, e(z) = e− ∂z x + y (x2 + y2)(e2Kx2 + y2) 2.7. COSMIC STRING IN SCHWARZSCHILD SPACETIME 39

2.7 Cosmic string in Schwarzschild spacetime

A cosmic string in the Schwarzschild spacetime represented by Schwarzschild coordinates (t,r,ϑ,ϕ) reads

rs 1 ds2 = 1 c2dt2 + dr2 + r2 dϑ 2 + β 2 sin2 ϑdϕ2, (2.7.1) − − r 1 rs/r −

2 where rs = 2GM/c is the Schwarzschild radius, G is Newton’s constant, c is the speed of light, M is the mass of the black hole, and β is the string parameter, compare Aryal et al[AFV86]. Christoffel symbols:

2 r c rs(r rs) t rs r rs Γtt = 3− , Γtr = , Γrr = , (2.7.2a) 2r 2r(r rs) −2r(r rs) − − ϑ 1 ϕ 1 r Γ = , Γr = , Γ = (r rs), (2.7.2b) rϑ r ϕ r ϑϑ − − ϕ r 2 2 ϑ 2 Γ = cotϑ, Γ = (r rs)β sin ϑ, Γ = β sinϑ cosϑ. (2.7.2c) ϑϕ ϕϕ − − ϕϕ − Partial derivatives

2 r (2r 3rs)c rs t (2r rs)rs r (2r rs)rs Γtt,r = − 4 , Γtr,r = 2 − 2 , Γrr,r = 2 − 2 , (2.7.3a) − 2r −2r (r rs) 2r (r rs) − − ϑ 1 ϕ 1 r Γ = , Γr r = , Γ = 1, (2.7.3b) rϑ,r −r2 ϕ, −r2 ϑϑ,r − 1 Γ ϕ = , Γ r = β 2 sin2 ϑ, Γ ϑ = β 2 cos(2ϑ), (2.7.3c) ϑϕ,ϑ −sin2 ϑ ϕϕ,r − ϕϕ,ϑ − r 2 Γ = (r rs)β sin(2ϑ). (2.7.3d) ϕϕ,ϑ − − Riemann-Tensor:

2 2 2 2 2 c rs 1 c (r rs)rs 1 c (r rs)rsβ sin ϑ Rtrtr = , Rt t = − , Rtϕtϕ = − , (2.7.4a) − r3 ϑ ϑ 2 r2 2 r2 2 2 1 rs 1 rsβ sin ϑ 2 2 Rrϑrϑ = , Rrϕrϕ = , Rϑϕϑϕ = rrsβ sin ϑ. (2.7.4b) −2 r rs −2 r rs − − The Ricci tensor as well as the Ricci scalar vanish identically. Hence, the Weyl tensor is identical to the Riemann tensor. The Kretschmann scalar reads r2 K = 12 s . (2.7.5) r6

Local tetrad: r 1 rs 1 1 e(t) = p ∂t , e(r) = 1 ∂r, e(ϑ) = ∂ϑ , e(ϕ) = ∂ϕ . (2.7.6) c 1 rs/r − r r rβ sinϑ − Dual tetrad: r (t) rs (r) dr (ϑ) (ϕ) θ = c 1 dt, θ = p , θ = r dϑ, θ = rβ sinϑ dϕ. (2.7.7) − r 1 rs/r − Ricci rotation coefﬁcients: r rs 1 rs cotϑ γ = , γ = γ = 1 , γ = . (2.7.8) (r)(t)(t) 2p (ϑ)(r)(ϑ) (ϕ)(r)(ϕ) (ϕ)(ϑ)(ϕ) 2r 1 rs/r r − r r − 40 CHAPTER 2. SPACETIMES

The contractions of the Ricci rotation coefﬁcients read

4r 3rs cotϑ γ = − , γ = . (2.7.9) (r) 2p (ϑ) 2r 1 rs/r r − Riemann-Tensor with respect to local tetrad:

rs R = R = , (2.7.10a) (t)(r)(t)(r) − (ϑ)(ϕ)(ϑ)(ϕ) −r3 rs R = R = R = R = . (2.7.10b) (t)(ϑ)(t)(ϑ) (t)(ϕ)(t)(ϕ) − (r)(ϑ)(r)(ϑ) − (r)(ϕ)(r)(ϕ) 2r3

Embedding: The embedding function for β 2 < 1 reads

r p 2 p 2 r 2 rs r/(r rs) β 1 β z = (r rs) β ln − − − − . (2.7.11) p 2 p 2 p 2 − r rs − − 2 1 β r/(r rs) β + 1 β − − − − − If β 2 = 1, we have the embedding function of the standard Schwarzschild metric, compare Eq.(2.2.15).

Euler-Lagrange: The Euler-Lagrangian formalism, Sec. 1.8.4, for geodesics in the ϑ = π/2 hyperplane yields

2 2 1 1 k 1 rs h r˙2 +V = , V = 1 κc2 (2.7.12) 2 eff 2 c2 eff 2 − r r2β 2 −

2 2 2 with the constants of motion k = (1 rs/r)c t˙ and h = r β ϕ˙. The maxima of the effective potential Veff −3 lead to the same critical orbits rpo = 2 rs and ritcg = 3rs as in the standard Schwarzschild metric. 2.8. EINSTEIN-ROSEN WAVE WITH WEBER-WHEELER-BONNOR PULSE 41

2.8 Einstein-Rosen wave with Weber-Wheeler-Bonnor pulse

The Einstein-Rosen wave in cylindrical coordinates (t,ρ,φ,z) is represented by the general line element [GM97]

2 2(γ ψ) 2 2 2 2ψ 2 2ψ 2 ds = e − dt + dρ + ρ e− dφ + e dz . (2.8.1) −

To be a vacuum spacetime, the potential functions γ = γ(t,ρ) and ψ = ψ(t,ρ) have to satisfy the constraint equations

2 2 " 2 2# ∂ ψ 1 ∂ψ ∂ ψ ∂γ ∂ψ ∂ψ ∂γ ∂ψ ∂ψ + ρ− = 0, = ρ + , = 2ρ . (2.8.2) ∂ρ2 ∂ρ − ∂t2 ∂ρ ∂ρ ∂t ∂t ∂ρ ∂t

A Weber-Wheeler-Bonnor pulse is realized for

sp (a2 + ρ2 t2)2 + 4a2t2 + a2 + ρ2 t2 ψ = √2c − − , (2.8.3a) (a2 + ρ2 t2)2 + 4a2t2 − ! c2 2a2ρ2 (a2 + ρ2 t2)2 4a2t2 ρ2 a2 t2 γ = 2 1 − − 2 + p − − . (2.8.3b) 2a − [(a2 + ρ2 t2)2 + 4a2t2] (a2 + ρ2 t2)2 + 4a2t2 − − Christoffel symbols:

t ρ t Γ = ∂t γ ∂t ψ, Γ = ∂ρ γ ∂ρ ψ, Γ = ∂ρ γ ∂ρ ψ, (2.8.4a) tt − tt − tρ − ρ φ z Γ = ∂t γ ∂t ψ, Γ = ∂t ψ, Γ = ∂t ψ, (2.8.4b) tρ − tφ − tz t ρ φ 1 ρ∂ρ ψ Γ = ∂t γ ∂t ψ, Γ = ∂ρ γ ∂ρ ψ, Γ = − , (2.8.4c) ρρ − ρρ − ρφ ρ z t 2 2γ ρ 2γ Γ = ∂ρ ψ, Γ = ρ e− ∂t ψ, Γ = ρe− 1 ρ∂ρ ψ , (2.8.4d) φz φφ − φφ − − t 4ψ 2γ ρ 4ρ 2γ Γ = e − ∂t ψ, Γ = e − ∂ρ ψ. (2.8.4e) zz zz − Local tetrad:

ψ γ ψ γ 1 ψ ψ e(t) = e − ∂t , e(ρ) = e − ∂ρ , e(φ) = ρ− e ∂φ , e(z) = e− ∂z. (2.8.5)

Dual tetrad:

(t) γ ψ (ρ) γ ψ (φ) ψ (z) ψ θ = e − dt, θ = e − dρ, θ = ρe− dφ, θ = e dz. (2.8.6) 42 CHAPTER 2. SPACETIMES

2.9 Ernst spacetime

“The Ernst metric is a static, axially symmetric, electro-vacuum solution of the Einstein-Maxwell equations with a black hole immersed in a magnetic ﬁeld.”[KV92] In spherical coordinates (t,r,ϑ,ϕ), the Ernst metric reads[Ern76] (G = c = 1)

2M dr2 r2 sin2 ϑ ds2 = Λ 2 1 dt2 + + r2 dϑ 2 + dϕ2, (2.9.1) − − r 1 2M/r Λ 2 −

where Λ = 1 + B2r2 sin2 ϑ. Here, M is the mass of the black hole and B the magnetic ﬁeld strength. Christoffel symbols:

2 2 2B2r3 sin ϑ 3MB2r2 sin ϑ + M (r 2M) 2(r 2M)B2 sinϑ cosϑ Γ r = − − , Γ ϑ = − , (2.9.2a) tt r3Λ tt rΛ 2B2r3 sin2 ϑ 3MB2r2 sin2 ϑ + M 2B2r2 sinϑ cosϑ Γ t = − , Γ t = , (2.9.2b) tr r (r 2M)Λ tϑ Λ − 2B2r3 sin2 ϑ 5MB2r2 sin2 ϑ M 2B2r sinϑ cosϑ Γ r = − − , Γ ϑ = , (2.9.2c) rr r (r 2M)Λ rr − (r 2M)Λ − − 2B2r2 sinϑ cosϑ 3B2r2 sin2 ϑ + 1 Γ r = , Γ ϑ = , (2.9.2d) rϑ Λ rϑ rΛ 2 1 B2r2 sin2 ϑ 3B2r2 sin ϑ + 1 (r 2M) Γ ϕ = − , Γ r = − , (2.9.2e) rϕ rΛ ϑϑ Λ 2B2r2 sinϑ cosϑ Ξ cosϑ Γ ϑ = , Γ ϕ = , (2.9.2f) ϑϑ Λ ϑϕ Λ (r 2M)Ξ sin2 ϑ Γ r = − , (2.9.2g) ϕϕ Λ 5 Ξ sinϑ cosϑ Γ ϑ = . (2.9.2h) ϕϕ Λ 5

with Ξ = 1 B2r2 sin2 ϑ. − Riemann-Tensor:

2 h 4 4 4 5 4 2 2 2 2 2 i Rtrtr = B r sin ϑ (3M r) M + 2r B sin ϑ cos ϑ + B r sin ϑ (r 2M) , (2.9.3a) r3 − − − 2 2 2 2 Rtrt = 2B sinϑ cosϑ (3B r sin ϑ (2M 3r) + r 2M , (2.9.3b) ϑ − − 1 4 4 4 2 3 2 Rt t = B r (r 2M)(4r 9M)sin ϑ + 2ΞB r (r 2M)cos ϑ + M(r 2M) , (2.9.3c) ϑ ϑ r2 − − − − 1 2 3 2 2 2 2 Rtϕtϕ = (2B r 3B Mr sin ϑ + M)Ξ(r 2M)sin ϑ , (2.9.3d) Λ 4r2 − − (2B2r3 3B2Mr2 sin2 ϑ + M)Ξ Rr r = − , (2.9.3e) ϑ ϑ − r 2M 2 − sin ϑ 4 4 4 2 2 2 2 3 2 Rrϕrϕ = B r (4r 9M)sin ϑ + 2B r (8M 4rϑ)sin ϑ + 2ΞB r cos ϑ + M , (2.9.3f) −Λ 4(r 2M) − − − 2B2r3 sin3 ϑ cosϑ 3B2r2 sin2 ϑ 5 Rr = − , (2.9.3g) ϕϑϕ − Λ 4 r sin2 ϑ R = 2B4r4(r 3M)sin4 ϑ + 4B2r3 cos2 ϑ(1 + Ξ) + 2B2r2 sin2 ϑ(2M r) + 2M. (2.9.3h) ϑϕϑϕ Λ 4 − − 2.9. ERNST SPACETIME 43

Ricci-Tensor: 4B2(r 2M)(r + 2M sin2 ϑ) 4B2[r cos2 ϑ (r 2M)sin2 ϑ] Rtt = − , Rrr = − − , (2.9.4a) r2Λ 2 − (r 2M)Λ 2 − 2 8B2r sinϑ cosϑ 4B2r r cos2 ϑ + (r 2M)sin ϑ R = , R = − , (2.9.4b) rϑ Λ 2 ϑϑ Λ 2 4B2r sin2 ϑ r + 2M sin2 ϑ R = . (2.9.4c) ϕϕ Λ 6 Ricci and Kretschmann scalars:

R = 0, (2.9.5a) 16 8 8 2 2 8 K = 3B r 4r 18Mr + 21M sin ϑ r6Λ 8 − + 2B4r4 31M2 37Mr 24B2r4 cos2 ϑ + 42B2Mr3 cos2 ϑ + 10r2 + 6B4r6 cos4 ϑ sin6 ϑ − − + 2B2r2 3Mr + 20B2r4 cos2 ϑ + 6M2 46B2Mr3 cos2 ϑ 12B4r6 cos4 ϑ sin4 ϑ − − − 6B6r6 6B2Mr3 cos2 ϑ + 4r2 4B2r4 cos2 ϑ + 18M2 17Mr − − − + 20B4r6 cos4 ϑ + 12B2Mr3 cos2 ϑ + 3M2 . (2.9.5b)

Static local tetrad: p 1 1 2m/r 1 Λ e(t) = p ∂t , e(r) = − ∂r, e(ϑ) = ∂ϑ , e(ϕ) = ∂ϕ . (2.9.6) Λ 1 2m/r Λ Λr r sinϑ − Dual tetrad: r (t) 2m (r) Λ (ϑ) (ϕ) r sinϑ θ = Λ 1 dt, θ = p dr, θ = Λr dϑ, θ = dϕ. (2.9.7) − r 1 2m/r Λ − Euler-Lagrange: The Euler-Lagrangian formalism, Sec. 1.8.4, for geodesics in the ϑ = π/2 hyperplane yields

2 2 h (1 rs/r) k 1 rs/r r˙2 + − + κ − = 0 (2.9.8) r2 − Λ 4 Λ 2

2 2 2 with constants of motion k = Λ (1 rs/r)t˙ and h = (r /Λ )ϕ˙. −

Further reading: Ernst[Ern76], Dhurandhar and Sharma[DS83], Karas and Vokrouhlicky[KV92], Stuchlík and Hledík[SH99]. 44 CHAPTER 2. SPACETIMES

2.10 Extreme Reissner-Nordstrøm dihole

The extreme Reissner-Nordstrøm (RN) dihole metric is a special case of the Majumdar-Papapetrou spacetimes (see 2.19.1) for N = 2. The two black holes have the masses M1 and M2 and are located at the T T + positions r1 = (0,0,+1) and r2 = (0,0, 1) . In cylindrical coordinates {t R,ρ R ,ϕ [0,2π),z R}, − ∈ ∈ ∈ ∈ the extreme RN dihole metric reads

c2dt2 ds2 = +U2(dρ2 + ρ2dϕ2 + dz2), (2.10.1) − U2

where

2 2 GM1/c GM2/c U(ρ,z) = 1 + p + p . (2.10.2) ρ2 + (z 1)2 ρ2 + (z + 1)2 −

The coordinate singularities (ρ = 0,z = 1) are the degenerated horizons of the two extreme RN black ± holes.

Derivations of U(ρ,z):

GM1 ρ GM2 ρ ∂ρU = · · , (2.10.3a) −c2 [ρ2 + (z 1)2]3/2 − c2 [ρ2 + (z + 1)2]3/2 − GM1 (z 1) GM2 (z + 1) ∂zU = · − · , (2.10.3b) −c2 [ρ2 + (z 1)2]3/2 − c2 [ρ2 + (z + 1)2]3/2 2 − 2 2 2 2 GM1 2ρ (z 1) GM2 2ρ (z + 1) ∂ρU = · − − + · − , (2.10.3c) c2 [ρ2 + (z 1)2]5/2 c2 [ρ2 + (z + 1)2]5/2 − 2 2 2 2 2 GM1 2(z 1) ρ GM2 2(z + 1) ρ ∂z U = · − − + · − , (2.10.3d) c2 [ρ2 + (z 1)2]5/2 c2 [ρ2 + (z + 1)2]5/2 − 3GM1 ρ(z 1) 3GM2 ρ(z + 1) ∂ρ ∂zU = · − + · . (2.10.3e) c2 [ρ2 + (z 1)2]5/2 c2 [ρ2 + (z + 1)2]5/2 −

1 2 2 The function U(ρ,z) fulﬁlls the Laplace-Equation ∆U = ρ ∂ρU +∂ρU +∂z U = 0, which will be used in the calculation of the following geometric quantities. (Note, that U is independent of ϕ.)

Christoffel symbols:

2 2 c ∂ρU c ∂zU ∂ρU ∂ρU Γ ρ = , Γ z = , Γ t = , Γ ρ = , (2.10.4a) tt − U5 tt − U5 tρ − U ρρ U 2 2 ∂zU 1 ∂ρU ρ ∂ρU ρ ∂zU Γ z = , Γ ϕ = + , Γ ρ = ρ , Γ z = , (2.10.4b) ρρ − U ρϕ ρ U ϕϕ − − U ϕϕ − U

t ∂zU ρ ∂zU z ∂ρU ϕ ∂zU Γ = , Γ z = , Γ = , Γ z = , (2.10.4c) tz − U ρ U ρz U ϕ U ∂ρU ∂zU Γ ρ = , Γ z = . (2.10.4d) zz − U zz U 2.10. EXTREME REISSNER-NORDSTRØM DIHOLE 45

Riemann-Tensor: 2 c h 2 2 2i 2 2 U Rtρtρ = 3(∂ρU) U∂ U (∂zU) , Rρzρz = (∂ρU) + (∂zU) + ∂ρU, (2.10.5a) U4 − ρ − ρ 2 2 c ρ h 2 2 U i 2 Rtϕtϕ = (∂ρU) + (∂zU) + ∂ρU , Rρϕϕz = ρ U∂ρ ∂zU 2∂ρU∂zU , (2.10.5b) − U4 ρ − 2 c h 2 2 2i 2 Rtz tz = 3(∂zU) U∂ U (∂ρU) , Rϕzρϕ = ρ U∂ρ ∂zU 2∂ρU∂zU , (2.10.5c) U4 − z − − 2 c 2h 2 2 2 i Rtρtz = 4∂ρU∂zU U∂ρ ∂zU , Rρϕρϕ = ρ (∂ρU) (∂zU) +U∂ U , (2.10.5d) U4 − − z 2 c 2h 2 2 2 i Rtz tρ = 4∂ρU∂zU U∂ρ ∂zU , Rϕzϕz = ρ (∂zU) (∂ρU) +U∂ U . (2.10.5e) U4 − − ρ Ricci-Tensor: 2 c 2 2 1 2 2 2∂ρU∂zU Rtt = (∂ρU) + (∂zU) , Rρρ = (∂zU) (∂ρU) , Rρz = , (2.10.6a) U6 U2 − − U2 2 ρ 2 2 1 2 2 Rϕϕ = (∂ρU) + (∂zU) , Rzz = (∂ρU) (∂zU) . (2.10.6b) U2 U2 − The Ricci scalar vanishes identically, also because the energy-momentum tensor of the electromagnetic ﬁeld is traceless. The Kretschmann scalar reads 4 2 4 2 4 2 K = 14ρ (∂zU) + 14ρ (∂ρU) 24ρ U∂zU∂ρU∂ρ ∂zU ρ2U8 − 2 2 2 2 2 2 2 12ρ U(∂ρU) ∂ U + 4ρ (∂zU) 7(∂ρU) 3U∂ U (2.10.7) − ρ − z 2h 2 2 2 2 2 2 2 2 2i +U ρ (∂z U) + 3(∂ρU) + 2ρ∂ρU∂ρU + ρ 4(∂ρ ∂zU) + 3(∂ρU)

Weyl-Tensor: 2 c h 2 2 2i 2 2 U Ctρtρ = 2(∂ρU) U∂ U (∂zU) , Cρzρz = (∂ρU) + (∂zU) + ∂ρU, (2.10.8a) U4 − ρ − ρ 2 2 c ρ h 2 2 U i 2 Ctϕtϕ = (∂ρU) + (∂zU) + ∂ρU , Cρϕϕz = ρ U∂ρ ∂zU 3∂ρU∂zU , (2.10.8b) − U4 ρ − 2 c h 2 2 2i 2 Ctz tz = 2(∂zU) U∂ U (∂ρU) , Cϕzρϕ = ρ U∂ρ ∂zU 3∂ρU∂zU , (2.10.8c) U4 − z − − 2 c 2h 2 2 2 i Ctρtz = 3∂ρU∂zU U∂ρ ∂zU , Cρϕρϕ = ρ (∂ρU) 2(∂zU) +U∂ U , (2.10.8d) U4 − − z 2 c 2h 2 2 2 i Ctz tρ = 3∂ρU∂zU U∂ρ ∂zU , Cϕzϕz = ρ (∂zU) 2(∂ρU) +U∂ U . (2.10.8e) U4 − − ρ Local tetrad: U 1 1 1 e = ∂t , e = ∂ , e = ∂ , e = ∂z. (2.10.9) (t) c (ρ) U ρ (ϕ) ρU ϕ (z) U Dual tetrad: c θ(t) = dt, θ(ρ) = Udρ, θ(ϕ) = ρUdϕ, θ(z) = Udz. (2.10.10) U Ricci rotation coefﬁcients:

∂zU 1 ∂ρU γ = γ = γ = , γ = + , (2.10.11a) (t)(z)(t) (ρ)(z)(ρ) (ϕ)(z)(ϕ) U2 (ϕ)(ρ)(ϕ) ρU U2 ∂ρU γ = γ = . (2.10.11b) (t)(ρ)(t) (z)(ρ)(z) U2 46 CHAPTER 2. SPACETIMES

The contractions of the Ricci rotation coefﬁcients read

∂ρU ∂zU γ = , γ = . (2.10.12) (ρ) U2 (z) U2

Riemann-Tensor with respect to local tetrad:

2 2 2 2 2 U 3(∂ρU) U∂ρU (∂zU) (∂ρU) + (∂zU) + ρ ∂ρU R = − − , R = , (2.10.13a) (t)(ρ)(t)(ρ) U4 (ρ)(z)(ρ)(z) U4 2 2 U (∂ρU) + (∂zU) + ρ ∂ρU U∂ρ ∂zU 2∂ρU∂zU R = , R = − , (2.10.13b) (t)(ϕ)(t)(ϕ) − U4 (ρ)(ϕ)(ϕ)(z) U4 2 2 2 3(∂zU) U∂ U (∂ρU) (U∂ρ ∂zU 2∂ρU∂zU R = − z − , R = − , (2.10.13c) (t)(z)(t)(z) U4 (ϕ)(z)(ρ)(ϕ) U4 2 2 2 4∂ρU∂zU U∂ρ ∂zU (∂ρU) (∂zU) +U∂ U R = − , R = − z , (2.10.13d) (t)(ρ)(t)(z) U4 (ρ)(ϕ)(ρ)(ϕ) U4 2 2 2 4∂ρU∂zU U∂ρ ∂zU (∂zU) (∂ρU) +U∂ρU R = − , R = − . (2.10.13e) (t)(z)(t)(ρ) U4 (ϕ)(z)(ϕ)(z) U4

Ricci-Tensor with respect to local tetrad:

2 2 2 2 (∂ρU) + (∂zU) (∂zU) (∂ρU) 2∂ρU∂zU R = , R = − , R = , (2.10.14a) (t)(t) U4 (ρ)(ρ) U4 (ρ)(z) − U4 2 2 2 2 (∂ρU) + (∂zU) (∂ρU) (∂zU) R = , R = − . (2.10.14b) (ϕ)(ϕ) U4 (z)(z) U4

Weyl-Tensor with respect to local tetrad:

2 2 2 2 2 U 2(∂ρU) U∂ρU (∂zU) (∂ρU) + (∂zU) + ρ ∂ρU C = − − , C = , (2.10.15a) (t)(ρ)(t)(ρ) U4 (ρ)(z)(ρ)(z) U4 2 2 U (∂ρU) + (∂zU) + ρ ∂ρU U∂ρ ∂zU 3∂ρU∂zU C = , C = − , (2.10.15b) (t)(ϕ)(t)(ϕ) − U4 (ρ)(ϕ)(ϕ)(z) U4 2 2 2 2(∂zU) U∂ U (∂ρU) U∂ρ ∂zU 3∂ρU∂zU C = − z − , C = − , (2.10.15c) (t)(z)(t)(z) U4 (ϕ)(z)(ρ)(ϕ) U4 2 2 2 3∂ρU∂zU U∂ρ ∂zU (∂ρU) 2(∂zU) +U∂ U C = − , C = − z , (2.10.15d) (t)(ρ)(t)(z) U4 (ρ)(ϕ)(ρ)(ϕ) U4 2 2 2 3∂ρU∂zU U∂ρ ∂zU (∂zU) 2(∂ρU) +U∂ρU C = − , C = − . (2.10.15e) (t)(z)(t)(ρ) U4 (ϕ)(z)(ϕ)(z) U4

Euler-Lagrange: The Euler-Lagrangian formalism, Sec. 1.8.4, yields

1 1 1 k2 1 L2 κc2 ρ˙ 2 + z˙2 +V (ρ,z) = , V (ρ,z) = z (2.10.16) 2 2 eff 2 c2 eff 2 ρ2U4 − U2

2 2 2 2 with constants of motion k = c t˙/U and Lz = ρ U ϕ˙. The quantity Lz is the angular momentum of a test particle with respect to the z-axis and k can be considered as a parameter for its energy. It is κ = 1,0 for − timelike or lightlike geodesics.

Further reading: Chandrasekhar[Cha89, Cha06], Hartle[HH72], Yurtsever[Yur95], Wünsch[WMW13], 2.11. FRIEDMAN-ROBERTSON-WALKER 47

2.11 Friedman-Robertson-Walker

The Friedman-Robertson-Walker metric describes a general homogeneous and isotropic universe. In a general form it reads:

ds2 = c2dt2 + R2dσ 2 (2.11.1) − with R = R(t) being an arbitrary function of time only and dσ 2 being a metric of a 3-space of constant curvature for which three explicit forms will be described here. In all formulas in this section a dot denotes differentiation with respect to t, e.g. R˙ = dR(t)/dt.

2.11.1 Form 1

dη2 ds2 = c2dt2 + R2 + η2 dϑ 2 + sin2 ϑdϕ2 (2.11.2) − 1 kη2 −

Christoffel symbols:

R˙ R˙ R˙ Γ η = , Γ ϑ = , Γ ϕ = , (2.11.3a) tη R tϑ R tϕ R RR˙ kη 1 Γ t = , Γ η = , Γ ϑ = , (2.11.3b) ηη c2(1 kη2) ηη 1 kη2 ηϑ η − − 1 Rη2R˙ Γ ϕ = , Γ t = , Γ η = (kη2 1)η, (2.11.3c) ηϕ η ϑϑ c2 ϑϑ − Rη2 sin2 ϑR˙ Γ ϕ = cotϑ, Γ t = , Γ η = (kη2 1)η sin2 ϑ, (2.11.3d) ϑϕ ϕϕ c2 ϕϕ − Γ ϑ = sinϑ cosϑ. (2.11.3e) ϕϕ − Riemann-Tensor: ¨ RR 2 Rtηtη = , Rt t = Rη R¨, (2.11.4a) kη2 1 ϑ ϑ − − 2 2 ˙2 2 2 2 R η R + kc Rtϕtϕ = Rη sin ϑR¨, R = , (2.11.4b) − ηϑηϑ − c2(kη2 1) − R2η2 sin2 ϑ R˙2 + kc2 R2η4 sin2 ϑ R˙2 + kc2 Rηϕηϕ = , R = . (2.11.4c) − c2(kη2 1) ϑϕϑϕ c2 − Ricci-Tensor: R¨ RR¨ + 2(R˙2 + kc2) Rtt = 3 , Rηη = , (2.11.5a) − R c2(1 kη2) − RR¨ + 2(R˙2 + kc2) RR¨ + 2(R˙2 + kc2) R = η2 , R = η2 sin2 ϑ . (2.11.5b) ϑϑ c2 ϕϕ c2 The Ricci scalar and Kretschmann scalar read:

RR¨ + R˙2 + kc2 R¨2R2 + R˙4 + 2R˙2kc2 + k2c4 R = 6 , K = 12 . (2.11.6) R2c2 R4c4

Local tetrad: p 1 1 kη2 1 1 e = ∂t , e = − ∂ , e = ∂ , e = ∂ . (2.11.7) (t) c (η) R η ϑ Rη ϑ ϕ Rη sinϑ ϕ 48 CHAPTER 2. SPACETIMES

Ricci rotation coefﬁcients: p R˙ 1 kη2 γ = γ = γ = γ = γ = − , (η)(t)(η) (ϑ)(t)(ϑ) (ϕ)(t)(ϕ) Rc (ϑ)(η)(ϑ) (ϕ)(η)(ϕ) Rη (2.11.8) cotϑ γ = . (ϕ)(ϑ)(ϕ) Rη The contractions of the Ricci rotation coefﬁcients read p 3R˙ 2 1 kη2 cotϑ γ = , γ = − , γ = . (2.11.9) (t) Rc (r) Rη (ϑ) Rη Riemann-Tensor with respect to local tetrad: R¨ R = R = R = (2.11.10a) (t)(η)(t)(η) (t)(ϑ)(t)(ϑ) (t)(ϕ)(t)(ϕ) −Rc2 R˙2 + kc2 R = R = R = . (2.11.10b) (η)(ϑ)(η)(ϑ) (η)(ϕ)(η)(ϕ) (ϑ)(ϕ)(ϑ)(ϕ) R2c2 Ricci-Tensor with respect to local tetrad: 3R¨ RR¨ + 2R˙2 + 2kc2 R = , R = R = R = . (2.11.11) (t)(t) −Rc2 (r)(r) (ϑ)(ϑ) (ϕ)(ϕ) R2c2

2.11.2 Form 2

R2 ds2 = c2dt2 + dr2 + r2(dϑ 2 + sin2 ϑdϕ2) (2.11.12) k 2 2 − (1 + 4 r )

Christoffel symbols: R˙ R˙ R˙ Γ r = , Γ ϑ = , Γ ϕ = , (2.11.13a) tr R tϑ R tϕ R RR˙ 2kr 4 kr2 Γ t = 16 , Γ r = , Γ ϑ = − , (2.11.13b) rr c2(4 + kr2)2 rr −4 + kr2 rϑ (4 + kr2)r 4 kr2 Rr2R˙ r(kr2 4) Γ ϕ = − , Γ t = 16 , Γ r = − , (2.11.13c) rϕ (4 + kr2)r ϑϑ c2(4 + kr2)2 ϑϑ 4 + kr2 Rr2 sin2 ϑR˙ Γ ϕ = cotϑ, Γ t = 16 , Γ ϑ = sinϑ cosϑ, (2.11.13d) ϑϕ ϕϕ c2(4 + kr2)2 ϕϕ − r sin2 ϑ(kr2 4) Γ r = − . (2.11.13e) ϕϕ 4 + kr2 Riemann-Tensor: RR¨ Rr2R¨ Rtrtr = 16 , Rt t = 16 , (2.11.14a) − (4 + kr2)2 ϑ ϑ − (4 + kr2)2 Rr2 sin2 ϑR¨ R2r2 R˙2 + kc2 Rtϕtϕ = 16 , Rr r = 256 , (2.11.14b) − (4 + kr2)2 ϑ ϑ c2(4 + kr2)4 R2r2 sin2 ϑ R˙2 + kc2 R2r4 sin2 ϑ R˙2 + kc2 Rr r = 256 , R = 256 . (2.11.14c) ϕ ϕ c2(4 + kr2)4 ϑϕϑϕ c2(4 + kr2)4 Ricci-Tensor: R¨ RR¨ + 2(R˙2 + kc2) Rtt = 3 , Rrr = 16 , (2.11.15a) − R c2(4 + kr2)2 RR¨ + 2(R˙2 + kc2) RR¨ + 2(R˙2 + kc2) R = 16r2 , R = 16r2 sin2 ϑ . (2.11.15b) ϑϑ c2(4 + kr2)2 ϕϕ c2(4 + kr2)2 2.11. FRIEDMAN-ROBERTSON-WALKER 49

The Ricci scalar and Kretschmann scalar read:

RR¨ + R˙2 + kc2 R¨2R2 + R˙4 + 2R˙2kc2 + k2c4 R = 6 , K = 12 . (2.11.16) R2c2 R4c4

Local tetrad:

k 2 k 2 k 2 1 1 + 4 r 1 + 4 r 1 + /4r e = ∂t , e = ∂r, e = ∂ , e = ∂ . (2.11.17) (t) c (r) R ϑ Rr ϑ ϕ Rr sinϑ ϕ

Ricci rotation coefﬁcients:

R˙ k r2 1 γ = γ = γ = γ = γ = 4 − , (2.11.18a) (r)(t)(r) (ϑ)(t)(ϑ) (ϕ)(t)(ϕ) Rc (ϑ)(r)(ϑ) (ϕ)(r)(ϕ) − Rr ( k r2 + 1)cotϑ γ = 4 . (2.11.18b) (ϕ)(ϑ)(ϕ) Rr

The contractions of the Ricci rotation coefﬁcients read

3R˙ 1 k r2 ( k r2 + 1)cotϑ γ = , γ = 2 − 4 , γ = 4 . (2.11.19) (t) Rc (r) Rr (ϑ) Rr

Riemann-Tensor with respect to local tetrad:

R¨ R = R = R = (2.11.20a) (t)(η)(t)(η) (t)(ϑ)(t)(ϑ) (t)(ϕ)(t)(ϕ) −Rc2 R˙2 + kc2 R = R = R = . (2.11.20b) (η)(ϑ)(η)(ϑ) (η)(ϕ)(η)(ϕ) (ϑ)(ϕ)(ϑ)(ϕ) R2c2

Ricci-Tensor with respect to local tetrad:

3R¨ RR¨ + 2R˙2 + 2kc2 R = , R = R = R = . (2.11.21) (t)(t) −Rc2 (r)(r) (ϑ)(ϑ) (ϕ)(ϕ) R2c2

2.11.3 Form 3 The following forms of the metric are obtained from 2.11.2 by setting η = sinψ,ψ,sinhψ for k = 1,0, 1 − respectively.

Positive Curvature

ds2 = c2dt2 + R2 dψ2 + sin2 ψ dϑ 2 + sin2 ϑdϕ2 (2.11.22) −

Christoffel symbols:

R˙ R˙ R˙ Γ ψ = , Γ ϑ = , Γ ϕ = , (2.11.23a) tψ R tϑ R tϕ R RR˙ Γ t = , Γ ϑ = cotψ, Γ ϕ = cotψ, (2.11.23b) ψψ c2 ψϑ ψϕ Rsin2 ψR˙ Γ t = , Γ ψ = sinψ cosψ, Γ ϕ = cot(ϑ), (2.11.23c) ϑϑ c2 ϑϑ − ϑϕ Rsin2 ψ sin2 ϑR˙ Γ t = , Γ ψ = sinψ cosψ sin2 ϑ, Γ ϑ = sinϑ cosϑ. (2.11.23d) ϕϕ c2 ϕϕ − ϕϕ − 50 CHAPTER 2. SPACETIMES

Riemann-Tensor: 2 Rtψtψ = RR¨, Rt t = Rsin ψR¨, (2.11.24a) − ϑ ϑ − 2 2 ˙2 2 2 2 R sin ψ R + c Rtϕtϕ = Rsin ψ sin ϑR¨, R = , (2.11.24b) − ψϑψϑ c2 R2 sin2 ψ sin2 ϑ R˙2 + c2 R2 sin4 ψ sin2 ϑ R˙2 + c2 R = , R = . (2.11.24c) ψϕψϕ c2 ϑϕϑϕ c2 Ricci-Tensor: R¨ RR¨ + 2(R˙2 + c2) Rtt = 3 , Rψψ = , (2.11.25a) − R c2 RR¨ + 2(R˙2 + c2) RR¨ + 2(R˙2 + c2) R = sin2 ψ , R = sin2 ϑ sin2 ψ . (2.11.25b) ϑϑ c2 ϕϕ c2 The Ricci scalar and Kretschmann read RR¨ + R˙2 + c2 R¨2R2 + R˙4 + 2R˙2c2 + c4 R = 6 , K = 12 . (2.11.26) R2c2 R4c4 Local tetrad: 1 1 1 1 e = ∂t , e = ∂ , e = ∂ , e = ∂ . (2.11.27) (t) c (ψ) R ψ ϑ Rsinψ ϑ ϕ Rsinψ sinϑ ϕ Ricci rotation coefﬁcients: R˙ cotψ γ = γ = γ = γ = γ = , (2.11.28a) (ψ)(t)(ψ) (ϑ)(t)(ϑ) (ϕ)(t)(ϕ) Rc (ϑ)(ψ)(ϑ) (ϕ)(ψ)(ϕ) R cotθ γ = . (2.11.28b) (ϕ)(ϑ)(ϕ) Rsinψ The contractions of the Ricci rotation coefﬁcients read 3R˙ cotψ cotϑ γ = , γ = 2 , γ = . (2.11.29) (t) Rc (r) R (ϑ) Rsinψ Riemann-Tensor with respect to local tetrad: R¨ R = R = R = , (2.11.30a) (t)(ψ)(t)(ψ) (t)(ϑ)(t)(ϑ) (t)(ϕ)(t)(ϕ) −Rc2 R˙2 + c2 R = R = R = . (2.11.30b) (ψ)(ϑ)(ψ)(ϑ) (ψ)(ϕ)(ψ)(ϕ) (ϑ)(ϕ)(ϑ)(ϕ) R2c2 Ricci-Tensor with respect to local tetrad: 3R¨ RR¨ + 2(R˙2 + c2) R = , R = R = R = . (2.11.31) (t)(t) −Rc2 (ψ)(ψ) (ϑ)(ϑ) (ϕ)(ϕ) R2c2

Vanishing Curvature

ds2 = c2dt2 + R2 dψ2 + ψ2 dϑ 2 + sin2 ϑdϕ2 (2.11.32) − Christoffel symbols: R˙ R˙ R˙ Γ ψ = , Γ ϑ = , Γ ϕ = , (2.11.33a) tψ R tϑ R tϕ R RR˙ 1 1 Γ t = , Γ ϑ = , Γ ϕ = , (2.11.33b) ψψ c2 ψϑ ψ ψϕ ψ Rψ2R˙ Γ t = , Γ ψ = ψ, Γ ϕ = cot(ϑ), (2.11.33c) ϑϑ c2 ϑϑ − ϑϕ Rψ2 sin2 ϑR˙ Γ t = , Γ ψ = ψ sin2 ϑ, Γ ϑ = sinϑ cosϑ. (2.11.33d) ϕϕ c2 ϕϕ − ϕϕ − 2.11. FRIEDMAN-ROBERTSON-WALKER 51

Riemann-Tensor: 2 Rtψtψ = RR¨, Rt t = Rψ R¨, (2.11.34a) − ϑ ϑ − 2 2 ˙2 2 2 R ψ R Rtϕtϕ = Rψ sin ϑR¨, R = , (2.11.34b) − ψϑψϑ c2 R2ψ2 sin2 ϑR˙2 R2ψ4 sin2 ϑR˙2 R = , R = . (2.11.34c) ψϕψϕ c2 ϑϕϑϕ c2 Ricci-Tensor: R¨ RR¨ + 2R˙2 Rtt = 3 , Rψψ = , (2.11.35a) − R c2 RR¨ + 2R˙2 RR¨ + 2R˙2 R = ψ2 , R = sin2 ϑψ2 . (2.11.35b) ϑϑ c2 ϕϕ c2 The Ricci scalar and Kretschmann read RR¨ + R˙2 R¨2R2 + R˙4 R = 6 , K = 12 . (2.11.36) R2c2 R4c4 Local tetrad: 1 1 1 1 e = ∂t , e = ∂ , e = ∂ , e = ∂ . (2.11.37) (t) c (ψ) R ψ ϑ Rψ ϑ ϕ Rψ sinϑ ϕ Ricci rotation coefﬁcients: R˙ 1 γ = γ = γ = γ = γ = , (2.11.38a) (ψ)(t)(ψ) (ϑ)(t)(ϑ) (ϕ)(t)(ϕ) Rc (ϑ)(ψ)(ϑ) (ϕ)(ψ)(ϕ) Rψ cot(ϑ) γ = . (2.11.38b) (ϕ)(ϑ)(ϕ) Rψ The contractions of the Ricci rotation coefﬁcients read 3R˙ 2 cotϑ γ = , γ = , γ = . (2.11.39) (t) Rc (r) Rψ (ϑ) Rψ Riemann-Tensor with respect to local tetrad: R¨ R = R = R = , (2.11.40a) (t)(ψ)(t)(ψ) (t)(ϑ)(t)(ϑ) (t)(ϕ)(t)(ϕ) −Rc2 R˙2 R = R = R = . (2.11.40b) (ψ)(ϑ)(ψ)(ϑ) (ψ)(ϕ)(ψ)(ϕ) (ϑ)(ϕ)(ϑ)(ϕ) R2c2 Ricci-Tensor with respect to local tetrad: 3R¨ RR¨ + 2R˙2 R = , R = R = R = . (2.11.41) (t)(t) −Rc2 (ψ)(ψ) (ϑ)(ϑ) (ϕ)(ϕ) R2c2

Negative Curvature

ds2 = c2dt2 + R2 dψ2 + sinh2 ψ dϑ 2 + sin2 ϑdϕ2 (2.11.42) − Christoffel symbols: R˙ R˙ R˙ Γ ψ = , Γ ϑ = , Γ ϕ = , (2.11.43a) tψ R tϑ R tϕ R RR˙ Γ t = , Γ ϑ = cothψ, Γ ϕ = cothψ, (2.11.43b) ψψ c2 ψϑ ψϕ Rsinh2 ψR˙ Γ t = , Γ ψ = sinhψ coshψ, Γ ϕ = cotϑ, (2.11.43c) ϑϑ c2 ϑϑ − ϑϕ Rsinh2 ψ sin2 ϑR˙ Γ t = , Γ ψ = sinhψ coshψ sin2 ϑ, Γ ϑ = sinϑ cosϑ. (2.11.43d) ϕϕ c2 ϕϕ − ϕϕ − 52 CHAPTER 2. SPACETIMES

Riemann-Tensor:

2 Rtψtψ = RR¨, Rt t = Rsinh ψR¨, (2.11.44a) − ϑ ϑ − 2 2 ˙2 2 2 2 R sinh ψ R c Rtϕtϕ = Rsinh ψ sin ϑR¨, R = − , (2.11.44b) − ψϑψϑ c2 R2 sinh2 ψ sin2 ϑ R˙2 c2 R2 sinhψ4 sin2 ϑ R˙2 c2 R = − , R = − . (2.11.44c) ψϕψϕ c2 ϑϕϑϕ c2

Ricci-Tensor: R¨ RR¨ + 2(R˙2 c2) Rtt = 3 , Rψψ = − , (2.11.45a) − R c2 RR¨ + 2(R˙2 c2) RR¨ + 2(R˙2 c2) R = sinh2 ψ − , R = sin2 ϑ sin2 ψ − . (2.11.45b) ϑϑ c2 ϕϕ c2 The Ricci scalar and Kretschmann read RR¨ + R˙2 c2 R¨2R2 + R˙4 2R˙2c2 + c4 R = 6 − , K = 12 − . (2.11.46) R2c2 R4c4

Local tetrad: 1 1 1 1 e = ∂t , e = ∂ , e = ∂ , e = ∂ . (2.11.47) (t) c (ψ) R ψ ϑ Rsinhψ ϑ ϕ Rsinhψ sinϑ ϕ

Ricci rotation coefﬁcients: R˙ cothψ γ = γ = γ = γ = γ = , (2.11.48a) (ψ)(t)(ψ) (ϑ)(t)(ϑ) (ϕ)(t)(ϕ) Rc (ϑ)(ψ)(ϑ) (ϕ)(ψ)(ϕ) R cotθ γ = . (2.11.48b) (ϕ)(ϑ)(ϕ) Rsinhψ

The contractions of the Ricci rotation coefﬁcients read 3R˙ cothψ cotϑ γ = , γ = 2 , γ = . (2.11.49) (t) Rc (r) R (ϑ) Rsinhψ

Riemann-Tensor with respect to local tetrad:

R¨ R = R = R = , (2.11.50a) (t)(ψ)(t)(ψ) (t)(ϑ)(t)(ϑ) (t)(ϕ)(t)(ϕ) −Rc2 R˙2 c2 R = R = R = − . (2.11.50b) (ψ)(ϑ)(ψ)(ϑ) (ψ)(ϕ)(ψ)(ϕ) (ϑ)(ϕ)(ϑ)(ϕ) R2c2

Ricci-Tensor with respect to local tetrad:

3R¨ RR¨ + 2(R˙2 c2) R = , R = R = R = − . (2.11.51) (t)(t) −Rc2 (ψ)(ψ) (ϑ)(ϑ) (ϕ)(ϕ) R2c2

Further reading: Rindler[Rin01] 2.12. GÖDEL UNIVERSE 53

2.12 Gödel Universe

Gödel introduced a homogeneous and rotating universe model in [Göd49]. We follow the notation of [KWSD04]

2.12.1 Cylindrical coordinates The Gödel metric in cylindrical coordinates is

dr2 r 2 c ds2 = c2dt2 + + r2 1 dϕ2 + dz2 2r2 dtdϕ, (2.12.1) − 1 + [r/(2a)]2 − 2a − √2a where 2a is the Gödel radius. Christoffel symbols:

t r 1 ϕ c 1 Γtr = , Γtr = , (2.12.2a) 2a2 1 + [r/(2a)]2 −√2ar 1 + [r/(2a)]2 2 r cr h r i r r 1 Γt = 1 + , Γrr = , (2.12.2b) ϕ √2a 2a −4a2 1 + [r/(2a)]2 3 t r 1 ϕ 1 1 Γr = , Γrϕ = , (2.12.2c) ϕ 4√2ca3 1 + [r/(2a)]2 r 1 + [r/(2a)]2 r 2 1 r 2 Γ r = r 1 + 1 . (2.12.2d) ϕϕ 2a − 2 a Riemann-Tensor: c2 1 cr2 1 Rtrtr = , Rtrrϕ = , (2.12.3a) 2a2 1 + [r/(2a)]2 −2√2a3 1 + [r/(2a)]2 c2r2 1 r2 1 + 3[r/(2a)]2 Rt t = , Rr r = . (2.12.3b) ϕ ϕ 2a2 1 + [r/(2a)]2 ϕ ϕ 2a2 1 + [r/(2a)]2 Ricci-Tensor: c2 r2c r4 Rtt = , Rtϕ = , Rϕϕ = . (2.12.4) a2 √2a3 2a4 Ricci and Kretschmann scalar 1 3 R = , K = . (2.12.5) −a2 a4 cosmological constant: R Λ = (2.12.6) 2 Killing vectors: µ µ µ ν σ An infinitesimal isometric transformation x0 = x +εξ (x ) leaves the metric unchanged, that is g0µν (x0 ) = σ µ gµν (x0 ). A killing vector field ξ is solution to the killing equation ξµ;ν +ξν;µ = 0. There exist five killing vector fields in Gödel’s spacetime:    r cosϕ    1 √2c 0 2 µ  0  µ 1  a 1 + [r/(2a)] sinϕ  µ  0  ξ =  , ξ =  , ξ =  , (2.12.7a) 0 p 2 a 2 1 a   b 1 + [r/(2a)]  r 1 + 2[r/(2a)] cosϕ  c   0 0 0    r sinϕ  0 √2c 2 µ  0  µ 1  a 1 + [r/(2a)] cosϕ  ξ =  , ξ =  . (2.12.7b) 0 p 2 −a 2 d   e 1 + [r/(2a)]  r 1 + 2[r/(2a)] sinϕ  1 0 54 CHAPTER 2. SPACETIMES

An arbitrary linear combination of killing vector ﬁelds is again a killing vector ﬁeld. Local tetrad: For the local tetrad in Gödel’s spacetime an ansatz similar to the local tetrad of a rotating spacetime in spherical coordinates (Sec. 1.4.7) can be used. After substituting ϑ z and swapping base vectors e → (2) and e(3) an orthonormalized and right-handed local tetrad is obtained. q 2 e(0) = Γ ∂t + ζ∂ϕ , e(1) = 1 + [r/(2a)] ∂r, e(2) = ∆Γ A∂t + B∂ϕ , e(3) = ∂z, (2.12.8a) where r2c ζr2c A = + ζr2 1 [r/(2a)]2, B = c2 + , (2.12.9a) −√2a − √2a 1 1 Γ = , ∆ = . (2.12.9b) q p 2 c2 + ζr2c√2/a ζ 2r2 (1 [r/(2a)]2) rc 1 + [r/(2a)] − − Transformation between local direction y(i) and coordinate direction yµ :

q y0 = y(0)Γ + y(2)∆Γ A, y1 = y(1) 1 + [r/(2a)]2, y2 = y(0)Γ ζ + y(2)∆Γ B, y3 = y(3). (2.12.10) with the above abbreviations.

2.12.2 Scaled cylindrical coordinates If we apply the simple transformation t r z T = , R = , φ = ϕ, Z = , (2.12.11) rG rG rG with rG = 2a, we ﬁnd a formulation for the metric scaling with rG, which is

dR2 ds2 = r2 c2dT 2 + + R2(1 R2)Dφ 2 + dZ2 2√2cR2 dTdφ . (2.12.12) G − 1 + R2 − −

Christoffel symbols:

2R √2c Γ T = , Γ φ = , (2.12.13a) TR 1 + R2 TR −R(1 + R2) R Γ R = √2cR(1 + R2), Γ R = , (2.12.13b) Tφ RR −1 + R2 √2R3 1 Γ T = , Γ φ = , (2.12.13c) Rφ c(1 + R2) Rφ R(1 + R2) Γ R = R(1 + R2)(2R2 1). (2.12.13d) φφ − Riemann-Tensor:

2 2 2 2 2rGc 2√2rGcR RTRTR = , RTRRφ = , (2.12.14a) 1 + R2 − 1 + R2 2 2 2 2 2 2 2 2rGR (1 + 3R ) RT T = 2c r R (1 + R ), RR R = . (2.12.14b) φ φ G φ φ 1 + R2 Ricci-Tensor:

2 2 4 RTT = 4c , RTφ = 4√2cR , Rφφ = 8R . (2.12.15) 2.12. GÖDEL UNIVERSE 55

Ricci and Kretschmann scalar 4 48 R = 2 , K = 4 . (2.12.16) −rG rG cosmological constant: R Λ = (2.12.17) 2

Killing vectors: The Killing vectors read

   R cosϕ    1 √2c 0 1 2 µ  0  µ 1  (1 + R )sinϕ  µ  0  ξ =  , ξ =  2 , ξ =  , (2.12.18a) 0 2 1 2 1 a   b √1 + R  2R (1 + 2R )cosϕ  c   0 0 0    R sinϕ  0 √2c 1 2 µ  0  µ 1  (1 + R )cosϕ  ξ =  , ξ =  2 . (2.12.18b) 0 2 −1 2 d   e √1 + R  2R (1 + 2R )sinϕ  1 0

Local tetrad: After the transformation to scaled cylindrical coordinates, the local tetrad reads

Γ 1 p 2 ∆Γ 1 e(0) = ∂T + ζ∂φ , e(1) = 1 + R ∂R, e(2) = A∂T + B∂φ , e(3) = ∂Z, (2.12.19a) rG rG rG rG where h i A = R2 √2c + (1 R2)ζ , B = c2 + √2R2cζ, (2.12.20a) − − 1 1 Γ = , ∆ = . (2.12.20b) q √ 2 c2 + 2√2R2cζ R2(1 R2)ζ 2 Rc 1 + R − − Transformation between local direction y(i) and coordinate direction yµ :

Γ ∆Γ A 1 p Γ ζ ∆Γ B 1 y0 = y(0) + y(2), y1 = 1 + R2y(1), y2 = y(0) + y(2), y3 = y(3), (2.12.21) rG rG rG rG rG rG and the back transformation is given by

0 2 2 0 (0) rG By Ay (1) rG 1 (2) rG y ζy (3) 3 y = − , y = y , y = − , y = rGy . (2.12.22a) Γ B ζA √1 + R2 ∆Γ B ζA − − 56 CHAPTER 2. SPACETIMES

2.13 Halilsoy standing wave

The standing wave metric by Halilsoy[Hal88] reads

1 ds2 = V e2K dρ2 dt2 + ρ2dϕ2 + (dz + Adϕ)2 , (2.13.1) − V where

2 2CJ (ρ)cos(t) 2 2CJ (ρ)cos(t) V = cosh αe− 0 + sinh αe 0 , (2.13.2a) 2 C 2 2 2 2 K = ρ J0(ρ) + J1(ρ) 2ρJ0(ρ)J1(ρ)cos t , (2.13.2b) 2 − A = 2C sinh(2α)ρJ1(ρ)sin(t). (2.13.2c) − with spherical Bessel functions J1,2 and parameters α and C. Local tetrad: K K e− e− 1 A e 0 = ∂t , e 1 = ∂ρ , e 2 = ∂ϕ ∂z, e 3 = √V∂z. (2.13.3) ( ) √V ( ) √V ( ) ρ√V − ρ√V ( ) dual tetrad: 1 θ(0) = √VeK dt, θ(2) = √VeK dρ, θ(2) = √Vρ dϕ, θ(3) = (dz + Adϕ). (2.13.4) √V 2.14. JANIS-NEWMAN-WINICOUR 57

2.14 Janis-Newman-Winicour

The Janis-Newman-Winicour[JNW68] spacetime in spherical coordinates (t,r,ϑ,ϕ) is represented by the line element

2 γ 2 2 γ 2 2 γ+1 2 2 2 ds = α c dt + α− dr + r α− dϑ + sin ϑdϕ , (2.14.1) −

2 where α = 1 rs/(γr). The Schwarzschild radius rs = 2GM/c is deﬁned by Newton’s constant G, the − speed of light c, and the mass parameter M. For γ = 1, we obtain the Schwarzschild metric (2.2.1). Christoffel symbols:

2 r rsc 2γ 1 t rs r rs Γ = α − , Γ = , Γ = , (2.14.2a) tt 2r2 tr 2r2α rr −2r2α ϑ 2γr rs(γ + 1) ϕ 2γr rs(γ + 1) r 2γr rs(γ + 1) Γ = − , Γr = − , Γ = − , (2.14.2b) rϑ 2γr2α ϕ 2γr2α ϑϑ − 2γ Γ r = Γ r sin2 ϑ, Γ ϕ = cotϑ, Γ ϑ = sinϑ cosϑ. (2.14.2c) ϕϕ ϑϑ ϑϕ ϕϕ − Riemann-Tensor: 2 γ 2 2 γ 1 rsc [2γr rs(γ + 1)]α − rsc [2γr rs(γ + 1)]α − Rtrtr = − , Rt t = − , (2.14.3a) − 2γr4 ϑ ϑ 4γr2 2 γ 1 2 2 r c [2γr r (γ + 1)]α sin ϑ rs 2γ r rs(γ + 1) R s s − R tϕtϕ = − 2 , rϑrϑ = 2−2 γ 1 , (2.14.3b) 4γr − 4γ r α − 2 2 2 2 2 rs 2γ r rs(γ + 1) sin ϑ rs 4γ r rs(γ + 1) sin ϑ R R rϕrϕ = − 2 2 γ 1 , ϑϕϑϕ = − 2 γ . (2.14.3c) − 4γ r α − 4γ α

Weyl-Tensor:

2 γ 2 2 γ 1 rsc α − β rsc α − β Ctrtr = , Ct t = , (2.14.4a) − 6γ2r4 ϑ ϑ 12γ2r2 2 γ 1 2 rsc α − β sin ϑ rsβ Ctϕtϕ = , Cr r = , (2.14.4b) 12γ2r2 ϑ ϑ −12γ2r2αγ+1 2 2 rsβ sin ϑ rsβ sin ϑ Crϕrϕ = , C = , (2.14.4c) −12γ2r2αγ+1 ϑϕϑϕ 6γ2αγ

2 where β = 6γ r rs(γ + 1)(2γ + 1). − Ricci-Tensor: 2 2 rs (1 γ ) Rrr = − . (2.14.5) 2γ2r4α2

The Ricci scalar reads r2(1 γ2)αγ 2 R = s − − , (2.14.6) 2γ2r4 whereas the Kretschmann scalar is given by

2 2γ 4 rs α − 2 2 2 4 2 2 2 K = 7γ r (2 + γ ) + 48γ r α + 8γrs(2γ + 1)(rs 2γr) + 3r . (2.14.7) 4γ4r8 s − s

Local tetrad:

(γ 1)/2 (γ 1)/2 1 γ/2 α − α − e = ∂t , e = α ∂r, e = ∂ , e = ∂ϕ . (2.14.8) (t) cαγ/2 (r) (ϑ) r ϑ (ϕ) r sinϑ 58 CHAPTER 2. SPACETIMES

Dual tetrad: dr r r sinϑ θ(t) = cαγ/2dt, θ(r) = , θ(ϑ) = dϑ, θ(ϕ) = dϕ. γ/2 (γ 1)/2 (γ 1)/2 (2.14.9) α α − α − Ricci rotation coefﬁcients:

rs 2γr rs(γ + 1) γ = α(γ 2)/2, γ = γ = − α(γ 2)/2, (2.14.10a) (r)(t)(t) 2r2 − (ϑ)(r)(ϑ) (ϕ)(r)(ϕ) 2γr2 − cotϑ γ = α(γ 1)/2. (2.14.10b) (ϕ)(ϑ)(ϕ) r − The contractions of the Ricci rotation coefﬁcients read

4γr rs(2 + γ) cotϑ γ = − α(γ 1)/2, γ = α(γ 1)/2. (2.14.11) (r) 2γr2 − (ϑ) r −

Structure coefﬁcients:

(t) rs (γ 2)/2 (ϑ) (ϕ) 2γr rs(γ + 1) (γ 2)/2 c = α − , c = c = − α − , (2.14.12a) (t)(r) 2r2 (r)(ϑ) (r)(ϕ) − 2γr2

(ϕ) cotϑ (γ 1)/2 c = α − . (2.14.12b) (ϑ)(ϕ) − r

Euler-Lagrange: The Euler-Lagrangian formalism, Sec. 1.8.4, for geodesics in the ϑ = π/2 hyperplane yields the effective potential

1 h2αγ 1 V = αγ − κc2 (2.14.13) eff 2 r2 −

2 γ+1 γ 2˙ 1 with the constants of motion h = r α− ϕ˙ and k = α c t. For null geodesics (κ = 0) and γ > 2 , there is an extremum at 1 + 2γ r = rs . (2.14.14) 2γ

Embedding: 2 2 The embedding function z = z(r) for r [rs(γ + 1) /(4γ ),∞) follows from ∈ s 2 2 dz rs [4rγ rs(1 + γ) ] = − . (2.14.15) dr 4r2γ2αγ+1

However, the analytic solution

2 1 γ + 1 1 1 rs rs(1 + γ) 2πγ 1 γ + 1 4γ z(r) = 2√rsr F1 ; , ; , , 2F1 , ;1; , (2.14.16) −2 2 −2 2 rγ 4rγ2 − γ + 1 −2 2 (γ + 1)2

depends on the Appell-F1- and the Hypergeometric-2F1-function. 2.15. KASNER 59 2.15 Kasner

The Kasner spacetime in Cartesian coordinates (t,x,y,z) is represented by the line element[MTW73, Kas21] (c = 1)

ds2 = dt2 +t2p1 dx2 +t2p2 dy2 +t2p3tz2, (2.15.1) − where p1, p2, p3 have to fulﬁll the two conditions 2 2 2 p1 + p2 + p3 = 1 and p1 + p2 + p3 = 1. (2.15.2) These two conditions can also be represented by the Khalatnikov-Lifshitz parameter u with u 1 + u u(1 + u) p1 = , p2 = , p3 = . (2.15.3) −1 + u + u2 1 + u + u2 1 + u + u2 Christoffel symbols: p1 y p2 p3 Γ x = , Γ = , Γ z = , (2.15.4a) tx t ty t tz t 2p 2p 2p p1t 1 p2t 2 p3t 3 Γ t = , Γ t = , Γ t = . (2.15.4b) xx t yy t zz t Partial derivatives p1 p2 p3 Γ x = , Γ t = , Γ z = , (2.15.5a) tx,t − t2 ty,t − t2 tz,t − t2 t 2p1 2 t 2p2 2 t 2p3 2 Γ = p1(2p1 1)t − , Γ = p2(2p2 1)t − , Γ = p3(2p3 1)t − . (2.15.5b) xx,t − yy,t − zz,t − Riemann-Tensor: 2p 2p 2p p1(1 p1)t 1 p2(1 p2)t 2 p3(1 p3)t 3 R = − , R = − , R = − , (2.15.6a) txtx t2 tyty t2 tztz t2 2p 2p 2p 2p 2p 2p p1 p2t 1t 2 p1 p3t 1t 3 p2 p3t 2t 3 R = , R = ,.R = . (2.15.6b) xyxy t2 xzxz t2 yzyz t2 The Ricci tensor as well as the Ricci scalar vanish identically. The Kretschmann scalar reads

4 2 3 4 2 3 4 2 2 2 3 4 2 2 2 2 K = p 2p + p + p 2p + p + p p + p 2p + p + p p + p p (2.15.7a) t4 1 − 1 1 2 − 2 2 1 3 3 − 3 3 1 2 2 3 16u2(1 + u)2 = . (2.15.7b) t4(1 + u + u2)3 Local tetrad:

p1 p2 p3 e(t) = ∂t , e(x) = t− ∂x, e(y) = t− ∂y, e(z) = t− ∂z. (2.15.8) Dual tetrad: θ(t) = dt, θ(x) = t p1 dx, θ(y) = t p2 dy, θ(z) = t p3 dz. (2.15.9)

Ricci rotation coefﬁcients: p1 p2 p3 γ = , γ = , γ = . (2.15.10) (t)(r)(r) t (t)(ϑ)(ϑ) t (t)(ϕ)(ϕ) t The contractions of the Ricci rotation coefﬁcients read 1 γ = . (2.15.11) (t) − t Riemann-Tensor with respect to local tetrad:

p1(1 p1) p2(1 p2) p3(1 p3) R = − , R = − , R = − , (2.15.12a) (t)(x)(y)(x) t2 (t)(y)(t)(y) t2 (t)(z)(t)(z) t2 p1 p2 p1 p3 p2 p3 R = , R = , R = . (2.15.12b) (x)(y)(x)(y) t2 (x)(z)(x)(z) t2 (y)(z)(y)(z) t2 60 CHAPTER 2. SPACETIMES

2.16 Kastor-Traschen

The Kastor-Traschen spacetime in Cartesian coordinates (t,x,y,z) is represented by the line element[KT93] (c = 1)

2 2 2 2 2 2 2 2 ds = Ω − dt + a Ω dx + dy + dz , (2.16.1) −

Ht mi p 2 2 2 p where a(t) = e , Ω = 1 + ∑ ar , ri = (x xi) + (y yi) + (z zi) , and H = Λ/3 with cosmological i i − − − ± constant Λ. Christoffel symbols:

∂t Ω ∂xΩ y ∂t Ω Γ t = , Γ x = , Γ = , (2.16.2a) tt − Ω tt −a2Ω 5 tt −a2Ω 5 ∂zΩ ∂xΩ a∂t Ω + Ω∂t a Γ z = , Γ t = , Γ x = , (2.16.2b) tt −a2Ω 5 tx − Ω tx aΩ ... (2.16.2c)

Local tetrad: 1 1 1 e = Ω∂t , e = ∂x, e = ∂y, e = ∂z. (2.16.3) (t) (x) aΩ (y) aΩ (z) aΩ Dual tetrad:

(t) 1 (x) (y) (z) θ = Ω − dt, θ = aΩdx, θ = aΩdy, θ = aΩdz. (2.16.4) 2.17. KERR 61 2.17 Kerr

The Kerr spacetime, found by Roy Kerr [Ker63] in 1963, describes a rotating black hole.

2.17.1 Boyer-Lindquist coordinates The Kerr metric in Boyer-Lindquist coordinates

2 rsr 2rsar sin ϑ Σ ds2 = 1 c2dt2 cdt dϕ + dr2 + Σdϑ 2 − − Σ − Σ ∆ 2 2 (2.17.1) rsa r sin ϑ + r2 + a2 + sin2 ϑdϕ2, Σ

2 2 2 2 2 2 with Σ = r + a cos ϑ, ∆ = r rsr + a , and rs = 2GM/c , is taken from Bardeen[BPT72]. M is the mass − and a = J/(Mc) is the angular momentum per unit mass of the black hole and scaled by the speed of light. The event horizon r+ is deﬁned by the outer root of ∆,

r 2 rs r r = + s a2, (2.17.2) + 2 4 − whereas the outer boundary r0 of the ergosphere follows from the outer root of Σ rsr, − r 2 rs rs 2 2 r0 = + a cos ϑ, (2.17.3) 2 4 −

y

x ergosphere r+ r0 Figure 2.1: Ergosphere and horizon rs (dashed circle) for a = 0.99 2 .

General local tetrad: r ∆ e = Γ ∂t + ζ∂ , e = ∂r, (2.17.4a) (0) ϕ (1) Σ 1 Γ gtϕ + ζgϕϕ gtt + ζgtϕ e(2) = ∂ϑ , e(3) = ∂t ∂ϕ , (2.17.4b) √Σ c ∓ √∆ sinϑ ± √∆ sinϑ

2 2 where Γ = gtt + 2ζgtϕ + ζ gϕϕ , − − 2 2 2 2 2 rsr 2rsar sin ϑ ζ 2 2 rsa r sin ϑ ζ 2 Γ − = 1 + r + a + sin ϑ (2.17.5) − Σ Σ c − Σ c2

Non-rotating local tetrad (ζ = ω): r r r A 1 ∆ 1 Σ 1 e(0) = ∂t + ω∂ϕ , e(1) = ∂r, e(2) = ∂ϑ , e(3) = ∂ϕ , (2.17.6) Σ∆ c Σ √Σ A sinϑ

2 22 2 2 2 2 2 2 where ω = gtϕ /gϕϕ = rsar/A, and A = r + a a ∆ sin ϑ = r + a Σ + rsa r sin ϑ. − − 62 CHAPTER 2. SPACETIMES

Dual tetrad: r r r Σ∆ Σ A θ(2) = cdt, θ(1) = dr, θ(2) = √Σdϑ, θ(3) = sinϑ (dϕ ω dϕ). (2.17.7) A ∆ Σ − The relation between the constants of motion E, L, Q, and µ (deﬁned in Bardeen[BPT72]) and the initial direction υ, compare Sec. (1.4.5), with respect to the LNRF reads (c = 1) r r (0) A rsra (1) ∆ υ = E L, υ = pr, (2.17.8a) Σ∆ − √AΣ∆ Σ s r 1 L2 Σ L υ(2) = Q cos2 ϑ a2 (µ2 E2) + , υ(3) = . (2.17.8b) √Σ − − sin2 ϑ A sinϑ

Static local tetrad (ζ = 0): r 1 ∆ 1 e(0) = p ∂t , e(1) = ∂r, e(2) = ∂ϑ , (2.17.9a) c 1 rsr/Σ Σ √Σ − p rsar sinϑ 1 rsr/Σ e(3) = p ∂t − ∂ϕ . (2.17.9b) ±c 1 rsr/Σ√∆Σ ∓ √∆ sinϑ − Christoffel symbols:

2 2 2 2 2 2 c rs∆(r a cos ϑ) c rsa r sinϑ cosϑ Γ r = − , Γ ϑ = , (2.17.10a) tt 2Σ 3 tt − Σ 3 2 2 2 2 2 2 2 2 rs(r + a )(r a cos ϑ) crsa(r a cos ϑ) Γ t = − , Γ ϕ = − , (2.17.10b) tr 2Σ 2∆ tr 2Σ 2∆ 2 rsa r sinϑ cosϑ crsar cotϑ Γ t = , Γ ϕ = , (2.17.10c) tϑ − Σ 2 tϑ − Σ 2 2 2 2 2 2 2 c∆rsasin ϑ(r a cos ϑ) crsar(r + a )sinϑ cosϑ Γ r = − , Γ ϑ = , (2.17.10d) tϕ − 2Σ 3 tϕ Σ 3 2 2 2 2 2 2 2ra sin ϑ rs(r a cos ϑ) a sinϑ cosϑ Γ r = − − , Γ ϑ = , (2.17.10e) rr 2Σ∆ rr Σ∆ a2 sinϑ cosϑ r Γ r = , Γ ϑ = , (2.17.10f) rϑ − Σ rϑ Σ r∆ a2 sinϑ cosϑ Γ r = , Γ ϑ = , (2.17.10g) ϑϑ − Σ ϑϑ − Σ 3 3 ϕ cotϑ 2 2 2 t rsa r sin ϑ cosϑ Γ = Σ + rsa r sin ϑ , Γ = , (2.17.10h) ϑϕ Σ 2 ϑϕ cΣ 2

2 2 2 2 2 2 2 2 rsasin ϑ a cos ϑ(a r ) r (a + 3r ) Γ t = − − , (2.17.10i) rϕ 2cΣ 2∆ 2 4 2 2 2 2 2 2rΣ + rs a sin ϑ cos ϑ r (Σ + r + a ) Γ ϕ = − , (2.17.10j) rϕ 2Σ 2∆ 2 r ∆ sin ϑ 2 2 2 2 2 2 Γ = 2rΣ + rsa sin ϑ(r a cos ϑ) , (2.17.10k) ϕϕ 2Σ 3 − − ϑ sinϑ cosϑ 2 2 2 2 Γ = AΣ + r + a rsa r sin ϑ , (2.17.10l) ϕϕ − Σ 3 Photon orbits: The direct(-) and retrograd(+) photon orbits have radius

2 2a rpo = rs 1 + cos arccos ∓ . (2.17.11) 3 rs 2.17. KERR 63

Marginally stable timelike circular orbits are deﬁned via rs p r = 3 + Z2 (3 Z1)(2 + Z1 + 2Z2) , (2.17.12) ms 2 ∓ − where " # 4a2 1/3 2a1/3 2a1/3 Z1 = 1 + 1 2 1 + + 1 , (2.17.13a) − rs rs − rs s 2 12a 2 Z2 = 2 + Z1 . (2.17.13b) rs

Euler-Lagrange: The Euler-Lagrangian formalism, Sec. 1.8.4, for geodesics in the ϑ = π/2 hyperplane yields 1 r˙2 +V = 0 (2.17.14) 2 eff with the effective potential

2 2 1 2 ahk k 3 2 κc ∆ V = h (r rs) + 2 rs r + a (r + rs) (2.17.15) eff 2r3 − c − c2 − r2 and the constants of motion 2 rs crsa rsa crsa k = 1 c2t˙+ ϕ˙, h = r2 + a2 + ϕ˙ t˙. (2.17.16) − r r r − r

Timelike circular orbits A timelike circular geodesic, see Sec. 1.9.1, is given by

4 2 2 p 7 rsar 3r + a A 2r rs β1 2 = ± , (2.17.17) , 2 5 2 2 r √∆ ( 2r + rsa r ) − where the positive sign is for contrarotating and the negative sign for corotating orbits, and β1,2 are w.r.t. the LNRF. Further reading: Boyer and Lindquist[BL67], Wilkins[Wil72], Brill[BC66]. 64 CHAPTER 2. SPACETIMES

2.18 Kottler spacetime

The Kottler spacetime is represented in spherical coordinates (t,r,ϑ,ϕ) by the line element[Per04]

2 2 rs Λr 2 2 1 2 2 2 ds = 1 c dt + 2 dr + r dΩ , (2.18.1) − − r − 3 1 rs/r Λr /3 − − 2 where rs = 2GM/c is the Schwarzschild radius, G is Newton’s constant, c is the speed of light, M is the mass of the black hole, and Λ is the cosmological constant. If Λ > 0 the metric is also known as Schwarzschild-deSitter metric, whereas if Λ < 0 it is called Schwarzschild-anti-deSitter. For the following, we deﬁne the two abbreviations 2 rs Λr rs 2Λ α = 1 and β = r2. (2.18.2) − r − 3 r − 3 The critical points of the Kottler metric follow from the roots of the cubic equation α = 0. These can be found by means of the parameters p = 1/Λ and q = 3rs/(2Λ). If Λ < 0, we have only one real root − 2 1 3rs r1 = sinh arsinh √ Λ . (2.18.3) √ Λ 3 2 − − If Λ > 0, we have to distinguish whether D q2 + p3 = 9r2/(4Λ 2) Λ 3 is positive or negative. If D > 0, ≡ s − − there is no real positive root. For D < 0, the two real positive roots read 2 π 1 3rs r = cos arccos √Λ (2.18.4) ± √Λ 3 ± 3 2 Christoffel symbols: c2αβ β β Γ r = , Γ t = , Γ r = , (2.18.5a) tt 2r tr 2rα rr −2rα ϑ 1 ϕ 1 r Γ = , Γr = , Γ = αr, (2.18.5b) rϑ r ϕ r ϑϑ − Γ ϕ = cotϑ, Γ r = αr sin2 ϑ, Γ ϑ = sinϑ cosϑ. (2.18.5c) ϑϕ ϕϕ − ϕϕ − Riemann-Tensor: 2 3 c 3rs +Λr 1 2 Rtrtr = , Rt t = c αβ, (2.18.6a) − 3r3 ϑ ϑ 2 1 2 2 β Rtϕtϕ = c αβ sin ϑ, Rr r = , (2.18.6b) 2 ϑ ϑ −2α 3 β 2 Λr 2 Rrϕrϕ = sin ϑ, R = r rs + sin ϑ. (2.18.6c) −2α ϑϕϑϕ 3

Ricci-Tensor:

2 Λ 2 2 2 Rtt = c αΛ, Rrr = , R = Λr , Rϕϕ = Λr sin ϑ. (2.18.7) − α ϑϑ The Ricci scalar and the Kretschmann scalar read r2 8Λ 2 R = 4Λ, K = 12 s + . (2.18.8) r6 3 Weyl-Tensor:

2 2 2 2 c rs c αrs c αrs sin ϑ Ctrtr = , Ct t = , Ctϕtϕ = , (2.18.9a) − r3 ϑ ϑ 2r 2r 2 rs rs sin ϑ 2 Cr r = , Crϕrϕ = , C = rrs sin ϑ. (2.18.9b) ϑ ϑ −2rα − 2rα ϑϕϑϕ 2.18. KOTTLER SPACETIME 65

Local tetrad: 1 1 1 e = ∂t , e = √α∂r, e = ∂ , e = ∂ . (2.18.10) (t) c√α (r) (ϑ) r ϑ (ϕ) r sinϑ ϕ

Dual tetrad: dr θ(t) = c√α dt, θ(r) = , θ(ϑ) = r dϑ, θ(ϕ) = r sinϑ dϕ. (2.18.11) √α

Ricci rotation coefﬁcients:

2 3 rs Λr √α cotϑ γ = − 3 , γ = γ = , γ = . (2.18.12) (r)(t)(t) 2r2√α (ϑ)(r)(ϑ) (ϕ)(r)(ϕ) r (ϕ)(ϑ)(ϕ) r

The contractions of the Ricci rotation coefﬁcients read

3 4r 3rs 2Λr cotϑ γ = − − , γ = . (2.18.13) (r) 2r2√α (ϑ) r

Riemann-Tensor with respect to local tetrad:

3 Λr + 3rs R = R = , (2.18.14a) (t)(r)(t)(r) − (ϑ)(ϕ)(ϑ)(ϕ) − 3r3 3 3rs 2Λr R = R = R = R = − . (2.18.14b) (t)(ϑ)(t)(ϑ) (t)(ϕ)(t)(ϕ) − (r)(ϑ)(r)(ϑ) − (r)(ϕ)(r)(ϕ) 6r3

Weyl-Tensor with respect to local tetrad:

rs C = C = , (2.18.15a) (t)(r)(t)(r) − (ϑ)(ϕ)(ϑ)(ϕ) −r3 rs C = C = C = C = . (2.18.15b) (t)(ϑ)(t)(ϑ) (t)(ϕ)(t)(ϕ) − (r)(ϑ)(r)(ϑ) − (r)(ϕ)(r)(ϕ) 2r3

Embedding: The embedding function follows from the numerical integration of s 2 dz rs/r +Λr /3 = 2 . (2.18.16) dr 1 rs/r Λr /3 − − Euler-Lagrange: The Euler-Lagrangian formalism[Rin01] yields the effective potential

2 2 1 rs Λr h V = 1 κc2 (2.18.17) eff 2 − r − 3 r2 −

2 2 2 with the constants of motion k = (1 rs/r Λr /3)c t˙, h = r ϕ˙, and κ as in Eq. (1.8.2). − − As in the Schwarzschild metric, the effective potential has only one extremum for null geodesics, the so 3 called photon orbit at r = 2 rs. For timelike geodesics, however, we have

2 2 2 3 dV h ( 6r + 9rs) + c r (3rs 2r Λ) ! eff = − − = 0. (2.18.18) dr 3r4 This polynomial of ﬁfth order might have up to ﬁve extrema.

Further reading: Kottler[Kot18], Weyl[Wey19], Hackmann[HL08], Cruz[COV05]. 66 CHAPTER 2. SPACETIMES

2.19 Majumdar-Papapetrou spacetimes

The Majumdar-Papapetrou (MP) metric[Cha89] describes an ensemble of N extreme Reissner-Nordstrøm (RN) black holes (see 2.25.3) with masses Mk at locations rk (k = 1,2,...,N) and charges of the same sign. Because of the charge-mass ratio of each black hole, their gravitational attraction is exactly compensated by their electrostatic repulsion. In Cartesian coordinates {t,x,y,z R} the MP metric reads ∈

c2dt2 ds2 = +U2(dx2 + dy2 + dz2), (2.19.1) − U2

where

N Rk/2 U(x,y,z) = 1 + ∑ (2.19.2) r rk k=1 | − |

2 T with Rk = 2GMk/c and r = (x,y,z) . The coordinate singularity rk is the degenerated horizon of the k-th extreme RN black hole. For N = 2, the MP spacetime is called extreme RN dihole metric (see 2.10.1).

Derivations of U(x,y,z):

N N 2 2 Rk xk x 2 Rk 3(x xk) r rk ∂xU = ∑ − 3 , ∂x U = ∑ − − | 5− | , (2.19.3a) 2 r rk 2 r rk k=1 | − | k=1 | − | N N 2 2 Rk yk y 2 Rk 3(y yk) r rk ∂yU = ∑ − 3 , ∂y U = ∑ − − | 5− | , (2.19.3b) 2 r rk 2 r rk k=1 | − | k=1 | − | N N 2 2 Rk zk z 2 Rk 3(z zk) r rk ∂zU = ∑ − 3 , ∂z U = ∑ − − | 5− | , (2.19.3c) 2 r rk 2 r rk k=1 | − | k=1 | − | N N 3Rk (xk x)(yk y) 3Rk (xk x)(zk z) ∂x∂yU = ∑ − 5− , ∂x∂zU = ∑ − 5− , (2.19.3d) 2 r rk 2 r rk k=1 | − | k=1 | − | N 3Rk (yk y)(zk z) ∂y∂zU = ∑ − 5− . (2.19.3e) 2 r rk k=1 | − |

The function U(x,y,z) fulﬁlls the Laplace-Equation ∆U = 0, which will be used in the calculation of the following geometric quantities.

Christoffel symbols:

2 2 2 c ∂xU y c ∂yU c ∂zU ∂xU Γ x = , Γ = , Γ z = , Γ t = , (2.19.4a) tt − U5 tt − U5 tt − U5 tx − U ∂xU ∂yU ∂zU ∂yU Γ x = , Γ y = , Γ z = , Γ t = , (2.19.4b) xx U xx − U xx − U ty − U ∂yU ∂xU ∂xU ∂yU Γ x = , Γ y = , Γ x = , Γ y = , (2.19.4c) xy U xy U yy − U yy U ∂zU ∂zU ∂zU ∂xU Γ z = , Γ t = , Γ x = , Γ z = , (2.19.4d) yy − U tz − U xz U xz U ∂zU ∂yU ∂xU ∂yU Γ y = , Γ z = , Γ x = , Γ y = , (2.19.4e) yz U yz U zz − U zz − U ∂zU Γ z = . (2.19.4f) zz U 2.19. MAJUMDAR-PAPAPETROU SPACETIMES 67

Riemann-Tensor:

2 c 2 2 2 Rtxtx = 4(∂xU) U∂ U (∇U) , Rxyxz = 2∂yU∂zU U∂y∂zU, (2.19.5a) U4 − x − − 2 c 2 2 2 Rtyty = 4(∂yU) U∂ U (∇U) , Rxzxy = 2∂yU∂zU U∂y∂zU, (2.19.5b) U4 − y − − 2 c 2 2 2 Rtztz = 4(∂zU) U∂ U (∇U) , Rxzyz = 2∂xU∂yU U∂x∂yU, (2.19.5c) U4 − z − − 2 c 2 2 2 Rtxty = (4∂xU∂yU U∂x∂yU), Rxyxy = (∇U) 2(∂zU) +U∂ U, (2.19.5d) U4 − − z 2 c 2 2 2 Rtytx = (4∂xU∂yU U∂x∂yU), Rxzxz = (∇U) 2(∂yU) +U∂ U, (2.19.5e) U4 − − y 2 c 2 2 2 Rtztx = (4∂xU∂zU U∂x∂zU), Ryzyz = (∇U) 2(∂xU) +U∂ U, (2.19.5f) U4 − − x c2 Rtxtz = (4∂xU∂zU U∂x∂zU), Ryzxz = 2∂xU∂yU U∂x∂yU, (2.19.5g) U4 − − c2 Rtytz = (4∂yU∂zU U∂y∂zU), Rxyyz = 2∂xU∂zU +U∂x∂zU, (2.19.5h) U4 − − c2 Rtzty = (4∂yU∂zU U∂y∂zU), Ryzxy = 2∂xU∂zU +U∂x∂zU. (2.19.5i) U4 − −

Ricci-Tensor:

2 2 2 (∇U) 2(∂xU) c 2 2∂xU∂yU Rxx = − , Rtt = (∇U) , Rxy = , (2.19.6a) U2 U6 − U2 2 2 (∇U) 2(∂yU) 2∂xU∂zU 2∂yU∂zU Ryy = − , Rxz = , Ryz = , (2.19.6b) U2 − U2 − U2 2 2 (∇U) 2(∂zU) R = − . (2.19.6c) zz U2

The Ricci scalar vanishes identically, also because the energy-momentum tensor of the electromagnetic ﬁeld is traceless. The Kretschmann scalar reads

4 4 2 22 K = 14(∂zU) + 14 (∂yU) + (∂xU) 24U∂zU (∂yU∂y∂zU + ∂xU∂x∂zU) U8 − 2 2 2 2 12U (∂xU) ∂ U + 2∂xU∂yU∂x∂yU + (∂yU) ∂ U − x y 2 2 2 2 2 2 2 2 +U (∂z U) + 4(∂y∂zU) + 3(∂y U) + 4(∂x∂zU) + 4(∂x∂yU) (2.19.7) 2 2 2 2 + 2∂y U∂x U + 3(∂x U) 2 2 2 2 + 4(∂zU) 7 (∂yU) + (∂xU) 3U∂ U . − z 68 CHAPTER 2. SPACETIMES

Weyl-Tensor:

2 c 2 2 2 Ctxtx = 3(∂xU) U∂ U (∇U) , Cxyxz = 3∂yU∂zU U∂y∂zU, (2.19.8a) U4 − x − − 2 c 2 2 2 Ctyty = 3(∂yU) U∂ U (∇U) , Cxzxy = 3∂yU∂zU U∂y∂zU, (2.19.8b) U4 − y − − 2 c 2 2 2 Ctztz = 3(∂zU) U∂ U (∇U) , Cxzyz = 3∂xU∂yU U∂x∂yU, (2.19.8c) U4 − z − − 2 c 2 2 2 Ctxty = (3∂xU∂yU U∂x∂yU), Cxyxy = (∇U) 3(∂zU) +U∂ U, (2.19.8d) U4 − − z 2 c 2 2 2 Ctytx = (3∂xU∂yU U∂x∂yU), Cxzxz = (∇U) 3(∂yU) +U∂ U, (2.19.8e) U4 − − y 2 c 2 2 2 Ctztx = (3∂xU∂zU U∂x∂zU), Cyzyz = (∇U) 3(∂xU) +U∂ U, (2.19.8f) U4 − − x c2 Ctxtz = (3∂xU∂zU U∂x∂zU), Cyzxy = 3∂xU∂zU +U∂x∂zU, (2.19.8g) U4 − − c2 Ctytz = (3∂yU∂zU U∂y∂zU), Cxyyz = 3∂xU∂zU +U∂x∂zU, (2.19.8h) U4 − − c2 Ctzty = (3∂yU∂zU U∂y∂zU), Cyzxz = 3∂xU∂yU U∂x∂yU. (2.19.8i) U4 − −

Local tetrad:

U 1 1 1 e = ∂ , e = ∂ , e = ∂ , e = ∂ . (2.19.9) (t) c t (x) U x (y) U y (z) U z

Dual tetrad:

c θ(t) = dt, θ(x) = Udx, θ(y) = Udy, θ(z) = Udz. (2.19.10) U

Ricci rotation coefﬁcients:

∂xU ∂yU γ = γ = γ = , γ = γ = γ = , (2.19.11a) (t)(x)(t) (y)(x)(y) (z)(x)(z) U2 (t)(y)(t) (x)(y)(x) (z)(y)(z) U2 ∂zU γ = γ = γ = . (2.19.11b) (t)(z)(t) (x)(z)(x) (y)(z)(y) U2

The contractions of the Ricci rotation coefﬁcients read

∂xU ∂yU ∂zU γ = , γ = , γ = . (2.19.12) (x) U2 (y) U2 (z) U2 2.19. MAJUMDAR-PAPAPETROU SPACETIMES 69

Riemann-Tensor with respect to local tetrad:

1 2 2 2 1 R = 4(∂xU) U∂ U (∇U) , R = (2∂yU∂zU U∂y∂zU), (2.19.13a) (t)(x)(t)(x) U4 − x − (x)(y)(x)(z) U4 − 1 2 2 2 1 R = 4(∂yU) U∂ U (∇U) , R = (2∂yU∂zU U∂y∂zU), (2.19.13b) (t)(y)(t)(y) U4 − y − (x)(z)(x)(y) U4 − 1 2 2 2 1 R = 4(∂zU) U∂ U (∇U) , R = (2∂xU∂yU U∂x∂yU), (2.19.13c) (t)(z)(t)(z) U4 − z − (x)(z)(y)(z) U4 − 1 2 2 2 1 R = (∇U) 2(∂zU) +U∂ U , R = (4∂xU∂yU U∂x∂yU), (2.19.13d) (x)(y)(x)(y) U4 − z (t)(x)(t)(y) U4 − 1 2 2 2 1 R = (∇U) 2(∂yU) +U∂ U , R = (4∂xU∂yU U∂x∂yU), (2.19.13e) (x)(z)(x)(z) U4 − y (t)(y)(t)(x) U4 − 1 2 2 2 1 R = (∇U) 2(∂xU) +U∂ U , R = (4∂xU∂zU U∂x∂zU), (2.19.13f) (y)(z)(y)(z) U4 − x (t)(z)(t)(x) U4 − 1 1 R = (4∂xU∂zU U∂x∂zU), R = (2∂xU∂yU U∂x∂yU), (2.19.13g) (t)(x)(t)(z) U4 − (y)(z)(x)(z) U4 − 1 1 R = (4∂yU∂zU U∂y∂zU), R = ( 2∂xU∂zU +U∂x∂zU), (2.19.13h) (t)(y)(t)(z) U4 − (x)(y)(y)(z) U4 − 1 1 R = (4∂yU∂zU U∂y∂zU), R = ( 2∂xU∂zU +U∂x∂zU). (2.19.13i) (t)(z)(t)(y) U4 − (y)(z)(x)(y) U4 −

Ricci-Tensor with respect to local tetrad:

2 2 2 (∇U) 2(∂xU) (∇U) 2∂xU∂yU R = − , R = , R = , (2.19.14a) (x)(x) U4 (t)(t) U4 (x)(y) − U4 2 2 (∇U) 2(∂yU) 2∂xU∂zU 2∂yU∂zU R = − , R = , R = , (2.19.14b) (y)(y) U4 (x)(z) − U4 (y)(z) − U4 2 2 (∇U) 2(∂zU) R = − . (2.19.14c) (z)(z) U4

Weyl-Tensor with respect to local tetrad:

1 2 2 2 1 C = 3(∂xU) U∂ U (∇U) , C = (3∂yU∂zU U∂y∂zU), (2.19.15a) (t)(x)(t)(x) U4 − x − (x)(y)(x)(z) U4 − 1 2 2 2 1 C = 3(∂yU) U∂ U (∇U) , C = (3∂yU∂zU U∂y∂zU), (2.19.15b) (t)(y)(t)(y) U4 − y − (x)(z)(x)(y) U4 − 1 2 2 2 1 C = 3(∂zU) U∂ U (∇U) , C = (3∂xU∂yU U∂x∂yU), (2.19.15c) (t)(z)(t)(z) U4 − z − (x)(z)(y)(z) U4 − 1 2 2 2 1 C = (∇U) 3(∂zU) +U∂ U , C = (3∂xU∂yU U∂x∂yU), (2.19.15d) (x)(y)(x)(y) U4 − z (t)(x)(t)(y) U4 − 1 2 2 2 1 C = (∇U) 3(∂yU) +U∂ U , C = (3∂xU∂yU U∂x∂yU), (2.19.15e) (x)(z)(x)(z) U4 − y (t)(y)(t)(x) U4 − 1 2 2 2 1 C = (∇U) 3(∂xU) +U∂ U , C = (3∂xU∂zU U∂x∂zU), (2.19.15f) (y)(z)(y)(z) U4 − x (t)(z)(t)(x) U4 − 1 1 C = (3∂xU∂zU U∂x∂zU), C = ( 3∂xU∂zU +U∂x∂zU), (2.19.15g) (t)(x)(t)(z) U4 − (y)(z)(x)(y) U4 − 1 1 C = (3∂yU∂zU U∂y∂zU), C = ( 3∂xU∂zU +U∂x∂zU), (2.19.15h) (t)(y)(t)(z) U4 − (x)(y)(y)(z) U4 − 1 1 C = (3∂yU∂zU U∂y∂zU), C = (3∂xU∂yU U∂x∂yU). (2.19.15i) (t)(z)(t)(y) U4 − (y)(z)(x)(z) U4 −

Further reading: Chandrasekhar[Cha89, Cha06], Hartle[HH72], Yurtsever[Yur95], Wünsch[WMW13], 70 CHAPTER 2. SPACETIMES

2.20 Melvin universe

The spacetime describing the Melvin universe is represented by the metric

2 2 B2 B2 − ds2 = 1 + ρ2 dt2 + dρ2 + dz2 + 1 + ρ2 ρ2dϕ2, (2.20.1) 4 − 4 where ... (see e.g. Grifﬁth [GP09]). Christoffel symbols:

B2ρ Γ ρ = 2 ,... (2.20.2a) tt B2ρ2 + 4

Local tetrad:

2 2 1 e(t) = 1 + B ρ /4 − ∂t ,... (2.20.3)

Dual tetrad:

θ(t) = ... (2.20.4) 2.21. MORRIS-THORNE 71

2.21 Morris-Thorne

The most simple wormhole geometry is represented by the metric of Morris and Thorne[MT88],

ds2 = c2dt2 + dl2 + (b2 + l2)dϑ 2 + sin2ϑ dϕ2, (2.21.1) − 0 where b0 is the throat radius and l is the proper radial coordinate; and t R,l R,ϑ (0,π),ϕ [0,2π) . { ∈ ∈ ∈ ∈ } Christoffel symbols: l l ϑ ϕ l l Γlϑ = 2 2 , Γlϕ = 2 2 , Γϑϑ = , (2.21.2a) b0 + l b0 + l − Γ ϕ = cotϑ, Γ l = l sin2ϑ, Γ ϑ = sinϑ cosϑ. (2.21.2b) ϑϕ ϕϕ − ϕϕ − Partial derivatives l2 b2 l2 b2 ϑ 0 ϕ 0 l 1 Γlϑ,l = 2 − 2 2 , Γlϕ,l = 2 − 2 2 , Γϑϑ,l = , (2.21.3a) −(b0 + l ) −(b0 + l ) − 1 Γ ϕ = , Γ l = sin2 ϑ, Γ l = l sin(2ϑ), (2.21.3b) ϑϕ,ϑ −sin2 ϑ ϕϕ,l − ϕϕ,ϑ − Γ ϑ = cos(2ϑ). (2.21.3c) ϕϕ,ϑ − Riemann-Tensor: b2 b2 sin2ϑ R 0 R 0 R b2 sin2 lϑlϑ = 2 2 , lϕlϕ = 2 2 , ϑϕϑϕ = 0 ϑ. (2.21.4) −b0 + l − b0 + l Ricci tensor, Ricci and Kretschmann scalar: 2 2 4 b0 b0 12b0 Rll = 2 , R = 2 , K = . (2.21.5) − 2 22 − 2 22 2 24 b0 + l b0 + l b0 + l

Weyl-Tensor:

2 2 2 2 2 2 2 2 c b0 1 c b0 1 c b0 sin ϑ Ctltl = , Ctϑtϑ = , Ctϕtϕ = , (2.21.6a) −3 2 22 3 b2 + l2 3 b2 + l2 b0 + l 0 0 1 b2 1 b2 sin2 ϑ 2 C 0 C 0 C b2 sin2 lϑlϑ = 2 2 , lϕlϕ = 2 2 , ϑϕϑϕ = 0 ϑ. (2.21.6b) −3 b0 + l −3 b0 + l 3

Local tetrad: 1 1 1 e(t) = ∂t , e(l) = ∂l, e(ϑ) = q ∂ϑ , e(ϕ) = q ∂ϕ . (2.21.7) c 2 2 2 2 b0 + l b0 + l sinϑ

Dual tetrad q q (t) (l) (ϑ) 2 2 (ϕ) 2 2 θ = cdt, θ = dl, θ = b0 + l dϑ, θ = b0 + l sinϑ dϕ. (2.21.8)

Ricci rotation coefﬁcients: l cotϑ γ γ γ (ϑ)(l)(ϑ) = (ϕ)(l)(ϕ) = 2 2 , (ϕ)(ϑ)(ϕ) = q . (2.21.9) b0 + l 2 2 b0 + l

The contractions of the Ricci rotation coefﬁcients read 2l cotϑ γ γ (l) = 2 2 , (ϑ) = q . (2.21.10) b0 + l 2 2 b0 + l 72 CHAPTER 2. SPACETIMES

Riemann-Tensor with respect to local tetrad:

2 b0 R(l)(ϑ)(l)(ϑ) = R(l)(ϕ)(l)(ϕ) = R(ϑ)(ϕ)(ϑ)(ϕ) = . (2.21.11) − − 2 22 b0 + l

Ricci-Tensor with respect to local tetrad:

2 2b0 R(l)(l) = . (2.21.12) − 2 22 b0 + l

Weyl-Tensor with respect to local tetrad:

2 2b0 C(t)(l)(t)(l) = C(ϑ)(ϕ)(ϑ)(ϕ) = , (2.21.13a) − − 2 22 3 b0 + l 2 b0 C(t)(ϑ)(t)(ϑ) = C(t)(ϕ)(t)(ϕ) = C(l)(ϑ)(l)(ϑ) = C(l)(ϕ)(l)(ϕ) = . (2.21.13b) − − 2 22 3 b0 + l

Embedding: The embedding function reads  s  r r 2 z(r) = b0 ln + 1 (2.21.14) ± b0 b0 −

2 2 2 with r = b0 + l .

Euler-Lagrange: The Euler-Lagrangian formalism, Sec. 1.8.4, for geodesics in the ϑ = π/2 hyperplane yields

1 1 k2 1 h2 l˙2 V V c2 + eff = 2 , eff = 2 2 κ , (2.21.15) 2 2 c 2 b0 + l −

2 2 2 with the constants of motion k = c t˙ and h = (b0 + l )ϕ˙. The shape of the effective potential Veff is inde- pendend of the geodesic type. The maximum of the effective potential is located at l = 0.

A geodesic that starts at l = li with direction y = e + cosξe + sinξe approaches the wormhole ± (t) (l) (ϕ) throat asymptotically for ξ = ξcrit with

b0 ξcrit = arcsin q . (2.21.16) 2 2 b0 + li

This critical angle is independent of the type of the geodesic.

Further reading: Ellis[Ell73], Visser[Vis95], Müller[Mül04, Mül08a] 2.22. OPPENHEIMER-SNYDER COLLAPSE 73

2.22 Oppenheimer-Snyder collapse

2.22.1 Outer metric

The metric of the outer spacetime, R > Rb, in comoving coordinates (τ,R,ϑ,ϕ) with (c = 1) is given by

4/3 2 2 R 2 3/2 3 2 2 2 ds = dτ + dR + R √rsτ dϑ + sin ϑdϕ . (2.22.1) − 3/2 3 2/3 − 2 R √rsτ − 2

Christoffel symbols:

1 √rs √rs Γ R = , Γ ϑ = , (2.22.2a) τR 3/2 3 τϑ 3/2 3 2 R √rsτ −R √rsτ − 2 − 2 ϕ √rs τ R√rs Γτϕ = 3 , ΓRR = , (2.22.2b) −R3/2 r 3/2 3 5/3 2 √ sτ 2 R √rsτ − − 2 3√rsτ √R Γ R = , Γ ϑ = , (2.22.2c) RR 3/2 3 Rϑ 3/2 3 −4 R √rsτ R R √rsτ − 2 − 2 1/3 ϕ √R τ 3/2 3 Γ = , Γ = √rs R √rsτ , (2.22.2d) Rϕ 3/2 3 ϑϑ R √rsτ − − 2 − 2 3/2 3 R √rsτ Γ R = − 2 , Γ ϕ = cotϑ, (2.22.2e) ϑϑ − √R ϑϕ 1/3 τ 3/2 3 2 ϑ Γ = √rs R √rsτ sin ϑ, Γ = sinϑ cosϑ, (2.22.2f) ϕϕ − − 2 ϕϕ − 3/2 3 2 R √rsτ sin ϑ Γ R = − 2 . (2.22.2g) ϕϕ − √R

Riemann-Tensor:

Rrs 1 rs RτRτR = , Rτϑτϑ = , (2.22.3a) − 3/2 3 8/3 2 3/2 3 2/3 R √rs τ R √rs τ − 2 − 2 2 1 rs sin ϑ 1 Rrs Rτϕτϕ = , RRϑRϑ = , (2.22.3b) 2 3/2 3 2/3 −2 3/2 3 4/3 R √rs τ R √rs τ − 2 − 2 2 2/3 1 Rrs sin ϑ 3/2 3 2 RRϕRϕ = , Rϑϕϑϕ = R √rs τ rs sin ϑ. (2.22.3c) −2 3/2 3 4/3 − 2 R √rs τ − 2 The Ricci tensor and the Ricci scalar vanish identically. Kretschmann scalar: r2 K = 12 s . (2.22.4) 3/2 3 4 R √rs τ − 2 Local tetrad:

3/2 3 1/3 R 2 √rsτ e = ∂τ , e R = − ∂R, (2.22.5a) (τ) ( ) √R 1 1 e(ϑ) = ∂ϑ , e(ϕ) = ∂ϕ . (2.22.5b) 3/2 3 2/3 3/2 3 2/3 R √rsτ R √rsτ sinϑ − 2 − 2 74 CHAPTER 2. SPACETIMES

Ricci rotation coefﬁcients:

√rs 2√rs γ = , γ = γ = , (τ)(R)(R) 3/2 (τ)(ϑ)(ϑ) (τ)(ϕ)(ϕ) 3/2 (2.22.6a) −2R 3√rsτ 2R 3√rsτ − − 2/3 3/2 3 − γ = γ = R √rsτ . (2.22.6b) (R)(ϕ)(ϕ) (R)(ϑ)(ϑ) − − 2

The contractions of the Ricci rotation coefﬁcients read 2/3 2/3 3√rs 3 − 3 − γ = , γ = 2 R3/2 r τ , γ = cotϑ R3/2 r τ . (τ) 3/2 (R) √ s (ϑ) √ s (2.22.7) −2R 3√rsτ − 2 − 2 − Riemann-Tensor with respect to local tetrad:

4rs R(τ)(R)(τ)(R) = R(ϑ)(ϕ)(ϑ)(ϕ) = , (2.22.8a) 3/2 2 − − 2R 3√rsτ − 2rs R(τ)(ϑ)(τ)(ϑ) = R(τ)(ϕ)(τ)(ϕ) = R(R)(ϑ)(R)(ϑ) = R(R)(ϕ)(R)(ϕ) = . (2.22.8b) 3/2 2 − − 2R 3√rsτ − The Ricci tensor with respect to the local tetrad vanishes identically.

2.22.2 Inner metric

The metric of the inside, R Rb, reads ≤

4/3 2 2 3 3/2 2 2 2 2 2 ds = dτ + 1 √rsR− τ dR + R dϑ + sin ϑdϕ . (2.22.9) − − 2 b

For the following components, we deﬁne

3 3/2 A := 1 √rsR− τ. (2.22.10) Oin − 2 b

Christoffel symbols:

3/2 3/2 3/2 R √rsR−b ϑ √rsRb− ϕ √rsRb− ΓτR = , Γτϑ = , Γτϕ = , (2.22.11a) − AOin − AOin − AOin τ 1/3 3/2 ϑ 1 ϕ 1 Γ = A √rsR− , Γ = , Γ = , (2.22.11b) RR − Oin b Rϑ R Rϕ R R ϕ τ 1/3 3/2 2 Γ = R, Γ = cotϑ, Γ = A √rsR− R , (2.22.11c) ϑϑ − ϑϕ ϑϑ − Oin b R 2 ϑ τ 1/3 3/2 2 2 Γ = Rsin ϑ, Γ = sinϑ cosϑ, Γ = A √rsR− R sin ϑ. (2.22.11d) ϕϕ − ϕϕ − ϕϕ − Oin b Riemann-Tensor:

2 2 2 1 rs 1 rsR 1 rsR sin ϑ RτRτR = , Rτϑτϑ = , Rτϕτϕ = , (2.22.12a) 2 3 2/3 2 3 2/3 2 3 2/3 RbAOin RbAOin RbAOin 2 2 2 4 2 rsR sin ϑ 2/3 rsR 2/3 rsR sin ϑ 2/3 RRϕRϕ = 3 AOin, RRϑRϑ = 3 AOin, Rϑϕϑϕ = 3 AOin. (2.22.12b) Rb Rb Rb

Ricci-Tensor:

2 2 2 3 rs 3 rs 3 rsR 3 rsR sin ϑ Rττ = , RRR = , Rϑϑ = , Rϕϕ = . (2.22.13) 2 R3A2 2 3 2/3 2 3 2/3 2 3 2/3 b Oin RbAOin RbAOin RbAOin 2.22. OPPENHEIMER-SNYDER COLLAPSE 75

The Ricci and Kretschmann scalars read:

2 3rs rs R = 3 2 , K = 15 6 4 . (2.22.14) RbAOin RbAOin

Local tetrad: 1 1 1 e = ∂ , e = ∂ , e = ∂ , e = ∂ . (τ) τ (R) 2/3 R (ϑ) 2/3 ϑ (ϕ) 2/3 ϕ (2.22.15) AOin RAOin AOinRsinϑ

Ricci rotation coefﬁcients:

3/2 √rsRb− γ(τ)(R)(R) = γ(τ)(ϑ)(ϑ) = γ(τ)(ϕ)(ϕ) = , (2.22.16a) AOin 1 γ = γ = , (2.22.16b) (R)(ϑ)(ϑ) (R)(ϕ)(ϕ) − 2/3 RAOin cotϑ γ = . (2.22.16c) (ϑ)(ϕ)(ϕ) − 2/3 RAOin The contractions of the Ricci rotation coefﬁcients read

3/2 3√rsR− 2 cotϑ γ = b , γ = , γ = . (τ) (R) 2/3 (ϑ) 2/3 (2.22.17) − AOin RAOin RAOin

Riemann-Tensor with respect to local tetrad:

3 rsRb− R(τ)(R)(τ)(R) = R(τ)(ϑ)(τ)(ϑ) = R(τ)(ϕ)(τ)(ϕ) = 2 , (2.22.18a) 2AOin 3 rsRb− R(R)(ϑ)(R)(ϑ) = R(R)(ϕ)(R)(ϕ) = R(ϑ)(ϕ)(ϑ)(ϕ) = 2 . (2.22.18b) AOin

Ricci-Tensor with respect to local tetrad:

3 3rsRb− R(τ)(τ) = R(R)(R) = R(ϑ)(ϑ) = R(ϕ)(ϕ) = 2 . (2.22.19) 2AOin

Further reading: Oppenheimer and Snyder[OS39]. 76 CHAPTER 2. SPACETIMES

2.23 Petrov-Type D – Levi-Civita spacetimes

The Petrov type D static vacuum spacetimes AI-C are taken from Stephani et al.[SKM+03], Sec. 18.6, with the coordinate and parameter ranges given in "Exact solutions of the gravitational ﬁeld equations" by Ehlers and Kundt [EK62].

2.23.1 Case AI In spherical coordinates, (t,r,ϑ,ϕ), the metric is given by the line element

r r b ds2 = r2 dϑ 2 + sin2 ϑdϕ2 + dr2 − dt2. (2.23.1) r b − r −

This is the well known Schwarzschild solution if b = rs, cf. Eq. (2.2.1). Coordinates and parameters are restricted to t R, 0 < ϑ < π, ϕ [0,2π), (0 < b < r) (b < 0 < r). ∈ ∈ ∨ Local tetrad: r r r r b 1 1 e = ∂t , e = − ∂r, e = ∂ , e = ∂ . (2.23.2) (t) r b (r) r (ϑ) r ϑ (ϕ) r sinϑ ϕ − Dual tetrad: r r b r r θ(t) = − dt, θ(r) = dr, θ(ϑ) = r dϑ, θ(ϕ) = r sinϑ dϕ. (2.23.3) r r b − Effective potential: 1 2 1 With the Hamilton-Jacobi formalism it is possible to obtain an effective potential fulﬁlling 2 r˙ + 2Veff(r) = 1 2 2C0 with r b r b V (r) = K − κ − (2.23.4) eff r3 − r and the constants of motion r b2 C2 = t˙2 − , (2.23.5a) 0 r K = ϑ˙ 2r4 + ϕ˙ 2r4 sin2 ϑ. (2.23.5b)

2.23.2 Case AII In cylindrical coordinates, the metric is given by the line element

z b z ds2 = z2 dr2 + sinh2 r dϕ2 + dz2 − dt2. (2.23.6) b z − z − Coordinates and parameters are restricted to t R, 0 < r, ϕ [0,2π), 0 < z < b. ∈ ∈ Local tetrad: r z 1 1 rb z e = ∂ , e = ∂ , e = ∂ , e = − ∂ . (2.23.7) (t) b z t (r) z r (ϕ) zsinhr ϕ (z) z z − Dual tetrad: rb z r z θ(t) = − dt, θ(r) = zdr, θ(ϕ) = zsinhr dϕ, θ(z) = dz. (2.23.8) z b z − 2.23. PETROV-TYPE D – LEVI-CIVITA SPACETIMES 77

2.23.3 Case AIII In cylindrical coordinates, the metric is given by the line element

1 ds2 = z2 dr2 + r2dϕ2 + zdz2 dt2. (2.23.9) − z

Coordinates and parameters are restricted to

t R, 0 < r, ϕ [0,2π), 0 < z. ∈ ∈ Local tetrad: 1 1 1 e = √z∂t , e = ∂r, e = ∂ϕ , e = ∂z. (2.23.10) (t) (r) z (ϕ) zr (z) √z

Dual tetrad: 1 θ(t) = dt, θ(r) = zdr, θ(ϕ) = zr dϕ, θ(z) = √zdz. (2.23.11) √z

2.23.4 Case BI In spherical coordinates, the metric is given by the line element

r r b ds2 = r2 dϑ 2 sin2 ϑdt2 + dr2 + − dϕ2. (2.23.12) − r b r − Coordinates and parameters are restricted to

t R, 0 < ϑ < π, ϕ [0,2π), (0 < b < r) (b < 0 < r). ∈ ∈ ∨ Local tetrad: r 1 r b 1 r r e = ∂t , e = − ∂r, e = ∂ , e = ∂ . (2.23.13) (t) r sinϑ (r) r (ϑ) r ϑ (ϕ) r b ϕ − Dual tetrad: r r r r b θ(t) = r sinϑ dt, θ(r) = dr, θ(ϑ) = r dϑ, θ(ϕ) = − dϕ. (2.23.14) r b r − Effective potential: With the Hamilton-Jacobi formalism, an effective potential for the radial coordinate can be calculated 1 2 1 1 2 fulﬁlling 2 r˙ + 2Veff(r) = 2C0 with r b r b V (r) = K − κ − (2.23.15) eff r3 − r and the constants of motion r b2 C2 = ϕ˙ 2 − , (2.23.16a) 0 r K = ϑ˙ 2r4 t˙2r4 sin2 ϑ. (2.23.16b) − Note that the metric is not spherically symmetric. Particles or light rays fall into one of the poles if they π are not moving in the ϑ = 2 plane. 78 CHAPTER 2. SPACETIMES

2.23.5 Case BII In cylindrical coordinates, the metric is given by the line element

z b z ds2 = z2 dr2 sinh2 r dt2 + dz2 + − dϕ2. (2.23.17) − b z z − Coordinates and parameters are restricted to t R, ϕ [0,2π), 0 < z < b, 0 < r. ∈ ∈ Local tetrad: 1 1 r z rb z e = ∂ , e = ∂ , e = ∂ , e = − ∂ . (2.23.18) (t) zsinhr t (r) z r (ϕ) b z ϕ (z) z z − Dual tetrad: rb z r z θ(t) = zsinhr dt, θ(r) = zdr, θ(ϕ) = − dϕ, θ(z) = dz. (2.23.19) z b z − 2.23.6 Case BIII In cylindrical coordinates, the metric is given by the line element

1 ds2 = z2 dr2 r2dt2 + zdz2 + dϕ2. (2.23.20) − z

Coordinates and parameters are restricted to t R, ϕ [0,2π), 0 < z, 0 < r. ∈ ∈ Local tetrad: 1 1 1 e = ∂t , e = ∂r, e = √z∂ϕ , e = ∂z. (2.23.21) (t) zr (r) z (ϕ) (z) √z Dual tetrad: 1 θ(t) = zr dt, θ(r) = zdr, θ(ϕ) = dϕ, θ(z) = √zdz. (2.23.22) √z

2.23.7 Case C The metric is given by the line element

1 1 1 ds2 = dx2 + f (x)dϕ2 dy2 + f ( y)dt2 (2.23.23) (x + y)2 f (x) − f ( y) − −

with f (u) := (u3 + au + b). Coordinates and parameters are restricted to ± 0 < x + y, f ( y) > 0, 0 > f (x). − Local tetrad:

1 p 3 e(t) = (x + y)p ∂t , e(x) = (x + y) x + ax + b∂x, (2.23.24a) y3 ay + b − − p 3 1 e(y) = (x + y) y ay + b∂y, e(ϕ) = (x + y) ∂ϕ , (2.23.24b) − − √x3 + ax + b 2.23. PETROV-TYPE D – LEVI-CIVITA SPACETIMES 79

Dual tetrad: 1 p 1 1 θ(t) = y3 ay + bdt, θ(x) = dx, (2.23.25a) x + y − − x + y √x3 + ax + b (y) 1 1 (ϕ) 1 p 3 θ = p dy, θ = x + ax + bdϕ, (2.23.25b) x + y y3 ay + b x + y − − A coordinate change can eliminate the linear term in the polynom f generating a quadratic term instead. This brings the line element to the form

1 1 1 ds2 = dx2 + f (x)dp2 dy2 + f ( y)dq2 (2.23.26) A(x + y)2 f (x) − f ( y) − − with f (u) := ( 2mAu3 u2 + 1) given in [PP01]. ± − − Furthermore, coordinates can be adapted to the boost-rotation symmetry with the line element in [PP01] from in [Bon83]

2 1 h ρ 2 2 λ 2i λ 2 2 ρ 2 ds = e r (zdt t dz) e (zdz t dt) e dr r e− dϕ (2.23.27) z2 t2 − − − − − − with 2 ρ R3 + R + Z3 r e = 2 − 2 , 4α (R1 + R + Z1 r ) − 2 2 2 λ 2α R(R + R1 + Z1) Z1r R1R3 + (R + Z1)(R + Z3) (Z1 + Z3)r e = − 2 − , RiR3 [R(R + R3 + Z3) Z3r ] − 1 R = z2 t2 + r2, 2 − q 2 2 Ri = (R + Zi) 2Zir , − Zi = zi z2, − 2 2 1 m α = 6 2 2 , 4 A (z2 z1) (z3 z1) − − 1 q = , 4α2 4 3 2 2 2 and z3 < z1 < z2 the roots of 2A z A z + m . − Local tetrad:

Case z2 t2 > 0: − 1 ρ/2 λ/2 λ/2 e t = qze− ∂t +te− ∂z, , e r = e− ∂r, (2.23.28a) ( ) √z2 t2 ( ) − 1 ρ/2 λ/2 ρ/2 e z = qte− ∂t + ze− ∂z, , e ϕ = re ∂ϕ . (2.23.28b) ( ) √z2 t2 ( ) − Case z2 t2 < 0: − 1 ρ/2 λ/2 λ/2 e t = qte− ∂t + ze− ∂z, , e r = e− ∂r, (2.23.29a) ( ) √t2 z2 ( ) − 1 ρ/2 λ/2 ρ/2 e z = qze− ∂t +te− ∂z, , e ϕ = re ∂ϕ . (2.23.29b) ( ) √t2 z2 ( ) − 80 CHAPTER 2. SPACETIMES

Dual tetrad:

Case z2 t2 > 0: − r eρ 1 θ(t) = (zdt +t dz), θ(r) = eλ dr, (2.23.30a) z2 t2 q s − eλ 1 θ(z) = (t dt + zdz), θ(ϕ) = dϕ. (2.23.30b) z2 t2 reρ − Case z2 t2 > 0: − s eλ θ(t) = (t dt + zdz), θ(r) = eλ dr, (2.23.31a) t2 z2 − r eρ 1 1 θ(z) = (zdt +t dz), θ(ϕ) = dϕ. (2.23.31b) t2 z2 q reρ − 2.24. PLANE GRAVITATIONAL WAVE 81

2.24 Plane gravitational wave

W. Rindler described in [Rin01] an exact plane gravitational wave which is bounded between two planes. The metric of the so called ’sandwich wave’ with u := t x reads −

ds2 = dt2 + dx2 + p2 (u)dy2 + q2 (u)dz2. (2.24.1) −

The functions p(u) and q(u) are given by   p0 = const. u < a q0 = const. u < a  −  − p(u) := 1 u 0 < u and q(u) := 1 u 0 < u , (2.24.2)  − m(u)  − m(u) L(u)e else L(u)e− else where a is the longitudinal extension of the wave. The functions L(u) and m(u) are s u3 u4 Z u2 + au L(u) = 1 u + + , m(u) = 2√3 du. (2.24.3) − a2 2a3 ± 2a3u 2au3 u4 2a3 − − − Christoffel symbols:

y 1 ∂ p ∂q 1 ∂q ∂ p Γ = Γ y = , Γ t = Γ x = q , Γ z = Γ z = , Γ t = Γ x = p . (2.24.4) ty − xy p ∂u zz zz ∂u tz − xz q ∂u yy yy ∂u

Riemann-Tensor: ∂ 2 p ∂ 2q Rtyty = Rxyxy = Rtyxy = p , Rtztz = Rxzxz = Rtzxz = q . (2.24.5) − − ∂u2 − − ∂u2

Local tetrad: 1 1 e = ∂ , e = ∂ , e = ∂ , e = ∂ . (2.24.6) (t) t (x) x (y) p y (z) q z

Dual tetrad:

θ(t) = dt, θ(x) = dx, θ(y) = pdy, θ(z) = qdz. (2.24.7) 82 CHAPTER 2. SPACETIMES

2.25 Reissner-Nordstrøm

The Reissner-Nordstrøm black hole in spherical coordinates t R,r R+,ϑ (0,π),ϕ [0,2π) is de- { ∈ ∈ ∈ ∈ } ﬁned by the metric[MTW73]

2 2 2 1 2 2 2 2 2 ds = A c dt + A− dr + r dϑ + sin ϑ dϕ , (2.25.1) − RN RN where 2 rs ρQ A = 1 + (2.25.2) RN − r r2 2 4 34 with rs = 2GM/c , the charge Q, and ρ = G/(ε0c ) 9.33 10− . As in the Schwarzschild case, there is a 2≈ 2 · true curvature singularity at r = 0. However, for Q < rs /(4ρ) there are also two critical points (horizons) at s 2 rs rs 4ρQ r = 1 2 . (2.25.3) 2 ± 2 − rs

2 2 2 2 Thus, for 0 Q < rs /(4ρ), the system is also called black hole and for Q = rs /(4ρ) extreme black hole. 2 2≤ For Q > rs /(4ρ), there are no horizons and the system is called naked singularity.

Christoffel symbols: 2 2 2 2 r ARNc (rsr 2ρQ ) t rsr 2ρQ r rsr 2ρQ Γtt = 3− , Γtr = −3 , Γrr = −3 , (2.25.4a) 2r 2r ARN − 2r ARN ϑ 1 ϕ 1 r Γ = , Γr = , Γ = rA , (2.25.4b) rϑ r ϕ r ϑϑ − RN Γ ϕ = cotϑ, Γ r = rA sin2 ϑ, Γ ϑ = sinϑ cosϑ. (2.25.4c) ϑϕ ϕϕ − RN ϕϕ − Riemann-Tensor: 2 2 2 2 c (rsr 3ρQ ) ARNc (rsr 2ρQ ) Rtrtr = − , Rt t = − , (2.25.5a) − r4 ϑ ϑ 2r2 2 2 2 2 ARNc (rsr 2ρQ )sin ϑ rsr 2ρQ Rtϕtϕ = − 2 , Rrϑrϑ = −2 , (2.25.5b) 2r − 2r ARN 2 2 (rsr 2ρQ )sin ϑ 2 2 Rrϕrϕ = − 2 , Rϑϕϑϕ = (rsr ρQ )sin ϑ. (2.25.5c) − 2r ARN − Ricci-Tensor: 2 2 2 2 2 2 c ρQ ARN ρQ ρQ ρQ sin ϑ Rtt = 4 , Rrr = 4 , Rϑϑ = 2 , Rϕϕ = 2 . (2.25.6) r −r ARN r r While the Ricci scalar vanishes identically, also because the energy-momentum tensor of the electromagnetic ﬁeld is traceless, the Kretschmann scalar reads 2 2 2 2 4 3r r 12rsrρQ + 14ρ Q K = 4 s − . (2.25.7) r8 Weyl-Tensor: 2 2 2 2 c (rsr 2ρQ ) ARNc (rsr 2ρQ ) Ctrtr = − , Ct t = − , (2.25.8a) − r4 ϑ ϑ 2r2 2 2 2 2 ARNc (rsr 2ρQ )sin ϑ rsr 2ρQ Ctϕtϕ = − 2 , Crϑrϑ = −2 , (2.25.8b) 2r − 2r ARN 2 2 (rsr 2ρQ )sin ϑ 2 2 Crϕrϕ = − 2 , Cϑϕϑϕ = (rsr 2ρQ )sin ϑ. (2.25.8c) − 2r ARN − 2.25. REISSNER-NORDSTRØM 83

Local tetrad: 1 p 1 1 e(t) = ∂t , e(r) = ARN∂r, e(ϑ) = ∂ϑ , e(ϕ) = ∂ϕ . (2.25.9) c√ARN r r sinϑ Dual tetrad:

(t) p (r) dr (ϑ) (ϕ) θ = c ARN dt, θ = , θ = r dϑ, θ = r sinϑ dϕ. (2.25.10) √ARN Ricci rotation coefﬁcients: rr 2ρQ2 √A cotϑ γ s γ γ RN γ (r)(t)(t) = 3− , (ϑ)(r)(ϑ) = (ϕ)(r)(ϕ) = , (ϕ)(ϑ)(ϕ) = . (2.25.11) 2r √ARN r r The contractions of the Ricci rotation coefﬁcients read 4r2 3rr + 2ρQ2 cotϑ γ s γ (r) = − 3 , (ϑ) = . (2.25.12) 2r √ARN r Riemann-Tensor with respect to local tetrad: 2 2 rsr 3ρQ rsr ρQ R = − , R = − , (2.25.13a) (t)(r)(t)(r) − r4 (ϑ)(ϕ)(ϑ)(ϕ) r4 2 rsr 2ρQ R = R = R = R = − . (2.25.13b) (t)(ϑ)(t)(ϑ) (t)(ϕ)(t)(ϕ) − (r)(ϑ)(r)(ϑ) − (r)(ϕ)(r)(ϕ) 2r4 Ricci-Tensor with respect to local tetrad: ρQ2 R = R = R = R = . (2.25.14) (t)(t) − (r)(r) (ϑ)(ϑ) (ϕ)(ϕ) r4 Weyl-Tensor with respect to local tetrad: 2 rsr 2ρQ C = C = − , (2.25.15a) (t)(r)(t)(r) − (ϑ)(ϕ)(ϑ)(ϕ) − r4 2 rsr 2ρQ C = C = C = C = − . (2.25.15b) (t)(ϑ)(t)(ϑ) (t)(ϕ)(t)(ϕ) − (r)(ϑ)(r)(ϑ) − (r)(ϕ)(r)(ϕ) 2r4 Embedding: The embedding function follows from the numerical integration of s dz 1 = 2 2 1. (2.25.16) dr 1 rs/r + ρQ /r − −

Euler-Lagrange: The Euler-Lagrangian formalism, Sec. 1.8.4, for geodesics in the ϑ = π/2 hyperplane yields 2 2 2 1 1 k 1 rs ρQ h r˙2 +V = , V = 1 + κc2 (2.25.17) 2 eff 2 c2 eff 2 − r r2 r2 − 2 2 with constants of motion k = ARNc t˙ and h = r ϕ˙. For null geodesics, κ = 0, there are two extremal points

s ! 3 32ρQ2 r = rs 1 1 2 , (2.25.18) ± 4 ± − 9rs where r+ is a maximum and r a minimum. −

Further reading: Eiroa[ERT02] 84 CHAPTER 2. SPACETIMES

2.26 de Sitter spacetime

The de Sitter spacetime with Λ > 0 is a solution of the Einstein ﬁeld equations with constant curvature. A detailed discussion can be found for example in Hawking and Ellis[HE99]. Here, we use the coordinate transformations given by Biˇcák[BK01].

2.26.1 Standard coordinates The de Sitter metric in standard coordinates τ R, χ [ π,π],ϑ (0,π),ϕ [0,2π) reads { ∈ ∈ − ∈ ∈ } τ ds2 = dτ2 + α2 cosh2 dχ2 + sin2 χ dϑ 2 + sin2 ϑ dϕ2, (2.26.1) − α

where α2 = 3/Λ. Christoffel symbols: 1 τ 1 τ 1 τ Γ χ = tanh , Γ ϑ = tanh , Γ ϕ = tanh , (2.26.2a) τχ α α τϑ α α τϕ α α τ τ Γ τ = α sinh cosh , Γ ϑ = cot χ, Γ ϕ = cot χ, (2.26.2b) χχ α α χϑ χϕ τ τ Γ τ = α sin2 χ sinh cosh , Γ χ = sin χ cos χ, Γ ϕ = cotϑ, (2.26.2c) ϑϑ α α ϑϑ − ϑϕ τ τ Γ τ = α sin2 χ sin2 ϑ sinh cosh ,Γ χ = sin2 ϑ sin χ cos χ,Γ ϑ = sinϑ cosϑ. (2.26.2d) ϕϕ α α ϕϕ − ϕϕ − Riemann-Tensor: 2 τ 2 τ 2 Rτχτχ = cosh , R = cosh sin χ, (2.26.3a) − α τϑτϑ − α 2 2 τ 2 2 2 2 τ 2 Rτϕτϕ = cosh sin χ sin ϑ, R = α 1 + sinh sin χ, (2.26.3b) − α χϑ χϑ α τ 2 τ 2 R = α2 1 + sinh2 sin2 χ sin2 ϑ, R = α2 1 + sinh2 sin4 χ sin2 ϑ. (2.26.3c) χϕχϕ α ϑϕϑϕ α Ricci-Tensor:

3 2 τ 2 τ 2 2 τ 2 2 Rττ = , Rχχ = 3cosh , R = 3cosh sin χ, Rϕϕ = 3cosh sin χ sin ϑ. (2.26.4) −α2 α ϑϑ α α Ricci and Kretschmann scalars: 12 24 R = , K = . (2.26.5) α2 α4 Local tetrad: 1 1 1 e(τ) = ∂τ , e(χ) = τ ∂χ , e(ϑ) = τ ∂ϑ , e(ϕ) = τ ∂ϕ . (2.26.6) α cosh α α cosh α sin χ α cosh α sin χ sinϑ Dual tetrad: τ τ τ θ(τ) = dτ, θ(χ) = α cosh dχ, θ(ϑ) = α cosh sin χ dϑ, θ(ϕ) = α cosh sin χ sinϑ dϕ. (2.26.7) α α α

2.26.2 Conformally Einstein coordinates In conformally Einstein coordinates η [0,π], χ [ π,π],ϑ [0,π],ϕ [0,2π) , the de Sitter metric { ∈ ∈ − ∈ ∈ } reads

α2 ds2 = dη2 + dχ2 + sin2 χ dϑ 2 + sin2 ϑ dϕ2. (2.26.8) sin2 η − 2.26. DE SITTER SPACETIME 85

It follows from the standard form (2.26.1) by the transformation η = 2arctan eτ/α . (2.26.9)

2.26.3 Conformally ﬂat coordinates Conformally ﬂat coordinates T R,r R,ϑ (0,π),ϕ [0,2π) follow from conformally Einstein co- { ∈ ∈ ∈ ∈ } ordinates by means of the transformations

α sinη α sin χ 2Tα 2rα T = , r = , or η = arctan , χ = arctan . (2.26.10) cos χ + cosη cos χ + cosη α2 T 2 + r2 α2 + T 2 r2 − − For the transformation (T,R) (η, χ), we have to take care of the coordinate domains. In that case, if → κ2 T 2 + r2 < 0, we have to map η η + π. On the other hand, if κ2 + T 2 r2 < 0, we have to consider − → − the sign of r. If r > 0, then χ χ + π, otherwise χ χ π. → → − The resulting metric reads

α2 ds2 = dT 2 + dr2 + r2 dϑ 2 + sin2 ϑ dϕ2. (2.26.11) T 2 −

Note that we identify points (r < 0,ϑ,ϕ) with (r > 0,π ϑ,ϕ π). − − Christoffel symbols:

2 T r ϑ ϕ T 1 ϑ ϕ 1 T r r Γ = Γ = Γ = Γ = Γ = , Γ = Γr = , Γ = , Γ = r, (2.26.12a) TT Tr Tϑ Tϕ rr −T rϑ ϕ r ϑϑ − T ϑϑ − r2 sin2 ϑ Γ ϕ = cotϑ, Γ T = , Γ r = r sin2 ϑ, Γ ϑ = sinϑ cosϑ. (2.26.12b) ϑϕ ϕϕ − T ϕϕ − ϕϕ −

Riemann-Tensor: α2 α2r2 α2r2 sin2 ϑ RTrTr = , RT T = , RTϕTϕ = , (2.26.13a) − T 4 ϑ ϑ − T 4 − T 4 α2r2 α2r2 sin2 ϑ α2r4 sin2 ϑ R = , R = , R = . (2.26.13b) rϑrϑ T 4 rϕrϕ T 4 ϑϕϑϕ T 4

Ricci-Tensor: 3 3 3r2 3r2 sin2 ϑ RTT = , Rrr = , R = , Rϕϕ = . (2.26.14) −T 2 T 2 ϑϑ T 2 T 2 The Ricci and Kretschmann scalar read: 12 24 R = , K = . (2.26.15) α2 α4

Local tetrad: T T T T e = ∂T , e = ∂r, e = ∂ , e = ∂ . (2.26.16) (T) α (r) α (ϑ) αr ϑ (ϕ) αr sinϑ ϕ

2.26.4 Static coordinates The de Sitter metric in static spherical coordinates t R,r R+,ϑ (0,π),ϕ [0,2π) reads { ∈ ∈ ∈ ∈ }

1 Λ Λ − ds2 = 1 r2 c2dt2 + 1 r2 dr2 + r2 dϑ 2 + sin2 ϑ dϕ2. (2.26.17) − − 3 − 3 86 CHAPTER 2. SPACETIMES

It follows from the conformally Einstein form (2.26.8) by the transformations

α cos χ cosη sin χ t = ln − , r = α . (2.26.18) 2 cos χ + cosη sinη

Christoffel symbols:

(Λr2 3) Λr Λr Γ r = − c2Λr, Γ t = , Γ r = , (2.26.19a) tt 9 tr Λr2 3 rr 3 Λr2 − − 1 1 (Λr2 3)r Γ ϑ = , Γ φ = , Γ r = − , (2.26.19b) rϑ r rφ r ϑϑ 3 Λr2 3 Γ φ = cot(ϑ), Γ r = − r sin2(ϑ), Γ ϑ = sin(ϑ)cos(ϑ). (2.26.19c) ϑφ φφ 3 φφ −

Riemann-Tensor:

2 2 Λ 2 3 Λr 2 2 3 Λr 2 2 2 Rtrtr = c , Rt t = − c Λr , Rtϕtϕ = − c Λr sin(ϑ) , (2.26.20a) − 3 ϑ ϑ − 9 − 9 Λr2 Λr2 sin(θ)2 r4 sin2(θ)Λ R = , Rr r = , R = . (2.26.20b) rϑrϑ Λr2 + 3 ϕ ϕ Λr2 + 3 ϑϕϑϕ 3 − − Ricci-Tensor:

2 Λr 3 2 3Λ 2 2 2 Rtt = − c Λ, Rrr = , R = Λr , R = r sin (ϑ)Λ. (2.26.21) 3 3 Λr2 ϑϑ ϕϕ − The Ricci scalar and Kretschmann scalar read: 8 R = 4Λ, K = Λ 2. (2.26.22) 3

Local tetrad:

r r 2 3 ∂t Λr 1 1 e = , e = 1 ∂r, e = ∂ , e = ∂ϕ . (2.26.23) (t) 3 Λr2 c (r) − 3 (ϑ) r ϑ (ϕ) r sin(ϑ) − Ricci rotation coefﬁcients:

Λr √9 3Λr2 cotϑ γ(t)(r)(t) = , γ(ϑ)(r)(ϑ) = γ(ϕ)(r)(ϕ) = − , γ(ϕ)(ϑ)(ϕ) = . (2.26.24) −√9 3Λr2 3r r − The contractions of the Ricci rotation coefﬁcients read

√9 3Λr2(Λr2 2) cotϑ γ = − − , γ = . (2.26.25) (r) (Λr2 3)r (ϑ) r − Riemann-Tensor with respect to local tetrad:

1 R = R = R = R = R = R = Λ. (2.26.26) − (t)(r)(t)(r) − (t)(ϑ)(t)(ϑ) − (t)(ϕ)(t)(ϕ) (r)(ϑ)(r)(ϑ) (r)(ϕ)(r)(ϕ) (ϑ)(ϕ)(ϑ)(ϕ) 3

Ricci-Tensor with respect to local tetrad:

R = R = R = R = Λ. (2.26.27) − (t)(t) (r)(r) (ϑ)(ϑ) (ϕ)(ϕ) 2.26. DE SITTER SPACETIME 87

2.26.5 Lemaître-Robertson form The de Sitter universe in the Lemaître-Robertson form reads

ds2 = c2dt2 + e2Ht dr2 + r2 dϑ 2 + sin2 ϑ dϕ2, (2.26.28) − q Λc2 c with Hubble’s Parameter H = 3 = α , which is assumed here to be time-independent. This a special case of the ﬁrst and second form of the Friedman-Robertson-Walker metric deﬁned in Eqs. (2.11.2) and (2.11.12) with R(t) = eHt and k = 0. Christoffel symbols:

r ϑ ϕ Γtr = H, Γtϑ = H, Γtϕ = H, (2.26.29a) e2Ht H 1 1 Γ t = , Γ ϑ = , Γ ϕ = , (2.26.29b) rr c2 rϑ r rϕ r e2Ht r2H Γ t = , Γ r = r, Γ ϕ = cot(ϑ), (2.26.29c) ϑϑ c2 ϑϑ − ϑϕ e2Ht r2 sin2(θ)H Γ t = , Γ r = r sin(ϑ)2, Γ ϑ = sin(ϑ)cos(ϑ). (2.26.29d) ϕϕ c2 ϕϕ − ϕϕ − Riemann-Tensor: 2Ht 2 2Ht 2 2 Rtrtr = e H , Rt t = e r H , (2.26.30a) − ϑ ϑ − 4Ht 2 2 2Ht 2 2 2 e r H Rtϕtϕ = e r sin (ϑ)H , Rr r = , (2.26.30b) − ϑ ϑ c2 e4Ht r2 sin2(ϑ)H2 e4Ht r4 sin2(ϑ)H2 R = , R = . (2.26.30c) rϕrϕ c2 ϑϕϑϕ c2 Ricci-Tensor: 2Ht 2 2Ht 2 2 2Ht 2 2 2 2 e H e r H e r sin (ϑ)H Rtt = 3H , Rrr = 3 , R = 3 , Rϕϕ = 3 . (2.26.31) − c2 ϑϑ c2 c2 Ricci and Kretschmann scalars: 12H2 24H4 R = , K = . (2.26.32) c2 c4 Local tetrad: Ht Ht 1 Ht e− e− e = ∂t , e = e ∂r, e = ∂ , e = ∂ . (2.26.33) (t) c (r) − (ϑ) r ϑ (ϕ) r sinϑ ϕ Ricci rotation coefﬁcients: H γ = γ = γ = (2.26.34a) (r)(t)(r) (ϑ)(t)(ϑ) (ϕ)(t)(ϕ) c 1 cot(θ) γ = γ = , γ = . (2.26.34b) (ϑ)(r)(ϑ) (ϕ)(r)(ϕ) eHt r (ϕ)(ϑ)(ϕ) eHt r The contractions of the Ricci rotation coefﬁcients read H 2 cot(θ) γ = 3 , γ = ,γ = . (2.26.35) (t) c (r) eHt r (ϑ) eHt r Riemann-Tensor with respect to local tetrad: H2 R = R = R = (2.26.36a) (t)(r)(t)(r) (t)(ϑ)(t)(ϑ) (t)(ϕ)(t)(ϕ) − c2 H2 R = R = R = . (2.26.36b) (r)(ϑ)(r)(ϑ) (r)(ϕ)(r)(ϕ) (ϑ)(ϕ)(ϑ)(ϕ) c2 88 CHAPTER 2. SPACETIMES

Ricci-Tensor with respect to local tetrad: H2 R = R = R = R = 3 . (2.26.37) − (t)(t) (r)(r) (ϑ)(ϑ) (ϕ)(ϕ) c2

2.26.6 Cartesian coordinates The de Sitter universe in Lemaître-Robertson form can also be expressed in Cartesian coordinates:

ds2 = c2dt2 + e2Ht dx2 + dy2 + dz2. (2.26.38) −

Christoffel symbols: x y z Γtx = H, Γty = H, Γtz = H, (2.26.39a) e2Ht H e2Ht H e2Ht H Γ t = , Γ t = , Γ t = . (2.26.39b) xx c2 yy c2 zz c2 (2.26.39c) Partial derivatives 2H2e2Ht Γ t = Γ t = Γ t = . (2.26.40) xx,t yy,t zz,t c2 Riemann-Tensor: 4Ht 2 2Ht 2 e H Rtxtx = Rtxtx = Rtztz = e H , Rxyxy = Rxzxz = Ryzyz = . (2.26.41) − c2 Ricci-Tensor: 2Ht 2 2 e H Rtt = 3H , Rxx = Ryy = Rzz = 3 . (2.26.42) − c2 The Ricci and Kretschmann scalar read: H2 H4 R = 12 , K = 24 . (2.26.43) c2 c4 Local tetrad: 1 e = ∂ , e = e Ht ∂ , e = e Ht ∂ , e = e Ht ∂ . (2.26.44) (t) c t (x) − x (y) − y (z) − z Ricci rotation coefﬁcients: H γ = γ = γ = . (2.26.45) (x)(t)(x) (y)(t)(y) (z)(t)(z) c The only non-vanishing contraction of the Ricci rotation coefﬁcients read H γ = 3 . (2.26.46) (t) c Riemann-Tensor with respect to local tetrad: H2 R = R = R = , (2.26.47a) (t)(x)(t)(x) (t)(y)(t)(y) (t)(z)(t)(z) − c2 H2 R = R = R = . (2.26.47b) (x)(y)(x)(y) (x)(z)(x)(z) (y)(z)(y)(z) c2 Ricci-Tensor with respect to local tetrad: H2 R = R = R = R = 3 . (2.26.48) − (t)(t) (x)(x) (y)(y) (z)(z) c2 Further reading: Tolman[Tol34, sec. 142], Biˇcák[BK01] 2.27. STATIONARY AXISYMMETRIC SPACETIMES IN WEYL COORDINATES 89

2.27 Stationary axisymmetric spacetimes in Weyl Coordinates

Stationary axisymmetric spacetimes in isotropic or Weyl coordinates (t,ρ,ϕ,z) read[SKM+03]eq(19.21)

2 2U(ρ,z) h 2k(ρ,z) 2 2 2 2i 2U(ρ,z) 2 ds = e− e dρ + dz + ρ dϕ e (dt + A(ρ,z)dϕ) , (2.27.1) − where (G = c = 1). Metric-Tensor:

2U(ρ,z) 2U(ρ,z) 2U(ρ,z) 2 2 2U(ρ,z) gtt = e , gtϕ = e A(ρ,z), gϕϕ = e A(ρ,z) + ρ e− , (2.27.2a) − − − 2k(ρ,z) 2U(ρ,z) gρρ = gzz = e − . (2.27.2b)

Christoffel symbols:

ρ 4U 2k z 4U 2k Γtt = e − ∂ρU, Γtt = e − ∂zU, (2.27.3a) 4U ϕ e t 1 2 4U Γ = ∂ρ A, Γ = 2ρ ∂ρU + Ae ∂ρ A , (2.27.3b) tρ −2ρ2 tρ 2ρ2 1 1 Γ ρ = e4U 2k 2A∂ U + ∂ A, Γ z = e4U 2k (2A∂ U + ∂ A), (2.27.3c) tϕ 2 − ρ ρ tϕ 2 − z z t 1 2 4U ϕ 1 4U Γ = 2ρ ∂zU + Ae ∂zA , Γ = e ∂zA, (2.27.3d) tz 2ρ2 tz −2ρ2 ρ z Γ = ∂ρU + ∂ρ k, Γ = ∂zU ∂zk, (2.27.3e) ρρ − ρρ − ρ z Γ z = ∂zU + ∂zk, Γ = ∂ρU + ∂ρ k, (2.27.3f) ρ − ρz − ϕ 1 4U 2 ϕ 1 2 4U Γ = Ae ∂ρ A + 2ρ ∂ρU 2ρ , Γ z = 2ρ ∂zU + Ae ∂zA , (2.27.3g) ρϕ −2ρ2 − ϕ −2ρ2 ρ z Γ = ∂ρU ∂ρ k, Γ = ∂zU + ∂zk, (2.27.3h) zz − zz − t 1 2 2 2 4U Γ = 4ρ A∂ρU + ρ ∂ρ A + A e ∂ρ A 2Aρ , (2.27.3i) ρϕ 2ρ2 − ρ 2k 2 2 4U 4U Γ = e− ρ ∂ρU ρ + A e ∂ρU + Ae ∂ρ A , (2.27.3j) ϕϕ − z 2k 2 2 4U 4U Γϕϕ = e− ρ ∂zU + A e ∂zU + Ae ∂zA , (2.27.3k) t 1 2 2 2 4U Γ = 4ρ A∂zU + ρ ∂zA + A e ∂zA . (2.27.3l) ϕz 2ρ2

Comoving local tetrad: s gϕϕ gtϕ U k 1 U k e(0) = 2 ∂t ∂ϕ , e(1) = e − ∂ρ , e(2) = ∂ϕ , e(3) = e − ∂z. (2.27.4) g gtt gϕϕ − gϕϕ √gϕϕ tϕ −

Static local tetrad: U U U k e U k e = e− ∂t , e = e − ∂ρ , e = A∂t + ∂ϕ , e = e − ∂z. (2.27.5) (0) (1) (2) ρ − (3) 90 CHAPTER 2. SPACETIMES

2.28 Straight spinning string

The metric of a straight spinning string in cylindrical coordinates (t,ρ,ϕ,z) reads

ds2 = (cdt adϕ)2 + dρ2 + k2ρ2dϕ2 + dz2, (2.28.1) − −

where a R and k > 0 are two parameters, see Perlick[Per04]. ∈ Metric-Tensor:

2 2 2 2 gtt = c , gtϕ = ac, gρρ = gzz = 1, gϕϕ = k ρ a . (2.28.2) − − Christoffel symbols: a 1 Γ t = , Γ ϕ = , Γ ρ = k2ρ. (2.28.3) ρϕ cρ ρϕ ρ ϕϕ −

Partial derivatives α 1 Γ t = , Γ ϕ = , Γ ρ = k2. (2.28.4) ρϕ,ρ −cρ2 ρϕ,ρ −ρ2 ϕϕ,ρ −

The Riemann-, Ricci-, and Weyl-tensors as well as the Ricci- and Kretschmann-scalar vanish identically. Static local tetrad: 1 1 a e = ∂t , e = ∂ , e = ∂t + ∂ , e = ∂z. (2.28.5) (0) c (1) ρ (2) kρ c ϕ (3)

Dual tetrad:

θ(0) = cdt adϕ, θ(1) = dρ, θ(2) = kρ dϕ, θ(3) = dz. (2.28.6) − Ricci rotation coefﬁcients and their contractions read 1 1 γ = , γ = γ = γ = 0, γ = . (2.28.7) (2)(1)(2) ρ (0) (2) (3) (1) ρ

Comoving local tetrad: p k2ρ2 a2 1 a e = − ∂t ∂ϕ , e = ∂ρ , (2.28.8a) (0) kρ c − k2ρ2 a2 (1) − 1 e(2) = p ∂ϕ , e(3) = ∂z. (2.28.8b) k2ρ2 a2 − Dual tetrad: q (0) kρ (1) (2) acdt 2 2 2 (3) θ = p cdt, θ = dρ, θ = p + k ρ a dϕ, θ = dz. (2.28.9) k2ρ2 a2 k2ρ2 a2 − − − Ricci rotation coefﬁcients and their contractions read a2 ak γ = , γ = γ = γ = , (2.28.10a) (0)(1)(0) ρ (k2ρ2 a2) (2)(1)(0) (0)(2)(1) (0)(1)(2) k2ρ2 a2 − − k2ρ γ = , (2.28.10b) (2)(1)(2) k2ρ2 a2 − 1 γ = . (2.28.10c) (1) ρ 2.28. STRAIGHT SPINNING STRING 91

Euler-Lagrange: The Euler-Lagrangian formalism, Sec. 1.8.4, for geodesics in the ϑ = π/2 hyperplane yields

2 2 2 1 ah1 2 h1 ρ˙ + h2 κc = , (2.28.11) k2ρ2 − c − c2

2 2 with the constants of motion h1 = c(ct˙ aϕ˙) and h2 = a(ct˙ aϕ˙) + k ρ ϕ˙. − − The point of closest approach ρ for a null geodesic that starts at ρ = ρi with y = e + cosξe + pca ± (0) (1) sinξe(2) with respect to the static tetrad is given by ρ = ρi sinξ. Hence, the ρpca is independent of a and k. The same is also true for timelike geodesics. 92 CHAPTER 2. SPACETIMES

2.29 Sultana-Dyer spacetime

The Sultana-Dyer metric represents a black hole in the Einstein-de Sitter universe. In spherical coordinates (t,r,ϑ,ϕ), the metric reads[SD05] (G = c = 1)

2M 4M 2M ds2 = t4 1 dt2 dt dr 1 + dr2 r2 dΩ 2 , (2.29.1) − r − r − r −

where M is the mass of the black hole and Ω 2 = dϑ 2 + sin2 ϑdϕ2 is the spherical surface element. Note that here, the signature of the metric is sign(g) = 2. − Christoffel symbols: 2 r3 + 4M2r + M2t M(r 2M)(4r +t) M(r + 2M)(4r +t) Γ t = , Γ r = − , Γ t = , (2.29.2a) tt tr3 tt tr3 tr tr3 2r3 4M2r M2t 2 2 Γ r = − − , Γ ϑ = , Γ ϕ = , (2.29.2b) tr tr3 tϑ t tϕ t 1 1 2r2 + 2Mr Mt Γ ϑ = , Γ ϕ = , Γ t = − , (2.29.2c) rϑ r rϕ r ϑϑ t 4Mr +tr 2Mt Γ r = − , Γ ϕ = cotϑ, Γ ϑ = sinϑ cosϑ, (2.29.2d) ϑϑ − t ϑϕ ϕϕ −

2r3 + 4Mr2 + 4M2r + M2t + Mtr M 4r2 + 8Mr + 2Mt +tr Γ t = , Γ r = , (2.29.2e) rr tr3 rr − tr3 2 2 r2 + 2Mr Mt sin ϑ (4Mr +tr 2Mt)sin2 ϑ Γ t = − , Γ r = − . (2.29.2f) ϕϕ t ϕϕ − t Riemann-Tensor: 2t2 2Mr2 r3 + Mt2 + 2Mtr R = − − , (2.29.3a) trtr r3 t2 2r4 + 16M2r2 + 4Mtr2 4M2r2t + Mt2r 2M2t2 Rr t = − − , (2.29.3b) ϑ ϑ − r2 2Mt2(4r +t)(r2 + 2Mr Mt) Rt r = − , (2.29.3c) ϑ ϑ − r2 t2 sin2 ϑ 2r4 + 16M2r2 + 4Mtr2 4M2r2t + Mt2r 2M2t2 Rrϕtϕ = − − , (2.29.3d) − r2 2Mt2 sin2 ϑ(4r +t)(r2 + 2Mr Mt) Rtϕrϕ = − , (2.29.3e) − r2 t2 4r4 + 16Mr4 4M2tr + 16M2r2 2M2t2 Mt2r Rr r = − − − , (2.29.3f) ϑ ϑ − r2 t2 sin2 ϑ 4r4 + 16Mr4 4M2tr + 16M2r2 2M2t2 Mt2r Rrϕrϕ = − − − , (2.29.3g) − r2 R = 2t2r sin2 ϑ 2r3 + 4Mr2 4Mtr + mt2. (2.29.3h) ϑϕϑϕ − − Ricci-Tensor: 2 3r2 + 12M2 + 2Mt 4M (3r +t + 6M) R = , R = , (2.29.4a) tt t2r2 tr t2r2 23r2 + 12Mr + 2Mt + 12M2 6r2 + 2Mr 2Mt R = , R = − , (2.29.4b) rr t2r2 ϑϑ t2 6r2 + 2Mr 2Mtsin2 ϑ R = − . (2.29.4c) ϕϕ t2 2.29. SULTANA-DYER SPACETIME 93

Ricci and Kretschmann scalars: 12r2 + 2Mr 2Mt R = − , (2.29.5a) − t6r2 48M2t4 + 20M2r4 + 20Mr5 + 8M2r2t2 4Mr4t 16M2r3t + 5r6 K = − − . (2.29.5b) t12r6

Comoving local tetrad: p 1 + 2M/r 2M/r 1 1 1 e(0) = 2 ∂t p ∂r, e(1) = p ∂r, e(2) = 2 ∂ϑ , e(3) = 2 ∂ϕ . (2.29.6) t − t2 1 + 2M/r t2 1 + 2M/r t r t r sinϑ

Static local tetrad: p 1 2M/r 1 2M/r 1 1 e(0) = p ∂t , e(1) = p ∂t + −2 ∂r, e(2) = 2 ∂ϑ , e(3) = 2 ∂ϕ . (2.29.7) t2 1 2M/r t2 1 2M/r t t r t r sinϑ − − Further reading: Sultana and Dyer[SD05]. 94 CHAPTER 2. SPACETIMES

2.30 TaubNUT

The TaubNUT metric in Boyer-Lindquist like spherical coordinates (t,r,ϑ,ϕ) reads[BCJ02] (G = c = 1)

∆ dr2 ds2 = (dt + 2`cosϑ dϕ)2 + Σ + dϑ 2 + sin2 ϑ dϕ2 , (2.30.1) − Σ ∆

where Σ = r2 + `2 and ∆ = r2 2Mr `2. Here, M is the mass of the black hole and ` the magnetic − − monopol strength. Christoffel symbols:

∆ρ ρ ∆ Γ r = , Γ t = , Γ t = 2`2 cosϑ , (2.30.2a) tt Σ 3 tr ∆Σ tϑ − Σ 2 `∆ 2`ρ∆ cosϑ `∆ sinϑ Γ ϕ = , Γ r = , Γ ϑ = , (2.30.2b) tϑ Σ 2 sinϑ tϕ Σ 3 tϕ − Σ 2 r ρ ϑ r ϕ r r r∆ Γ = , Γ = , Γr = , Γ = , (2.30.2c) rr −Σ∆ rϑ Σ ϕ Σ ϑϑ − Σ

2`(r3 3Mr2 3r`2 + M`2)cosϑ Γ t = − − − , (2.30.2d) rϕ Σ∆ `cos2 ϑ 6r2`2 8`2Mr 3`4 + r4 + Σ 2 Γ t = − − , (2.30.2e) ϑϕ − Σ 2 sinϑ ∆ h i Γ r = cos2 ϑ 9r`4 + 4`2Mr2 4`4M + r5 + 2r3`2 rΣ 2 , (2.30.2f) ϕϕ Σ 3 − − 4r2`2 4Mr`2 `4 + r4cotϑ Γ ϕ = − − , (2.30.2g) ϑϕ Σ 2 6r2`2 8Mr`2 3`4 + r4sinϑ cosϑ Γ ϑ = − − , (2.30.2h) ϕϕ − Σ 2 where ρ = 2r`2 + Mr2 M`2. − Static local tetrad: r r Σ ∆ 1 2`cotϑ 1 e(0) = ∂t , e(1) = ∂r, e(2) = ∂ϑ , e(3) = ∂t + ∂ϕ . (2.30.3) ∆ Σ √Σ − √Σ √Σ sinϑ Dual tetrad: r r ∆ Σ θ(0) = (dt + 2`cosϑ dϕ), θ(1) = dr, θ(2) = √Σdϑ, θ(3) = √Σ sinϑ dϕ. (2.30.4) Σ ∆

Euler-Lagrange: The Euler-Lagrangian formalism, Sec. 1.8.4, for geodesics in the ϑ = π/2 hyperplane yields

1 1 k2 1 ∆ h2 r˙2 +V = , V = κ (2.30.5) 2 eff 2 c2 eff 2 Σ Σ −

with the constants of motion k = (∆/Σ)t˙ and h = Σϕ˙. For null geodesics, we obtain a photon orbit at r = rpo with p 2 2 1 M rpo = M + 2 M + ` cos arccos (2.30.6) 3 √M2 + `2

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