arXiv:0904.4184v3 [gr-qc] 4 Nov 2010

Catalogue of Spacetimes

e2 e1 x1 = 2

q

x1 = 1

x2 = 2

∂ 11 00 00 11 ∂x11 00 00 11

x2

x2 = 1

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://www.vis.uni-stuttgart.de/~muelleta/CoS

Date:

04. Nov 2010

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

Contents 1

2

Introduction and Notation 1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Basic objects of a metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Natural local tetrad and initial conditions for geodesics . . . . . . . . . 1.4.1 Orthonormality condition . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Tetrad transformations . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Ricci rotation-, connection-, and structure coefficients . . . . . . 1.4.4 Riemann-, Ricci-, and Weyl-tensor with respect to a local tetrad 1.4.5 Null or timelike directions . . . . . . . . . . . . . . . . . . . . . . 1.4.6 Local tetrad for diagonal metrics . . . . . . . . . . . . . . . . . . 1.4.7 Local tetrad for stationary axisymmetric spacetimes . . . . . . . 1.5 Newman-Penrose tetrad and spin-coefficients . . . . . . . . . . . . . . . 1.6 Coordinate relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Spherical and Cartesian coordinates . . . . . . . . . . . . . . . . 1.6.2 Cylindrical and Cartesian coordinates . . . . . . . . . . . . . . . 1.7 Embedding diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Equations of motion and transport equations . . . . . . . . . . . . . . . 1.8.1 Geodesic equation . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.2 Fermi-Walker transport . . . . . . . . . . . . . . . . . . . . . . . 1.8.3 Parallel transport . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.4 Euler-Lagrange formalism . . . . . . . . . . . . . . . . . . . . . . 1.8.5 Hamilton formalism . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.1 Maple/GRTensorII . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.2 Mathematica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.3 Maxima . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1 1 1 2 3 3 4 4 5 5 5 6 6 7 7 8 8 9 9 9 9 10 10 10 10 10 11 13

Spacetimes 2.1 Minkowski . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Cartesian coordinates . . . . . . . . . . . . . . . 2.1.2 Cylindrical coordinates . . . . . . . . . . . . . . 2.1.3 Spherical coordinates . . . . . . . . . . . . . . . . 2.1.4 Conform-compactified coordinates . . . . . . . . 2.1.5 Rotating coordinates . . . . . . . . . . . . . . . . 2.1.6 Rindler coordinates . . . . . . . . . . . . . . . . . 2.2 Schwarzschild spacetime . . . . . . . . . . . . . . . . . . 2.2.1 Schwarzschild coordinates . . . . . . . . . . . . 2.2.2 Schwarzschild in pseudo-Cartesian coordinates 2.2.3 Isotropic coordinates . . . . . . . . . . . . . . . . 2.2.4 Eddington-Finkelstein . . . . . . . . . . . . . . . 2.2.5 Kruskal-Szekeres . . . . . . . . . . . . . . . . . . 2.2.6 Tortoise coordinates . . . . . . . . . . . . . . . .

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14 14 14 14 15 15 16 17 18 18 20 20 22 23 24

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ii

CONTENTS

2.3 2.4 2.5 2.6 2.7 2.8 2.9

2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18

2.19 2.20 2.21

2.22 2.23 2.24

2.2.7 Painlevé-Gullstrand . . . . . . . . 2.2.8 Israel coordinates . . . . . . . . . . Alcubierre Warp . . . . . . . . . . . . . . . Barriola-Vilenkin monopol . . . . . . . . . Bertotti-Kasner . . . . . . . . . . . . . . . Bessel gravitational wave . . . . . . . . . 2.6.1 Cylindrical coordinates . . . . . . 2.6.2 Cartesian coordinates . . . . . . . Cosmic string in Schwarzschild spacetime Ernst spacetime . . . . . . . . . . . . . . . Friedman-Robertson-Walker . . . . . . . . 2.9.1 Form 1 . . . . . . . . . . . . . . . . 2.9.2 Form 2 . . . . . . . . . . . . . . . . 2.9.3 Form 3 . . . . . . . . . . . . . . . . Gödel Universe . . . . . . . . . . . . . . . 2.10.1 Cylindrical coordinates . . . . . . 2.10.2 Scaled cylindrical coordinates . . . Halilsoy standing wave . . . . . . . . . . Janis-Newman-Winicour . . . . . . . . . . Kasner . . . . . . . . . . . . . . . . . . . . Kerr . . . . . . . . . . . . . . . . . . . . . . 2.14.1 Boyer-Lindquist coordinates . . . Kottler spacetime . . . . . . . . . . . . . . Morris-Thorne . . . . . . . . . . . . . . . . Oppenheimer-Snyder collapse . . . . . . . 2.17.1 Outer metric . . . . . . . . . . . . . 2.17.2 Inner metric . . . . . . . . . . . . . Petrov-Type D – Levi-Civita spacetimes . 2.18.1 Case AI . . . . . . . . . . . . . . . . 2.18.2 Case AII . . . . . . . . . . . . . . . 2.18.3 Case AIII . . . . . . . . . . . . . . . 2.18.4 Case BI . . . . . . . . . . . . . . . . 2.18.5 Case BII . . . . . . . . . . . . . . . 2.18.6 Case BIII . . . . . . . . . . . . . . . 2.18.7 Case C . . . . . . . . . . . . . . . . Plane gravitational wave . . . . . . . . . . Reissner-Nordstrøm . . . . . . . . . . . . de Sitter spacetime . . . . . . . . . . . . . 2.21.1 Standard coordinates . . . . . . . . 2.21.2 Conformally Einstein coordinates 2.21.3 Conformally flat coordinates . . . 2.21.4 Static coordinates . . . . . . . . . . 2.21.5 Lemaître-Robertson form . . . . . 2.21.6 Cartesian coordinates . . . . . . . Straight spinning string . . . . . . . . . . Sultana-Dyer spacetime . . . . . . . . . . TaubNUT . . . . . . . . . . . . . . . . . . .

Bibliography

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25 27 28 29 31 33 33 33 34 36 38 38 39 40 44 44 45 47 48 50 51 51 54 56 58 58 59 61 61 61 62 62 63 63 63 66 67 69 69 69 70 70 72 73 74 76 78 79

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 visualized 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:

ζµ ζ µ ≡

3

∑ ζµ ζ µ .

(1.1.1)

µ =0

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

 1 a µ bν + aν b µ , 2

a[ µ bν ] =

 1 a µ bν − aν b µ 2

(1.1.2)

1.2 General remarks The Einstein field equation in the most general form reads[MTW73] Gµν = κTµν − Λg µν ,

κ=

8π G , c4

(1.2.1)

with the symmetric and divergence-free Einstein tensor Gµν = Rµν − 21 Rgµν , the Ricci tensor Rµν , the 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 divergencefree, the conservation equation T µν ;ν = 0 is automatically fulfilled. 1

2

CHAPTER 1. INTRODUCTION AND NOTATION

A solution to the field equation is given by the line element ds2 = gµν dxµ dxν

(1.2.2)

with the symmetric, covariant metric tensor gµν . The contravariant metric tensor gµν is related to the covariant tensor via g µν gνλ = δµλ with the Kronecker-δ . Even though gµν is only a component of the metric tensor g = gµν dxµ ⊗ dxν , we will also call g µν 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 Γνλ µ =

 1 g µν ,λ + gµλ ,ν − gνλ ,µ 2

(1.3.1)

with the relation

gνλ ,µ = Γµνλ + Γµλ ν

(1.3.2)

Christoffel symbols of the second kind:  1 µ Γνλ = gµρ gρν ,λ + gρλ ,ν − gνλ ,ρ 2

(1.3.3)

which are related to the Christoffel symbols of the first kind via µ

Γνλ = gµρ Γνλ ρ

(1.3.4)

Riemann tensor: µ

µ

µ

µ

λ λ Rµ νρσ = Γνσ ,ρ − Γνρ ,σ + Γρλ Γνσ − Γσ λ Γνρ

(1.3.5)

λ λ Rµνρσ = gµλ Rλ νρσ = Γνσ µ ,ρ − Γνρ µ ,σ + Γνρ Γµσ λ − Γνσ Γµσ λ

(1.3.6)

or

with symmetries Rµνρσ = −Rµνσ ρ ,

Rµνρσ = −Rν µρσ ,

Rµνρσ = Rρσ µν

(1.3.7)

and Rµνρσ + Rµρσ ν + Rµσ νρ = 0

(1.3.8)

Ricci tensor: Rµν = gρσ Rρ µσ ν = Rρ µρν

(1.3.9)

Ricci and Kretschmann scalar: R = gµν Rµν = Rµ µ , 1 The

K = Rαβ γδ Rαβ γδ = Rγδ αβ Rαβ γδ

= ΓCoS notation of the Christoffel symbols of the first kind differs from the one used by Rindler[Rin01], ΓRindler µνλ νλ µ .

(1.3.10)

1.4. NATURAL LOCAL TETRAD AND INITIAL CONDITIONS FOR GEODESICS

3

Weyl tensor: Cµνρσ = Rµνρσ −

 1 1 gµ [ ρ Rσ ]ν − gν [ ρ Rσ ]µ + R gµ [ ρ gσ ]ν 2 3

(1.3.11)

If we change the signature of a metric, these basic objects transform as follows: µ

µ

Γνλ 7→ Γνλ ,

Rµνρσ 7→ −Rµνρσ ,

Rµν 7→ Rµν ,

R 7→ −R,

Cµνρσ 7→ −Cµνρσ , K 7→ K .

(1.3.12a) (1.3.12b)

Covariant derivative ∇λ gµν = gµν ;λ = 0.

(1.3.13)

Covariant derivative of the vector field ψ µ : µ

µ

∇ν ψ µ = ψ;ν = ∂ν ψ µ + Γνλ ψ λ

(1.3.14)

Covariant derivative of a r-s-tensor field: ∇c T a1 ...ar b1 ...bs = ∂c T a1 ...ar b1 ...bs + Γadc1 T d...ar b1 ...bs + . . . + Γadcr T a1 ...ar−1 d b1 ...bs − Γdb1 c T a1 ...ar d...bs − . . . − Γdbs c T a1 ...ar b1 ...bs−1 d

(1.3.15)

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 The four base vectors e(i) = e(i) ∂µ are given with respect to coordinate directions ∂ /∂ xµ = ∂µ , compare Nakahara[Nak90] or Chandrasekhar[Cha06] for an introduction to the tetrad formalism. The inverse or (i) dual tetrad is given by θ (i) = θµ dxµ with (i) µ

(i)

θ µ e( j) = δ( j)

and

θµ eν(i) = δµν . (i)

(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

 ! µ e(i) , e( j) g = g e(i) , e( j) = gµν e(i) eν( j) = η(i)( j) ,

(1.4.2)

ds2 = η(i)( j) θ (i) θ ( j) = η(i)( j) θµ θν dxµ dxν .

(1.4.3)

where η(i)( j) = diag(∓1, ±1, ±1, ±1) depending on the signature sign(g) = ±2 of the metric. Thus, the line element of a metric can be written as (i) ( j)

To obtain a local tetrad e(i) , we could first determine the dual tetrad θ (i) via Eq. (1.4.3). If we combine all four dual tetrad vectors into one matrix Θ, we only have to determine its inverse Θ−1 to find the tetrad vectors,   0  (0) (0) (0) (0)  e(0) e0(1) e0(2) e0(3) θ0 θ1 θ2 θ3 1 1 1  e1  (1) (1) (1) (1)  θ1 θ2 θ3   (0) e(1) e(2) e(3)  θ −1 (1.4.4) ⇒ Θ = Θ =  0(2) .   2 2 2 2 (2) (2) (2) e(0) e(1) e(2) e(3)  θ0 θ1 θ2 θ3  (3) (3) (3) (3) e3(0) e3(1) e3(2) e3(3) θ θ θ θ 0

1

2

3

4

CHAPTER 1. INTRODUCTION AND NOTATION

There are also several useful relations: µ η(a)(b) = e(a) e(b)µ ,

e(a)µ = gµν eν(a) , (b)

(a)

θµ = η (a)(b) e(a)µ ,

gµν = e(a)µ θν ,

(a)

e(b)µ = η(a)(b) θµ ,

(1.4.5a)

η (a)(b) = θµ θν gµν .

(1.4.5b)

(a) (b)

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 eˆ (i) = Aki e(k) ,

(1.4.6)

where A is an element of the Lorentz group O(1, 3). Hence AT η A = η and (det A)2 = 1. p Lorentz-transformation in the direction na = (sin χ cos ξ , sin χ sin ξ , cos ξ )T = na with γ = 1/ 1 − β 2, Λ00 = γ ,

Λ0a = −β γ na ,

Λa0 = −β γ na ,

Λab = (γ − 1)na nb + δba .

(1.4.7)

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   µ µ µ β γ(i)( j)(k) := gµλ e(i) ∇e(k) eλ( j) = gµλ e(i) eν(k) ∇ν eλ( j) = gµλ e(i) eν(k) ∂ν eλ( j) + Γλνβ e( j) .

(1.4.8)

They are antisymmetric in the first two indices, γ(i)( j)(k) = −γ( j)(i)(k) , which follows from the definition, Eq. (1.4.8), and the relation   β (1.4.9) 0 = ∂µ η(i)( j) = ∇µ gβ ν e(i) eν( j) , where ∇µ gβ ν = 0, compare [Cha06]. Otherwise, we have

γ (i)( j)(k) = θλ eν(k) ∇ν eλ( j) = −eλ( j) eν(k) ∇ν θλ . (i)

(i)

(1.4.10)

The contraction of the first and the last index is given by

γ( j) = γ (k)( j)(k) = η (k)(i) γ(i)( j)(k) = −γ(0)( j)(0) + γ(1)( j)(1) + γ(2)( j)(2) + γ(3)( j)(3) = ∇ν eν( j) .

(1.4.11)

(m)

The connection coefficients ω ( j)(n) with respect to the local tetrad e(i) are defined by   (m) (m) (m) (m) µ µ µ µ β ω ( j)(n) := θµ ∇e( j) e(n) = θµ eα( j) ∇α e(n) = θµ eα( j) ∂α e(n) + Γαβ e(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)

Furthermore, the local tetrad has a non-vanishing Lie-bracket [X,Y ]ν = X µ ∂µ Y ν − Y µ ∂µ X ν . Thus,   (k) e(i) , e( j) = c(i)( j) e(k)

or (k)

  (k) c(i)( j) = θ (k) e(i) , e( j) .

(1.4.14)

The structure coefficients c(i)( j) are related to the connection coefficients or the Ricci rotation coefficients via  (k) (k) (k) c(i)( j) = ω (i)( j) − ω ( j)(i) = η (k)(m) γ(m)( j)(i) − γ(m)(i)( j) = γ (k)( j)(i) − γ (k)(i)( j) . (1.4.15)

1.4. NATURAL LOCAL TETRAD AND INITIAL CONDITIONS FOR GEODESICS

5

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

ρ

R(a)(b)(c)(d) = Rµνρσ e(a) eν(b) e(c) eσ(d) ,

(1.4.16a)

µ

R(a)(b) = Rµν e(a) eν(b) ,

(1.4.16b)

ρ µ C(a)(b)(c)(d) = Cµνρσ e(a) eν(b) e(c) eσ(d)

= R(a)(b)(c)(d) −

 R 1 η(a)[ (c) R(d) ](b) − η(b)[ (c) R(d) ](a) + η(a)[ (c) η(d) ](b) . 2 3

(1.4.16c)

1.4.5 Null or timelike directions A null or timelike direction υ = υ (i) e(i) with respect to a local tetrad e(i) can be written as  υ = υ (0) e(0) + ψ sin χ cos ξ e(1) + sin χ sin ξ e(2) + cos χ e(3) = υ (0) e(0) + ψ n.

(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 + β n), where

 sign(g) = ±2. (1.4.18) u2 = hu, uig = c2 γ 2 e(0) + β n, e(0) + β n = c2 γ 2 −1 + β 2 = ∓c2 , Thus, ψ = cβ γ and υ 0 = ±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 with respect to the local tetrad e(i) .

e(1)

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

and

υ (i) = θµ υ µ . (i)

(1.4.19)

1.4.6 Local tetrad for diagonal metrics If a spacetime is represented by a diagonal metric ds2 = g00 (dx0 )2 + g11(dx1 )2 + g22(dx2 )2 + g33(dx3 )2 ,

(1.4.20)

the natural local tetrad reads 1 e(0) = √ ∂0 , g00

1 e(1) = √ ∂1 , g11

1 e(2) = √ ∂2 , g22

1 e(3) = √ ∂3 , g33

(1.4.21)

given that the metric coefficients are well behaved. Analogously, the dual tetrad reads

θ (0) =



g00 dx0 ,

θ (1) =

√ g11 dx1 ,

θ (2) =

√ g22 dx2 ,

θ (3) =

√ g33 dx3 .

(1.4.22)

6

CHAPTER 1. INTRODUCTION AND NOTATION

1.4.7 Local tetrad for stationary axisymmetric spacetimes The line element of a stationary axisymmetric spacetime is given by ds2 = gtt dt 2 + 2gt ϕ dt d ϕ + gϕϕ d ϕ 2 + grr dr2 + gϑ ϑ d ϑ 2 ,

(1.4.23)

where the metric components are functions of r and ϑ only. The local tetradfor 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(2) = √ e(1) = √ ∂r , ∂ϑ , grr gϑ ϑ   e(3) = ∆Γ ±(gt ϕ + ζ gϕϕ )∂t ∓ (gtt + ζ gt ϕ )∂ϕ ,

(1.4.24a)

(1.4.24b)

where

1 Γ= q  − gtt + 2ζ gt ϕ + ζ 2 gϕϕ

and

1 . ∆= q 2 gt ϕ − gtt gϕϕ

(1.4.25)

The angular velocity ζ is limited due to gtt + 2ζ gt ϕ + ζ 2 gϕϕ < 0 r r gtt gtt 2 ζmin = ω − ω − and ζmax = ω + ω 2 − gϕϕ gϕϕ

(1.4.26)

with ω = −gt ϕ /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: ζmin = 0 ⇒ gtt = 0. The transformation between the local direction υ (i) and the coordinate direction υ µ reads     υ (1) υ (2) υ 0 = Γ υ (0) ± υ (3) ∆w1 , υ1 = √ , υ2 = √ , υ 3 = Γ υ (0) ζ ∓ υ (3)∆w2 , (1.4.27) grr gϑ ϑ with

w1 = gt ϕ + ζ gϕϕ

and

w2 = gtt + ζ gt ϕ .

(1.4.28)

The back transformation reads

υ (0) =

1 υ 0 w2 + υ 3w1 , Γ ζ w1 + w2

υ (1) =

√ grr υ 1 ,

υ (2) =

√ gϑ ϑ υ 2 ,

υ (3) = ±

1 ζ υ0 − υ3 . ∆Γ ζ w1 + w2

(1.4.29)

  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 ¯ where l and n are real and m The Newman-Penrose tetrad consists of four null vectors e⋆(i) = {l, n, m, m}, ¯ are complex conjugates; see Penrose and Rindler[PR84] or Chandrasekhar[Cha06] for a thorough and m discussion. The Newman-Penrose (NP) tetrad has to fulfill the orthonormality relation   0 1 0 0 D E  1 0 0 0   e⋆(i) , e⋆( j) = η ⋆(i)( j) with η ⋆(i)( j) =  (1.5.1)  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 l = ∓ √ e(0) + e(1) , 2

 1 n = ∓ √ e(0) − e(1) , 2

 1 m = ∓ √ e(2) + ie(3) , 2

(1.5.2)

1.6. COORDINATE RELATIONS

7

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:

κ = γ(2)(1)(1) ,

ρ = γ(2)(0)(3),

σ = γ(2)(0)(2) ,

µ = γ(1)(3)(2),

λ = γ(1)(3)(3) ,

τ = γ(2)(0)(1),

ν = γ(1)(3)(1) ,

π = γ(1)(3)(0),

1 2 1 γ= 2 1 α= 2 1 β= 2

ε=

 γ(1)(0)(0) + γ(2)(3)(0) ,

(1.5.3a)

 γ(1)(0)(1) + γ(2)(3)(1) ,

(1.5.3b)

 γ(1)(0)(2) + γ(2)(3)(2) .

(1.5.3d)

 γ(1)(0)(3) + γ(2)(3)(3) ,

(1.5.3c)

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 sin ϑ cos ϕ ,

y = r sin ϑ sin ϕ ,

z = r cos ϑ ,

(1.6.1)

and r=

p ϑ = arctan 2( x2 + y2 , z),

p x2 + y2 + z2 ,

ϕ = arctan 2(y, x),

(1.6.2)

where arctan 2() ensures that ϕ ∈ [0, 2π ) and ϑ ∈ (0, π ). z

ϑ r ϕ

y Figure 1.2: Relation between spherical and Cartesian coordinates.

x

The total differentials of the spherical coordinates read dr =

x dx + y dy + z dz , r

dϑ =

xz dx + yz dy − (x2 + y2 )dz p , r 2 x2 + y2

dϕ =

−y dx + x dy , x2 + y2

(1.6.3)

whereas the coordinate derivatives read

∂r =

∂x ∂y ∂z ∂x + ∂y + ∂z ∂r ∂r ∂r

∂ϑ =

∂x ∂y ∂z ∂x + ∂y + ∂z = r cos ϑ cos ϕ ∂x + r cos ϑ sin ϕ ∂y − r sin ϑ ∂z , ∂ϑ ∂ϑ ∂ϑ

(1.6.4b)

∂ϕ =

∂x ∂y ∂z ∂x + ∂y + ∂z = −r sin ϑ sin ϕ ∂x + r sin ϑ cos ϕ ∂y , ∂ϕ ∂ϕ ∂ϕ

(1.6.4c)

= sin ϑ cos ϕ ∂x + sin ϑ sin ϕ ∂y + cos ϑ ∂z ,

(1.6.4a)

8

CHAPTER 1. INTRODUCTION AND NOTATION

and

∂x =

sin ϕ cos ϑ cos ϕ ∂r ∂ϑ ∂ϕ ∂r + ∂ϑ + ∂ϕ = sin ϑ cos ϕ ∂r + ∂ϑ − ∂ϕ , ∂x ∂x ∂x r r sin ϑ

(1.6.5a)

∂y =

∂r ∂ϑ ∂ϕ cos ϕ cos ϑ sin ϕ ∂r + ∂ϑ + ∂ϕ = sin ϑ sin ϕ ∂r + ∂ϑ + ∂ϕ , ∂y ∂y ∂y r r sin ϑ

(1.6.5b)

∂z =

sin ϑ ∂r ∂ϑ ∂ϕ ∂r + ∂ϑ + ∂ϕ = cos ϑ ∂r − ∂ϑ . ∂z ∂z ∂z r

(1.6.5c)

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 cos ϕ , y = r sin ϕ , and r = x2 + y2 , ϕ = arctan 2(y, x), (1.6.6) where arctan 2() again ensures that the angle ϕ ∈ [0, 2π ). z

z

ϕ

y Figure 1.3: Relation between cylindrical and Cartesian coordinates.

r x

The total differentials of the spherical coordinates are given by dr =

x dx + y dy , r

dϕ =

−y dx + x dy , r2

(1.6.7)

and dx = cos ϕ dr − r sin ϕ d ϕ ,

dy = sin ϕ dr + r cos ϕ d ϕ .

(1.6.8)

The coordinate derivatives are

∂r =

∂x ∂y ∂x + ∂y = cos ϕ ∂x + sin ϕ ∂y , ∂r ∂r

(1.6.9a)

∂ϕ =

∂x ∂y ∂x + ∂y = −r sin ϕ ∂x + r cos ϕ ∂y m ∂ϕ ∂ϕ

(1.6.9b)

∂x =

∂r ∂ϕ sin ϕ ∂r + ∂ϕ = cos ϕ ∂r − ∂y , ∂x ∂x r

(1.6.10a)

∂y =

cos ϕ ∂r ∂ϕ ∂r + ∂ϕ = sin ϕ ∂r + ∂y . ∂y ∂y r

(1.6.10b)

and

1.7 Embedding diagram A two-dimensional hypersurface with line segment d σ 2 = grr (r)dr2 + gϕϕ (r)d ϕ 2

(1.7.1)

1.8. EQUATIONS OF MOTION AND TRANSPORT EQUATIONS can be embedded in a three-dimensional Euclidean space with cylindrical coordinates, "  2 # dz 2 dσ = 1 + d ρ 2 + ρ 2d ϕ 2 . dρ With ρ (r)2 = gϕϕ (r) and dr = (dr/d ρ )d ρ , we obtain for the embedding function z = z(r), s  √  d gϕϕ 2 dz = ± grr − . dr dr

9

(1.7.2)

(1.7.3)

√ If gϕϕ (r) = r2 , then d gϕϕ /dr = 1.

1.8 Equations of motion and transport equations 1.8.1 Geodesic equation The geodesic equation reads ρ σ d 2 xµ D2 x µ µ dx dx = + Γρσ =0 2 2 dλ dλ dλ dλ

(1.8.1)

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 /d λ )(λ = 0). This initial direction has to fulfill the constraint equation gµν

dxµ dxν = κ c2 , dλ dλ

(1.8.2)

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 dxµ µ (1.8.3) = υ µ = υ (i) e(i) . d λ λ =0

1.8.2 Fermi-Walker transport

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

dX µ 1 µ + Γρσ uρ X σ + 2 (uσ aµ − aσ uµ ) gρσ X ρ . dτ c

(1.8.4)

The four-acceleration follows from the four-velocity via aµ =

Duµ D2 x µ duµ µ = = + Γρσ uρ uσ . dτ 2 dτ dτ

(1.8.5)

1.8.3 Parallel transport If the four-acceleration vanishes, the Fermi-Walker transport simplifies to the parallel transport Pu X µ = 0 with Pu X µ :=

DX µ dX µ µ = + Γρσ uρ X σ . dτ dτ

(1.8.6)

10

CHAPTER 1. INTRODUCTION AND NOTATION

1.8.4 Euler-Lagrange formalism A detailed discussion of the Euler-Lagrange formalism can be found, e.g., in Rindler[Rin01]. The Lagrangian L is defined as L := gµν x˙µ x˙ν ,

!

L = κ c2 ,

(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. The Euler-Lagrange equations read d ∂L ∂L − = 0. d λ ∂ x˙µ ∂ xµ

(1.8.8)

If L is independent of xρ , then xρ is a cyclic variable and pρ = gρν x˙ν = const. Note that [L ]U =

length2 time2

(1.8.9)

for timelike and [L ]U = 1 for lightlike geodesics, see Sec. 1.9.

1.8.5 Hamilton formalism The super-Hamiltonian H is defined as ! 1 H = κ c2 , 2

1 H := gµν pµ pν , 2

(1.8.10)

where p µ = gµν x˙ν are the canonical momenta, see e.g. MTW[MTW73], para. 21.1. As in classical mechanics, we have

∂H dxµ = dλ ∂ pµ

and

d pµ ∂H =− µ . dλ ∂x

(1.8.11)

1.9 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 [G]U =

length3 , mass · time2

(1.9.1)

where [·]U indicates that we evaluate the units of the enclosed expression. Further examples are [ds]U = length,

[u]U =

length , time

Schwarzschild [Rtrtr ]U =

1 , time2

h i RSchwarzschild = length2 . ϑ ϕϑ ϕ U

(1.9.2)

1.10 Tools 1.10.1 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: 2 The commercial software Maple can be found here: http://www.maplesoft.com. http://grtensor.phy.queensu.ca.

The GRTensorII-package is free:

1.10. TOOLS

11

> 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) 1) 2) 3) 4) 5) 6) > 0:

Use the metric WITHOUT saving it, Save the metric as it is, Correct an element of the metric, Re-enter the metric, Add/change constraint equations, Add a text description, or Abandon this metric and return to Maple.

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 nonholonomic basis. The various basic objects can be determined via µ

Christoffel symbols Γνρ µ partial derivatives Γνρ ,σ Riemann tensor Rµνρσ Ricci tensor Rµν Ricci scalar R Kretschmann scalar K

grcalc(Chr2); grcalc(Riemman); grcalc(Ricci); grcalc(Ricciscalar); grcalc(RiemSq);

grcalc(Chr(dn,dn,up)); grcalc(Chr(dn,dn,up,pdn)); grcalc(R(dn,dn,dn,dn)); grcalc(R(dn,dn));

1.10.2 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, 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

12

CHAPTER 1. INTRODUCTION AND NOTATION

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}]], {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[

1.10. TOOLS

13

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.

1.10.3 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; cmetric(); /* calculate the christoffel symbols of the second kind */ christof(mcs); /* calculate the riemann tensor */ lriemann(mcs); /* calculate the ricci tensor */ ricci(mcs); /* calculate the ricci scalar */ scurvature(); /* calculate the Kretschmann scalar */ uriemann(mcs); rinvariant(); ratsimp(%);

As you may have noticed, the Schwarzschild metric must be given with negative signature.

Chapter 2

Spacetimes 2.1 Minkowski 2.1.1 Cartesian coordinates The Minkowski metric in Cartesian coordinates {t, x, y, z ∈

R} reads

ds2 = −c2 dt 2 + 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(t) = ∂t , c

e(x) = ∂x ,

e(y) = ∂y ,

e(z) = ∂z ,

(2.1.2)

with dual

θ (t) = c dt,

θ (x) = dx,

θ (y) = dy,

θ (z) = dz.

2.1.2 Cylindrical coordinates The Minkowski metric in cylindrical coordinates {t ∈

(2.1.3)

R, r ∈ R+ , ϕ ∈ [0, 2π ), z ∈ R},

ds2 = −c2 dt 2 + dr2 + r2 d ϕ 2 + dz2 ,

(2.1.4)

has the natural local tetrad 1 e(t) = ∂t , c

e(r) = ∂r ,

1 e(ϕ ) = ∂ϕ , r

e(z) = ∂z .

(2.1.5)

Christoffel symbols: 1 ϕ Γrϕ = . r

r Γϕϕ = −r,

(2.1.6)

Partial derivatives ϕ

Γrϕ ,r = −

1 , r2

Γrϕϕ ,r = −1.

(2.1.7)

Ricci rotation coefficients:

γ(ϕ )(r)(ϕ ) =

1 r

and

1 γ(r) = . r

(2.1.8)

14

2.1. MINKOWSKI

15

2.1.3 Spherical coordinates

R, r ∈ R+, ϑ ∈ (0, π ), ϕ ∈ [0, 2π )}, the Minkowski metric reads

In spherical coordinates {t ∈

 ds2 = −c2 dt 2 + dr2 + r2 d ϑ 2 + sin2 ϑ d ϕ 2 .

(2.1.9)

Christoffel symbols: Γrϑ ϑ = −r,

Γrϕϕ = −r sin2 ϑ ,

1 Γϑrϑ = , r

ϑ Γϕϕ = − sin ϑ cos ϑ ,

1 ϕ Γrϕ = , r

Γϑ ϕ = cot ϑ .

ϕ

(2.1.10a) (2.1.10b)

Partial derivatives 1 , r2 1 =− 2 , sin ϑ = − sin(2ϑ ).

ϕ

1 , r2

Γϑrϑ ,r = −

Γrϕ ,r = −

ϕ

r 2 Γϕϕ ,r = − sin ϑ ,

Γϑ ϕ ,ϑ r Γϕϕ ,ϑ

Γrϑ ϑ ,r = −1,

(2.1.11a)

Γϑϕϕ ,ϑ = − cos(2ϑ ),

(2.1.11b) (2.1.11c)

Local tetrad: 1 e(t) = ∂t , c

1 e(ϑ ) = ∂ϑ , r

e(r) = ∂r ,

e(ϕ ) =

1 ∂ϕ . r sin ϑ

(2.1.12)

Ricci rotation coefficients: 1 γ(ϑ )(r)(ϑ ) = γ(ϕ )(r)(ϕ ) = , r

γ(ϕ )(ϑ )(ϕ ) =

cot ϑ . r

(2.1.13)

The contractions of the Ricci rotation coefficients read 2 γ(r) = , r

γ(ϑ ) =

cot ϑ . r

(2.1.14)

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 = −d ψ 2 + d ξ 2 + sin2 ξ d ϑ 2 + sin2 ϑ d ϕ 2 .

(2.1.15)

This form follows from the spherical Minkowski metric (2.1.9) by means of the coordinate transformation ct + r = tan

ψ +ξ , 2

ct − r = tan

ψ −ξ , 2

(2.1.16)

resulting in the metric d s˜2 =

−d ψ 2 + d ξ 2

ξ 4 cos2 ψ + 2

ξ cos2 ψ − 2

+

sin2 ξ ξ 4 cos2 ψ + 2

 d ϑ 2 + sin2 ϑ d ϕ 2 ,

ξ cos2 ψ − 2

(2.1.17)

ξ 2 ψ −ξ . and by the conformal transformation ds2 = Ω2 d s˜2 with Ω2 = 4 cos2 ψ + 2 cos 2 Christoffel symbols:

Γϑξϑ = cot ξ , ϕ

Γϑ ϕ = cot ϑ ,

ϕ

Γξ ϕ = cot ξ , ξ

Γϕϕ = − sin ξ cos ξ sin2 ϑ ,

ξ

Γϑ ϑ = − sin ξ cos ξ ,

ϑ Γϕϕ = − sin ϑ cos ϑ .

(2.1.18a) (2.1.18b)

16

CHAPTER 2. SPACETIMES

Partial derivatives 1 1 ϕ ξ , Γξ ϕ ,ξ = − 2 , Γϑ ϑ ,ξ = − cos(2ξ ), 2 sin ξ sin ξ 1 ξ =− 2 , Γϕϕ ,ξ = − cos(2ξ ) sin2 ϑ , Γϑϕϕ ,ϑ = − cos(2ϑ ), sin ϑ 1 = − sin(2ξ ) sin(2ϑ ). 2

Γϑξϑ ,ξ = − ϕ

Γϑ ϕ ,ϑ ξ

Γϕϕ ,ϑ

(2.1.19a) (2.1.19b) (2.1.19c)

Riemann-Tensor: Rξ ϑ ξ ϑ = sin2 ξ ,

Rξ ϕξ ϕ = sin2 ξ sin2 ϑ ,

Rϑ ϕϑ ϕ = sin4 ξ sin2 ϑ .

(2.1.20)

Ricci-Tensor: Rϑ ϑ = 2 sin2 ξ ,

Rξ ξ = 2,

Rϕϕ = 2 sin2 ξ sin2 ϑ .

(2.1.21)

Ricci and Kretschmann scalars: R = 6,

(2.1.22)

K = 12.

The Weyl tensor vanishs identically. Local tetrad: e(ψ ) = ∂ψ ,

e(ξ ) = ∂ξ ,

e(ϑ ) =

1 ∂ϑ , sin ξ

e(ϕ ) =

1 ∂ϕ . sin ξ sin ϑ

(2.1.23)

Ricci rotation coefficients:

γ(ϑ )(ξ )(ϑ ) = γ(ϕ )(ξ )(ϕ ) = cot ξ ,

γ(ϕ )(ϑ )(ϕ ) =

cot ϑ . sin ξ

(2.1.24)

The contractions of the Ricci rotation coefficients read

γ(ξ ) = 2 cot ξ ,

γ(ϑ ) =

cot ϑ . sin ξ

(2.1.25)

Riemann-Tensor with respect to local tetrad: (2.1.26)

R(ξ )(ϑ )(ξ )(ϑ ) = R(ξ )(ϕ )(ξ )(ϕ ) = R(ϑ )(ϕ )(ϑ )(ϕ ) = 1. Ricci-Tensor with respect to local tetrad:

(2.1.27)

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

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

R

R

R

  ω 2 r2 r2 [c dt − Ω(r)d ϕ ]2 + dr2 + d ϕ 2 + dz2 ds2 = − 1 − 2 c 1 − ω 2r2 /c2

(2.1.28)

with Ω(r) = (r2 ω /c)/(1 − ω 2r2 /c2 ). Metric-Tensor: gtt = −c2 + ω 2 r2 ,

gt ϕ = ω r2 ,

grr = gzz = 1,

gϕϕ = r2 .

(2.1.29)

2.1. MINKOWSKI

17

Christoffel symbols: ϕ

Γttr = −ω 2 r,

Γtr =

ω , r

1 ϕ Γrϕ = , r

Γtrϕ = −ω r,

r Γϕϕ = −r.

(2.1.30)

r Γϕϕ ,r = −1.

(2.1.31)

Partial derivatives r Γtt,r = −ω 2 ,

ϕ

Γtr,r = −

ω , r2

ϕ

Γtrϕ ,r = −ω ,

Γrϕ ,r = −

1 , r2

The local tetrad of the comoving observer is 1 ω e(t) = ∂t − ∂ϕ , c c

1 e(ϕ ) = ∂ϕ , r

e(r) = ∂r ,

e(z) = ∂z ,

(2.1.32)

whereas the static observer has the local tetrad 1 ∂t , e(t) = p c 1 − ω 2r2 /c2

e(r) = ∂r ,

ωr e(ϕ ) = p ∂t + 2 c 1 − ω 2r2 /c2

e(z) = ∂z ,

p 1 − ω 2r2 /c2 ∂ϕ . r

(2.1.33a) (2.1.33b)

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 x=

αt′ c2 cosh , α c

ct =

αt′ c2 sinh , α c

(2.1.34)

where t ′ 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 (ct, x) 7→ (τ , ρ ) :

ct =

1 sinh τ , ρ

x=

1 cosh τ , ρ

(2.1.35)

where ρ = α /c2 , the Rindler metric reads ds2 = −

1 2 1 d τ + 4 d ρ 2 + dy2 + dz2 . ρ2 ρ

(2.1.36)

Christoffel symbols: ρ

Γττ = −ρ ,

1 τ Γτρ =− , ρ

2 ρ Γρρ = − . ρ

(2.1.37)

Partial derivatives ρ

Γττ ,ρ = −1,

τ Γτρ ,ρ =

1 , ρ2

ρ

Γρρ ,ρ =

2 . ρ2

(2.1.38)

The Riemann and Ricci tensors as well as the Ricci and Kretschmann scalar vanish identically. Local tetrad: e(τ ) = ρ∂τ ,

e(ρ ) = ρ 2 ∂ρ ,

e(y) = ∂y ,

e(z) = ∂z .

(2.1.39)

Ricci rotation coefficients:

γ(τ )(ρ )(τ ) = ρ ,

and

γ(ρ ) = −ρ .

(2.1.40)

18

CHAPTER 2. SPACETIMES

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

  1 rs  2 2 dr2 + r2 d ϑ 2 + sin2 ϑ d ϕ 2 , c dt + ds2 = − 1 − r 1 − rs/r

(2.2.1)

where rs = 2GM/c2 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: c2 rs (r − rs ) , 2r3 1 = , r = cot ϑ ,

Γϑrϑ ϕ

Γϑ ϕ

rs , 2r(r − rs ) 1 = , r = −(r − rs ) sin2 ϑ ,

Γttr =

Γttr =

ϕ

Γrϕ Γrϕϕ

Γrrr = −

rs , 2r(r − rs )

(2.2.2a)

Γrϑ ϑ = −(r − rs ),

(2.2.2b)

ϑ Γϕϕ = − sin ϑ cos ϑ .

(2.2.2c)

Partial derivatives (2r − 3rs )c2 rs , 2r4 1 Γϑrϑ ,r = − 2 , r 1 ϕ Γϑ ϕ ,ϑ = − 2 , sin ϑ r Γϕϕ ,ϑ = −(r − rs ) sin(2ϑ ). r Γtt,r =−

(2r − rs )rs , 2r2 (r − rs )2 1 ϕ Γrϕ ,r = − 2 , r

Γϑr ϑ ,r = −1,

(2.2.3b)

Γrϕϕ ,r = − sin2 ϑ ,

ϑ Γϕϕ ,ϑ = − cos(2ϑ ),

(2.2.3c)

Γttr,r = −

Γrrr,r =

(2r − rs )rs , 2r2 (r − rs )2

(2.2.3a)

(2.2.3d)

Riemann-Tensor: c2 rs , r3 1 rs , =− 2 r − rs

Rtrtr = − Rr ϑ r ϑ

1 c2 (r − rs ) rs , 2 r2 1 rs sin2 ϑ , =− 2 r − rs

Rt ϑ t ϑ = Rr ϕ r ϕ

Rt ϕ t ϕ =

1 c2 (r − rs ) rs sin2 ϑ , 2 r2

Rϑ ϕϑ ϕ = rrs sin2 ϑ .

(2.2.4a) (2.2.4b)

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 K = 12

rs2 . r6

(2.2.5)

Here, it becomes clear that at r = rs there is no real singularity. Local tetrad: r 1 1 rs e(r) = 1 − ∂r , e(ϑ ) = ∂ϑ , e(t) = p ∂t , r r c 1 − rs /r

1 ∂ϕ . r sin ϑ

(2.2.6)

θ (ϕ ) = r sin ϑ d ϕ .

(2.2.7)

cot ϑ . r

(2.2.8)

e(ϕ ) =

Dual tetrad:

r rs θ (t) = c 1 − dt, r Ricci rotation coefficients:

dr θ (r) = p , 1 − rs/r

r p s γ(r)(t)(t) = , 2 2r 1 − rs/r

θ (ϑ ) = r d ϑ ,

1 γ(ϑ )(r)(ϑ ) = γ(ϕ )(r)(ϕ ) = r

r rs 1− , r

γ(ϕ )(ϑ )(ϕ ) =

2.2. SCHWARZSCHILD SPACETIME

19

The contractions of the Ricci rotation coefficients read

γ(r) =

4r − 3rs p , 1 − rs/r

2r2

γ(ϑ ) =

cot ϑ . r

(2.2.9)

Structure coefficients: (t) c(t)(r)

rs p , = 2 2r 1 − rs/r

(ϑ ) c(r)(ϑ )

=

(ϕ ) c(r)(ϕ )

1 =− r

r rs 1− , r

(ϕ )

c(ϑ )(ϕ ) =

cot ϑ . r

(2.2.10)

Riemann-Tensor with respect to local tetrad: R(t)(r)(t)(r) = −R(ϑ )(ϕ )(ϑ )(ϕ ) = −

rs , r3

(2.2.11a)

R(t)(ϑ )(t)(ϑ ) = R(t)(ϕ )(t)(ϕ ) = −R(r)(ϑ )(r)(ϑ ) = −R(r)(ϕ )(r)(ϕ ) =

rs . 2r3

The covariant derivatives of the Riemann tensor read 3rs p R(t)(r)(t)(r);(r) = −R(ϑ )(ϕ )(ϑ )(ϕ );(r) = 5 r(r − rs ), r 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)(ϕ )(ϑ )(ϕ );(ϑ ) = − 5 r(r − rs ), 2r 3rs p R(r)(ϑ )(r)(ϑ );(r) = R(r)(ϑ )(ϑ )(ϕ );(ϕ ) = R(r)(ϕ )(r)(ϕ );(r) = 5 r(r − rs ). 2r

(2.2.11b)

(2.2.12a)

(2.2.12b) (2.2.12c)

Newman-Penrose tetrad:  1 l = √ e(t) + e(r) , 2

 1 n = √ e(t) − e(r) , 2

 1 m = √ e(ϑ ) + ie(ϕ ) . 2

Non-vanishing spin coefficients: r 1 rs rs ρ = µ = −√ 1− , γ = ε = √ p , r 2r 4 2r2 1 − rs/r

cot ϑ α = −β = − √ . 2 2r

Embedding: The embedding function reads √ √ z = 2 rs r − rs .

Euler-Lagrange: The Euler-Lagrangian formalism, Sec. 1.8.4, for geodesics in the ϑ = π /2 hyperplane yields   rs  h2 1 1 k2 1 2 2 1 − , V = − r˙ + Veff = κ c eff 2 2 c2 2 r r2

(2.2.13)

(2.2.14)

(2.2.15)

(2.2.16)

with the constants of motion k = (1 − rs /r)c2t˙, h = r2 ϕ˙ , and κ as in Eq. (1.8.2). For timelike geodesics, the effective potential has the extremal points p h2 ± h h2 − 3c2rs2 r± = , (2.2.17) c2 rs

where r+ is a maximum and r− is a minimum. The innermost timelike circular geodesic follows from h2 = 3c2 rs2 and reads ritcg = 3rs . Null geodesics, however, have only a maximum at rpo = 32 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].

20

CHAPTER 2. SPACETIMES

2.2.2 Schwarzschild in pseudo-Cartesian coordinates The Schwarzschild spacetime in pseudo-Cartesian coordinates (t, x, y, z) reads ds2

  2   2  rs  2 2 x2 y2 2 2 2 dx 2 dy = − 1− c dt + + x + +y +z +z 1− r2 1 − rs /r r2  rs/r2  r 2 dz 2r z s + 2 (xy dxdy + xz dxdz + yz dydz), + x2 + y2 + 1 − rs/r r2 r (r − rs )

(2.2.18)

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 , c 1 − rs /r

e(1) = A∂x ,

1 A= √ , gxx

−gxy q , gxx −g2xy /gxx + gyy

B=

gxy gyz − gxz gyy √ , NW

D=

e(2) = B∂x + C∂y ,

gxz gxy − gxx gyz √ , NW

E=

e(3) = D∂x + E ∂y + F ∂z .

1 C= q , −g2xy /gxx + gyy √ N F=√ , W

(2.2.19)

(2.2.20a)

(2.2.20b)

with N = gxx gyy − g2xy ,

(2.2.21a)

gxx gyy gzz − g2xz gyy + 2gxzgxy gyz − g2xy gzz − gxx g2yz .

W=

(2.2.21b)

2.2.3 Isotropic coordinates Spherical isotropic coordinates The Schwarzschild metric (2.2.1) in spherical isotropic coordinates (t, ρ , ϑ , ϕ ) reads 1 − ρs /ρ ds = − 1 + ρs /ρ 2



2

   ρs 4  2 c dt + 1 + d ρ + ρ 2 d ϑ 2 + sin2 ϑ d ϕ 2 , ρ 2

2

(2.2.22)

where   ρs 2 r = ρ 1+ ρ

ρ=

or

 p 1 2r − rs ± 2 r(r − rs ) 4

(2.2.23)

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 ρitcg = 5 + 2 6 ρs . and (2.2.24) Christoffel symbols: ρ

2ρs 2(ρ − ρs)ρ 4 ρs c2 , , Γtt ρ = 2 (ρ + ρs )7 ρ − ρs2 ρ − ρs ρ − ρs ϕ = , Γρϕ = , (ρ + ρs )ρ (ρ + ρs )ρ

Γtt = ϑ Γρϑ ϕ

Γϑ ϕ = cot ϑ ,

ρ

Γϕϕ = −

ρ

2ρs , (ρ + ρs)ρ ρ − ρs , = −ρ ρ + ρs

Γρρ = − ρ

Γϑ ϑ

(ρ − ρs )ρ sin2 ϑ ϑ , Γϕϕ = − sin ϑ cos ϑ . ρ + ρs

(2.2.25a) (2.2.25b) (2.2.25c)

2.2. SCHWARZSCHILD SPACETIME

21

Riemann-Tensor: Rt ρ t ρ = −4 Rt ϕ t ϕ = 2

(ρ − ρs )2 ρs c2 , (ρ + ρs)4 ρ

(ρ − ρs)2 ρ c2 ρs sin2 ϑ , (ρ + ρs)4

Rρϕρϕ = −2

(ρ − ρs )2 ρρs c2 , (ρ + ρs )4

(2.2.26a)

(ρ + ρs )2 ρs , ρ3

(2.2.26b)

4(ρ + ρs )2 ρs sin2 ϑ . ρ

(2.2.26c)

Rt ϑ t ϑ = 2

Rρϑ ρϑ = −2

(ρ + ρs )2 ρs sin2 ϑ , ρ3

Rϑ ϕϑ ϕ =

The Ricci tensor and the Ricci scalar vanish identically. Kretschmann scalar: K = 192

rs2

ρ 6 (1 + ρs/ρ )12

= 12

rs2 . r(ρ )6

(2.2.27)

Local tetrad: e(t) = e(ϑ ) =

1 + ρs/ρ ∂t , 1 − ρs /ρ c 1

ρ [1 + ρs/ρ ]2

e(r) =

∂ϑ ,

e(ϕ ) =

1 [1 + ρs/ρ ]2 1

∂ρ ,

ρ [1 + ρs/ρ ]2 sin2 ϑ

(2.2.28a)

∂ϕ .

(2.2.28b)

Ricci rotation coefficients: 2ρs ρ 2 , (ρ + ρs)3 (ρ − ρs) ρ cot ϑ γ(ϕ )(ϑ )(ϕ ) = . (ρ + ρs)2

γ(ρ )(t)(t) =

γ(ϑ )(ρ )(ϑ ) = γ(ϕ )(ρ )(ϕ ) =

ρ (ρ − ρs ) , (ρ + ρs)3

(2.2.29a) (2.2.29b)

The contractions of the Ricci rotation coefficients read

γ(ρ ) =

2ρ (ρ 2 − ρρs + ρs2) , (ρ + ρs)3 (ρ − ρs)

γ(ϑ ) =

ρ cot ϑ . (ρ + ρs)2

(2.2.30)

Riemann-Tensor with respect to local tetrad: R(t)(ρ )(t)(ρ ) = −R(ϑ )(ϕ )(ϑ )(ϕ ) = −

rs , r(ρ )3

R(t)(ϑ )(t)(ϑ ) = R(t)(ϕ )(t)(ϕ ) = −R(ρ )(ϑ )(ρ )(ϑ ) = −R(ρ )(ϕ )(ρ )(ϕ ) =

(2.2.31a) rs . 2r(ρ )3

(2.2.31b)

Further reading: Buchdahl[Buc85]. Cartesian isotropic coordinates The Schwarzschild metric (2.2.1) in Cartesian isotropic coordinates (t, x, y, z) reads, 1 − ρs /ρ ds = − 1 + ρs /ρ 2



2

   ρs 4  2 c dt + 1 + dx + dy2 + dz2 , ρ 2

2

(2.2.32)

where ρ 2 = x2 + y2 + z2 and, as before,   ρs 2 r = ρ 1+ . ρ

(2.2.33)

22

CHAPTER 2. SPACETIMES

Christoffel symbols: Γttx =

2c2 ρ 3 ρs (ρ − ρs) x

(ρ + ρs) 2ρs x Γttx = 3 , ρ [1 − ρs2/ρ 2 ]

Γtty =

,

2c2 ρ 3 ρs (ρ − ρs) y

( ρ + ρs ) ρ y 2 s Γtty = 3 , ρ [1 − ρs2/ρ 2 ] x 2ρs , Γxxx = Γyxy = Γzxz = −Γxyy = −Γxzz = − 3 ρ 1 + ρs /ρ y 2ρs , Γyxx = −Γxxy = −Γyyy = −Γzyz = Γyzz = 3 ρ 1 + ρs /ρ 2ρs z Γzxx = −Γxxz = Γzyy = −Γyyz = −Γzzz = 3 . ρ 1 + ρs /ρ 7

7

Γttz =

,

Γttz =

2c2 ρ 3 ρs (ρ − ρs) z (ρ + ρs )7

,

2ρs z , ρ 3 [1 − ρs2/ρ 2 ]

(2.2.34a) (2.2.34b) (2.2.34c) (2.2.34d) (2.2.34e)

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 (2.2.35)

cv = ct + r + rs ln(r − rs ) leads to the ingoing Eddington-Finkelstein[Edd24, Fin58] metric with coordinates (v, r, ϑ , ϕ ),   rs  2 2 c dv + 2c dv dr + r2 d ϑ 2 + sin2 ϑ d ϕ 2 . ds2 = − 1 − r

(2.2.36)

Metric-Tensor:  rs  , gvv = −c2 1 − r

gvr = c,

gϑ ϑ = r 2 ,

gϕϕ = r2 sin2 ϑ .

(2.2.37)

Christoffel symbols: c2 rs (r − rs ) crs 1 crs , Γrvv = , Γrvr = − 2 , Γϑrϑ = , 2 2r 2r3 2r r 1 r ϕ v r Γϑ ϑ = − , Γϑ ϑ = −(r − rs ), = , Γϑ ϕ = cot ϑ , r c 2 r sin ϑ ϑ r =− = −(r − rs ) sin2 ϑ , Γϕϕ = − sin ϑ cos ϑ . , Γϕϕ c

Γvvv = ϕ

Γrϕ Γvϕϕ

(2.2.38a) (2.2.38b) (2.2.38c)

Partial derivatives crs , r3 1 Γvϑ ϑ ,r = − , c 2 sin ϑ v Γϕϕ , ,r = − c

(2.2.39b)

Γrϑ ϑ ,r = −1,

(2r − 3rs )c2 rs , 2r4 1 ϕ Γrϕ ,r = − 2 , r 1 ϕ Γϑ ϕ ,ϑ = − 2 , sin ϑ

Γvϕϕ ,ϑ = −

Γrϕϕ ,r = − sin2 ϑ ,

Γϑϕϕ ,ϑ = − cos(2ϑ ),

(2.2.39d)

crs , r3 1 Γϑrϑ ,r = − 2 , r Γvvv,r = −

r Γϕϕ ,ϑ

r sin(2ϑ ) , c = −(r − rs ) sin(2ϑ ).

Γrvv,r = −

Γrvr,r =

(2.2.39a)

(2.2.39c)

(2.2.39e)

Riemann-Tensor: c2 rs , r3 c2 rs (r − rs ) sin2 ϑ , = 2r2

Rvrvr = − Rvϕ vϕ

c2 rs (r − rs ) , 2r2 crs sin2 ϑ , =− 2r

Rvϑ vϑ = Rvϕ rϕ

Rvϑ rϑ = −

crs , 2r

Rϑ ϕϑ ϕ = rrs sin2 ϑ .

(2.2.40a) (2.2.40b)

2.2. SCHWARZSCHILD SPACETIME

23

While the Ricci tensor and the Ricci scalar vanish identically, the Kretschmann scalar is K = 12rs2 /r6 . Static local tetrad: 1 ∂v , e(v) = p c 1 − rs/r

1 ∂v + e(r) = p c 1 − rs/r

Dual tetrad:

θ

(v)

r dr rs = c 1 − dv − p , r 1 − rs /r

dr , θ (r) = p 1 − rs /r

Ricci rotation coefficients:

rs p γ(r)(v)(v) = , 2 2r 1 − rs/r

r rs 1 − ∂r , r

1 γ(ϑ )(r)(ϑ ) = γ(ϕ )(r)(ϕ ) = r

1 e(ϑ ) = ∂ϑ , r

θ (ϑ ) = r d ϑ ,

r rs 1− , r

e(ϕ ) =

1 ∂ϕ . r sin ϑ

(2.2.41)

θ (ϕ ) = r sin ϑ d ϕ .

(2.2.42)

cot ϑ . r

(2.2.43)

γ(ϕ )(ϑ )(ϕ ) =

The contractions of the Ricci rotation coefficients read

γ(r) =

4r − 3rs p , 1 − rs/r

γ(ϑ ) =

2r2

cot ϑ . r

(2.2.44)

Riemann-Tensor with respect to local tetrad: R(v)(r)(v)(r) = −R(ϑ )(ϕ )(ϑ )(ϕ ) = −

rs , r3

(2.2.45a)

R(v)(ϑ )(v)(ϑ ) = R(v)(ϕ )(v)(ϕ ) = −R(r)(ϑ )(r)(ϑ ) = −R(r)(ϕ )(r)(ϕ ) =

rs . 2r3

(2.2.45b)

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

ds2 =

 4rs3 −r/rs e −dT 2 + dX 2 + r2 dΩ2 , r

(2.2.46)

where r ∈ R+ \ {0} is given by means of the LambertW-function W , 

 r − 1 er/rs = X 2 − T 2 rs

or

  2   X −T2 +1 . r = rs W e

(2.2.47)

The Schwarzschild coordinate time t in terms of the Kruskal coordinates T and X reads T t = 2rs arctanh , X X t = 2rs arctanh , T t = ∞,

r > rs ,

(2.2.48a)

r < rs ,

(2.2.48b)

r = rs .

(2.2.48c)

The transformations between Kruskal- and Schwarzschild coordinates read r r r r/(2rs ) r ct ct sinh , T = 1 − er/(2rs ) cosh , 0 < r < r2 , X = 1− e rs 2rs rs 2rs r r r r ct ct − 1 er/(2rs ) cosh , T= − 1 er/(2rs ) sinh , r ≥ rs . X= rs 2rs rs 2rs

(2.2.49a) (2.2.49b)

24

CHAPTER 2. SPACETIMES

Christoffel symbols: Trs (r + rs ) −r/rs e , r2 Xrs (r + rs ) −r/rs e , =− r2

ΓTT T = ΓXT X = ΓTXX = ΓXT T = ΓTT X = ΓXXX ΓϑT ϑ =

2r2 T − s2 e−r/rs , r

(2.2.50a) (2.2.50b) ΓϑX ϑ =

r T, 2rs r = − T sin2 ϑ , 2rs = cot ϑ ,

2rs2 X −r/rs e , r2

(2.2.50c)

r X, 2rs r X sin2 ϑ , = 2rs = − sin ϑ cos ϑ .

ΓTϑ ϑ = −

ΓXϑ ϑ =

(2.2.50d)

ΓTϑ ϑ

ΓXϑ ϑ

(2.2.50e)

ϕ

Γϑ ϕ

ϑ Γϕϕ

(2.2.50f)

Riemann-Tensor: rs7 −2r/rs e , r5

2rs4 −r/rs e , r2 2r4 = − 2s e−r/rs , r

RT ϑ T ϑ =

(2.2.51a)

RT ϕ T ϕ =

RX ϑ X ϑ

(2.2.51b)

RX ϕ X ϕ

Rϑ ϕϑ ϕ = rrs sin2 ϑ .

RT XT X = −16

2rs4 −r/rs 2 e sin ϑ , r2 2r4 = − 2s e−r/rs sin2 ϑ , r

(2.2.51c)

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

12rs2 . r6

(2.2.52)

Local tetrad: e(T ) =

√ r √ er/(2rs ) ∂T , 2rs rs

e(X) =

√ r √ er/(2rs ) ∂X , 2rs rs

1 e(ϑ ) = ∂ϑ , r

e(ϕ ) =

1 ∂ϕ r sin ϑ

(2.2.53)

Riemann-Tensor with respect to local tetrad: R(T )(X)(T )(X) = R(X)(ϑ )(X)(ϑ ) = R(X)(ϕ )(X)(ϕ ) = −R(ϑ )(ϕ )(ϑ )(ϕ ) = − R(T )(ϑ )(T )(ϑ ) = R(T )(ϕ )(T )(ϕ ) =

rs . 2r3

rs , r3

(2.2.54a) (2.2.54b)

2.2.6 Tortoise coordinates The Schwarzschild metric represented by tortoise coordinates (t, ρ , ϑ , ϕ ) reads      rs rs ds2 = − 1 − c2 dt 2 + 1 − d ρ 2 + r(ρ )2 d ϑ 2 + sin2 ϑ d ϕ 2 , r(ρ ) r(ρ )

(2.2.55)

where rs = 2GM/c2 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

ρ = r + rs ln



r −1 rs



or

    ρ r = rs 1 + W exp −1 . rs

(2.2.56)

2.2. SCHWARZSCHILD SPACETIME

25

Christoffel symbols: ρ

c2 rs , 2r(ρ )2 1 1 = − , r(ρ ) rs

Γtt = ϑ Γρϑ ϕ

Γϑ ϕ = cot ϑ ,

rs , 2r(ρ )2 1 1 = − , r(ρ ) rs

ρ

Γtt ρ = ϕ

Γρϕ ρ

Γρρ =

rs , 2r(ρ )2

(2.2.57a)

ρ

Γϕϕ = −r(ρ ) sin2 ϑ ,

Γϑ ϑ = −r(ρ ),

(2.2.57b)

ϑ Γϕϕ = − sin ϑ cos ϑ .

(2.2.57c)

Riemann-Tensor:   rs 2 , 1− r(ρ )   c2 sin2 ϑ rs rs = 1− , 2 r(ρ ) r(ρ )   rs rs sin2 ϑ 1− , =− 2 r(ρ ) r(ρ )

Rt ρ t ρ = − Rt ϕ t ϕ Rρϕρϕ

c2 rs r(ρ )3

Rρϑ ρϑ

c2 2

  rs rs 1− , r(ρ ) r(ρ )   1 rs rs =− 1− 2 r(ρ ) r(ρ )

Rt ϑ t ϑ =

(2.2.58a) (2.2.58b)

Rϑ ϕϑ ϕ = r(ρ )rs sin2 ϑ .

(2.2.58c)

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 K = 12

rs2 . r(ρ )6

(2.2.59)

Local tetrad: 1 ∂ρ , e(ρ ) = p 1 − rs /r(ρ )

1 ∂t , e(t) = p c 1 − rs /r(ρ )

Dual tetrad:

θ

(t)

r rs = c 1− dt, r(ρ )

θ

(ρ )

r rs = 1− dρ , r(ρ )

e(ϑ ) =

1 ∂ϑ , r(ρ )

θ (ϑ ) = r(ρ ) d ϑ ,

e(ϕ ) =

1 ∂ϕ . r(ρ ) sin ϑ

θ (ϕ ) = r(ρ ) sin ϑ d ϕ .

(2.2.60)

(2.2.61)

Riemann-Tensor with respect to local tetrad: R(t)(ρ )(t)(ρ ) = −R(ϑ )(ϕ )(ϑ )(ϕ ) = −

rs , r(ρ )3

R(t)(ϑ )(t)(ϑ ) = R(t)(ϕ )(t)(ϕ ) = −R(ρ )(ϑ )(ρ )(ϑ ) = −R(ρ )(ϕ )(ρ )(ϕ ) =

(2.2.62a) rs . 2r(ρ )3

(2.2.62b)

Further reading: MTW[MTW73]

2.2.7 Painlevé-Gullstrand The Schwarzschild metric expressed in Painlevé-Gullstrand coordinates[MP01] reads r 2   rs c dT + r2 d ϑ 2 + sin2 ϑ d ϕ 2 , ds2 = −c2 dT 2 + dr + r where the new time coordinate T follows from the Schwarzschild time t in the following way: p ! r r 1 r/rs − 1 + ln p . cT = ct + 2rs rs 2 r/rs + 1

(2.2.63)

(2.2.64)

26

CHAPTER 2. SPACETIMES

Metric-Tensor: gT T = −c

2



rs  1− , r

gTr = c

Christoffel symbols: r crs rs T ΓT T = 2 , 2r r r crs rs ΓrTr = − 2 , 2r r

r

rs , r

gϑ ϑ = r 2 ,

grr = 1,

c2 rs (r − rs ) , 2r3 r r rs , ΓTrr = 2 2cr rs

ΓrT T =

1 Γϑrϑ = , r

1 ϕ Γrϕ = , r

Γϑr ϑ = −(r − rs ),

Γϑ ϕ = cot ϑ ,

Γrϕϕ = −(r − rs ) sin2 ϑ ,

ϑ Γϕϕ = − sin ϑ cos ϑ .

ϕ

gϕϕ = r2 sin2 ϑ .

rs , 2r2 rs Γrrr = − 2 , 2r r r rs T Γϑ ϑ = − , c r r r rs 2 T sin ϑ , Γϕϕ =− c r ΓTTr =

(2.2.65)

(2.2.66a) (2.2.66b) (2.2.66c) (2.2.66d) (2.2.66e)

Riemann-Tensor: RTrTr

c2 rs =− 3 , r

RT ϕ T ϕ =

RT ϑ T ϑ

c2 rs (r − rs ) sin2 ϑ , 2r2

Rr ϕ r ϕ = −

rs sin2 ϑ , 2r

RT ϕ r ϕ

r c2 rs (r − rs ) crs rs , RT ϑ r ϑ = − , = 2r2 2r r r crs rs 2 rs =− sin ϑ , Rrϑ rϑ = − , 2r r 2r

Rϑ ϕϑ ϕ = rrs sin2 ϑ .

(2.2.67a) (2.2.67b) (2.2.67c)

The Ricci tensor and the Ricci scalar vanish identically. Kretschmann scalar: K = 12rs2 /r6 .

(2.2.68)

For the Painlevé-Gullstrand coordinates, we can define two natural local tetrads. Static local tetrad: r √ rs rs 1 1 1 ∂T , eˆ (r) = √ ∂T + 1 − ∂r , eˆ (ϑ ) = ∂ϑ , eˆ (ϕ ) = eˆ (T ) = p ∂ϕ , c r − rs r r r sin ϑ c 1 − rs /r

(2.2.69)

Dual tetrad:

r dr rs (T ) , θˆ = c 1 − dT − p r r/rs − 1

Freely falling local tetrad: r rs 1 ∂r , e(T ) = ∂T − c r

dr (r) , θˆ = p 1 − rs/r

e(r) = ∂r ,

1 e(ϑ ) = ∂ϑ , r

(ϑ ) θˆ = r d ϑ ,

(ϕ ) θˆ = r sin ϑ d ϕ .

(2.2.70)

1 ∂ϕ . r sin ϑ

(2.2.71)

θ (ϕ ) = r sin ϑ d ϕ .

(2.2.72)

e(ϕ ) =

Dual tetrad:

θ (T ) = c dT,

θ (r) = c

r

rs dT + dr, r

θ (ϑ ) = r d ϑ ,

Riemann-Tensor with respect to local tetrad: R(T )(r)(T )(r) = −R(ϑ )(ϕ )(ϑ )(ϕ ) = −

rs , r3

R(T )(ϑ )(T )(ϑ ) = R(T )(ϕ )(T )(ϕ ) = −R(r)(ϑ )(r)(ϑ ) = −R(r)(ϕ )(r)(ϕ ) =

(2.2.73a) rs . 2r3

(2.2.73b)

2.2. SCHWARZSCHILD SPACETIME

27

2.2.8 Israel coordinates The Schwarzschild metric in Israel coordinates (x, y, ϑ , ϕ ) reads[SKM+03] 2

ds =

rs2



    y2 dx 2 2 2 2 4dx dy + , + (1 + xy) d ϑ + sin ϑ d ϕ 1 + xy

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

(2.2.74)

(2.2.75)

Christoffel symbols: y(2 + xy) , (1 + xy)2 y = , 1 + xy x = , 1 + xy

y3 (3 + xy) , (1 + xy)3 y = , 1 + xy x = − (1 + xy), 2 x = − (1 + xy) sin2 ϑ , 2

Γxxx = −

Γyxx =

Γϑxϑ

Γxϕ

ϕ

Γxϕ

ϕ

Γxϑ ϑ

ϕ

Γϑ ϕ = cot ϑ ,

Γxϕϕ

ϑ Γϕϕ = − sin ϑ cos ϑ .

y(2 + xy) , (1 + xy)2 x = , 1 + xy y = − (1 − xy), 2 y = − (1 − xy) sin2 ϑ , 2

Γyxy =

(2.2.76a)

Γϑyϑ

(2.2.76b)

Γyϑ ϑ Γyϕϕ

(2.2.76c) (2.2.76d) (2.2.76e)

Riemann-Tensor: Rxyxy = −4 Rxϕ xϕ = −2

rs2 , (1 + xy)3

Rxϑ xϑ = −2

rs2 y2 sin2 ϑ , (1 + xy)2

Rxϕ yϕ = −

y2 rs2 rs2 , R , = − x ϑ y ϑ (1 + xy)2 1 + xy

rs2 sin2 ϑ , 1 + xy

Rϑ ϕϑ ϕ = (1 + xy)rs2 sin2 ϑ .

(2.2.77a) (2.2.77b)

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 K =

12 rs4 (1 + xy)6

(2.2.78)

.

Local tetrad: √ 1 + xy y e(0) = − ∂x + √ ∂y , 2rs y rs 1 + xy 1 e(2) = ∂ϑ , rs (1 + xy)

√ 1 + xy ∂x , e(1) = 2rs y 1 ∂ϕ . e(3) = rs (1 + xy) sin ϑ

(2.2.79a) (2.2.79b)

Dual tetrad:

θ

(0)

√ rs 1 + xy = dy, y

θ (2) = rs (1 + xy) d ϑ ,

θ

(1)

√ 2rs y rs 1 + xy =√ dy, dx + y 1 + xy

θ (3) = rs (1 + xy) sin ϑ d ϕ .

(2.2.80a) (2.2.80b)

28

CHAPTER 2. SPACETIMES

2.3 Alcubierre Warp The Warp metric given by Miguel Alcubierre[Alc94] reads ds2 = −c2 dt 2 + (dx − vs f (rs )dt)2 + dy2 + dz2

(2.3.1)

where vs = rs (t) = f (rs ) =

dxs (t) , qdt

(2.3.2a) (2.3.2b)

(x − xs (t))2 + y2 + z2 ,

tanh(σ (rs + R)) − tanh(σ (rs − R)) . 2 tanh(σ R)

(2.3.2c)

The parameter R > 0 defines the radius of the warp bubble and the parameter σ > 0 its thickness. Metric-Tensor: gtt = −c2 + v2s f (rs )2 ,

gtx = −vs f (rs ),

(2.3.3)

gxx = gyy = gzz = 1.

Christoffel symbols: f 2 fx v3s , Γttz = − f fz v2s , Γtty = − f fy v2s , c2 f f x v2 f 2 f x v3 f 3 fx v4s − c2 f fx v2s − c2 ft vs x , Γttx = − 2 s , Γtx =− 2 s, Γttx = 2 c c c 2 f f f v f v y vs y s z s y z , Γtx = , Γtty = − , Γtx = 2 2 2c2 f 2 fy v3s + c2 fy vs f fz v2s f 2 fz v3s + c2 fz vs t x x , Γ = − , Γ = − , Γty =− tz tz 2c2 2c2 2c2 2 f y vs f fx v f x vs Γxxx = 2 s , Γtxy = 2 , Γtxx = 2 , c c 2c 2 f f v f v f fz v2s y z s s t x , Γ = , Γ = , Γxxy = xz xz 2c2 2c2 2c2 with derivatives i d f (rs ) −vs σ (x − xs(t)) h ft = sech2 (σ (rs + R)) − sech2 (σ (rs − R)) = dt 2rs tanh(σ R) i σ (x − xs(t)) h d f (rs ) fx = sech2 (σ (rs + R)) − sech2 (σ (rs − R)) = dx 2rs tanh(σ R) i h d f (rs ) σy fy = sech2 (σ (rs + R)) − sech2 (σ (rs − R)) = dy 2rs tanh(σ R) i h d f (rs ) σz fz = = sech2 (σ (rs + R)) − sech2 (σ (rs − R)) dz 2rs tanh(σ R) Γttt =

(2.3.4a) (2.3.4b) (2.3.4c) (2.3.4d) (2.3.4e) (2.3.4f)

(2.3.5a) (2.3.5b) (2.3.5c) (2.3.5d)

Riemann- and Ricci-tensor as well as Ricci- and Kretschman-scalar are shown only in the Maple worksheet. Comoving local tetrad: e(0) =

1 (∂t + vs f ∂x ) , c

e(1) = ∂x ,

e(2) = ∂y ,

e(3) = ∂z .

(2.3.6)

Static local tetrad: 1

e(0) = p ∂t , c2 − v2s f 2

p c2 − v2s f 2 ∂t + ∂x , e(1) = p c c c2 − v2s f 2 vs f

Further reading: Pfenning[PF97], Clark[CHL99], Van Den Broeck[Bro99]

e(2) = ∂y ,

e(3) = ∂z .

(2.3.7)

2.4. BARRIOLA-VILENKIN MONOPOL

29

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 = −c2 dt 2 + dr2 + k2 r2 d ϑ 2 + sin2 ϑ d ϕ 2 ,

(2.4.1)

where k is the scaling factor responsible for the deficit/surplus angle. Christoffel symbols: Γrϑ ϑ = −k2 r,

r Γϕϕ = −k2 r sin2 ϑ ,

1 Γϑrϑ = , r

ϑ = − sin ϑ cos ϑ , Γϕϕ

1 ϕ Γrϕ = , r

Γϑ ϕ = cot ϑ .

(2.4.2a)

ϕ

(2.4.2b)

Partial derivatives 1 , r2 1 =− 2 , sin ϑ = −k2 r sin(2ϑ ).

ϕ

1 , r2

Γϑrϑ ,r = −

Γrϕ ,r = −

ϕ

Γrϕϕ ,r = −k2 sin2 ϑ ,

Γϑ ϕ ,ϑ Γrϕϕ ,ϑ

Γϑr ϑ ,r = −k2 ,

(2.4.3a)

ϑ Γϕϕ ,ϑ = − cos(2ϑ ),

(2.4.3b) (2.4.3c)

Riemann-Tensor: Rϑ ϕϑ ϕ = (1 − k2)k2 r2 sin2 ϑ .

(2.4.4)

Ricci tensor, Ricci and Kretschmann scalar: Rϕϕ = (1 − k2 ) sin2 ϑ ,

Rϑ ϑ = (1 − k2),

R=2

1 − k2 , k2 r 2

K =4

(1 − k2)2 . k4 r 4

(2.4.5)

Weyl-Tensor: c2 (1 − k2) c2 c2 2 = = , C (1 − k ), C (1 − k2 ) sin2 ϑ , ϕ t ϕ t ϑ t ϑ t 3k2 r2 6 6 1 1 k2 r 2 = − (1 − k2), Crϕ rϕ = − (1 − k2 ) sin2 ϑ , Cϑ ϕϑ ϕ = (1 − k2) sin2 ϑ . 6 6 3

Ctrtr = − Crϑ rϑ

(2.4.6a) (2.4.6b)

Local tetrad: 1 e(t) = ∂t , c

e(r) = ∂r ,

e(ϑ ) =

1 ∂ϑ , kr

e(ϕ ) =

1 ∂ϕ . kr sin ϑ

(2.4.7)

Dual tetrad:

θ (t) = c dt,

θ (r) = dr,

θ (ϑ ) = kr d ϑ ,

θ (ϕ ) = kr sin ϑ d ϕ .

(2.4.8)

Ricci rotation coefficients: 1 γ(ϑ )(r)(ϑ ) = γ(ϕ )(r)(ϕ ) = , r

γ(ϕ )(ϑ )(ϕ ) =

cot ϑ . kr

(2.4.9)

The contractions of the Ricci rotation coefficients read 2 γ(r) = , r

γ(ϑ ) =

cot ϑ . kr

(2.4.10)

30

CHAPTER 2. SPACETIMES

Riemann-Tensor with respect to local tetrad: R(ϑ )(ϕ )(ϑ )(ϕ ) =

1 − k2 . k2 r 2

(2.4.11)

Ricci-Tensor with respect to local tetrad: R(ϑ )(ϑ ) = R(ϕ )(ϕ ) =

1 − k2 . k2 r 2

(2.4.12)

Weyl-Tensor with respect to local tetrad: C(t)(r)(t)(r) = −C(ϑ )(ϕ )(ϑ )(ϕ ) = −

1 − k2 , 3k2 r2

C(t)(ϑ )(t)(ϑ ) = C(t)(ϕ )(t)(ϕ ) = −C(r)(ϑ )(r)(ϑ ) = −C(r)(ϕ )(r)(ϕ ) = Embedding: The embedding function, see Sec. 1.7, for k < 1 reads p z = 1 − k2 r.

(2.4.13a) 1 − k2 . 6k2 r2

(2.4.13b)

(2.4.14)

Euler-Lagrange: The Euler-Lagrangian formalism, Sec. 1.8.4, for geodesics in the ϑ = π /2 hyperplane yields 1 h21 1 2 , r˙ + Veff = 2 2 c2

1 Veff = 2



 h22 2 − κc , k2 r 2

(2.4.15)

with the constants of motion h1 = c2t˙ and h2 = k2 r2 ϕ˙ . The point of closest approach rpca for a null geodesic that starts at r = ri with y = ±e(t) +cos ξ e(r) +sin ξ e(ϕ ) is given by r = ri sin ξ . Hence, the rpca is independent of k. The same is also true for timelike geodesics. Further reading: Barriola and Vilenkin[BV89], Perlick[Per04].

2.5. BERTOTTI-KASNER

31

2.5 Bertotti-Kasner The Bertotti-Kasner spacetime in spherical coordinates (t, r, ϑ , ϕ ) reads[Rin98] ds2 = −c2 dt 2 + e2

√ Λct

dr2 +

 1 d ϑ 2 + sin2 ϑ d ϕ 2 , Λ

where the cosmological constant Λ must be positive. Christoffel symbols: √ √ Λ 2√Λct ϕ r Γtrr = e Γtr = c Λ, , Γϑ ϕ = cot ϑ , c

(2.5.1)

ϑ Γϕϕ = − sin ϑ cos ϑ .

(2.5.2)

Partial derivatives Γtrr,t = 2Λe2

√ Λct

ϕ

Γϑ ϕ ,ϑ = −

,

1 , sin2 ϑ

Γϑϕϕ ,ϑ = − cos(2ϑ ).

(2.5.3)

Riemann-Tensor: Rtrtr = −Λc2 e2

√ Λct

Rϑ ϕϑ ϕ =

,

sin2 ϑ . Λ

(2.5.4)

Ricci-Tensor: Rtt = −Λc2 ,

Rrr = Λe2

√ Λct

,

Rϑ ϑ = 1,

Rϕϕ = sin2 ϑ .

(2.5.5)

The Ricci and Kretschmann scalars read R = 4Λ,

K = 8Λ2 .

(2.5.6)

Weyl-Tensor: √ 2 Ctrtr = − Λc2 e2 Λct , 3 1 √ Crϑ rϑ = − e2 Λct , 3

c2 , 3 1 √ = − e2 Λct sin2 ϑ , 3

Ct ϑ t ϑ = Crϕ rϕ

1 √ Ct ϕ t ϕ = − e2 Λct , 3 2 sin2 ϑ Cϑ ϕϑ ϕ = . 3 Λ

(2.5.7a) (2.5.7b)

Local tetrad: 1 e(t) = ∂t , c

e(r) = e

∂r ,

√ e(ϑ ) = Λ∂ϑ ,

dr,

1 θ (ϑ ) = √ d ϑ , Λ

√ − Λct

√ Λ ∂ϕ . e(ϕ ) = sin ϑ

(2.5.8)

Dual tetrad:

θ (t) = c dt,

θ (r) = e

√ Λct

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

sin ϑ θ (ϕ ) = √ d ϕ . Λ

(2.5.9)

(2.5.10)

(2.5.11)

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

(2.5.12)

32

CHAPTER 2. SPACETIMES

Ricci-Tensor with respect to local tetrad: (2.5.13)

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

2Λ , 3

(2.5.14a)

C(t)(ϑ )(t)(ϑ ) = C(t)(ϕ )(t)(ϕ ) = −C(r)(ϑ )(r)(ϑ ) = −C(r)(ϕ )(r)(ϕ ) =

Λ . 3

(2.5.14b)

Euler-Lagrange: The Euler-Lagrangian formalism, Sec. 1.8.4, for geodesics in the ϑ = π /2 hyperplane yields c2t˙2 = h21 e−2

√ Λ ct

+ Λh22 − κ

with the constants of motion h1 = r˙e

(2.5.15) √ 2 Λ ct

and h2 = ϕ˙ /Λ. Thus,

  1 1 + q(t) 1 − q(ti) λ= √ q ln , 1 − q(t) 1 + q(ti) c Λ Λh22 − κ



h2 e−2 Λ ct + 1, q(t) = 1 2 Λh2 − κ

where ti is the initial time. We can also solve the orbital equation: q √ h21 e−2 Λ ct + Λh22 − κ √ r(t) = w(t) − w(ti ) + ri , w(t) = − , h1 Λ

(2.5.16)

(2.5.17)

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).”

2.6. BESSEL GRAVITATIONAL WAVE

33

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    ds2 = e−2U e2K d ρ 2 − dt 2 + ρ 2d ϕ 2 + e2U dz2 .

(2.6.1)

The functions U and K are given by

U := CJ0 (ρ ) cos (t) , o i n h 1 K := C2 ρ ρ J0 (ρ )2 + J1 (ρ )2 − 2J0 (ρ ) J1 (ρ ) cos2 (t) , 2

(2.6.2) (2.6.3)

where Jn (ρ ) are the Bessel functions of the first kind. Christoffel symbols:

∂U ∂ K + , ∂t ∂t ∂U ∂ K ρ ρ + , Γtt = Γtt ρ = Γρρ = − ∂ρ ∂ρ   ∂U ρ −1 , Γϕϕ = ρ e−2K ρ ∂ρ ρ

Γttt = Γt ρ = Γtρρ = −

∂U , ∂t 1 ∂U ϕ Γρϕ = − , ρ ∂ρ ∂U Γρz z = , ∂ρ

∂U , ∂t ∂U Γρzz = −e4U−2K , ∂ρ ∂U Γtzz = e4U−2K . ∂t

ϕ

Γtϕϕ = −e−2K ρ 2

z =− Γt ϕ = Γtz

(2.6.4a) (2.6.4b) (2.6.4c)

Local tetrad: e(t) = eU−K ∂t ,

e(ρ ) = eU−K ∂ρ ,

e(ϕ ) =

1 U e ∂ϕ , ρ

e(z) = e−U ∂z .

(2.6.5)

Dual tetrad:

θ (t) = eK−U dt,

θ (ρ ) = eK−U d ρ ,

θ (ϕ ) = ρ e−U d ϕ ,

θ (z) = eU dz.

(2.6.6)

2.6.2 Cartesian coordinates In Cartesian coordinates with ρ =

p x2 + y2 the metric (2.6.1) reads

   e−2U ds = −e dt + 2 e2K x2 + y2 dx2 + 2xy e2K − 1 dxdy x + y2   2 2 2K 2 + x + e y dy + e2U dz2 . 2

2(K−U)

2

(2.6.7)

Local tetrad: e(t) = eU−K ∂t , e(y) = eU−K

s

e(x) = eU

s

x2 + y2 ∂x , e2K x2 + y2

 eU−K e2K − 1 e2K x2 + y2 ∂y + xy p ∂x , x2 + y2 (x2 + y2 ) (e2K x2 + y2 )

(2.6.8) e(z) = e−U ∂z

34

CHAPTER 2. SPACETIMES

2.7 Cosmic string in Schwarzschild spacetime A cosmic string in the Schwarzschild spacetime represented by Schwarzschild coordinates (t, r, ϑ , ϕ ) reads   rs  2 2 1 ds2 = − 1 − dr2 + r2 d ϑ 2 + β 2 sin2 ϑ d ϕ 2 , c dt + r 1 − rs/r

(2.7.1)

where rs = 2GM/c2 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: c2 rs (r − rs ) , 2r3 1 = , r = cot ϑ ,

Γϑrϑ ϕ

Γϑ ϕ

rs , 2r(r − rs ) 1 = , r = −(r − rs )β 2 sin2 ϑ ,

Γttr =

Γttr =

ϕ

Γrϕ Γrϕϕ

Γrrr = −

rs , 2r(r − rs )

(2.7.2a)

Γrϑ ϑ = −(r − rs ),

(2.7.2b)

ϑ Γϕϕ = −β 2 sin ϑ cos ϑ .

(2.7.2c)

Partial derivatives (2r − rs )rs (2r − rs )rs (2r − 3rs )c2 rs , Γttr,r = − 2 , Γrrr,r = 2 , 2r4 2r (r − rs )2 2r (r − rs )2 1 1 ϕ Γϑrϑ ,r = − 2 , Γrϕ ,r = − 2 , Γϑr ϑ ,r = −1, r r 1 ϕ 2 2 r Γϑ ϕ ,ϑ = − 2 , Γϕϕ Γϑϕϕ ,ϑ = −β 2 cos(2ϑ ), ,r = −β sin ϑ , sin ϑ Γrϕϕ ,ϑ = −(r − rs )β 2 sin(2ϑ ). r Γtt,r =−

(2.7.3a) (2.7.3b) (2.7.3c) (2.7.3d)

Riemann-Tensor: 1 c2 (r − rs ) rs 1 c2 (r − rs ) rs β 2 sin2 ϑ c2 rs = = , R , R , t ϕ t ϕ t ϑ t ϑ r3 2 r2 2 r2 1 rs 1 rs β 2 sin2 ϑ , Rr ϕ r ϕ = − , Rϑ ϕϑ ϕ = rrs β 2 sin2 ϑ . =− 2 r − rs 2 r − rs

Rtrtr = − Rr ϑ r ϑ

(2.7.4a) (2.7.4b)

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 K = 12

rs2 . r6

(2.7.5)

Local tetrad: r rs e(r) = 1 − ∂r , r

1 e(ϑ ) = ∂ϑ , r

e(ϕ ) =

1 ∂ϕ . rβ sin ϑ

(2.7.6)

dr θ (r) = p , 1 − rs/r

θ (ϑ ) = r d ϑ ,

θ (ϕ ) = rβ sin ϑ d ϕ .

(2.7.7)

1 e(t) = p ∂t , c 1 − rs /r

Dual tetrad:

θ

(t)

r rs = c 1 − dt, r

Ricci rotation coefficients:

r p s γ(r)(t)(t) = , 2 2r 1 − rs/r

1 γ(ϑ )(r)(ϑ ) = γ(ϕ )(r)(ϕ ) = r

r rs 1− , r

γ(ϕ )(ϑ )(ϕ ) =

cot ϑ . r

(2.7.8)

2.7. COSMIC STRING IN SCHWARZSCHILD SPACETIME

35

The contractions of the Ricci rotation coefficients read

γ(r) =

4r − 3rs p , 1 − rs/r

2r2

γ(ϑ ) =

cot ϑ . r

(2.7.9)

Riemann-Tensor with respect to local tetrad: R(t)(r)(t)(r) = −R(ϑ )(ϕ )(ϑ )(ϕ ) = −

rs , r3

R(t)(ϑ )(t)(ϑ ) = R(t)(ϕ )(t)(ϕ ) = −R(r)(ϑ )(r)(ϑ ) = −R(r)(ϕ )(r)(ϕ ) =

(2.7.10a) rs . 2r3

(2.7.10b)

Embedding: The embedding function for β 2 < 1 reads r z = (r − rs )

p p rs r r/(r − rs ) − β 2 − 1 − β 2 2 p −β − p ln p . r − rs 2 1−β2 r/(r − rs ) − β 2 + 1 − β 2

(2.7.11)

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 1 k2 1 2 , r˙ + Veff = 2 2 c2

Veff =

  rs  h2 1 2 1− − κ c 2 r r2 β 2

(2.7.12)

with the constants of motion k = (1 − rs/r)c2t˙ and h = r2 β 2 ϕ˙ . The maxima of the effective potential Veff lead to the same critical orbits rpo = 32 rs and ritcg = 3rs as in the standard Schwarzschild metric.

36

CHAPTER 2. SPACETIMES

2.8 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)     r2 sin2 ϑ 2 dr2 2M dt 2 + dϕ , + r2 d ϑ 2 + ds2 = Λ2 − 1 − r 1 − 2M/r Λ2

(2.8.1)

where Λ = 1 + B2r2 sin2 ϑ . Here, M is the mass of the black hole and B the magnetic field strength. Christoffel symbols:  2B2 r3 sin2 ϑ − 3MB2r2 sin2 ϑ + M (r − 2M) 2 (r − 2M) B2 sin ϑ cos ϑ ϑ = = , Γ , tt r3 Λ rΛ 2B2 r2 sin ϑ cos ϑ 2B2 r3 sin2 ϑ − 3MB2r2 sin2 ϑ + M , Γtt ϑ = , Γttr = r (r − 2M) Λ Λ Γttr

Γrrr = Γrrϑ = ϕ

Γrϕ = Γϑϑ ϑ = Γrϕϕ = ϑ Γϕϕ =

2B2 r3 sin2 ϑ − 5MB2r2 sin2 ϑ − M , r (r − 2M) Λ 2B2 r2 sin ϑ cos ϑ , Λ 1 − B2r2 sin2 ϑ , rΛ 2B2 r2 sin ϑ cos ϑ , Λ (r − 2M) Ξ sin2 ϑ , Λ5 Ξ sin ϑ cos ϑ . Λ5

Γϑrr = −

2B2 r sin ϑ cos ϑ , (r − 2M) Λ

3B2 r2 sin2 ϑ + 1 , rΛ  3B2 r2 sin2 ϑ + 1 (r − 2M) = , Λ Ξ cos ϑ , = Λ

(2.8.2a) (2.8.2b) (2.8.2c)

Γϑrϑ =

(2.8.2d)

Γϑr ϑ

(2.8.2e)

ϕ

Γϑ ϕ

(2.8.2f) (2.8.2g) (2.8.2h)

with Ξ = 1 − B2r2 sin2 ϑ . Riemann-Tensor: i 2h 4 4 4 5 4 2 2 2 2 2 B r sin ϑ (3M − r) − M + 2r B sin ϑ cos ϑ + B r sin ϑ (r − 2M) , r3   = 2B2 sin ϑ cos ϑ (3B2 r2 sin2 ϑ (2M − 3r) + r − 2M ,  1 = 2 B4 r4 (r − 2M)(4r − 9M) sin4 ϑ + 2ΞB2r3 (r − 2M) cos2 ϑ + M(r − 2M) , r  1  = 4 2 (2B2 r3 − 3B2Mr2 sin2 ϑ + M)Ξ(r − 2M) sin2 ϑ , Λ r (2B2 r3 − 3B2Mr2 sin2 ϑ + M)Ξ =− , r − 2M  sin2 ϑ  4 4 B r (4r − 9M) sin4 ϑ + 2B2r2 (8M − 4rϑ ) sin2 ϑ + 2ΞB2r3 cos2 ϑ + M , =− 4 Λ (r − 2M)  2B2r3 sin3 ϑ cos ϑ 3B2 r2 sin2 ϑ − 5 =− , Λ4  r sin2 ϑ  4 4 2B r (r − 3M) sin4 ϑ + 4B2r3 cos2 ϑ (1 + Ξ) + 2B2r2 sin2 ϑ (2M − r) + 2M . = 4 Λ

Rtrtr =

(2.8.3a)

Rtrt ϑ

(2.8.3b)

Rt ϑ t ϑ Rt ϕ t ϕ Rr ϑ r ϑ Rr ϕ r ϕ Rrϕϑ ϕ Rϑ ϕϑ ϕ

(2.8.3c) (2.8.3d) (2.8.3e) (2.8.3f) (2.8.3g) (2.8.3h)

2.8. ERNST SPACETIME

37

Ricci-Tensor: 4B2[r cos2 ϑ − (r − 2M) sin2 ϑ ] 4B2 (r − 2M)(r + 2M sin2 ϑ ) , Rrr = − , 2 2 r Λ (r − 2M)Λ2   4B2 r r cos2 ϑ + (r − 2M) sin2 ϑ 8B2 r sin ϑ cos ϑ , Rϑ ϑ = , = Λ2 Λ2  4B2 r sin2 ϑ r + 2M sin2 ϑ = . Λ6

Rtt = Rr ϑ Rϕϕ

(2.8.4a) (2.8.4b) (2.8.4c)

Ricci and Kretschmann scalars: (2.8.5a)

R = 0,   16 K = 6 8 3B8 r8 4r2 − 18Mr + 21M 2 sin8 ϑ r Λ   + 2B4r4 31M 2 − 37Mr − 24B2r4 cos2 ϑ + 42B2Mr3 cos2 ϑ + 10r2 + 6B4r6 cos4 ϑ sin6 ϑ   + 2B2r2 −3Mr + 20B2r4 cos2 ϑ + 6M 2 − 46B2Mr3 cos2 ϑ − 12B4r6 cos4 ϑ sin4 ϑ  − 6B6r6 6B2 Mr3 cos2 ϑ + 4r2 − 4B2r4 cos2 ϑ + 18M 2 − 17Mr  2 3 2 2 4 6 4 + 20B r cos ϑ + 12B Mr cos ϑ + 3M .

(2.8.5b)

Static local tetrad: e(t) =

1 p ∂t , Λ 1 − 2m/r

e(r) =

p 1 − 2m/r ∂r , Λ

1 ∂ϑ , Λr

e(ϕ ) =

Λ ∂ϕ . r sin ϑ

(2.8.6)

θ (ϑ ) = Λr d ϑ ,

θ (ϕ ) =

r sin ϑ dϕ . Λ

(2.8.7)

e(ϑ ) =

Dual tetrad:

θ

(t)



r

1−

2m dt, r

Λ dr, θ (r) = p 1 − 2m/r

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

h2 (1 − rs/r) k2 1 − rs /r − 4 +κ =0 2 r Λ Λ2

(2.8.8)

with constants of motion k = Λ2 (1 − rs/r)t˙ and h = (r2 /Λ2 )ϕ˙ . Further reading: Ernst[Ern76], Dhurandhar and Sharma[DS83], Karas and Vokrouhlicky[KV92], Stuchlík and Hledík[SH99].

38

CHAPTER 2. SPACETIMES

2.9 Friedman-Robertson-Walker The Friedman-Robertson-Walker metric describes a general homogeneous and isotropic universe. In a general form it reads: ds2 = −c2 dt 2 + R2 d σ 2

(2.9.1)

with R = R(t) being an arbitrary function of time only and d σ 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.

2.9.1 Form 1 2

2

2

ds = −c dt + R

2



  dη 2 2 2 2 2 + η d ϑ + sin ϑ d ϕ 1 − kη 2

(2.9.2)

Christoffel symbols: Γtηη = Γtηη = ϕ

Γηϕ =

R˙ , R

R˙ , R kη , = 1 − kη 2 Rη 2 R˙ = , c2 Rη 2 sin2 ϑ R˙ , = c2

ϕ

Γtϑϑ = RR˙

c2 (1 − kη 2)

η , Γηη

1 , η

Γtϑ ϑ

ϕ

Γϑ ϕ = cot ϑ ,

Γtϕϕ

ϑ = − sin ϑ cos ϑ . Γϕϕ

R˙ , R 1 = , η

Γt ϕ = Γϑηϑ

(2.9.3a) (2.9.3b)

Γϑη ϑ = (kη 2 − 1)η ,

(2.9.3c)

Γηϕϕ = (kη 2 − 1)η sin2 ϑ ,

(2.9.3d) (2.9.3e)

Riemann-Tensor: Rt η t η =

RR¨ , kη 2 − 1

¨ Rt ϑ t ϑ = −Rη 2 R,

¨ Rt ϕ t ϕ = −Rη 2 sin2 ϑ R, Rηϕηϕ

(2.9.4a)

R2 η 2 R˙ 2 + kc , c2 (kη 2 − 1)  R2 η 4 sin2 ϑ R˙ 2 + kc2  2

Rηϑ ηϑ = −

 R2 η 2 sin2 ϑ R˙ 2 + kc2 , =− c2 (kη 2 − 1)

Rϑ ϕϑ ϕ =

c2

(2.9.4b) .

(2.9.4c)

Ricci-Tensor: R¨ Rtt = −3 , R R R¨ + 2(R˙ 2 + kc2 ) Rϑ ϑ = η 2 , c2

RR¨ + 2(R˙ 2 + kc2 ) , c2 (1 − kη 2 ) RR¨ + 2(R˙ 2 + kc2 ) = η 2 sin2 ϑ . c2

Rηη =

(2.9.5a)

Rϕϕ

(2.9.5b)

The Ricci scalar and Kretschmann scalar read:

R=6

RR¨ + R˙ 2 + kc2 , R 2 c2

K = 12

R¨ 2R2 + R˙ 4 + 2R˙ 2kc2 + k2 c4 . R 4 c4

(2.9.6)

Local tetrad: 1 e(t) = ∂t , c

p 1 − kη 2 e(η ) = ∂η , R

eϑ =

1 ∂ϑ , Rη

eϕ =

1 ∂ϕ . Rη sin ϑ

(2.9.7)

2.9. FRIEDMAN-ROBERTSON-WALKER

39

Ricci rotation coefficients: p 1 − kη 2 γ(ϑ )(η )(ϑ ) = γ(ϕ )(η )(ϕ ) = , Rη

R˙ γ(η )(t)(η ) = γ(ϑ )(t)(ϑ ) = γ(ϕ )(t)(ϕ ) = Rc cot ϑ γ(ϕ )(ϑ )(ϕ ) = . Rη

The contractions of the Ricci rotation coefficients read p 3R˙ 2 1 − kη 2 cot ϑ γ(t) = γ(r) = , γ(ϑ ) = . , Rc Rη Rη

(2.9.8)

(2.9.9)

Riemann-Tensor with respect to local tetrad: R¨ Rc2 R˙ 2 + kc2 . R(η )(ϑ )(η )(ϑ ) = R(η )(ϕ )(η )(ϕ ) = R(ϑ )(ϕ )(ϑ )(ϕ ) = R 2 c2

(2.9.10a)

R(t)(η )(t)(η ) = R(t)(ϑ )(t)(ϑ ) = R(t)(ϕ )(t)(ϕ ) = −

(2.9.10b)

Ricci-Tensor with respect to local tetrad: R(t)(t) = −

3R¨ , Rc2

R(r)(r) = R(ϑ )(ϑ ) = R(ϕ )(ϕ ) =

RR¨ + 2R˙ 2 + 2kc2 . R 2 c2

(2.9.11)

2.9.2 Form 2 ds2 = −c2 dt 2 +

 2 R2 dr + r2 (d ϑ 2 + sin2 ϑ d ϕ 2 ) k 2 2 (1 + 4 r )

(2.9.12)

Christoffel symbols: r Γtr =

R˙ , R

Γtrr = 16 ϕ

Γrϕ =

Γtϑϑ = RR˙ c2 (4 + kr2 )2

ϕ

Γϑ ϕ = cot ϑ , r = Γϕϕ

Γrrr = −

,

4 − kr2 , (4 + kr2 )r

R˙ , R

ϕ

2kr , 4 + kr2 Rr2 R˙

R˙ , R 4 − kr2 = , (4 + kr2 )r

Γt ϕ =

(2.9.13a)

Γϑrϑ

(2.9.13b)

, Γϑr ϑ =

r(kr2 − 4) , 4 + kr2

Γtϑ ϑ = 16

c2 (4 + kr2)2

Γtϕϕ = 16

Rr2 sin2 ϑ R˙ ϑ , Γϕϕ = − sin ϑ cos ϑ , c2 (4 + kr2)2

(2.9.13c) (2.9.13d)

r sin2 ϑ (kr2 − 4) . 4 + kr2

(2.9.13e)

Riemann-Tensor: Rtrtr = −16 Rt ϕ t ϕ = −16 Rrϕ rϕ = 256

RR¨ , (4 + kr2)2 Rr2 sin2 ϑ R¨ (4 + kr2)2

,

Rr2 R¨ , (4 + kr2)2  R2r2 R˙ 2 + kc2

Rt ϑ t ϑ = −16

(2.9.14a)

Rrϑ rϑ = 256

(2.9.14b)

 R2r2 sin2 ϑ R˙ 2 + kc2 , Rϑ ϕϑ ϕ = 256 c2 (4 + kr2 )4

, c2 (4 + kr2 )4  R2r4 sin2 ϑ R˙ 2 + kc2 c2 (4 + kr2 )4

.

(2.9.14c)

Ricci-Tensor: R¨ Rtt = −3 , R Rϑ ϑ = 16r2

RR¨ + 2(R˙ 2 + kc2 ) , c2 (4 + kr2)2

RR¨ + 2(R˙ 2 + kc2 ) , c2 (4 + kr2 )2 RR¨ + 2(R˙ 2 + kc2 ) = 16r2 sin2 ϑ . c2 (4 + kr2)2

Rrr = 16 Rϕϕ

(2.9.15a) (2.9.15b)

40

CHAPTER 2. SPACETIMES

The Ricci scalar and Kretschmann scalar read:

R=6

RR¨ + R˙ 2 + kc2 , R 2 c2

K = 12

R¨ 2R2 + R˙ 4 + 2R˙ 2kc2 + k2 c4 . R 4 c4

(2.9.16)

Local tetrad: 1 e(t) = ∂t , c

e(r) =

1 + 4k r2 ∂r , R

1 + 4k r2 ∂ϑ , Rr

eϑ =

eϕ =

1 + k/4r2 ∂ϕ . Rr sin ϑ

(2.9.17)

Ricci rotation coefficients:

γ(r)(t)(r) = γ(ϑ )(t)(ϑ ) = γ(ϕ )(t)(ϕ ) = γ(ϕ )(ϑ )(ϕ ) =

R˙ Rc

k 2 r −1

γ(ϑ )(r)(ϑ ) = γ(ϕ )(r)(ϕ ) = − 4

Rr

( 4k r2 + 1) cot ϑ . Rr

,

(2.9.18a) (2.9.18b)

The contractions of the Ricci rotation coefficients read

γ(t) =

3R˙ , Rc

γ(r) = 2

1 − 4k r2 , Rr

γ(ϑ ) =

( 4k r2 + 1) cot ϑ . Rr

(2.9.19)

Riemann-Tensor with respect to local tetrad: R¨ Rc2 R˙ 2 + kc2 R(η )(ϑ )(η )(ϑ ) = R(η )(ϕ )(η )(ϕ ) = R(ϑ )(ϕ )(ϑ )(ϕ ) = . R 2 c2

(2.9.20a)

R(t)(η )(t)(η ) = R(t)(ϑ )(t)(ϑ ) = R(t)(ϕ )(t)(ϕ ) = −

(2.9.20b)

Ricci-Tensor with respect to local tetrad: R(t)(t) = −

3R¨ , Rc2

R(r)(r) = R(ϑ )(ϑ ) = R(ϕ )(ϕ ) =

RR¨ + 2R˙ 2 + 2kc2 . R 2 c2

(2.9.21)

2.9.3 Form 3 The following forms of the metric are obtained from 2.9.2 by setting η = sin ψ , ψ , sinh ψ for k = 1, 0, −1 respectively. Positive Curvature   ds2 = −c2 dt 2 + R2 d ψ 2 + sin2 ψ d ϑ 2 + sin2 ϑ d ϕ 2

(2.9.22)

Christoffel symbols: ψ

R˙ , Γtϑϑ R RR˙ = 2, Γϑψϑ c R sin2 ψ R˙ ψ , Γϑ ϑ = c2 R sin2 ψ sin2 ϑ R˙ ψ = , Γϕϕ c2

Γt ψ = Γtψψ Γtϑ ϑ Γtϕϕ

=

R˙ , R

ϕ

Γt ϕ =

R˙ , R

ϕ

= cot ψ ,

Γψϕ = cot ψ ,

= − sin ψ cos ψ ,

Γϑ ϕ = cot(ϑ ),

ϕ

ϑ = − sin ψ cos ψ sin2 ϑ , Γϕϕ = − sin ϑ cos ϑ .

(2.9.23a) (2.9.23b) (2.9.23c) (2.9.23d)

2.9. FRIEDMAN-ROBERTSON-WALKER

41

Riemann-Tensor: ¨ Rt ϑ t ϑ = −R sin2 ψ R,  2 2 R sin ψ R˙ 2 + c2 2 2 ¨ = −R sin ψ sin ϑ R, , Rψϑ ψϑ = c2   R2 sin2 ψ sin2 ϑ R˙ 2 + c2 R2 sin4 ψ sin2 ϑ R˙ 2 + c2 = , Rϑ ϕϑ ϕ = . c2 c2

¨ Rt ψ t ψ = −RR,

(2.9.24a)

Rt ϕ t ϕ

(2.9.24b)

Rψϕψϕ

(2.9.24c)

Ricci-Tensor: R¨ Rtt = −3 , R

RR¨ + 2(R˙ 2 + c2 ) , c2 RR¨ + 2(R˙ 2 + c2 ) = sin2 ϑ sin2 ψ . c2

Rψψ =

RR¨ + 2(R˙ 2 + c2 ) , Rϕϕ c2 The Ricci scalar and Kretschmann read R¨ 2R2 + R˙ 4 + 2R˙ 2 c2 + c4 RR¨ + R˙ 2 + c2 , K = 12 . R=6 R 2 c2 R 4 c4 Rϑ ϑ = sin2 ψ

Local tetrad: 1 e(t) = ∂t , c

e(ψ ) =

1 ∂ψ , R

eϑ =

1 ∂ϑ , R sin ψ

eϕ =

(2.9.25a) (2.9.25b)

(2.9.26)

1 ∂ϕ . R sin ψ sin ϑ

(2.9.27)

Ricci rotation coefficients:

γ(ψ )(t)(ψ ) = γ(ϑ )(t)(ϑ ) = γ(ϕ )(t)(ϕ ) = γ(ϕ )(ϑ )(ϕ ) =

cot θ . R sin ψ

R˙ Rc

γ(ϑ )(ψ )(ϑ ) = γ(ϕ )(ψ )(ϕ ) =

cot ψ , R

(2.9.28a) (2.9.28b)

The contractions of the Ricci rotation coefficients read 3R˙ cot ψ cot ϑ γ(t) = γ(r) = 2 γ(ϑ ) = . , , Rc R R sin ψ

(2.9.29)

Riemann-Tensor with respect to local tetrad: R¨ , Rc2 R˙ 2 + c2 R(ψ )(ϑ )(ψ )(ϑ ) = R(ψ )(ϕ )(ψ )(ϕ ) = R(ϑ )(ϕ )(ϑ )(ϕ ) = 2 2 . R c R(t)(ψ )(t)(ψ ) = R(t)(ϑ )(t)(ϑ ) = R(t)(ϕ )(t)(ϕ ) = −

(2.9.30a) (2.9.30b)

Ricci-Tensor with respect to local tetrad: R(t)(t) = −

3R¨ , Rc2

R(ψ )(ψ ) = R(ϑ )(ϑ ) = R(ϕ )(ϕ ) =

RR¨ + 2(R˙ 2 + c2 ) . R 2 c2

(2.9.31)

Vanishing Curvature   ds2 = −c2 dt 2 + R2 d ψ 2 + ψ 2 d ϑ 2 + sin2 ϑ d ϕ 2

Christoffel symbols: R˙ ψ Γt ψ = , R R R˙ Γtψψ = 2 , c Rψ 2 R˙ , Γtϑ ϑ = c2 Rψ 2 sin2 ϑ R˙ , Γtϕϕ = c2

R˙ , R 1 = , ψ

Γtϑϑ = Γϑψϑ

ϕ

R˙ , R 1 = , ψ

Γt ϕ = ϕ

Γψϕ

(2.9.33a) (2.9.33b)

ψ

Γϑ ϕ = cot(ϑ ),

(2.9.33c)

ψ

ϑ Γϕϕ = − sin ϑ cos ϑ .

(2.9.33d)

Γϑ ϑ = −ψ , Γϕϕ = −ψ sin2 ϑ ,

ϕ

(2.9.32)

42

CHAPTER 2. SPACETIMES

Riemann-Tensor: ¨ Rt ψ t ψ = −RR, ¨ Rt ϕ t ϕ = −Rψ 2 sin2 ϑ R, Rψϕψϕ =

R2 ψ 2 sin2 ϑ R˙ 2 , c2

¨ Rt ϑ t ϑ = −Rψ 2 R, 2 2 R ψ R˙ 2 Rψϑ ψϑ = , c2 R2 ψ 4 sin2 ϑ R˙ 2 Rϑ ϕϑ ϕ = . c2

(2.9.34a) (2.9.34b) (2.9.34c)

Ricci-Tensor: R¨ RR¨ + 2R˙ 2 Rtt = −3 , , Rψψ = R c2 ¨ ˙2 RR¨ + 2R˙ 2 2 2 RR + 2 R = sin ϑ ψ , R . Rϑ ϑ = ψ 2 ϕϕ c2 c2 The Ricci scalar and Kretschmann read R¨ 2 R2 + R˙ 4 RR¨ + R˙ 2 K = 12 . R=6 2 2 , R c R 4 c4 Local tetrad: 1 e(t) = ∂t , c

e(ψ ) =

1 ∂ψ , R

eϑ =

1 ∂ϑ , Rψ

eϕ =

(2.9.35a) (2.9.35b)

(2.9.36)

1 ∂ϕ . Rψ sin ϑ

(2.9.37)

Ricci rotation coefficients:

γ(ψ )(t)(ψ ) = γ(ϑ )(t)(ϑ ) = γ(ϕ )(t)(ϕ ) = γ(ϕ )(ϑ )(ϕ ) =

R˙ Rc

γ(ϑ )(ψ )(ϑ ) = γ(ϕ )(ψ )(ϕ ) =

1 , Rψ

cot(ϑ ) . Rψ

(2.9.38a) (2.9.38b)

The contractions of the Ricci rotation coefficients read 3R˙ 2 cot ϑ γ(t) = γ(r) = , γ(ϑ ) = . , Rc Rψ Rψ

(2.9.39)

Riemann-Tensor with respect to local tetrad: R¨ , Rc2 R˙ 2 R(ψ )(ϑ )(ψ )(ϑ ) = R(ψ )(ϕ )(ψ )(ϕ ) = R(ϑ )(ϕ )(ϑ )(ϕ ) = 2 2 . R c

(2.9.40a)

R(t)(ψ )(t)(ψ ) = R(t)(ϑ )(t)(ϑ ) = R(t)(ϕ )(t)(ϕ ) = −

(2.9.40b)

Ricci-Tensor with respect to local tetrad: R(t)(t) = −

3R¨ , Rc2

R(ψ )(ψ ) = R(ϑ )(ϑ ) = R(ϕ )(ϕ ) =

RR¨ + 2R˙ 2 . R 2 c2

(2.9.41)

Negative Curvature   ds2 = −c2 dt 2 + R2 d ψ 2 + sinh2 ψ d ϑ 2 + sin2 ϑ d ϕ 2

Christoffel symbols: R˙ ψ Γt ψ = , Γtϑϑ R RR˙ Γϑψϑ Γtψψ = 2 , c R sinh2 ψ R˙ ψ , Γϑ ϑ Γtϑ ϑ = c2 R sinh2 ψ sin2 ϑ R˙ ψ , Γϕϕ Γtϕϕ = c2

=

R˙ , R

(2.9.42)

ϕ

Γt ϕ =

R˙ , R

ϕ

= coth ψ ,

Γψϕ = coth ψ ,

= − sinh ψ cosh ψ ,

Γϑ ϕ = cot ϑ ,

ϕ

ϑ = − sinh ψ cosh ψ sin2 ϑ , Γϕϕ = − sin ϑ cos ϑ .

(2.9.43a) (2.9.43b) (2.9.43c) (2.9.43d)

2.9. FRIEDMAN-ROBERTSON-WALKER

43

Riemann-Tensor: ¨ Rt ϑ t ϑ = −R sinh2 ψ R,  2 2 R sinh ψ R˙ 2 − c2 2 2 ¨ Rψϑ ψϑ = = −R sinh ψ sin ϑ R, , c2   R2 sinh2 ψ sin2 ϑ R˙ 2 − c2 R2 sinh ψ 4 sin2 ϑ R˙ 2 − c2 , Rϑ ϕϑ ϕ = . = c2 c2

¨ Rt ψ t ψ = −RR,

(2.9.44a)

Rt ϕ t ϕ

(2.9.44b)

Rψϕψϕ

(2.9.44c)

Ricci-Tensor: R¨ Rtt = −3 , R Rϑ ϑ = sinh2 ψ

RR¨ + 2(R˙ 2 − c2 ) c2

,

RR¨ + 2(R˙ 2 − c2 ) , c2 RR¨ + 2(R˙ 2 − c2 ) = sin2 ϑ sin2 ψ . c2

Rψψ =

(2.9.45a)

Rϕϕ

(2.9.45b)

The Ricci scalar and Kretschmann read R=6

RR¨ + R˙ 2 − c2 , R 2 c2

K = 12

R¨ 2R2 + R˙ 4 − 2R˙ 2 c2 + c4 . R 4 c4

(2.9.46)

Local tetrad: 1 e(t) = ∂t , c

e(ψ ) =

1 ∂ψ , R

eϑ =

1 ∂ϑ , R sinh ψ

eϕ =

1 ∂ϕ . R sinh ψ sin ϑ

(2.9.47)

Ricci rotation coefficients:

γ(ψ )(t)(ψ ) = γ(ϑ )(t)(ϑ ) = γ(ϕ )(t)(ϕ ) = γ(ϕ )(ϑ )(ϕ ) =

cot θ . R sinh ψ

R˙ Rc

γ(ϑ )(ψ )(ϑ ) = γ(ϕ )(ψ )(ϕ ) =

coth ψ , R

(2.9.48a) (2.9.48b)

The contractions of the Ricci rotation coefficients read

γ(t) =

3R˙ , Rc

γ(r) = 2

coth ψ , R

γ(ϑ ) =

cot ϑ . R sinh ψ

(2.9.49)

Riemann-Tensor with respect to local tetrad: R¨ , Rc2 R˙ 2 − c2 R(ψ )(ϑ )(ψ )(ϑ ) = R(ψ )(ϕ )(ψ )(ϕ ) = R(ϑ )(ϕ )(ϑ )(ϕ ) = 2 2 . R c R(t)(ψ )(t)(ψ ) = R(t)(ϑ )(t)(ϑ ) = R(t)(ϕ )(t)(ϕ ) = −

(2.9.50a) (2.9.50b)

Ricci-Tensor with respect to local tetrad: R(t)(t) = −

3R¨ , Rc2

Further reading: Rindler[Rin01]

R(ψ )(ψ ) = R(ϑ )(ϑ ) = R(ϕ )(ϕ ) =

RR¨ + 2(R˙ 2 − c2 ) . R 2 c2

(2.9.51)

44

CHAPTER 2. SPACETIMES

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

2.10.1 Cylindrical coordinates The Gödel metric in cylindrical coordinates is ds2 = −c2 dt 2 +

  r 2  dr2 c 2 1 − d ϕ 2 + dz2 − 2r2 √ dtd ϕ , + r 2 1 + [r/(2a)] 2a 2a

(2.10.1)

where 2a is the Gödel radius. Christoffel symbols: r 1 , 2 2a 1 + [r/(2a)]2  r i2 cr h =√ , 1+ 2a 2a 1 r3 , = √ 3 4 2ca 1 + [r/(2a)]2      r 2 1  r 2 = r 1+ 1− . 2a 2 a

Γttr = Γtrϕ Γtrϕ Γrϕϕ

c 1 ϕ Γtr = − √ , 2ar 1 + [r/(2a)]2 r 1 Γrrr = − 2 , 4a 1 + [r/(2a)]2 ϕ

Γrϕ =

1 1 , r 1 + [r/(2a)]2

(2.10.2a) (2.10.2b) (2.10.2c) (2.10.2d)

Riemann-Tensor:

Rt ϕ t ϕ

c2 1 , 2 2a 1 + [r/(2a)]2

cr2 1 Rtrrϕ = − √ , 3 2 2a 1 + [r/(2a)]2 c2 r 2 r2 1 + 3[r/(2a)]2 1 = 2 = , R . r ϕ r ϕ 2a 1 + [r/(2a)]2 2a2 1 + [r/(2a)]2

Rtrtr =

(2.10.3a) (2.10.3b)

Ricci-Tensor: Rtt =

c2 , a2

r2 c Rt ϕ = √ , 2a3

Ricci and Kretschmann scalar 3 1 K = 4. R = − 2, a a cosmological constant: Λ=

R 2

Rϕϕ =

r4 . 2a4

(2.10.4)

(2.10.5)

(2.10.6)

Killing vectors: An infinitesimal isometric transformation x′µ = xµ + εξ µ (xν ) leaves the metric unchanged, that is g′µν (x′σ ) = gµν (x′σ ). A killing vector field ξ µ is solution to the killing equation ξµ ;ν + ξν ;µ = 0. There exist five killing vector fields in Gödel’s spacetime:       √r cos ϕ 1 0 2c  0     a 1 + [r/(2a)]2 sin ϕ  1 , ξ µ =  0 , , ξµ = p   ξµ =  (2.10.7a) a 2       2 0 1 a c 1 + [r/(2a)] b r 1 + 2[r/(2a)] cos ϕ 0 0 0     √r sin ϕ 0 2c     1 0  −a 1 + [r/(2a)]2 cos ϕ  µ µ  .  p (2.10.7b) ξ =  , ξ = a 0 e 1 + [r/(2a)]2  r 1 + 2[r/(2a)]2 sin ϕ  d 1 0

2.10. GÖDEL UNIVERSE

45

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.  e(0) = Γ ∂t + ζ ∂ϕ ,

e(1) =

where

q 1 + [r/(2a)]2∂r ,

 r2 c A = − √ + ζ r2 1 − [r/(2a)]2 , 2a 1 Γ= q , √ c2 + ζ r2 c 2/a − ζ 2r2 (1 − [r/(2a)]2)

 e(2) = ∆Γ A∂t + B∂ϕ ,

e(3) = ∂z ,

ζ r2 c B = c2 + √ , 2a 1 ∆= p . rc 1 + [r/(2a)]2

Transformation between local direction y(i) and coordinate direction yµ : q y0 = y(0) Γ + y(2)∆ΓA, y1 = y(1) 1 + [r/(2a)]2, y2 = y(0) Γζ + y(2)∆ΓB,

(2.10.8a)

(2.10.9a) (2.10.9b)

y3 = y(3) .

(2.10.10)

with the above abbreviations.

2.10.2 Scaled cylindrical coordinates If we apply the simple transformation t , rG

T=

R=

r , rG

φ = ϕ,

Z=

z , rG

(2.10.11)

with rG = 2a, we find a formulation for the metric scaling with rG , which is 2

ds =

2 rG

  √ dR2 2 2 2 2 2 2 2 −c dT + + R (1 − R )Dφ + dZ − 2 2cR dT d φ . 1 + R2

(2.10.12)

Christoffel symbols: √ 2c , R(1 + R2) R ΓRRR = − , 1 + R2 1 φ ΓRφ = , R(1 + R2)

2R , 1 + R2 √ = 2cR(1 + R2), √ 3 2R , = c(1 + R2)

φ

ΓTT R = ΓRT φ ΓTRφ

ΓT R = −

ΓφRφ = R(1 + R2)(2R2 − 1).

(2.10.13a) (2.10.13b) (2.10.13c) (2.10.13d)

Riemann-Tensor: 2r2 c2 RT RT R = G 2 , 1+R

RT RRφ

2 2 RT φ T φ = 2c2 rG R (1 + R2),

RR φ R φ

√ 2 2 cR 2 2rG , =− 1 + R2 2r2 R2 (1 + 3R2) . = G 1 + R2

(2.10.14a) (2.10.14b)

Ricci-Tensor: RT T = 4c2 ,

√ RT φ = 4 2cR2 ,

Rφ φ = 8R4 .

(2.10.15)

46

CHAPTER 2. SPACETIMES

Ricci and Kretschmann scalar R=−

4 , 2 rG

K =

48 . 4 rG

(2.10.16)

cosmological constant: Λ=

R 2

(2.10.17)

Killing vectors: The Killing vectors read   √R cos ϕ 1 2c  0   1 (1 + R2) sin ϕ , ξµ = √ 1  2 ξµ =  1  0  (1 + 2R2) cos ϕ a 1 + R2  2R b 0 0    R √ sin ϕ 0 2c 1    1 0  (1 + R2) cos ϕ − µ µ   2 ξ =  , ξ = √ 1  2 2 0 e 1+R d 2R (1 + 2R ) sin ϕ 1 0 

 0  0   ξµ =   1 , c 0 



 , 



(2.10.18a)

 . 

(2.10.18b)

Local tetrad: After the transformation to scaled cylindrical coordinates, the local tetrad reads e(0) = where

 Γ ∂T + ζ ∂φ , rG

e(1) =

1p 1 + R2 ∂R , rG

h √ i A = R2 − 2c + (1 − R2)ζ ,

e(2) =

 ∆Γ A∂T + B∂φ , rG

e(3) =

1 ∂Z , rG

√ B = c2 + 2R2 cζ ,

1 , Γ= q √ c2 + 2 2R2 cζ − R2(1 − R2)ζ 2

∆=

(2.10.19a)

(2.10.20a)

1 √ . Rc 1 + R2

(2.10.20b)

Transformation between local direction y(i) and coordinate direction yµ : y0 =

Γ (0) ∆ΓA (2) y + y , rG rG

y1 =

1p 1 + R2y(1) , rG

y2 =

Γζ (0) ∆ΓB (2) y + y , rG rG

y3 =

1 (3) y , (2.10.21) rG

and the back transformation is given by y(0) =

rG By0 − Ay2 , Γ B−ζA

rG y(1) = √ y1 , 1 + R2

y(2) =

rG y2 − ζ y0 , ∆Γ B − ζ A

y(3) = rG y3 .

(2.10.22a)

2.11. HALILSOY STANDING WAVE

47

2.11 Halilsoy standing wave The standing wave metric by Halilsoy[Hal88] reads   1  ds2 = V e2K d ρ 2 − dt 2 + ρ 2d ϕ 2 + (dz + A d ϕ )2 , V

(2.11.1)

where V = cosh2 α e−2CJ0 (ρ ) cos(t) + sinh2 α e2CJ0 (ρ ) cos(t) ,   C2  2 ρ J0 (ρ )2 + J1 (ρ )2 − 2ρ J0(ρ )J1 (ρ ) cos2 t , K= 2 A = −2C sinh(2α )ρ J1 (ρ ) sin(t).

(2.11.2a) (2.11.2b) (2.11.2c)

with spherical Bessel functions J1,2 and parameters α and C. Local tetrad: e−K e(0) = √ ∂t , V

e−K e(1) = √ ∂ρ , V

1 A e(2) = √ ∂ϕ − √ ∂z , ρ V ρ V

e(3) =

√ V ∂z .

(2.11.3)

dual tetrad:

θ (0) =

√ K V e dt,

√ θ (2) = V eK d ρ ,

√ θ (2) = V ρ d ϕ ,

1 θ (3) = √ (dz + A d ϕ ). V

(2.11.4)

48

CHAPTER 2. SPACETIMES

2.12 Janis-Newman-Winicour The Janis-Newman-Winicour[JNW68] spacetime in spherical coordinates (t, r, ϑ , ϕ ) is represented by the line element  ds2 = −α γ c2 dt 2 + α −γ dr2 + r2 α −γ +1 d ϑ 2 + sin2 ϑ d ϕ 2 ,

(2.12.1)

where α = 1 − rs/(γ r). The Schwarzschild radius rs = 2GM/c2 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: rs c2 2γ −1 , α 2r2 2γ r − rs (γ + 1) = , 2γ r 2 α

Γttr = Γϑrϑ

Γrϕϕ = Γrϑ ϑ sin2 ϑ ,

rs , 2γ r 2 α 2γ r − rs (γ + 1) = , 2γ r 2 α

Γttr = ϕ

Γrϕ ϕ

rs , 2γ r 2 α 2γ r − rs (γ + 1) =− , 2γ

Γrrr = − Γϑr ϑ

ϑ Γϕϕ = − sin ϑ cos ϑ .

Γϑ ϕ = cot ϑ ,

(2.12.2a) (2.12.2b) (2.12.2c)

Riemann-Tensor: Rtrtr = −

rs c2 [2γ r − rs (γ + 1)] α γ −2 , 2γ r 4

rs c2 [2γ r − rs (γ + 1)] α γ −1 sin2 ϑ , Rr ϑ r ϑ 4γ r 2  2  2 rs 2γ r − rs (γ + 1) sin ϑ , Rϑ ϕϑ ϕ =− 4γ 2 r2 α γ −1

Rt ϕ t ϕ = Rr ϕ r ϕ

rs c2 [2γ r − rs (γ + 1)] α γ −1 , 4γ r 2   2 rs 2γ r − rs (γ + 1) , =− 4γ 2 r2 α γ −1   2 rs 4γ r − rs (γ + 1)2 sin2 ϑ = . 4γ 2 α γ

Rt ϑ t ϑ =

(2.12.3a) (2.12.3b) (2.12.3c)

Weyl-Tensor: Ctrtr = − Ct ϕ t ϕ =

rs c2 α γ −2 β , 6γ 2 r 4

Ct ϑ t ϑ =

rs c2 α γ −1 β sin2 ϑ , 12γ 2 r2

Crϕ rϕ = −

rs β sin2 ϑ , 12γ 2 r2 α γ −1

rs c2 α γ −1 β , 12γ 2 r2

Crϑ rϑ = − Cϑ ϕϑ ϕ =

(2.12.4a)

rs β , 12γ 2 r2 α γ −1

(2.12.4b)

rs β sin2 ϑ , 6γ 2 α γ

(2.12.4c)

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

rs2 (1 − γ 2 ) . 2γ 2 r 4 α 2

(2.12.5)

The Ricci scalar reads R=

rs2 (1 − γ 2)α γ −2 , 2γ 2 r 4

(2.12.6)

whereas the Kretschmann scalar is given by K =

 rs2 α 2γ −4  2 2 7γ rs (2 + γ 2) + 48γ 4r2 α + 8γ rs(2γ 2 + 1)(rs − 2γ r) + 3rs2 . 4γ 4 r 8

(2.12.7)

Local tetrad: e(t) =

1 ∂t , cα γ /2

e(r) = α γ /2 ∂r ,

e(ϑ ) =

α (γ −1)/2 ∂ϑ , r

e(ϕ ) =

α (γ −1)/2 ∂ϕ . r sin ϑ

(2.12.8)

2.12. JANIS-NEWMAN-WINICOUR

49

Dual tetrad:

θ (t) = cα γ /2 dt,

θ (r) =

dr , α γ /2

θ (ϑ ) =

r

α (γ −1)/2

dϑ ,

θ (ϕ ) =

r sin ϑ dϕ . α (γ −1)/2

(2.12.9)

Ricci rotation coefficients: rs (γ −2)/2 α , 2r2 cot ϑ (γ −1)/2 γ(ϕ )(ϑ )(ϕ ) = α . r

γ(r)(t)(t) =

γ(ϑ )(r)(ϑ ) = γ(ϕ )(r)(ϕ ) =

2γ r − rs (γ + 1) (γ −2)/2 α , 2γ r 2

(2.12.10a) (2.12.10b)

The contractions of the Ricci rotation coefficients read

γ(r) =

4γ r − rs (2 + γ ) (γ −1)/2 α , 2γ r 2

γ(ϑ ) =

cot ϑ (γ −1)/2 α . r

(2.12.11)

Structure coefficients: rs (γ −2)/2 2γ r − rs (γ + 1) (γ −2)/2 (ϑ ) (ϕ ) α α , c(r)(ϑ ) = c(r)(ϕ ) = − , 2r2 2γ r 2 cot ϑ (γ −1)/2 (ϕ ) α . c(ϑ )(ϕ ) = − r (t)

c(t)(r) =

(2.12.12a) (2.12.12b)

Euler-Lagrange: The Euler-Lagrangian formalism, Sec. 1.8.4, for geodesics in the ϑ = π /2 hyperplane yields the effective potential 1 Veff = α γ 2



h2 α γ −1 − κ c2 r2



(2.12.13)

with the constants of motion h = r2 α −γ +1 ϕ˙ and k = α γ c2t˙. For null geodesics (κ = 0) and γ > 21 , there is an extremum at r = rs

1 + 2γ . 2γ

Embedding: The embedding function z = z(r) for r ∈ [rs (γ + 1)2 /(4γ 2 ), ∞) follows from s dz rs [4rγ 2 − rs (1 + γ )2] = . dr 4r2 γ 2 α γ +1

(2.12.14)

(2.12.15)

However, the analytic solution     √ 2πγ 1 γ + 1 1 1 rs rs (1 + γ )2 1 γ +1 4γ − z(r) = 2 rs r F1 − ; − , F ,− ; , , , ; 1; 2 1 2 2 2 2 rγ 4rγ 2 γ +1 2 2 (γ + 1)2 depends on the Appell-F1- and the Hypergeometric-2 F1 -function.

(2.12.16)

50

CHAPTER 2. SPACETIMES

2.13 Kasner The Kasner spacetime in Cartesian coordinates (t, x, y, z) is represented by the line element[MTW73, Kas21] (c = 1) ds2 = −dt 2 + t 2p1 dx2 + t 2p2 dy2 + t 2p3 tz2 ,

(2.13.1)

where p1 , p2 , p3 have to fulfill the two conditions p1 + p2 + p3 = 1

and

p21 + p22 + p23 = 1.

(2.13.2)

These two conditions can also be represented by the Khalatnikov-Lifshitz parameter u with p1 = −

u , 1 + u + u2

p2 =

1+u , 1 + u + u2

p3 =

u(1 + u) . 1 + u + u2

Christoffel symbols: p2 p3 p1 y z x Γty = , Γtz = , Γtx = , t t t 2p1 2p2 2p3 p t p t p 1 2 3t Γtxx = , Γtyy = , Γtzz = . t t t Partial derivatives p2 p1 x Γtty,t = − 2 , Γtx,t =− 2 , t t Γtxx,t = p1 (2p1 − 1)t 2p1−2 , Γtyy,t = p2 (2p2 − 1)t 2p2−2 ,

(2.13.3)

(2.13.4a) (2.13.4b) p3 , t2 = p3 (2p3 − 1)t 2p3−2 .

z Γtz,t =−

(2.13.5a)

Γtzz,t

(2.13.5b)

Riemann-Tensor: p1 (1 − p1)t 2p1 p2 (1 − p2)t 2p2 p3 (1 − p3)t 2p3 , R = , R = , tyty tztz t2 t2 t2 p1 p3t 2p1 t 2p3 p2 p3t 2p2 t 2p3 p1 p2t 2p1 t 2p2 , R = , .R = . Rxyxy = xzxz yzyz t2 t2 t2 The Ricci tensor as well as the Ricci scalar vanish identically. The Kretschmann scalar reads  4 K = 4 p21 − 2p31 + p41 + p22 − 2p32 + p42 + p21 p23 + p23 − 2p33 + p43 + p21 p22 + p22 p23 t 16u2(1 + u)2 . = 4 t (1 + u + u2)3 Rtxtx =

(2.13.6a) (2.13.6b)

(2.13.7a) (2.13.7b)

Local tetrad: e(t) = ∂t ,

e(x) = t −p1 ∂x ,

e(y) = t −p2 ∂y ,

e(z) = t −p3 ∂z .

(2.13.8)

θ (x) = t p1 dx,

θ (y) = t p2 dy,

θ (z) = t p3 dz.

(2.13.9)

Dual tetrad:

θ (t) = dt,

Ricci rotation coefficients: p1 p2 p3 γ (t)(r)(r) = , γ (t)(ϑ )(ϑ ) = , γ (t)(ϕ )(ϕ ) = . t t t The contractions of the Ricci rotation coefficients read 1 γ (t) = − . t

(2.13.10)

(2.13.11)

Riemann-Tensor with respect to local tetrad: p1 (1 − p1) , t2 p1 p2 R(x)(y)(x)(y) = 2 , t R(t)(x)(y)(x) =

p2 (1 − p2) , t2 p1 p3 R(x)(z)(x)(z) = 2 , t R(t)(y)(t)(y) =

p3 (1 − p3) , t2 p2 p3 R(y)(z)(y)(z) = 2 . t R(t)(z)(t)(z) =

(2.13.12a) (2.13.12b)

2.14. KERR

51

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

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

ds2

 rs r  2 2 2rs ar sin2 ϑ Σ = − 1− c dt − c dt d ϕ + dr2 + Σd ϑ 2 Σ Σ ∆   rs a2 r sin2 ϑ sin2 ϑ d ϕ 2 , + r 2 + a2 + Σ

(2.14.1)

with Σ = r2 + a2 cos2 ϑ , ∆ = r2 − rs r + a2 , and rs = 2GM/c2 , is taken from Bardeen[BPT72]. M is the mass and a is the angular momentum per unit mass of the black hole. The contravariant form of the metric reads

∂s2 = −

∆ 2 1 2 ∆ − a2 sin2 ϑ 2 A 2 2rs ar ∂ϕ , ∂ ∂ ∂ + ∂ + ∂ + − t ϕ t c2 Σ∆ cΣ∆ Σ r Σ ϑ Σ∆ sin2 ϑ

where A = r2 + a2

2

(2.14.2)

 − a2∆ sin2 ϑ = r2 + a2 Σ + rs a2 r sin2 ϑ .

The event horizon r+ is defined by the outer root of ∆, rs r+ = + 2

r

rs2 − a2 , 4

(2.14.3)

whereas the outer boundary r0 of the ergosphere follows from the outer root of Σ − rs r, r

rs2 − a2 cos2 ϑ , 4

(2.14.4)

y

ergosphere

rs r0 = + 2

r+

x

r0

Figure 2.1: Ergosphere and horizon (dashed circle) for a = 0.99 r2s .

52

CHAPTER 2. SPACETIMES

Christoffel symbols: c2 rs ∆(r2 − a2 cos2 ϑ ) , 2Σ3 rs (r2 + a2)(r2 − a2 cos2 ϑ ) , Γttr = 2Σ2 ∆ 2 rs a r sin ϑ cos ϑ Γtt ϑ = − , Σ2 c∆rs a sin2 ϑ (r2 − a2 cos2 ϑ ) , Γtrϕ = − 2Σ3 2ra2 sin2 ϑ − rs (r2 − a2 cos2 ϑ ) Γrrr = , 2Σ∆ a2 sin ϑ cos ϑ , Γrrϑ = − Σ r∆ Γrϑ ϑ = − , Σ  ϑ cot ϕ Γϑ ϕ = 2 Σ2 + rs a2 r sin2 ϑ , Σ Γttr =

Γtrϕ ϕ

Γrϕ r Γϕϕ

ϑ Γϕϕ

c2 rs a2 r sin ϑ cos ϑ , Σ3 crs a(r2 − a2 cos2 ϑ ) ϕ Γtr = , 2Σ2 ∆ crs ar cot ϑ ϕ Γt ϑ = − , Σ2 crs ar(r2 + a2 ) sin ϑ cos ϑ Γtϑϕ = , Σ3 a2 sin ϑ cos ϑ Γϑrr = , Σ∆ r Γϑrϑ = , Σ a2 sin ϑ cos ϑ ϑ Γϑ ϑ = − , Σ rs a3 r sin3 ϑ cos ϑ , Γtϑ ϕ = cΣ2 Γttϑ = −

  rs a sin2 ϑ a2 cos2 ϑ (a2 − r2 ) − r2 (a2 + 3r2) = , 2cΣ2 ∆   2rΣ2 + rs a4 sin2 ϑ cos2 ϑ − r2 (Σ + r2 + a2 ) = , 2Σ2 ∆  ∆ sin2 ϑ  = −2rΣ2 + rs a2 sin2 ϑ (r2 − a2 cos2 ϑ ) , 3 2Σ   sin ϑ cos ϑ  =− AΣ + r2 + a2 rs a2 r sin2 ϑ , Σ3

(2.14.5a) (2.14.5b) (2.14.5c) (2.14.5d) (2.14.5e) (2.14.5f) (2.14.5g) (2.14.5h)

(2.14.5i) (2.14.5j) (2.14.5k) (2.14.5l)

General local tetrad:

∆ ∂r , Σ   gt ϕ + ζ gϕϕ gtt + ζ gt ϕ Γ ∓ √ ∂t ± √ ∂ϕ , e(3) = c ∆ sin ϑ ∆ sin ϑ

e(0) = Γ ∂t + ζ ∂ϕ , 1 e(2) = √ ∂ϑ , Σ



e(1) =

r

(2.14.6a) (2.14.6b)

where −Γ−2 = gtt + 2ζ gt ϕ + ζ 2 gϕϕ , Γ

−2

   rs a2 r sin2 ϑ ζ 2 2 rs r  2rs ar sin2 ϑ ζ 2 2 + sin ϑ − r +a + = 1− Σ Σ c Σ c2

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

1 e(2) = √ ∂ϑ , Σ

(2.14.7)

Σ 1 ∂ϕ , A sin ϑ

(2.14.8)

A sin ϑ (d ϕ − ω d ϕ ) . Σ

(2.14.9)

e(3) =

r

where ω = −gt ϕ /gϕϕ = rs ar/A.

Dual tetrad:

θ

(2)

=

r

Σ∆ c dt, A

θ

(1)

=

r

Σ dr, ∆

θ

(2)

√ = Σd ϑ ,

θ

(3)

=

r

2.14. KERR

53

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 rs ra A ∆ (0) (1) E−√ pr , (2.14.10a) L, υ = υ = Σ∆ Σ AΣ∆ s r   1 L2 Σ L (2) υ =√ Q − cos2 ϑ a2 ( µ 2 − E 2 ) + 2 υ (3) = . (2.14.10b) , A sin ϑ sin ϑ Σ Static local tetrad (ζ = 0): 1 e(0) = p ∂t , c 1 − rs r/Σ

e(1) =

r

∆ ∂r , Σ

1 e(2) = √ ∂ϑ , Σ

p 1 − rsr/Σ rs ar sin ϑ √ ∂t ∓ √ e(3) = ± p ∂ϕ . ∆ sin ϑ c 1 − rsr/Σ ∆Σ

Photon orbits: The direct(-) and retrograd(+) photon orbits have radius    2 ∓2a rpo = rs 1 + cos . arccos 3 rs Marginally stable timelike circular orbits are defined via  p rs  3 + Z2 ∓ (3 − Z1)(2 + Z1 + 2Z2 ) , rms = 2

(2.14.11a) (2.14.11b)

(2.14.12)

(2.14.13)

where

1/3 "    #  2a 1/3 2a 1/3 4a2 1+ + 1− , Z1 = 1 + 1 − 2 rs rs rs s 12a2 + Z12 . Z2 = rs2

(2.14.14a) (2.14.14b)

Euler-Lagrange: The Euler-Lagrangian formalism, Sec. 1.8.4, for geodesics in the ϑ = π /2 hyperplane yields 1 2 r˙ + Veff = 0 2

(2.14.15)

with the effective potential 1 Veff = 3 2r

   ahk k2  3 κ c2 ∆ 2 2 h (r − rs ) + 2 rs − 2 r + a (r + rs ) − 2 c c r

(2.14.16)

and the constants of motion

 rs  2 crs a ϕ˙ , k = 1− c t˙ + r r

  rs a2 crs a h = r 2 + a2 + ϕ˙ − t˙. r r

Further reading: Boyer and Lindquist[BL67], Wilkins[Wil72], Brill[BC66].

(2.14.17)

54

CHAPTER 2. SPACETIMES

2.15 Kottler spacetime The Kottler spacetime is represented in spherical coordinates (t, r, ϑ , ϕ ) by the line element[Per04]   rs Λr2 2 2 1 ds = − 1 − − c dt + dr2 + r2 dΩ2 , r 3 1 − rs /r − Λr2 /3 2

(2.15.1)

where rs = 2GM/c2 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

α = 1−

rs Λr2 − r 3

and

β=

rs 2Λ 2 − r . r 3

(2.15.2)

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    3rs √ 2 1 −Λ . (2.15.3) r1 = √ sinh arsinh 3 2 −Λ

If Λ > 0, we have to distinguish whether D ≡ q2 + p3 = 9rs2 /(4Λ2 ) − Λ−3 is positive or negative. If D > 0, there is no real positive root. For D < 0, the two real positive roots read    2 3rs √ π 1 Λ (2.15.4) r± = √ cos ± arccos 3 3 2 Λ Christoffel symbols: c2 αβ , 2r 1 = , r = cot ϑ ,

Γttr = Γϑrϑ ϕ Γϑ ϕ

β , 2rα 1 = , r = −α r sin2 ϑ ,

Γttr = ϕ

Γrϕ r Γϕϕ

Γrrr = −

β , 2rα

Γϑr ϑ = −α r,

ϑ Γϕϕ = − sin ϑ cos ϑ .

(2.15.5a) (2.15.5b) (2.15.5c)

Riemann-Tensor: Rtrtr Rt ϕ t ϕ Rr ϕ r ϕ

 c2 3rs + Λr3 =− , 3r3 1 = c2 αβ sin2 ϑ , 2 β =− sin2 ϑ , 2α

1 Rt ϑ t ϑ = c2 αβ , 2 β Rr ϑ r ϑ = − , 2α   Λr3 Rϑ ϕϑ ϕ = r rs + sin2 ϑ . 3

(2.15.6a) (2.15.6b) (2.15.6c)

Ricci-Tensor: Rtt = −c2 α Λ,

Rrr =

Λ , α

Rϕϕ = Λr2 sin2 ϑ .

Rϑ ϑ = Λr2 ,

(2.15.7)

The Ricci scalar and the Kretschmann scalar read R = 4Λ,

K = 12

rs2 8Λ2 + . r6 3

(2.15.8)

Weyl-Tensor: c2 rs , r3 rs =− , 2rα

Ctrtr = − Crϑ rϑ

c2 α rs , 2r rs sin2 ϑ =− , 2rα

Ct ϑ t ϑ = Crϕ rϕ

Ct ϕ t ϕ =

c2 α rs sin2 ϑ , 2r

Cϑ ϕϑ ϕ = rrs sin2 ϑ .

(2.15.9a) (2.15.9b)

2.15. KOTTLER SPACETIME

55

Local tetrad: 1 e(t) = √ ∂t , c α

√ α∂r ,

1 e(ϑ ) = ∂ϑ , r

e(ϕ ) =

1 ∂ϕ . r sin ϑ

(2.15.10)

dr θ (r) = √ , α

θ (ϑ ) = r d ϑ ,

θ (ϕ ) = r sin ϑ d ϕ .

(2.15.11)

e(r) =

Dual tetrad: √ θ (t) = c α dt,

Ricci rotation coefficients:

γ (r)(t)(t) =

rs − 32 Λr3 √ , 2r2 α

γ (ϑ )(r)(ϑ ) = γ (ϕ )(r)(ϕ ) =



α , r

γ (ϕ )(ϑ )(ϕ ) =

cot ϑ . r

(2.15.12)

The contractions of the Ricci rotation coefficients read

γ (r) =

4r − 3rs − 2Λr3 √ , 2r2 α

γ (ϑ ) =

cot ϑ . r

(2.15.13)

Riemann-Tensor with respect to local tetrad: R(t)(r)(t)(r) = −R(ϑ )(ϕ )(ϑ )(ϕ ) = −

Λr3 + 3rs , 3r3

R(t)(ϑ )(t)(ϑ ) = R(t)(ϕ )(t)(ϕ ) = −R(r)(ϑ )(r)(ϑ ) = −R(r)(ϕ )(r)(ϕ ) =

(2.15.14a) 3rs − 2Λr3 . 6r3

(2.15.14b)

Weyl-Tensor with respect to local tetrad: C(t)(r)(t)(r) = −C(ϑ )(ϕ )(ϑ )(ϕ ) = −

rs , r3

C(t)(ϑ )(t)(ϑ ) = C(t)(ϕ )(t)(ϕ ) = −C(r)(ϑ )(r)(ϑ ) = −C(r)(ϕ )(r)(ϕ ) =

(2.15.15a) rs . 2r3

Embedding: The embedding function follows from the numerical integration of s dz rs /r + Λr2 /3 = . dr 1 − rs/r − Λr2 /3

(2.15.15b)

(2.15.16)

Euler-Lagrange: The Euler-Lagrangian formalism[Rin01] yields the effective potential Veff =

   2 rs Λr2 h 1 2 κ c 1− − − 2 r 3 r2

(2.15.17)

with the constants of motion k = (1 − rs/r − Λr2 /3)c2t˙, h = r2 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 called photon orbit at r = 32 rs . For timelike geodesics, however, we have dVeff h2 (−6r + 9rs ) + c2r2 (3rs − 2r3 Λ) ! = 0. = dr 3r4 This polynomial of fifth order might have up to five extrema. Further reading: Kottler[Kot18], Weyl[Wey19], Hackmann[HL08], Cruz[COV05].

(2.15.18)

56

CHAPTER 2. SPACETIMES

2.16 Morris-Thorne The most simple wormhole geometry is represented by the metric of Morris and Thorne[MT88],  ds2 = −c2 dt 2 + dl 2 + (b20 + l 2 ) d ϑ 2 + sin2 ϑ d ϕ 2 ,

(2.16.1)

where b0 is the throat radius and l is the proper radial coordinate; and {t ∈ Christoffel symbols: Γϑl ϑ = ϕ

l b20 + l 2

l

ϕ

Γl ϕ =

,

Γϑ ϕ = cot ϑ ,

b20 + l 2 2

,

Γlϕϕ = −l sin ϑ ,

R, l ∈ R, ϑ ∈ (0, π ), ϕ ∈ [0, 2π )}.

Γϑl ϑ = −l,

(2.16.2a)

ϑ Γϕϕ = − sin ϑ cos ϑ .

(2.16.2b)

Partial derivatives l 2 − b20 , (b20 + l 2 )2 1 =− 2 , sin ϑ = − cos(2ϑ ).

ϕ

Γϑl ϑ ,l = − ϕ

Γϑ ϕ ,ϑ ϑ Γϕϕ ,ϑ

Γl ϕ ,l = −

l 2 − b20 , (b20 + l 2 )2

Γlϕϕ ,l = − sin2 ϑ ,

Γϑl ϑ ,l = −1,

(2.16.3a)

l Γϕϕ ,ϑ = −l sin(2ϑ ),

(2.16.3b) (2.16.3c)

Riemann-Tensor: Rl ϑ l ϑ = −

b20 , 2 b0 + l 2

Rl ϕ l ϕ = −

b20 sin2 ϑ , b20 + l 2

Rϑ ϕϑ ϕ = b20 sin2 ϑ .

(2.16.4)

Ricci tensor, Ricci and Kretschmann scalar: Rll = −2

b20 b20 + l 2

Weyl-Tensor: Ctltl = − Cl ϑ l ϑ = −

2 ,

R = −2

2 c2 b20  , 3 b2 + l 2 2 0

b20 b20 + l 2

Ct ϑ t ϑ =

1 b20 , 3 b20 + l 2

2 ,

K =

1 c2 b20 , 3 b20 + l 2

Cl ϕ l ϕ = −

1 b20 sin2 ϑ , 3 b20 + l 2

12b40 b20 + l 2

Ct ϕ t ϕ =

4 .

1 c2 b20 sin2 ϑ , 3 b20 + l 2

2 Cϑ ϕϑ ϕ = b20 sin2 ϑ . 3

(2.16.5)

(2.16.6a) (2.16.6b)

Local tetrad: 1 e(t) = ∂t , c

1 ∂ϑ , e(ϑ ) = q 2 b0 + l 2

e(l) = ∂l ,

Dual tetrad

θ (t) = c dt,

θ (ϑ ) =

θ (l) = dl,

Ricci rotation coefficients:

γ (ϑ )(r)(ϑ ) = γ (ϕ )(r)(ϕ ) =

l b20 + l 2

,

q b20 + l 2 d ϑ ,

1 e(ϕ ) = q ∂ϕ . 2 b0 + l 2 sin ϑ

θ (ϕ ) =

cot ϑ . γ (ϕ )(ϑ )(ϕ ) = q b20 + l 2

q b20 + l 2 sin ϑ d ϕ .

(2.16.7)

(2.16.8)

(2.16.9)

The contractions of the Ricci rotation coefficients read

γ (r) =

2l b20 + l 2

,

cot ϑ γ (ϑ ) = q . b20 + l 2

(2.16.10)

2.16. MORRIS-THORNE

57

Riemann-Tensor with respect to local tetrad: R(l)(ϑ )(l)(ϑ ) = R(l)(ϕ )(l)(ϕ ) = −R(ϑ )(ϕ )(ϑ )(ϕ ) = −

b20 b20 + l 2

Ricci-Tensor with respect to local tetrad: R(l)(l) = −

2b20 b20 + l 2

(2.16.11)

2 .

(2.16.12)

2 .

Weyl-Tensor with respect to local tetrad: C(t)(l)(t)(l) = −C(ϑ )(ϕ )(ϑ )(ϕ ) = −

2b20 3 b20 + l 2

2 ,

C(t)(ϑ )(t)(ϑ ) = C(t)(ϕ )(t)(ϕ ) = −C(l)(ϑ )(l)(ϑ ) = −C(l)(ϕ )(l)(ϕ ) = Embedding: The embedding function reads   s  2 r r − 1 z(r) = ±b0 ln  + b0 b0

(2.16.13a) b20 3 b20 + l 2

2 .

(2.16.13b)

(2.16.14)

with r2 = b20 + l 2 .

Euler-Lagrange: The Euler-Lagrangian formalism, Sec. 1.8.4, for geodesics in the ϑ = π /2 hyperplane yields 1 k2 1 ˙2 , l + Veff = 2 2 c2

1 Veff = 2



 h2 2 − κc , b20 + l 2

(2.16.15)

with the constants of motion k = c2t˙ and h = (b20 + l 2 )ϕ˙ . The shape of the effective potential Veff is independend 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(t) + cos ξ e(l) + sin ξ e(ϕ ) approaches the wormhole throat asymptotically for ξ = ξcrit with b0 ξcrit = arcsin q . b20 + li2

This critical angle is independent of the type of the geodesic. Further reading: Ellis[Ell73], Visser[Vis95], Müller[Mül04, Mül08a]

(2.16.16)

58

CHAPTER 2. SPACETIMES

2.17 Oppenheimer-Snyder collapse 2.17.1 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   R 3√ 2 3/2 d ϑ 2 + sin2 ϑ d ϕ 2 . ds = −d τ + 2/3 dR + R − 2 rs τ 3√ 3/2 R − 2 rs τ 2

2

(2.17.1)

Christoffel symbols: √ rs 1 √ , 2 R3/2 − 32 rs τ √ rs ϕ Γτϕ = − 3/2 3 √ , R − 2 rs τ √ 3 rs τ ΓRRR = − √  , 4 R3/2 − 23 rs τ R √ R ϕ ΓRϕ = 3/2 3 √ , R − 2 rs τ √ R3/2 − 23 rs τ √ ΓRϑ ϑ = − , R 1/3  √ 3√ τ 3/2 Γϕϕ = − rs R − sin2 ϑ , rs τ 2 √  R3/2 − 32 rs τ sin2 ϑ R √ Γϕϕ = − . R ΓτRR =



rs √ , R3/2 − 23 rs τ √ R rs τ ΓRR = √ 5/3 , 2 R3/2 − 23 rs τ √ R ϑ ΓRϑ = 3/2 3 √ , R − 2 rs τ 1/3  √ 3√ , Γτϑ ϑ = − rs R3/2 − rs τ 2 Γϑτϑ = −

ϕ

(2.17.2a) (2.17.2b) (2.17.2c) (2.17.2d)

Γϑ ϕ = cot ϑ ,

(2.17.2e)

ϑ Γϕϕ = − sin ϑ cos ϑ ,

(2.17.2f) (2.17.2g)

Riemann-Tensor: Rτ R τ R = − Rτϕτϕ =

1 2

RR ϕ R ϕ = −

Rrs √ 8/3 , 3/2 R − 32 rs τ rs sin2 ϑ √ 2/3 , R3/2 − 32 rs τ

1 Rrs sin2 ϑ , 2 R3/2 − 3 √r τ 4/3 s 2

Rτϑ τϑ =

1 rs , 2 R3/2 − 3 √rs τ 2/3 2

Rrs 1 , 2 R3/2 − 3 √r τ 4/3 s 2  2/3 3√ = R3/2 − rs τ rs sin2 ϑ . 2

(2.17.3a)

RR ϑ R ϑ = −

(2.17.3b)

Rϑ ϕϑ ϕ

(2.17.3c)

The Ricci tensor and the Ricci scalar vanish identically. Kretschmann scalar: K = 12

Local tetrad:

rs2

e(τ ) = ∂τ , e(ϑ ) =

(2.17.4)

√ 4 . R3/2 − 32 rs τ

1

R3/2 −

2/3 ∂ϑ , 3√ 2 rs τ

√ 1/3 R3/2 − 32 rs τ √ e(R) = ∂R , R 1 ∂ϕ . e(ϕ ) = 2/3 √ 3 R3/2 − 2 rs τ sin ϑ

(2.17.5a) (2.17.5b)

2.17. OPPENHEIMER-SNYDER COLLAPSE

59

Ricci rotation coefficients: √ √ rs 2 rs γ(τ )(R)(R) = − 3/2 √ , γ(τ )(ϑ )(ϑ ) = γ(τ )(ϕ )(ϕ ) = 3/2 √ , 2R − 3 rs τ 2R − 3 rs τ −2/3  3√ 3/2 . γ(R)(ϕ )(ϕ ) = γ(R)(ϑ )(ϑ ) = − R − rs τ 2 The contractions of the Ricci rotation coefficients read  −2/3 √ 3 rs 3√ 3/2 γ(τ ) = − 3/2 rs τ , √ , γ(R) = 2 R − 2 2R − 3 rs τ

(2.17.6a) (2.17.6b)



γ(ϑ ) = cot ϑ R

3/2

3√ rs τ − 2

−2/3

. (2.17.7)

Riemann-Tensor with respect to local tetrad: R(τ )(R)(τ )(R) = −R(ϑ )(ϕ )(ϑ )(ϕ ) = −

4rs

(2.17.8a)

√ 2 , 2R3/2 − 3 rs τ

R(τ )(ϑ )(τ )(ϑ ) = R(τ )(ϕ )(τ )(ϕ ) = −R(R)(ϑ )(R)(ϑ ) = −R(R)(ϕ )(R)(ϕ ) =

2rs √ 2 . 3/2 2R − 3 rs τ

(2.17.8b)

The Ricci tensor with respect to the local tetrad vanishes identically.

2.17.2 Inner metric The metric of the inside, R ≤ Rb , reads    3 √ −3/2 4/3  2 dR + R2 d ϑ 2 + sin2 ϑ d ϕ 2 . ds2 = −d τ 2 + 1 − rs Rb τ 2

(2.17.9)

For the following components, we define AOin := 1 −

3 √ −3/2 rs Rb τ . 2

(2.17.10)

Christoffel symbols: √ −3/2 √ −3/2 rs Rb rs Rb , Γϑτϑ = − , AOin AOin 1 1/3√ −3/2 ΓτRR = −AOin rs Rb , ΓϑRϑ = , R ϕ ΓϑR ϑ = −R, Γϑ ϕ = cot ϑ , ΓτRR = −

ϕ

Γτϕ = − ϕ

(2.17.11a)

1 , R 1/3√ −3/2 = −AOin rs Rb R2 , 1/3√ −3/2 = −AOin rs Rb R2 sin2 ϑ .

ΓRϕ =

(2.17.11b)

Γϑτ ϑ

(2.17.11c)

ϑ τ Γϕϕ = − sin ϑ cos ϑ , Γϕϕ

ΓRϕϕ = −R sin2 ϑ ,

√ −3/2 rs Rb , AOin

(2.17.11d)

Riemann-Tensor: Rτ R τ R = RR ϕ R ϕ =

1 rs , 2 R3 A2/3 rs

b Oin R2 sin2 ϑ

R3b

Rτϑ τϑ = 2/3

AOin ,

RR ϑ R ϑ =

1 rs R2 , 2 R3 A2/3 b Oin rs R2 2/3 A , R3b Oin

Rτϕτϕ = Rϑ ϕϑ ϕ =

1 rs R2 sin2 ϑ , 2 R3 A2/3 rs

b Oin R4 sin2 ϑ

R3b

2/3

AOin .

(2.17.12a) (2.17.12b)

Ricci-Tensor: Rττ =

3 rs , 2 R3b A2Oin

RRR =

3 rs , 2 R3 A2/3 b Oin

Rϑ ϑ =

3 rs R2 , 2 R3 A2/3 b Oin

Rϕϕ =

3 rs R2 sin2 ϑ . 2 R3 A2/3 b Oin

(2.17.13)

60

CHAPTER 2. SPACETIMES

The Ricci and Kretschmann scalars read: R=

3rs , R3b A2Oin

K = 15

rs2 . 6 Rb A4Oin

(2.17.14)

Local tetrad: e(τ ) = ∂τ ,

e(R) =

1

∂ , 2/3 R AOin

e(ϑ ) =

1

∂ , 2/3 ϑ RAOin

e(ϕ ) =

1

∂ϕ . 2/3 AOin R sin ϑ

(2.17.15)

Ricci rotation coefficients:

γ(τ )(R)(R) = γ(τ )(ϑ )(ϑ ) = γ(τ )(ϕ )(ϕ ) = γ(R)(ϑ )(ϑ ) = γ(R)(ϕ )(ϕ ) = − γ(ϑ )(ϕ )(ϕ ) = −

cot ϑ 2/3

1 2/3

√ −3/2 rs Rb , AOin

(2.17.16a) (2.17.16b)

,

RAOin

(2.17.16c)

.

RAOin

The contractions of the Ricci rotation coefficients read √ −3/2 3 rs Rb γ(τ ) = − , AOin

γ(R) =

2 2/3 RAOin

,

γ(ϑ ) =

cot ϑ 2/3

.

(2.17.17)

RAOin

Riemann-Tensor with respect to local tetrad: R(τ )(R)(τ )(R) = R(τ )(ϑ )(τ )(ϑ ) = R(τ )(ϕ )(τ )(ϕ ) =

rs R−3 b , 2A2Oin

R(R)(ϑ )(R)(ϑ ) = R(R)(ϕ )(R)(ϕ ) = R(ϑ )(ϕ )(ϑ )(ϕ ) =

rs R−3 b . A2Oin

(2.17.18a) (2.17.18b)

Ricci-Tensor with respect to local tetrad: R(τ )(τ ) = R(R)(R) = R(ϑ )(ϑ ) = R(ϕ )(ϕ ) = Further reading: Oppenheimer and Snyder[OS39].

3rs R−3 b . 2A2Oin

(2.17.19)

2.18. PETROV-TYPE D – LEVI-CIVITA SPACETIMES

61

2.18 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].

2.18.1 Case AI In spherical coordinates, (t, r, ϑ , ϕ ), the metric is given by the line element  ds2 = r2 d ϑ 2 + sin2 ϑ d ϕ 2 +

r r−b 2 dr2 − dt . r−b r

(2.18.1)

This is the well known Schwarzschild solution if b = rs , cf. Eq. (2.2.1). Coordinates and parameters are restricted to t∈

R,

Local tetrad: r r ∂t , e(t) = r−b

Dual tetrad: r

θ

(t)

=

r−b dt, r

0 < ϑ < π,

e(r) =

θ

(r)

r

=

ϕ ∈ [0, 2π ),

r−b ∂r , r

r

1 e(ϑ ) = ∂ϑ , r

r dr, r−b

(0 < b < r) ∨ (b < 0 < r).

e(ϕ ) =

θ (ϑ ) = r d ϑ ,

1 ∂ϕ . r sin ϑ

θ (ϕ ) = r sin ϑ d ϕ .

(2.18.2)

(2.18.3)

Effective potential: With the Hamilton-Jacobi formalism it is possible to obtain an effective potential fulfilling 12 r˙2 + 21 Veff (r) = 1 2 2 C0 with Veff (r) = K

r−b r−b −κ r3 r

(2.18.4)

and the constants of motion  2 2 2 r−b ˙ C0 = t , r K = ϑ˙ 2 r4 + ϕ˙ 2r4 sin2 ϑ .

(2.18.5a) (2.18.5b)

2.18.2 Case AII In cylindrical coordinates, the metric is given by the line element  ds2 = z2 dr2 + sinh2 r d ϕ 2 +

z b−z 2 dz2 − dt . b−z z

(2.18.6)

Coordinates and parameters are restricted to t∈ Local tetrad: r z e(t) = ∂t , b−z Dual tetrad: r

θ

(t)

=

b−z dt, z

R,

1 e(r) = ∂r , z

θ

(r)

= z dr,

0 < r,

ϕ ∈ [0, 2π ),

1 ∂ϕ , e(ϕ ) = z sinh r

θ

(ϕ )

= z sinh r d ϕ ,

0 < z < b.

e(z) =

θ

(z)

r

=

b−z ∂z . z

r

z dz. b−z

(2.18.7)

(2.18.8)

62

CHAPTER 2. SPACETIMES

2.18.3 Case AIII In cylindrical coordinates, the metric is given by the line element  1 ds2 = z2 dr2 + r2 d ϕ 2 + zdz2 − dt 2 . z

(2.18.9)

Coordinates and parameters are restricted to t∈

R,

ϕ ∈ [0, 2π ),

0 < r,

0 < z.

Local tetrad: e(t) =



1 e(r) = ∂r , z

z∂t ,

e(ϕ ) =

1 ∂ϕ , zr

1 e(z) = √ ∂z . z

(2.18.10)

Dual tetrad: 1 θ (t) = √ dt, z

θ (ϕ ) = zr d ϕ ,

θ (r) = z dr,

θ (z) =

√ z dz.

(2.18.11)

2.18.4 Case BI In spherical coordinates, the metric is given by the line element  ds2 = r2 d ϑ 2 − sin2 ϑ dt 2 +

r r−b 2 dr2 + dϕ . r−b r

(2.18.12)

Coordinates and parameters are restricted to t∈

R,

0 < ϑ < π,

ϕ ∈ [0, 2π ),

(0 < b < r) ∨ (b < 0 < r).

1 e(ϑ ) = ∂ϑ , r

e(ϕ ) =

Local tetrad: e(t) =

1 ∂t , r sin ϑ

e(r) =

r

r−b ∂r , r

r

r ∂ϕ . r−b

(2.18.13)

r

(2.18.14)

Dual tetrad:

θ

(t)

= r sin ϑ dt,

θ

(r)

=

r

r dr, r−b

θ

(ϑ )

= r dϑ ,

θ

(ϕ )

=

r−b dϕ . r

Effective potential: With the Hamilton-Jacobi formalism, an effective potential for the radial coordinate can be calculated fulfilling 12 r˙2 + 12 Veff (r) = 21 C02 with Veff (r) = K

r−b r−b −κ r3 r

and the constants of motion   r−b 2 , C02 = ϕ˙ 2 r K = ϑ˙ 2 r4 − t˙2 r4 sin2 ϑ .

(2.18.15)

(2.18.16a) (2.18.16b)

Note that the metric is not spherically symmetric. Particles or light rays fall into one of the poles if they are not moving in the ϑ = π2 plane.

2.18. PETROV-TYPE D – LEVI-CIVITA SPACETIMES

63

2.18.5 Case BII In cylindrical coordinates, the metric is given by the line element  ds2 = z2 dr2 − sinh2 r dt 2 +

z b−z 2 dz2 + dϕ . b−z z

(2.18.17)

Coordinates and parameters are restricted to t∈

R,

ϕ ∈ [0, 2π ),

0 < z < b,

0 < r.

Local tetrad: e(t) =

1 ∂t , z sinh r

r

1 e(r) = ∂r , z

e(ϕ ) =

θ (r) = z dr,

θ (ϕ ) =

z ∂ϕ , b−z

e(z) =

r

b−z ∂z . z

(2.18.18)

Dual tetrad:

θ (t) = z sinh r dt,

r

b−z dϕ , z

θ (z) =

r

z dz. b−z

(2.18.19)

2.18.6 Case BIII In cylindrical coordinates, the metric is given by the line element  1 ds2 = z2 dr2 − r2 dt 2 + zdz2 + d ϕ 2 . z

(2.18.20)

Coordinates and parameters are restricted to t∈

R,

ϕ ∈ [0, 2π ),

0 < z,

0 < r.

Local tetrad: e(t) =

1 ∂t , zr

√ z ∂ϕ ,

1 e(r) = ∂r , z

e(ϕ ) =

θ (r) = z dr,

1 θ (ϕ ) = √ d ϕ , z

1 e(z) = √ ∂z . z

(2.18.21)

Dual tetrad:

θ (t) = zr dt,

θ (z) =

√ z dz.

(2.18.22)

2.18.7 Case C The metric is given by the line element ds2 =

1 (x + y)2



1 1 dx2 + f (x)d ϕ 2 − dy2 + f (−y)dt 2 f (x) f (−y)



(2.18.23)

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

f (−y) > 0,

0 > f (x).

Local tetrad: 1 e(t) = (x + y) p ∂t , 3 −y − ay + b p e(y) = (x + y) −y3 − ay + b ∂y ,

p e(x) = (x + y) x3 + ax + b ∂x ,

1 ∂ϕ , e(ϕ ) = (x + y) √ 3 x + ax + b

(2.18.24a) (2.18.24b)

64

CHAPTER 2. SPACETIMES

Dual tetrad: 1 p 3 −y − ay + bdt, x+y 1 1 p θ (y) = dy, x + y −y3 − ay + b

θ (t) =

1 1 √ dx, 3 x + y x + ax + b 1 p 3 θ (ϕ ) = x + ax + bd ϕ , x+y

θ (x) =

(2.18.25a) (2.18.25b)

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 ds2 =

1 A(x + y)2



1 1 dx2 + f (x)d p2 − dy2 + f (−y)dq2 f (x) f (−y)



(2.18.26)

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]

ds2 =

i 1 h ρ 2 2 2 λ − eλ dr2 − r2 e−ρ d ϕ 2 r (z dt − t dz) − e (z dz − t dt) e z2 − t 2

with eρ =

(2.18.27)

R3 + R + Z3 − r2 , 2 4α (R1 + R + Z1 − r2 )

   2α 2 R(R + R1 + Z1 ) − Z1 r2 R1 R3 + (R + Z1)(R + Z3 ) − (Z1 + Z3 )r2 e = , Ri R3 [R(R + R3 + Z3 ) − Z3 r2 ] λ

 1 2 2 z − t + r2 , 2 q Ri = (R + Zi )2 − 2Zi r2 , R=

Zi = zi − z2 ,

α2 =

m2 1 , 6 4 A (z2 − z1 )2 (z3 − z1)2 q=

1 , 4α 2

and z3 < z1 < z2 the roots of 2A4 z3 − A2z2 + m2 . Local tetrad:

Case z2 − t 2 > 0:   1 e(t) = √ qze−ρ /2 ∂t + te−λ /2 ∂z , , z2 − t 2   1 qte−ρ /2 ∂t + ze−λ /2 ∂z , , e(z) = √ z2 − t 2

e(r) = e−λ /2 ∂r ,

(2.18.28a)

e(ϕ ) = reρ /2 ∂ϕ .

(2.18.28b)

e(r) = e−λ /2 ∂r ,

(2.18.29a)

e(ϕ ) = reρ /2 ∂ϕ .

(2.18.29b)

Case z2 − t 2 < 0:   1 qte−ρ /2 ∂t + ze−λ /2 ∂z , , e(t) = √ t 2 − z2   1 e(z) = √ qze−ρ /2 ∂t + te−λ /2 ∂z , , t 2 − z2

2.18. PETROV-TYPE D – LEVI-CIVITA SPACETIMES

65

Dual tetrad: Case z2 − t 2 > 0: r

θ (t) =

θ (z) =

s

eρ 1 (z dt + t dz) , z2 − t 2 q eλ

z2 − t 2

Case z2 − t 2 > 0: s

θ (t) =

θ

(z)

=

r



t 2 − z2 eρ

t 2 − z2

θ (r) = eλ dr, 1 dϕ . reρ

(t dt + z dz) ,

θ (ϕ ) =

(t dt + z dz) ,

θ (r) = eλ dr,

1 (z dt + t dz) , q

θ (ϕ ) =

1 dϕ . reρ

(2.18.30a) (2.18.30b)

(2.18.31a) (2.18.31b)

66

CHAPTER 2. SPACETIMES

2.19 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 = −dt 2 + dx2 + p2 (u) dy2 + q2 (u) dz2 . The functions p (u) and q (u) are given by    p0 = const. u < −a and p (u) := 1 − u 0
(2.19.1)

  q0 = const. q (u) := 1 − u   L (u) e−m(u)

u < −a 0
(2.19.2)

where a is the longitudinal extension of the wave. The functions L (u) and m (u) are s √ Z u3 u4 u2 + au L (u) = 1 − u + 2 + 3 , m (u) = ±2 3 du. a 2a 2a3 u − 2au3 − u4 − 2a3

(2.19.3)

Christoffel symbols: y Γty = −Γyxy =

1 ∂p , p ∂u

Γtzz = Γxzz = q

∂q , ∂u

z Γtz = −Γzxz =

1 ∂q , q ∂u

Γtyy = Γxyy = p

∂p . ∂u

(2.19.4)

Riemann-Tensor: Rtyty = Rxyxy = −Rtyxy = −p

∂2p , ∂ u2

Rtztz = Rxzxz = −Rtzxz = −q

∂ 2q . ∂ u2

(2.19.5)

Local tetrad: e(t) = ∂t ,

e(x) = ∂x ,

e(y) =

1 ∂y , p

1 e(z) = ∂z . q

(2.19.6)

Dual tetrad:

θ (t) = dt,

θ (x) = dx,

θ (y) = pdy,

θ (z) = qdz.

(2.19.7)

2.20. REISSNER-NORDSTRØM

67

2.20 Reissner-Nordstrøm The Reissner-Nordstrøm black hole in spherical coordinates {t ∈ fined by the metric[MTW73]

where

R, r ∈ R+, ϑ ∈ (0, π ), ϕ ∈ [0, 2π )} is de-

 2 2 2 2 2 ds2 = −ARN c2 dt 2 + A−1 RN dr + r d ϑ + sin ϑ d ϕ ,

ARN = 1 −

(2.20.1)

rs ρ Q2 + 2 r r

(2.20.2)

with rs = 2GM/c2 , the charge Q, and ρ = G/(ε0 c4 ) ≈ 9.33 · 10−34. As in the Schwarzschild case, there is a true curvature singularity at r = 0. However, for Q2 < rs2 /(4ρ ) there are also two critical points at rs rs r= ± 2 2

s

1−

4 ρ Q2 . rs2

(2.20.3)

Christoffel symbols: ARN c2 (rs r − 2ρ Q2 ) , 2r3 1 = , r = cot ϑ ,

Γttr = Γϑrϑ ϕ Γϑ ϕ

rs r − 2ρ Q2 , 2r3 ARN 1 = , r = −rARN sin2 ϑ ,

Γttr = ϕ

Γrϕ Γrϕϕ

Γrrr = −

rs r − 2ρ Q2 , 2r3 ARN

Γϑr ϑ = −rARN ,

ϑ Γϕϕ = − sin ϑ cos ϑ .

(2.20.4a) (2.20.4b) (2.20.4c)

Riemann-Tensor: c2 (rs r − 3ρ Q2) , r4 ARN c2 (rs r − 2ρ Q2 ) sin2 ϑ , = 2r2 (rs r − 2ρ Q2) sin2 ϑ , =− 2r2 ARN

Rtrtr = − Rt ϕ t ϕ Rr ϕ r ϕ

ARN c2 (rsr − 2ρ Q2) , 2r2 rs r − 2ρ Q2 , =− 2r2 ARN

Rt ϑ t ϑ =

(2.20.5a)

Rr ϑ r ϑ

(2.20.5b)

Rϑ ϕϑ ϕ = (rs r − ρ Q2 ) sin2 ϑ .

(2.20.5c)

Ricci-Tensor: Rtt =

c2 ρ Q2 ARN , r4

Rrr = −

ρ Q2 , r4 ARN

Rϑ ϑ =

ρ Q2 , r2

Rϕϕ =

ρ Q2 sin2 ϑ . r2

(2.20.6)

While the Ricci scalar vanishes identically, the Kretschmann scalar reads K =4

3rs2 r2 − 12rs rρ Q2 + 14ρ 2Q4 . r8

(2.20.7)

Weyl-Tensor: c2 (rs r − 2ρ Q2 ) , r4 ARN c2 (rs r − 2ρ Q2 ) sin2 ϑ = , 2r2 (rs r − 2ρ Q2 ) sin2 ϑ , =− 2r2 ARN

Ctrtr = − Ct ϕ t ϕ Crϕ rϕ

ARN c2 (rsr − 2ρ Q2) , 2r2 rs r − 2ρ Q2 =− , 2r2 ARN

Ct ϑ t ϑ = −

(2.20.8a)

Crϑ rϑ

(2.20.8b)

Cϑ ϕϑ ϕ = (rs r − 2ρ Q2) sin2 ϑ .

(2.20.8c)

68

CHAPTER 2. SPACETIMES

Local tetrad: 1 ∂t , e(t) = √ c ARN

p ARN ∂r ,

1 e(ϑ ) = ∂ϑ , r

e(ϕ ) =

1 ∂ϕ . r sin ϑ

(2.20.9)

dr θ (r) = √ , ARN

θ (ϑ ) = r d ϑ ,

θ (ϕ ) = r sin ϑ d ϕ .

(2.20.10)

cot ϑ . r

(2.20.11)

e(r) =

Dual tetrad: p θ (t) = c ARN dt,

Ricci rotation coefficients:

γ (r)(t)(t) =

rrs − 2ρ Q2 √ , 2r3 ARN

γ (ϑ )(r)(ϑ ) = γ (ϕ )(r)(ϕ ) =

√ ARN , r

γ (ϕ )(ϑ )(ϕ ) =

The contractions of the Ricci rotation coefficients read

γ (r) =

4r2 − 3rrs + 2ρ Q2 √ , 2r3 ARN

γ (ϑ ) =

cot ϑ . r

(2.20.12)

Riemann-Tensor with respect to local tetrad: R(t)(r)(t)(r) = −

rs r − 3ρ Q2 , r4

R(ϑ )(ϕ )(ϑ )(ϕ ) =

rs r − ρ Q2 , r4

R(t)(ϑ )(t)(ϑ ) = R(t)(ϕ )(t)(ϕ ) = −R(r)(ϑ )(r)(ϑ ) = −R(r)(ϕ )(r)(ϕ ) =

(2.20.13a) rs r − 2ρ Q2 . 2r4

(2.20.13b)

Ricci-Tensor with respect to local tetrad:

ρ Q2 . r4

(2.20.14)

rs r − 2ρ Q2 , r4

(2.20.15a)

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

C(t)(ϑ )(t)(ϑ ) = C(t)(ϕ )(t)(ϕ ) = −C(r)(ϑ )(r)(ϑ ) = −C(r)(ϕ )(r)(ϕ ) =

rs r − 2ρ Q2 . 2r4

Embedding: The embedding function follows from the numerical integration of s dz 1 − 1. = dr 1 − rs/r + ρ Q2 /r2

Euler-Lagrange: The Euler-Lagrangian formalism, Sec. 1.8.4, for geodesics in the ϑ = π /2 hyperplane yields    2 rs ρ Q2 1 h 1 k2 1 2 2 1− + 2 , Veff = − κc r˙ + Veff = 2 2 c2 2 r r r2

(2.20.15b)

(2.20.16)

(2.20.17)

with constants of motion k = ARN c2t˙ and h = r2 ϕ˙ . For null geodesics, κ = 0, there are two extremal points s ! 32ρ Q2 3 , (2.20.18) r± = rs 1 ± 1 − 4 9rs2 where r+ is a maximum and r− a minimum. Further reading: Eiroa[ERT02]

2.21. DE SITTER SPACETIME

69

2.21 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].

2.21.1 Standard coordinates The de Sitter metric in standard coordinates {τ ∈ ds2 = −d τ 2 + α 2 cosh2 where α 2 = 3/Λ. Christoffel symbols: χ

R, χ ∈ [−π , π ], ϑ ∈ (0, π ), ϕ ∈ [0, 2π )} reads

 τ  2 d χ + sin2 χ d ϑ 2 + sin2 ϑ d ϕ 2 , α

τ 1 tanh , Γϑτϑ α α τ τ = α sinh cosh , Γϑχϑ α α τ τ χ = α sin2 χ sinh cosh , Γϑ ϑ α α τ τ χ = α sin2 χ sin2 ϑ sinh cosh , Γϕϕ α α

τ 1 tanh , α α

(2.21.1)

ϕ

Γτϕ =

τ 1 tanh , α α

Γτ χ =

=

Γτχ χ

= cot χ ,

Γχϕ = cot χ ,

= − sin χ cos χ ,

Γϑ ϕ = cot ϑ ,

Γϑτ ϑ τ Γϕϕ

ϕ

(2.21.2a) (2.21.2b)

ϕ

(2.21.2c)

ϑ = − sin2 ϑ sin χ cos χ ,Γϕϕ = − sin ϑ cos ϑ .

(2.21.2d)

Riemann-Tensor:

τ , α τ = − cosh2 sin2 χ sin2 ϑ , α  τ 2 2 2 = α 1 + sinh2 sin χ sin2 ϑ , α

Rτ χτ χ = − cosh2 Rτϕτϕ Rχϕ χϕ

Ricci-Tensor: Rττ = −

3 , α2

Rχ χ = 3 cosh2

τ , α

τ Rτϑ τϑ = − cosh2 sin2 χ , α  τ 2 2 sin χ , Rχϑ χϑ = α 2 1 + sinh2 α   τ 2 4 Rϑ ϕϑ ϕ = α 2 1 + sinh2 sin χ sin2 ϑ . α

Rϑ ϑ = 3 cosh2

τ sin2 χ , α

Rϕϕ = 3 cosh2

τ sin2 χ sin2 ϑ . α

(2.21.3a) (2.21.3b) (2.21.3c)

(2.21.4)

Ricci and Kretschmann scalars: R=

12 , α2

K =

24 . α4

(2.21.5)

Local tetrad: e(τ ) = ∂τ ,

e( χ ) =

1 ∂χ , α cosh ατ

Dual tetrad:

θ (τ ) = d τ ,

θ (χ ) = α cosh

τ dχ, α

e(ϑ ) =

1 ∂ϑ , α cosh ατ sin χ

θ (ϑ ) = α cosh

τ sin χ d ϑ , α

e(ϕ ) =

1 ∂ϕ . α cosh ατ sin χ sin ϑ

θ (ϕ ) = α cosh

τ sin χ sin ϑ d ϕ . α

(2.21.6)

(2.21.7)

2.21.2 Conformally Einstein coordinates In conformally Einstein coordinates {η ∈ [0, π ], χ ∈ [−π , π ], ϑ ∈ [0, π ], ϕ ∈ [0, 2π )}, the de Sitter metric reads ds2 =

 α2  −d η 2 + d χ 2 + sin2 χ d ϑ 2 + sin2 ϑ d ϕ 2 . 2 sin η

(2.21.8)

70

CHAPTER 2. SPACETIMES

It follows from the standard form (2.21.1) by the transformation   η = 2 arctan eτ /α .

(2.21.9)

2.21.3 Conformally flat coordinates

R

R

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

α sin η , cos χ + cos η

r=

α sin χ , cos χ + cos η

η = arctan

or

2T α , α 2 − T 2 + r2

χ = arctan

2rα . (2.21.10) α 2 + T 2 − r2

For the transformation (T, R) → (η , χ ), we have to take care of the coordinate domains. In that case, if κ 2 − T 2 + r2 < 0, we have to map η → η + π . On the other hand, if κ 2 + 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 ds2 =

 α2  −dT 2 + dr2 + r2 d ϑ 2 + sin2 ϑ d ϕ 2 . 2 T

(2.21.11)

Note that we identify points (r < 0, ϑ , ϕ ) with (r > 0, π − ϑ , ϕ − π ). Christoffel symbols: 1 ϕ ϕ ΓTT T = ΓrTr = ΓϑT ϑ = ΓT ϕ = ΓTrr = − , Γϑrϑ = Γrϕ = T r2 sin2 ϑ ϕ T r Γϑ ϕ = cot ϑ , Γϕϕ =− = −r sin2 ϑ , , Γϕϕ T

1 , r

ΓϑT ϑ = −

r2 , T

Γϑr ϑ = −r,

ϑ Γϕϕ = − sin ϑ cos ϑ .

(2.21.12a) (2.21.12b)

Riemann-Tensor:

α2 , T4 α 2 r2 = 4 , T

RTrTr = − Rr ϑ r ϑ

α 2 r2 , T4 α 2 r2 sin2 ϑ = , T4

RT ϑ T ϑ = − Rr ϕ r ϕ

α 2 r2 sin2 ϑ , T4 α 2 r4 sin2 ϑ = . T4

RT ϕ T ϕ = −

(2.21.13a)

Rϑ ϕϑ ϕ

(2.21.13b)

Ricci-Tensor: RT T = −

3 , T2

3 , T2

Rrr =

Rϑ ϑ =

3r2 , T2

Rϕϕ =

3r2 sin2 ϑ . T2

(2.21.14)

The Ricci and Kretschmann scalar read: R=

12 , α2

K =

24 . α4

(2.21.15)

Local tetrad: e(T ) =

T ∂T , α

e(r) =

T ∂r , α

e(ϑ ) =

T ∂ϑ , αr

2.21.4 Static coordinates The de Sitter metric in static spherical coordinates {t ∈

e(ϕ ) =

T ∂ϕ . α r sin ϑ

(2.21.16)

R, r ∈ R+ , ϑ ∈ (0, π ), ϕ ∈ [0, 2π )} reads

  −1   Λ Λ dr2 + r2 d ϑ 2 + sin2 ϑ d ϕ 2 . ds2 = − 1 − r2 c2 dt 2 + 1 − r2 3 3

(2.21.17)

2.21. DE SITTER SPACETIME

71

It follows from the conformally Einstein form (2.21.8) by the transformations t=

α cos χ − cos η , ln 2 cos χ + cos η

r=α

sin χ . sin η

(2.21.18)

Christoffel symbols: (Λr2 − 3) 2 c Λr, 9 1 = , r

Γttr = Γϑrϑ

Λr , Λr2 − 3 1 = , r Λr2 − 3 r sin2 (ϑ ), = 3

Γttr = φ

Γrφ

φ

Γϑ φ = cot(ϑ ),

Γrφ φ

Λr , 3 − Λr2 (Λr2 − 3)r = , 3

Γrrr = Γϑr ϑ

(2.21.19a) (2.21.19b)

Γφϑφ = − sin(ϑ ) cos(ϑ ).

(2.21.19c)

Riemann-Tensor: Λ Rtrtr = − c2 , 3 Λr2 , Rr ϑ r ϑ = −Λr2 + 3

3 − Λr2 2 2 c Λr , 9 Λr2 sin(θ )2 = , −Λr2 + 3

Rt ϑ t ϑ = − Rr ϕ r ϕ

3 − Λr2 2 2 c Λr sin(ϑ )2 , 9 r4 sin2 (θ )Λ = . 3

Rt ϕ t ϕ = − Rϑ ϕϑ ϕ

(2.21.20a) (2.21.20b)

Ricci-Tensor: Rtt =

Λr2 − 3 2 c Λ, 3

Rrr =

3Λ , 3 − Λr2

Rϑ ϑ = Λr2 ,

Rϕϕ = r2 sin2 (ϑ )Λ.

(2.21.21)

The Ricci scalar and Kretschmann scalar read: R = 4Λ, Local tetrad: r e(t) =

8 K = Λ2 . 3

3 ∂t , 3 − Λr2 c

(2.21.22)

e(r) =

r

1−

Λr2 ∂r , 3

1 e(ϑ ) = ∂ϑ , r

e(ϕ ) =

1 ∂ϕ . r sin(ϑ )

(2.21.23)

Ricci rotation coefficients: Λr , γ(t)(r)(t) = − √ 9 − 3Λr2

√ 9 − 3Λr2 , γ(ϑ )(r)(ϑ ) = γ(ϕ )(r)(ϕ ) = 3r

The contractions of the Ricci rotation coefficients read √ cot ϑ 9 − 3Λr2(Λr2 − 2) , . γ(r) = γ(ϑ ) = (Λr2 − 3)r r

γ(ϕ )(ϑ )(ϕ ) =

cot ϑ . r

(2.21.24)

(2.21.25)

Riemann-Tensor with respect to local tetrad: 1 −R(t)(r)(t)(r) = −R(t)(ϑ )(t)(ϑ ) = −R(t)(ϕ )(t)(ϕ ) = R(r)(ϑ )(r)(ϑ ) = R(r)(ϕ )(r)(ϕ ) = R(ϑ )(ϕ )(ϑ )(ϕ ) = Λ. (2.21.26) 3 Ricci-Tensor with respect to local tetrad: −R(t)(t) = R(r)(r) = R(ϑ )(ϑ ) = R(ϕ )(ϕ ) = Λ.

(2.21.27)

72

CHAPTER 2. SPACETIMES

2.21.5 Lemaître-Robertson form The de Sitter universe in the Lemaître-Robertson form reads   ds2 = −c2 dt 2 + e2Ht dr2 + r2 d ϑ 2 + sin2 ϑ d ϕ 2 ,

(2.21.28)

q 2 with Hubble’s Parameter H = Λc3 = αc , 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. (2.9.2) and (2.9.12) with R(t) = eHt and k = 0. Christoffel symbols: ϕ

Γtϑϑ = H,

Γt ϕ = H,

(2.21.29a)

1 Γϑrϑ = , r

1 ϕ Γrϕ = , r

(2.21.29b)

Γtϑ ϑ =

Γrϑ ϑ = −r,

Γϑ ϕ = cot(ϑ ),

(2.21.29c)

Γtϕϕ

r = −r sin(ϑ )2 , Γϕϕ

ϑ Γϕϕ = − sin(ϑ ) cos(ϑ ).

(2.21.29d)

r Γtr = H,

Γtrr =

e2Ht H c2

,

e2Ht r2 H , c2 e2Ht r2 sin2 (θ )H = , c2

ϕ

Riemann-Tensor: Rtrtr = −e2Ht H 2 ,

Rt ϑ t ϑ = −e2Ht r2 H 2 ,

Rt ϕ t ϕ = −e2Ht r2 sin2 (ϑ )H 2 , Rr ϕ r ϕ =

e4Ht r2 sin2 (ϑ )H 2 , c2

(2.21.30b)

, c2 e4Ht r4 sin2 (ϑ )H 2 . = c2

Rr ϑ r ϑ = Rϑ ϕϑ ϕ

(2.21.30a)

e4Ht r2 H 2

(2.21.30c)

Ricci-Tensor: Rtt = −3H 2 ,

Rrr = 3

e2Ht H 2 , c2

Rϑ ϑ = 3

e2Ht r2 H 2 , c2

Rϕϕ = 3

e2Ht r2 sin2 (ϑ )H 2 . c2

(2.21.31)

Ricci and Kretschmann scalars: R=

12H 2 , c2

K =

24H 4 . c4

(2.21.32)

Local tetrad: 1 e(t) = ∂t , c

e(r) = e−Ht ∂r ,

e(ϑ ) =

e−Ht ∂ϑ , r

e(ϕ ) =

e−Ht ∂ϕ . r sin ϑ

(2.21.33)

Ricci rotation coefficients:

γ(r)(t)(r) = γ(ϑ )(t)(ϑ ) = γ(ϕ )(t)(ϕ ) = γ(ϑ )(r)(ϑ ) = γ(ϕ )(r)(ϕ ) =

1 , eHt r

H c

γ(ϕ )(ϑ )(ϕ ) =

(2.21.34a) cot(θ ) . eHt r

(2.21.34b)

The contractions of the Ricci rotation coefficients read

γ(t) = 3

H , c

γ(r) =

2 eHt r

, γ(ϑ ) =

cot(θ ) . eHt r

(2.21.35)

Riemann-Tensor with respect to local tetrad: H2 c2 H2 R(r)(ϑ )(r)(ϑ ) = R(r)(ϕ )(r)(ϕ ) = R(ϑ )(ϕ )(ϑ )(ϕ ) = 2 . c R(t)(r)(t)(r) = R(t)(ϑ )(t)(ϑ ) = R(t)(ϕ )(t)(ϕ ) = −

(2.21.36a) (2.21.36b)

2.21. DE SITTER SPACETIME

73

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

H2 . c2

(2.21.37)

2.21.6 Cartesian coordinates The de Sitter universe in Lemaître-Robertson form can also be expressed in Cartesian coordinates:   ds2 = −c2 dt 2 + e2Ht dx2 + dy2 + dz2 .

(2.21.38)

Christoffel symbols: x Γtx = H,

Γtxx =

e2Ht H c2

y Γty = H,

,

Γtyy =

e2Ht H c2

z Γtz = H,

Γtzz =

,

e2Ht H c2

(2.21.39a) (2.21.39b)

.

(2.21.39c) Partial derivatives Γtxx,t = Γtyy,t = Γtzz,t =

2H 2 e2Ht . c2

(2.21.40)

Riemann-Tensor: Rtxtx = Rtxtx = Rtztz = −e2Ht H 2 ,

Rxyxy = Rxzxz = Ryzyz =

e4Ht H 2 . c2

(2.21.41)

Ricci-Tensor: Rtt = −3H 2 ,

Rxx = Ryy = Rzz = 3

e2Ht H 2 . c2

(2.21.42)

The Ricci and Kretschmann scalar read: R = 12

H2 , c2

Local tetrad: 1 e(t) = ∂t , c

K = 24

H4 . c4

e(x) = e−Ht ∂x ,

(2.21.43)

e(y) = e−Ht ∂y ,

e(z) = e−Ht ∂z .

(2.21.44)

Ricci rotation coefficients: H . c The only non-vanishing contraction of the Ricci rotation coefficients read

γ(x)(t)(x) = γ(y)(t)(y) = γ(z)(t)(z) =

γ(t) = 3

H . c

(2.21.45)

(2.21.46)

Riemann-Tensor with respect to local tetrad: H2 , c2 H2 R(x)(y)(x)(y) = R(x)(z)(x)(z) = R(y)(z)(y)(z) = 2 . c R(t)(x)(t)(x) = R(t)(y)(t)(y) = R(t)(z)(t)(z) = −

(2.21.47a) (2.21.47b)

Ricci-Tensor with respect to local tetrad: −R(t)(t) = R(x)(x) = R(y)(y) = R(z)(z) = 3 Further reading: Tolman[Tol34, sec. 142], Biˇcák[BK01]

H2 . c2

(2.21.48)

74

CHAPTER 2. SPACETIMES

2.22 Straight spinning string The metric of a straight spinning string in cylindrical coordinates (t, ρ , ϕ , z) reads ds2 = − (c dt − a d ϕ )2 + d ρ 2 + k2 ρ 2 d ϕ 2 + dz2 ,

(2.22.1)

R

where a ∈ and k > 0 are two parameters, see Perlick[Per04]. Metric-Tensor: gtt = −c2 ,

gt ϕ = ac,

gρρ = gzz = 1,

gϕϕ = k2 ρ 2 − a2.

(2.22.2)

Christoffel symbols: Γtρϕ =

a , cρ

ϕ

Γρϕ =

1 , ρ

ρ

Γϕϕ = −k2 ρ .

(2.22.3)

1 , ρ2

(2.22.4)

Partial derivatives Γtρϕ ,ρ = −

α , cρ 2

ϕ

Γρϕ ,ρ = −

ρ

Γϕϕ ,ρ = −k2 .

The Riemann-, Ricci-, and Weyl-tensors as well as the Ricci- and Kretschmann-scalar vanish identically. Static local tetrad:  1 1 a e(1) = ∂ρ , (2.22.5) e(3) = ∂z . e(0) = ∂t , e(2) = ∂t + ∂ϕ , c kρ c Dual tetrad:

θ (0) = c dt − a d ϕ ,

θ (1) = d ρ ,

θ (2) = kρ d ϕ ,

θ (3) = dz.

(2.22.6)

Ricci rotation coefficients and their contractions read

γ(2)(1)(2) =

1 , ρ

γ(0) = γ(2) = γ(3) = 0,

Comoving local tetrad: p   k 2 ρ 2 − a2 1 a e(0) = ∂t − 2 2 ∂ϕ , kρ c k ρ − a2 1 ∂ϕ , e(2) = p k 2 ρ 2 − a2

1 . ρ

(2.22.7)

e(1) = ∂ρ ,

(2.22.8a)

e(3) = ∂z .

(2.22.8b)

γ(1) =

Dual tetrad:

kρ c dt, θ (0) = p 2 k ρ 2 − a2

θ (1) = d ρ ,

θ (2) = p

ac dt k 2 ρ 2 − a2

+

q k 2 ρ 2 − a2 d ϕ ,

θ (3) = dz.

(2.22.9)

Ricci rotation coefficients and their contractions read

γ(0)(1)(0) =

a2 , ρ (k2 ρ 2 − a2 )

k2 ρ , k 2 ρ 2 − a2 1 γ(1) = . ρ

γ(2)(1)(2) =

γ(2)(1)(0) = γ(0)(2)(1) = γ(0)(1)(2) =

ak , k 2 ρ 2 − a2

(2.22.10a) (2.22.10b) (2.22.10c)

2.22. STRAIGHT SPINNING STRING

75

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

ρ˙ 2 +

1 k2 ρ 2

  h2 ah1 2 − κ c2 = 21 , h2 − c c

with the constants of motion h1 = c(ct˙ − aϕ˙ ) and h2 = a(ct˙ − aϕ˙ ) + k2 ρ 2 ϕ˙ .

(2.22.11)

The point of closest approach ρpca for a null geodesic that starts at ρ = ρi with y = ±e(0) + cos ξ e(1) + sin ξ e(2) with respect to the static tetrad is given by ρ = ρi sin ξ . Hence, the ρpca is independent of a and k. The same is also true for timelike geodesics.

76

CHAPTER 2. SPACETIMES

2.23 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 2 2 2 2 1− ds = t dt − dr − r dΩ , dt dr − 1 + r r r 2

4

(2.23.1)

where M is the mass of the black hole and Ω2 = d ϑ 2 + sin2 ϑ d ϕ 2 is the spherical surface element. Note that here, the signature of the metric is sign(g) = −2. Christoffel symbols:  2 r3 + 4M 2 r + M 2t M(r − 2M)(4r + t) M(r + 2M)(4r + t) Γttt = , Γttr = , Γttr = , (2.23.2a) 3 tr3 tr tr3  2 r3 − 4M 2 r − M 2t 2 2 ϕ r , Γtϑϑ = , Γt ϕ = , (2.23.2b) Γtr = 3 tr t t  2 r2 + 2Mr − Mt 1 1 ϕ Γrϕ = , Γtϑ ϑ = , (2.23.2c) Γϑrϑ = , r r t 4Mr + tr − 2Mt ϕ ϑ Γϑr ϑ = − Γϕϕ = − sin ϑ cos ϑ , (2.23.2d) , Γϑ ϕ = cot ϑ , t   2 r3 + 4Mr2 + 4M 2 r + M 2t + Mtr M 4r2 + 8Mr + 2Mt + tr r , Γ = − , rr tr3 tr3 2 r2 + 2Mr − Mt sin2 ϑ (4Mr + tr − 2Mt)sin2 ϑ r = =− , Γϕϕ . t t

Γtrr = Γtϕϕ

(2.23.2e) (2.23.2f)

Riemann-Tensor:  2t 2 −2Mr2 − r3 + Mt 2 + 2Mtr , r3  t 2 2r4 + 16M 2 r2 + 4Mtr2 − 4M 2 r2t + Mt 2 r − 2M 2t 2 =− , r2 2Mt 2 (4r + t)(r2 + 2Mr − Mt) , =− r2  t 2 sin2 ϑ 2r4 + 16M 2 r2 + 4Mtr2 − 4M 2r2 t + Mt 2 r − 2M 2t 2 , =− r2 2Mt 2 sin2 ϑ (4r + t)(r2 + 2Mr − Mt) =− , r2  t 2 4r4 + 16Mr4 − 4M 2tr + 16M 2 r2 − 2M 2t 2 − Mt 2 r =− , r2  t 2 sin2 ϑ 4r4 + 16Mr4 − 4M 2tr + 16M 2 r2 − 2M 2t 2 − Mt 2 r , =− r2  = −2t 2r sin2 ϑ 2r3 + 4Mr2 − 4Mtr + mt 2 .

Rtrtr = Rr ϑ t ϑ Rt ϑ rϑ Rr ϕ t ϕ Rt ϕ rϕ Rr ϑ r ϑ Rr ϕ r ϕ Rϑ ϕϑ ϕ

(2.23.3a) (2.23.3b) (2.23.3c) (2.23.3d) (2.23.3e) (2.23.3f) (2.23.3g) (2.23.3h)

Ricci-Tensor:

 2 3r2 + 12M 2 + 2Mt 4M (3r + t + 6M) , Rtr = , Rtt = 2 2 t r t 2 r2   2 3r2 + 12Mr + 2Mt + 12M 2 6 r2 + 2Mr − 2Mt , Rϑ ϑ = , Rrr = t 2 r2  t2 6 r2 + 2Mr − 2Mt sin2 ϑ . Rϕϕ = t2

(2.23.4a) (2.23.4b) (2.23.4c)

2.23. SULTANA-DYER SPACETIME

77

Ricci and Kretschmann scalars:  12 r2 + 2Mr − 2Mt R=− , t 6 r2  48 M 2t 4 + 20M 2 r4 + 20Mr5 + 8M 2r2 t 2 − 4Mr4t − 16M 2r3t + 5r6 K = . t 1 2r6 Comoving local tetrad: p 1 + 2M/r 2M/r ∂t − p ∂r , e(0) = 2 t2 t 1 + 2M/r

e(1) =

t2

Static local tetrad:

1 e(0) = p ∂t , 2 t 1 − 2M/r

2M/r ∂t + e(1) = p 2 t 1 − 2M/r

Further reading: Sultana and Dyer[SD05].

1 p ∂r , 1 + 2M/r p 1 − 2M/r ∂r , t2

(2.23.5a) (2.23.5b)

e(2) =

1 ∂ϑ , t 2r

e(3) =

1 ∂ϕ . (2.23.6) t 2 r sin ϑ

e(2) =

1 ∂ϑ , t 2r

e(3) =

1 ∂ϕ . (2.23.7) t 2 r sin ϑ

78

CHAPTER 2. SPACETIMES

2.24 TaubNUT The TaubNUT metric in Boyer-Lindquist like spherical coordinates (t, r, ϑ , ϕ ) reads[BCJ02] (G = c = 1)  2  dr ∆ + d ϑ 2 + sin2 ϑ d ϕ 2 , ds2 = − (dt + 2ℓ cos ϑ d ϕ )2 + Σ Σ ∆

(2.24.1)

where Σ = r2 + ℓ2 and ∆ = r2 − 2Mr − ℓ2 . Here, M is the mass of the black hole and ℓ the magnetic monopol strength. Christoffel symbols: ∆ρ , Γttr = Σ3 ℓ∆ ϕ , Γtrϕ = Γt ϑ = 2 Σ sin ϑ ρ Γϑrϑ = Γrrr = − , Σ∆ Γttr =

ρ ∆ , Γtt ϑ = −2ℓ2 cos ϑ 2 , ∆Σ Σ 2ℓρ ∆ cos ϑ ℓ∆ sin ϑ , Γtϑϕ = − , Σ3 Σ2 r r r∆ ϕ , Γrϕ = , Γϑr ϑ = − , Σ Σ Σ

(2.24.2a) (2.24.2b) (2.24.2c)

−2ℓ(r3 − 3Mr2 − 3rℓ2 + Mℓ2 ) cos ϑ , Σ∆  2   2 2 2 4 ℓ cos ϑ 6r ℓ − 8ℓ Mr − 3ℓ + r4 + Σ2 =− , Σ2 sin ϑ i   ∆ h = 3 cos2 ϑ 9rℓ4 + 4ℓ2Mr2 − 4ℓ4M + r5 + 2r3 ℓ2 − rΣ2 , Σ  4r2 ℓ2 − 4Mrℓ2 − ℓ4 + r4 cot ϑ = , Σ2  6r2 ℓ2 − 8Mrℓ2 − 3ℓ4 + r4 sin ϑ cos ϑ =− , Σ2

Γtrϕ =

(2.24.2d)

Γtϑ ϕ

(2.24.2e)

r Γϕϕ

ϕ

Γϑ ϕ ϑ Γϕϕ

where ρ = 2rℓ2 + Mr2 − Mℓ2 . Static local tetrad: r r Σ ∆ ∂t , e(1) = ∂r , e(0) = ∆ Σ

1 e(2) = √ ∂ϑ , Σ

(2.24.2f) (2.24.2g) (2.24.2h)

2ℓ cot ϑ 1 e(3) = − √ ∂t + √ ∂ϕ . Σ Σ sin ϑ

(2.24.3)

Dual tetrad:

θ

(0)

=

r

∆ (dt + 2ℓ cos ϑ d ϕ ) , Σ

θ

(1)

=

r

Σ dr, ∆

θ (2) =

√ Σd ϑ ,

θ (3) =

√ Σ sin ϑ d ϕ .

Euler-Lagrange: The Euler-Lagrangian formalism, Sec. 1.8.4, for geodesics in the ϑ = π /2 hyperplane yields   1 ∆ h2 1 k2 1 2 , V = r˙ + Veff = − κ eff 2 2 c2 2Σ Σ

(2.24.4)

(2.24.5)

with the constants of motion k = (∆/Σ)t˙ and h = Σϕ˙ . For null geodesics, we obtain a photon orbit at r = rpo with   p 1 M rpo = M + 2 M 2 + ℓ2 cos arccos √ (2.24.6) 3 M 2 + ℓ2 Further reading: Bini et al.[BCdMJ03].

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Nov 4, 2010 - 2.10 Gödel Universe . ...... With the Hamilton-Jacobi formalism it is possible to obtain an effective potential fulfilling 1. 2. ˙r2 + 1. 2. Veff(r)=. 1. 2.

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