Catalogue of Spacetimes

Transcript

1 Catalogue of Spacetimes axiv: v3 [g-qc] 4 Nov 00 x = x = q 0 x 0 x 00 0 x = 0 x = 0 x = Authos: Thomas Mülle Visualisieungsinstitut de Univesität Stuttgat VISUS) Allmanding 9, Stuttgat, Gemany Thomas.Muelle@vis.uni-stuttgat.de e e M x = Fank Gave fomely, Univesität Stuttgat, Institut fü Theoetische Physik ITP) Pfaffenwalding 57 // IV, Stuttgat, Gemany Fank.Gave@vis.uni-stuttgat.de URL: Date: 04. Nov 00

2 Co-authos Andeas Lemme, fomely, Institut fü Theoetische Physik ITP), Univesität Stuttgat Alcubiee Wap Sebastian Boblest, Institut fü Theoetische Physik ITP), Univesität Stuttgat desitte, Fiedmann-Robetson-Walke Felix Beslmeisl, Institut fü Theoetische Physik ITP), Univesität Stuttgat Petov-Type D Heiko Munz, Institut fü Theoetische Physik ITP), Univesität Stuttgat Bessel and plane wave

3 Contents Intoduction and Notation. Notation Geneal emaks Basic objects of a metic Natual local tetad and initial conditions fo geodesics Othonomality condition Tetad tansfomations Ricci otation-, connection-, and stuctue coefficients Riemann-, Ricci-, and Weyl-tenso with espect to a local tetad Null o timelike diections Local tetad fo diagonal metics Local tetad fo stationay axisymmetic spacetimes Newman-Penose tetad and spin-coefficients Coodinate elations Spheical and Catesian coodinates Cylindical and Catesian coodinates Embedding diagam Equations of motion and tanspot equations Geodesic equation Femi-Walke tanspot Paallel tanspot Eule-Lagange fomalism Hamilton fomalism Units Tools Maple/GRTensoII Mathematica Maxima Spacetimes 4. Minkowski Catesian coodinates Cylindical coodinates Spheical coodinates Confom-compactified coodinates Rotating coodinates Rindle coodinates Schwazschild spacetime Schwazschild coodinates Schwazschild in pseudo-catesian coodinates Isotopic coodinates Eddington-Finkelstein Kuskal-Szekees Totoise coodinates i

4 ii CONTENTS..7 Painlevé-Gullstand Isael coodinates Alcubiee Wap Baiola-Vilenkin monopol Betotti-Kasne Bessel gavitational wave Cylindical coodinates Catesian coodinates Cosmic sting in Schwazschild spacetime Enst spacetime Fiedman-Robetson-Walke Fom Fom Fom Gödel Univese Cylindical coodinates Scaled cylindical coodinates Halilsoy standing wave Janis-Newman-Winicou Kasne Ke Boye-Lindquist coodinates Kottle spacetime Mois-Thone Oppenheime-Snyde collapse Oute metic Inne metic Petov-Type D Levi-Civita spacetimes Case AI Case AII Case AIII Case BI Case BII Case BIII Case C Plane gavitational wave Reissne-Nodstøm de Sitte spacetime Standad coodinates Confomally Einstein coodinates Confomally flat coodinates Static coodinates Lemaîte-Robetson fom Catesian coodinates Staight spinning sting Sultana-Dye spacetime TaubNUT Bibliogaphy 79

5 Chapte Intoduction and Notation The Catalogue of Spacetimes is a collection of fou-dimensional Loentzian spacetimes in the context of the Geneal Theoy of Relativity GR). The aim of the catalogue is to give a quick efeence fo students who need some basic facts of the most well-known spacetimes in GR. Fo a detailed discussion of a metic, the eade is efeed to the standad liteatue o the oiginal aticles. Impotant esouces fo exact solutions ae the book by Stephani et al[skm + 03] and the book by Giffiths and Podolský[GP09]. Most of the metics in this catalogue ae implemented in the Motion4D-libay[MG09] and can be visualized using the GeodesicViewe[MG0]. Except fo the Minkowski and Schwazschild spacetimes, the metics ae soted by thei names.. Notation The notation we use in this catalogue is as follows: Indices: Coodinate indices ae epesented eithe by Geek lettes o by coodinate names. Tetad indices ae indicated by Latin lettes o coodinate names in backets. Einstein sum convention: When an index appeas twice in a single tem, once as lowe index and once as uppe index, we build the sum ove all indices: ζ µ ζ µ 3 µ=0 ζ µ ζ µ...) Vectos: A coodinate vecto in x µ diection is epesented as x µ µ. Fo abitay vectos, we use boldface symbols. Hence, a vecto a in coodinate epesentation eads a=a µ µ. Deivatives: Patial deivatives ae indicated by a comma, ψ/ x µ µ ψ ψ,µ, wheeas covaiant deivatives ae indicated by a semicolon, ψ = ψ ;µ. Symmetization and Antisymmetization backets: a µ b ν) = aµ b ν + a ν b µ ), a[ µ b ν] =. Geneal emaks aµ b ν a ν b µ ) The Einstein field equation in the most geneal fom eads[mtw73] G µν =κt µν Λg µν,..) κ = 8πG c 4,..) with the symmetic and divegence-fee Einstein tenso G µν = R µν Rg µν, the Ricci tenso R µν, the Ricci scala R, the metic tenso g µν, the enegy-momentum tenso T µν, the cosmological constant Λ, Newton s gavitational constant G, and the speed of light c. Because the Einstein tenso is divegencefee, the consevation equation T µν ;ν = 0 is automatically fulfilled.

6 CHAPTER. INTRODUCTION AND NOTATION A solution to the field equation is given by the line element ds = g µν dx µ dx ν..) with the symmetic, covaiant metic tenso g µν. The contavaiant metic tenso g µν is elated to the covaiant tenso via g µν g νλ = δ λ µ with the Konecke-δ. Even though g µν is only a component of the metic tenso g=g µν dx µ dx ν, we will also call g µν the metic tenso. Note that, in this catalogue, we mostly use the convention that the signatue of the metic is +. In geneal, we will also keep the physical constants c and G within the metics..3 Basic objects of a metic The basic objects of a metic ae the Chistoffel symbols, the Riemann and Ricci tensos as well as the Ricci and Ketschmann scalas which ae defined as follows: Chistoffel symbols of the fist kind: Γ νλ µ = gµν,λ + g µλ,ν g νλ,µ ).3.) with the elation g νλ,µ = Γ µνλ + Γ µλ ν.3.) Chistoffel symbols of the second kind: Γ µ νλ = gµρ g ρν,λ + g ρλ,ν g νλ,ρ ).3.3) which ae elated to the Chistoffel symbols of the fist kind via Γ µ νλ = gµρ Γ νλ ρ.3.4) Riemann tenso: o R µ νρσ = Γ µ νσ,ρ Γ µ νρ,σ + Γ µ ρλ Γλ νσ Γ µ σλ Γλ νρ.3.5) R µνρσ = g µλ R λ νρσ = Γ νσ µ,ρ Γ νρµ,σ + Γ λ νρ Γ µσλ Γ λ νσ Γ µσλ.3.6) with symmeties and R µνρσ = R µνσρ, R µνρσ = R νµρσ, R µνρσ = R ρσ µν.3.7) R µνρσ + R µρσν + R µσνρ = 0.3.8) Ricci tenso: R µν = g ρσ R ρµσν = R ρ µρν.3.9) Ricci and Ketschmann scala: R = g µν R µν = R µ µ, K = R αβ γδ R αβ γδ = R γδ αβ Rαβ γδ.3.0) The notation of the Chistoffel symbols of the fist kind diffes fom the one used by Rindle[Rin0], Γ Rindle µνλ = Γ CoS νλ µ.

7 .4. NATURAL LOCAL TETRAD AND INITIAL CONDITIONS FOR GEODESICS 3 Weyl tenso: C µνρσ = R µνρσ gµ[ ρ R σ ]ν g ν[ ρ R σ ]µ ) + 3 Rg µ[ ρg σ ]ν.3.) If we change the signatue of a metic, these basic objects tansfom as follows: Γ µ νλ Γµ νλ, R µνρσ R µνρσ, C µνρσ C µνρσ,.3.a) R µν R µν, R R, K K..3.b) Covaiant deivative λ g µν = g µν;λ = ) Covaiant deivative of the vecto field ψ µ : ν ψ µ = ψ µ ;ν = ν ψ µ + Γ µ νλ ψλ.3.4) Covaiant deivative of a -s-tenso field: c T a...a b...b s = c T a...a b...b s + Γ a dc T d...a b...b s Γ a dc T a...a d b...b s Γ d b c T a...a d...bs... Γ d b s c T a...a b...b s d.3.5) Killing equation: ξ µ;ν + ξ ν;µ = ).4 Natual local tetad and initial conditions fo geodesics We will call a local tetad natual if it is adapted to the symmeties o the coodinates of the spacetime. The fou base vectos e i) = e µ i) µ ae given with espect to coodinate diections / x µ = µ, compae Nakahaa[Nak90] o Chandasekha[Cha06] fo an intoduction to the tetad fomalism. The invese o dual tetad is given by θ i) = θ i) µ dx µ with θ i) µ e µ j) = δi) j) and θ i) µ e ν i) = δ ν µ..4.) Note that we us Latin indices in backets fo tetads and Geek indices fo coodinates..4. Othonomality condition To be applicable as a local efeence fame Minkowski fame), a local tetad e i) has to fulfill the othonomality condition ei),e j) g = g e i),e j) ) = gµν e µ i) eν j)! = η i) j),.4.) whee η i) j) = diag,±,±,±) depending on the signatue signg) = ± of the metic. Thus, the line element of a metic can be witten as ds = η i) j) θ i) θ j) = η i) j) θ i) µ θ j) ν dx µ dx ν..4.3) To obtain a local tetad e i), we could fist detemine the dual tetad θ i) via Eq..4.3). If we combine all fou dual tetad vectos into one matix Θ, we only have to detemine its invese Θ to find the tetad vectos, θ 0) 0 θ 0) θ 0) θ 0) e 0 e 3 θ ) Θ= 0 θ ) θ ) θ ) 0) 0 e ) 0 e ) 0 3) 3 θ ) 0 θ ) θ ) θ ) e Θ 0) e ) e ) e 3) = e 3 0) e ) e ) e 3)..4.4) θ 3) 0 θ 3) θ 3) θ 3) e 3 3 0) e 3 ) e 3 ) e 3 3)

8 4 CHAPTER. INTRODUCTION AND NOTATION Thee ae also seveal useful elations: e a)µ = g µν e ν a), η a)b) = e µ a) e b)µ, e b)µ = η a)b) θ a) µ,.4.5a) µ = η a)b) e a)µ, g µν = e a)µ θ a) ν, η a)b) = θ a) µ θ b) ν g µν..4.5b) θ b).4. Tetad tansfomations Instead of the above found local tetad that was diectly constucted fom the spacetime metic, we can also use any othe local tetad ê i) = A k i e k),.4.6) whee A is an element of the Loentz goup O,3). Hence A T ηa=η and deta) =. Loentz-tansfomation in the diection n a =sin χ cosξ,sin χ sinξ,cosξ) T = n a with γ = / β, Λ 0 0 = γ, Λ0 a = β γn a, Λ a 0 = β γna, Λ a b =γ )na n b + δ a b..4.7).4.3 Ricci otation-, connection-, and stuctue coefficients The Ricci otation coefficients γ i) j)k) with espect to the local tetad e i) ae defined by γ i) j)k) := g µλ e µ i) e k) e λ j) = g µλ e µ i) eν k) νe λ j) = g µλ e µ i) eν k) ) ν e λ j) + Γλ νβ eβ..4.8) j) They ae antisymmetic in the fist two indices, γ i) j)k) = γ j)i)k), which follows fom the definition, Eq..4.8), and the elation ) 0= µ η i) j) = µ g β ν e β i) eν j),.4.9) whee µ g β ν = 0, compae [Cha06]. Othewise, we have γ i) j)k) = θi) λ eν k) νe λ j) = eλ j) eν k) νθ i) λ..4.0) The contaction of the fist and the last index is given by γ j) = γ k) j)k) = ηk)i) γ i) j)k) = γ 0) j)0) + γ ) j)) + γ ) j)) + γ 3) j)3) = ν e ν j)..4.) The connection coefficients ω m) j)n) with espect to the local tetad e i) ae defined by ) ω m) := θm) j)n) µ e j) e µ n) = θm) µ e α j) αe µ n) = θm) µ e α j) α e µ n) + Γµ αβ eβ,.4.) n) compae Nakahaa[Nak90]. They ae elated to the Ricci otation coefficients via γ i) j)k) = η i)m) ω m) k) j)..4.3) Futhemoe, the local tetad has a non-vanishing Lie-backet[X,Y] ν = X µ µ Y ν Y µ µ X ν. Thus, [ ei),e j) ] = c k) i) j) e k) o c k) i) j) = θk)[ e i),e j) ]..4.4) The stuctue coefficients c k) via i) j) ae elated to the connection coefficients o the Ricci otation coefficients c k) i) j) = ωk) i) j) ωk) j)i) = ηk)m) γ m) j)i) γ m)i) j) ) = γ k) j)i) γ k) i) j)..4.5)

9 .4. NATURAL LOCAL TETRAD AND INITIAL CONDITIONS FOR GEODESICS Riemann-, Ricci-, and Weyl-tenso with espect to a local tetad The tansfomations between the coodinate epesentations of the Riemann-, Ricci-, and Weyl-tensos and thei epesentation with espect to a local tetad e i) ae given by R a)b)c)d) = R µνρσ e µ a) eν b) eρ c) eσ d), R a)b) = R µν e µ a) eν b),.4.6a).4.6b) C a)b)c)d) = C µνρσ e µ a) eν b) eρ c) eσ d) = R a)b)c)d) ) R ηa)[c) R d)]b) η b)[c) R d)]a) + 3 η a)[c)η d)]b)..4.6c).4.5 Null o timelike diections A null o timelike diection υ = υ i) e i) with espect to a local tetad e i) can be witten as υ = υ 0) e 0) + ψ sin χ cosξ e ) + sin χ sinξ e ) + cosχ e 3) ) = υ 0) e 0) + ψn..4.7) In the case of a null diection we have ψ = and υ 0) =±. A timelike diection can be identified with an initial fou-velocity u=cγe 0 + β n), whee u = u,u g = c γ e 0) + β n,e 0) + β n = c γ +β ) = c, signg)=±..4.8) Thus, ψ = cβ γ and υ 0 =±cγ. The sign of υ 0) detemines the time diection. e 3) χ ψ υ ξ e ) e ) Figue.: Null o timelike diection υ with espect to the local tetad e i). The tansfomations between a local diection υ i) and its coodinate epesentation υ µ ead υ µ = υ i) e µ i) and υ i) = θ i) µ υ µ..4.9).4.6 Local tetad fo diagonal metics If a spacetime is epesented by a diagonal metic ds = g 00 dx 0 ) + g dx ) + g dx ) + g 33 dx 3 ),.4.0) the natual local tetad eads e 0) = g00 0, e ) = g, e ) = g, e 3) = g33 3,.4.) given that the metic coefficients ae well behaved. Analogously, the dual tetad eads θ 0) = g 00 dx 0, θ ) = g dx, θ ) = g dx, θ 3) = g 33 dx 3..4.)

10 6 CHAPTER. INTRODUCTION AND NOTATION.4.7 Local tetad fo stationay axisymmetic spacetimes The line element of a stationay axisymmetic spacetime is given by ds = g tt dt + g tϕ dt dϕ+ g ϕϕ dϕ + g d + g ϑϑ dϑ,.4.3) whee the metic components ae functions of and ϑ only. The local tetad fo an obseve on a stationay cicula obit, = const,ϑ = const), with fou velocity u=cγ t + ζ ϕ ) can be defined as, compae Bini[BJ00], e 0) = Γ t + ζ ϕ ), e) = g, e ) = gϑϑ ϑ, e 3) = Γ [ ±g tϕ + ζg ϕϕ ) t g tt + ζg tϕ ) ϕ ],.4.4a).4.4b) whee Γ= ) and =..4.5) g tt + ζg tϕ + ζ g ϕϕ g tϕ g tt g ϕϕ The angula velocity ζ is limited due to g tt + ζg tϕ + ζ g ϕϕ < 0 ζ min = ω ω g tt and ζ max = ω+ ω g g tt.4.6) ϕϕ g ϕϕ with ω = g tϕ /g ϕϕ. Fo ζ = 0, the obseve is static with espect to spatial infinity. The locally non-otating fame LNRF) has angula velocity ζ = ω, see also MTW[MTW73], execise Static limit: ζ min = 0 g tt = 0. The tansfomation between the local diection υ i) and the coodinate diection υ µ eads with υ 0 = Γ υ 0) ± υ 3) w ), υ = υ) g, υ = υ) gϑϑ, υ 3 = Γ υ 0) ζ υ 3) w ),.4.7) w = g tϕ + ζg ϕϕ and w = g tt + ζg tϕ..4.8) The back tansfomation eads υ 0) = υ 0 w + υ 3 w, υ ) = g υ, υ ) = g ϑϑ υ, υ 3) =± ζυ 0 υ ) Γ ζw + w Γ ζw + w ) Note, to obtain a ight-handed local tetad, det e µ > 0, the uppe sign has to be used. i).5 Newman-Penose tetad and spin-coefficients The Newman-Penose tetad consists of fou null vectos e i) ={l,n,m, m}, whee l and n ae eal and m and m ae complex conjugates; see Penose and Rindle[PR84] o Chandasekha[Cha06] fo a thoough discussion. The Newman-Penose NP) tetad has to fulfill the othonomality elation e i),e j) = η i) j) with η i) j) = ) A staightfowad elation between the NP tetad and the natual local tetad, as discussed in Sec..4, is given by l= e0) + e ) ), n= e0) e ) ), m= e) + ie 3) ),.5.)

11 .6. COORDINATE RELATIONS 7 whee the uppe/lowe sign has to be used fo metics with positive/negative signatue. The Ricci otation-coefficients of a NP tetad ae now called spin coefficients and ae designated by specific symbols: κ = γ ))), ρ = γ )0)3), ε = ) γ)0)0) + γ )3)0),.5.3a) σ = γ )0)), µ = γ )3)), γ = ) γ)0)) + γ )3)),.5.3b) λ = γ )3)3), τ = γ )0)), α = ) γ)0)3) + γ )3)3),.5.3c) ν = γ )3)), π = γ )3)0), β = ) γ)0)) + γ )3))..5.3d).6 Coodinate elations.6. Spheical and Catesian coodinates The well-known elation between the spheical coodinates, ϑ, ϕ) and the Catesian coodinatesx, y, z), compae Fig.., ae and x= sin ϑ cosϕ, y= sin ϑ sinϕ, z= cosϑ,.6.) = x + y + z, ϑ = actan x + y,z), ϕ = actany,x),.6.) whee actan) ensues that ϕ [0,π) and ϑ 0,π). z ϑ ϕ y x Figue.: Relation between spheical and Catesian coodinates. The total diffeentials of the spheical coodinates ead d= xdx+ydy+zdz, dϑ = xzdx+yzdy x + y )dz, dϕ = ydx+xdy x + y x + y,.6.3) wheeas the coodinate deivatives ead = x x+ y y+ z z = sin ϑ cosϕ x + sinϑ sinϕ y + cosϑ z,.6.4a) ϑ = x ϑ x+ y ϑ y+ z ϑ z = cosϑ cosϕ x + cosϑ sinϕ y sinϑ z,.6.4b) ϕ = x ϕ x+ y ϕ y+ z ϕ z = sinϑ sinϕ x + sinϑ cosϕ y,.6.4c)

12 8 CHAPTER. INTRODUCTION AND NOTATION and x = x + ϑ x ϑ + ϕ x ϕ = sinϑ cosϕ + y = y + ϑ y ϑ + ϕ y ϕ = sinϑ sinϕ + z = z + ϑ z ϑ + ϕ z ϕ = cosϑ sinϑ ϑ. cosϑ cosϕ ϑ sinϕ sinϑ ϕ, cosϑ sin ϕ ϑ + cosϕ sin ϑ ϕ,.6.5a).6.5b).6.5c).6. Cylindical and Catesian coodinates The elation between cylindical coodinates,ϕ,z) and Catesian coodinatesx,y,z) is given by x= cosϕ, y= sinϕ, and = x + y, ϕ = actany,x),.6.6) whee actan ) again ensues that the angle ϕ [0, π). z ϕ z y x Figue.3: Relation between cylindical and Catesian coodinates. The total diffeentials of the spheical coodinates ae given by and d= xdx+ydy, dϕ = ydx+xdy,.6.7) dx=cosϕ d sinϕ dϕ, dy=sin ϕ d+ cosϕ dϕ..6.8) The coodinate deivatives ae and = x x+ y y = cosϕ x + sinϕ y,.6.9a) ϕ = x ϕ x+ y ϕ y = sinϕ x + cosϕ y m x = x + ϕ x ϕ = cosϕ sinϕ y, y = y + ϕ y ϕ = sinϕ + cosϕ y..7 Embedding diagam A two-dimensional hypesuface with line segment.6.9b).6.0a).6.0b) dσ = g )d + g ϕϕ )dϕ.7.)

13 .8. EQUATIONS OF MOTION AND TRANSPORT EQUATIONS 9 can be embedded in a thee-dimensional Euclidean space with cylindical coodinates, [ ) ] dz dσ = + dρ + ρ dϕ..7.) dρ With ρ) = g ϕϕ ) and d=d/dρ)dρ, we obtain fo the embedding function z=z), ) dz d gϕϕ d =± g..7.3) d If g ϕϕ )=, then d g ϕϕ /d=..8 Equations of motion and tanspot equations.8. Geodesic equation The geodesic equation eads D x µ dλ = d x µ dλ + Γµ ρσ dx ρ dλ dx σ dλ = 0.8.) with the affine paamete λ. Fo timelike geodesics, howeve, we eplace the affine paamete by the pope time τ. The geodesic equation.8.) is a system of odinay diffeential equations of second ode. Hence, to solve these diffeential equations, we need an initial position x µ λ = 0) as well as an initial diection dx µ /dλ)λ = 0). This initial diection has to fulfill the constaint equation dx µ dx ν g µν dλ dλ = κc,.8.) whee κ = 0 fo lightlike and κ =,signg) = ±), fo timelike geodesics. The initial diection can also be detemined by means of a local efeence fame, compae sec..4.5, that automatically fulfills the constaint equation.8.). If we use the natual local tetad as local efeence fame, we have dx µ dλ = υ µ = υ i) e µ i)..8.3) λ=0.8. Femi-Walke tanspot The Femi-Walke tanspot, see e.g. Stephani[SS90], of a vecto X = X µ µ along the woldline x µ τ) with fou-velocity u=u µ τ) µ is given by F u X µ = 0 with F u X µ := dx µ dτ + Γµ ρσu ρ X σ + c uσ a µ a σ u µ )g ρσ X ρ..8.4) The fou-acceleation follows fom the fou-velocity via a µ = D x µ dτ = Duµ dτ.8.3 Paallel tanspot = duµ dτ + Γµ ρσu ρ u σ..8.5) If the fou-acceleation vanishes, the Femi-Walke tanspot simplifies to the paallel tanspotp u X µ = 0 with P u X µ := DX µ dτ = dx µ dτ + Γµ ρσu ρ X σ..8.6)

14 0 CHAPTER. INTRODUCTION AND NOTATION.8.4 Eule-Lagange fomalism A detailed discussion of the Eule-Lagange fomalism can be found, e.g., in Rindle[Rin0]. The Lagangian L is defined as L := g µν ẋ µ ẋ ν, L! = κc,.8.7) whee x µ ae the coodinates of the metic, and the dot means diffeentiation with espect to the affine paamete λ. Fo timelike geodesics, κ = depending on the signatue of the metic, signg)=±. Fo lightlike geodesics, κ = 0. The Eule-Lagange equations ead d L dλ ẋ µ L x µ = 0. If L is independent of x ρ, then x ρ is a cyclic vaiable and p ρ = g ρν ẋ ν = const..8.8).8.9) Note that [L] U = length time fo timelike and [L] U = fo lightlike geodesics, see Sec Hamilton fomalism The supe-hamiltonian H is defined as H := gµν p µ p ν, H! = κc,.8.0) whee p µ = g µν ẋ ν ae the canonical momenta, see e.g. MTW[MTW73], paa... As in classical mechanics, we have dx µ dλ = H p µ.9 Units and d p µ dλ = H x µ..8.) A fist test in analyzing whethe an equation is coect is to check the units. Newton s gavitational constant G, fo example, has the following units [G] U = length3 mass time, whee[ ] U indicates that we evaluate the units of the enclosed expession. Futhe examples ae [ds] U = length,.0 Tools [u] U = length time,.0. Maple/GRTensoII [RSchwazschild tt ] U = [ time, R Schwazschild ϑϕϑϕ ] U.9.) = length..9.) The Chistoffel symbols, the Riemann- and Ricci-tensos as well as the Ricci and Ketschmann scalas in this catalogue wee detemined by means of the softwae Maple togethe with the GRTensoII package by Musgave, Pollney, and Lake. A typical woksheet to ente a new metic may look like this: The commecial softwae Maple can be found hee: The GRTensoII-package is fee:

15 .0. TOOLS > gtw); > makegschwazschild); Makeg.0: GRTenso metic/basis enty utility To quit makeg, type exit at any pompt. Do you wish to ente a ) metic [gdn,dn)], ) line element [ds], 3) non-holonomic basis [e)...en)], o 4) NP tetad [l,n,m,mba]? > : Ente coodinates as a LIST eg. [t,,theta,phi]): > [t,,theta,phi]: Ente the line element using d[cood] to indicate diffeentials. fo example, ^*d[theta]^ + sintheta)^*d[phi]^) [Type exit to quit makeg] ds^ = If thee ae any complex valued coodinates, constants o functions fo this spacetime, please ente them as a SET eg. { z, psi } ). Complex quantities [default={}]: > {}: You may choose to 0) Use the metic WITHOUT saving it, ) Save the metic as it is, ) Coect an element of the metic, 3) Re-ente the metic, 4) Add/change constaint equations, 5) Add a text desciption, o 6) Abandon this metic and etun to Maple. > 0: The woksheets fo some of the metics in this catalogue can be found on the authos homepage. To detemine the objects that ae defined with espect to a local tetad, the metic must be given as nonholonomic basis. The vaious basic objects can be detemined via Chistoffel symbols Γ νρ µ gcalcch); gcalcchdn,dn,up)); patial deivatives Γ νρ,σ µ gcalcchdn,dn,up,pdn)); Riemann tenso R µνρσ gcalcriemman); gcalcrdn,dn,dn,dn)); Ricci tenso R µν gcalcricci); gcalcrdn,dn)); Ricci scala R gcalcricciscala); Ketschmann scala K gcalcriemsq);.0. Mathematica The calculation of the Chistoffel symbols, the Riemann- o Ricci-tenso within Mathematica could ead like this: Cleaing the values of symbols: In[]:= Clea[cood, metic, invesemetic, affine, t,, Theta, Phi] Setting the dimension: In[]:= n := 4 Defining a list of coodinates: In[3]:= cood := {t,, Theta, Phi} Defining the metic: In[4]:= metic := {{- - s/) c^, 0, 0, 0}, {0, / - s/), 0, 0}, {0, 0, ^, 0}, {0, 0, 0, ^ Sin[Theta]^}} In[5]:= metic // MatixFom

16 CHAPTER. INTRODUCTION AND NOTATION Calculating the invese metic: In[6]:= invesemetic := Simplify[Invese[metic]] In[7]:= invesemetic // MatixFom Calculating the Chistoffel symbols of the second kind: In[8]:= affine := affine = Simplify[ Table[/) Sum[invesemetic[[Mu, Rho]] D[metic[[Rho, Nu]], cood[[lambda]]] + D[metic[[Rho, Lambda]], cood[[nu]]] - D[metic[[Nu, Lambda]], cood[[rho]]]), {Rho,, n}], {Nu,, n}, {Lambda,, n}, {Mu,, n}]] Displaying the Chistoffel symbols of the second kind: In[9]:= listaffine := Table[If[UnsameQ[affine[[Nu, Lambda, Mu]], 0], {Style[ Subsupescipt[\[CapitalGamma], Row[{cood[[Nu]], cood[[lambda]]}], cood[[mu]]], 8], "=", Style[affine[[Nu, Lambda, Mu]], 4]}], {Lambda,, n}, {Nu,, Lambda}, {Mu,, n}] In[0]:= TableFom[Patition[DeleteCases[Flatten[listaffine], Null], 3], TableSpacing -> {, }] Defining the Riemann tenso: In[]:= iemann := iemann = Table[D[affine[[Nu, Sigma, Mu]], cood[[rho]]] - D[affine[[Nu, Rho, Mu]], cood[[sigma]]] + Sum[affine[[Rho, Lambda, Mu]] affine[[nu, Sigma, Lambda]] - affine[[sigma, Lambda, Mu]] affine[[nu, Rho, Lambda]], {Lambda,, n}], {Mu,, n}, {Nu,, n}, {Rho,, n}, {Sigma,, n}] Defining the Riemann tenso with lowe indices: In[]:= iemanndn := iemanndn = Table[Simplify[ Sum[metic[[Mu, Kappa]] iemann[[kappa, Nu, Rho, Sigma]], {Kappa,, n}]], {Mu,, n}, {Nu,, n}, {Rho,, n}, {Sigma,, n}] In[3]:= listriemann := Table[If[UnsameQ[iemannDn[[Mu, Nu, Rho, Sigma]], 0], {Style[Subscipt[R, Row[{cood[[Mu]], cood[[nu]], cood[[rho]], cood[[sigma]]}]], 6], "=", iemanndn[[mu, Nu, Rho, Sigma]]}], {Nu,, n}, {Mu,, Nu}, {Sigma,, n}, {Rho,, Sigma}] In[4]:= TableFom[Patition[DeleteCases[Flatten[listRiemann], Null], 3], TableSpacing -> {, }] Defining the Ricci tenso: In[5]:= icci := icci = Table[Simplify[ Sum[iemann[[Rho, Mu, Rho, Nu]], {Rho,, n}]], {Mu,, n}, {Nu,, n}] In[6]:= listricci := Table[If[UnsameQ[icci[[Mu, Nu]], 0], {Style[Subscipt[R, Row[{cood[[Mu]], cood[[nu]]}]], 6], "=", Style[icci[[Mu, Nu]], 6]}], {Nu,, 4}, {Mu,, Nu}] In[7]:= TableFom[Patition[DeleteCases[Flatten[listRicci], Null], 3], TableSpacing -> {, }] Defining the Ricci scala: In[8]:= icciscala := icciscala = Simplify[Sum[

17 .0. TOOLS 3 Sum[invesemetic[[Mu, Nu]] icci[[nu, Mu]], {Mu,, n}], {Nu,, n}]] Defining the Ketschmann scala: In[9]:= iemannup := iemannup = Table[Simplify[ Sum[invesemetic[[Nu, Kappa]] iemann[[mu, Kappa, Rho, Sigma]], {Kappa,, n}]], {Mu,, n}, {Nu,, n}, {Rho,, n}, {Sigma,, n}] In[0]:= ketschmann := ketschmann = Simplify[Sum[ Sum[Sum[Sum[ iemannup[[mu, Nu, Rho, Sigma]] iemannup[[rho, Sigma, Mu, Nu]], {Mu,, n}], {Nu,, n}], {Rho,, n}], {Sigma,, n}]] Some example notebooks can be found on the authos homepage..0.3 Maxima Instead of using commecial softwae like Maple o Mathematica, Maxima also offes a tenso package that helps to calculate the Chistoffel symbols etc. The above example fo the Schwazschild metic can be witten as a maxima woksheet as follows: /* load ctenso package */ loadctenso); /* define coodinates to use */ ct_coods:[t,,theta,phi]; /* stat with the identity metic */ lg:ident4); lg[,]:c^*-s/); lg[,]:-/-s/); lg[3,3]:-^; lg[4,4]:-^*sintheta)^; cmetic); /* calculate the chistoffel symbols of the second kind */ chistofmcs); /* calculate the iemann tenso */ liemannmcs); /* calculate the icci tenso */ iccimcs); /* calculate the icci scala */ scuvatue); /* calculate the Ketschmann scala */ uiemannmcs); invaiant); atsimp%); As you may have noticed, the Schwazschild metic must be given with negative signatue.

18 Chapte Spacetimes. Minkowski.. Catesian coodinates The Minkowski metic in Catesian coodinates{t, x, y, z Ê} eads ds = c dt + dx + dy + dz...) All Chistoffel symbols as well as the Riemann- and Ricci-tenso vanish identically. The natual local tetad is tivial, with dual e t) = c t, e x) = x, e y) = y, e z) = z,..) θ t) = cdt, θ x) = dx, θ y) = dy, θ z) = dz...3).. Cylindical coodinates The Minkowski metic in cylindical coodinates {t Ê, Ê +,ϕ [0,π),z Ê}, ds = c dt + d + dϕ + dz,..4) has the natual local tetad e t) = c t, e ) =, e ϕ) = ϕ, e z) = z...5) Chistoffel symbols: Γ ϕϕ =, Γ ϕ ϕ =...6) Patial deivatives Γϕ, ϕ =, Γ ϕϕ, =...7) Ricci otation coefficients: γ ϕ))ϕ) = and γ ) =...8) 4

19 .. MINKOWSKI 5..3 Spheical coodinates In spheical coodinates{t Ê, Ê +,ϑ 0,π),ϕ [0,π)}, the Minkowski metic eads ds = c dt + d + dϑ + sin ϑdϕ )...9) Chistoffel symbols: Γ ϑϑ =, Γ ϕϕ = sin ϑ, Γ ϑ ϑ =,..0a) Γϕϕ ϑ = sinϑ cosϑ, Γϕ ϕ =, Γϕ ϑϕ = cotϑ...0b) Patial deivatives Γϑ, ϑ =, Γϕ ϕ, =, Γ ϑϑ, =,..a) Γ ϕ ϑϕ,ϑ = sin ϑ, Γ ϕϕ, = sin ϑ, Γ ϑ ϕϕ,ϑ = cosϑ),..b) Γ ϕϕ,ϑ = sinϑ)...c) Local tetad: e t) = c t, e ) =, e ϑ) = ϑ, e ϕ) = sin ϑ ϕ...) Ricci otation coefficients: γ ϑ))ϑ) = γ ϕ))ϕ) =, The contactions of the Ricci otation coefficients ead γ ) =, γ ϕ)ϑ)ϕ) = cotϑ...3) γ ϑ) = cotϑ...4)..4 Confom-compactified coodinates The Minkowski metic in confom-compactified coodinates{ψ [ π,π],ξ 0,π),ϑ 0,π),ϕ [0,π)} eads[he99] ds = dψ + dξ + sin ξ dϑ + sin ϑdϕ )...5) This fom follows fom the spheical Minkowski metic..9) by means of the coodinate tansfomation ct+ =tan ψ+ ξ esulting in the metic d s = dψ + dξ 4cos ψ+ξ cos ψ ξ, ct =tan ψ ξ,..6) + sin ξ dϑ 4cos ψ+ξ cos ψ ξ + sin ϑdϕ ),..7) and by the confomal tansfomation ds = Ω d s with Ω = 4cos ψ+ξ cos ψ ξ. Chistoffel symbols: Γ ϑ ξ ϑ = cotξ, Γϕ ξ ϕ = cotξ, Γξ ϑϑ = sinξ cosξ,..8a) Γ ϕ ϑϕ = cotϑ, Γξ ϕϕ = sinξ cosξ sin ϑ, Γϕϕ ϑ = sinϑ cosϑ...8b)

20 6 CHAPTER. SPACETIMES Patial deivatives Γ ϑ ξ ϑ,ξ = sin ξ, Γϕ ξ ϕ,ξ = sin ξ, Γξ = cosξ),..9a) ϑϑ,ξ Γ ϕ ϑϕ,ϑ = sin ϑ, Γξ ϕϕ,ξ = cosξ)sin ϑ, Γ ϑ ϕϕ,ϑ = cosϑ),..9b) Γ ξ ϕϕ,ϑ = sinξ)sinϑ). Riemann-Tenso:..9c) R ξ ϑξϑ = sin ξ, R ξ ϕξ ϕ = sin ξ sin ϑ, R ϑϕϑϕ = sin 4 ξ sin ϑ...0) Ricci-Tenso: R ξ ξ =, R ϑϑ = sin ξ, R ϕϕ = sin ξ sin ϑ...) Ricci and Ketschmann scalas: R = 6, K =...) The Weyl tenso vanishs identically. Local tetad: e ψ) = ψ, e ξ) = ξ, e ϑ) = sinξ ϑ, e ϕ) = sinξ sinϑ ϕ...3) Ricci otation coefficients: γ ϑ)ξ)ϑ) = γ ϕ)ξ)ϕ) = cotξ, γ ϕ)ϑ)ϕ) = cotϑ sinξ...4) The contactions of the Ricci otation coefficients ead γ ξ) = cotξ, Riemann-Tenso with espect to local tetad: γ ϑ) = cotϑ sinξ...5) R ξ)ϑ)ξ)ϑ) = R ξ)ϕ)ξ)ϕ) = R ϑ)ϕ)ϑ)ϕ) =...6) Ricci-Tenso with espect to local tetad: R ξ)ξ) = R ϑ)ϑ) = R ϕ)ϕ) =...7)..5 Rotating coodinates The tansfomation dϕ dϕ + ω dt bings the Minkowski metic..4) into the otating fom[rin0] with coodinates{t Ê, Ê +,ϕ [0,π),z Ê}, ds = ω c ) [cdt Ω)dϕ] + d + ω /c dϕ + dz..8) with Ω)= ω/c)/ ω /c ). Metic-Tenso: g tt = c + ω, g tϕ = ω, g = g zz =, g ϕϕ =...9)

21 .. MINKOWSKI 7 Chistoffel symbols: Γtt = ω, Γt ϕ = ω, Γ tϕ = ω, Γϕ ϕ =, Γ ϕϕ =...30) Patial deivatives Γ tt, = ω, Γ ϕ t, = ω, Γ tϕ, = ω, Γ ϕ ϕ, =, Γ ϕϕ, =...3) The local tetad of the comoving obseve is e t) = c t ω c ϕ, e ) =, e ϕ) = ϕ, e z) = z,..3) wheeas the static obseve has the local tetad e t) = c ω /c t, e ) =, e z) = z,..33a) ω ω e ϕ) = c ω /c t+ /c ϕ...33b)..6 Rindle coodinates The woldline of an obseve in the Minkowski spacetime who moves with constant pope acceleation α along the x diection eads x= c αt cosh α c, c αt ct = sinh α c,..34) whee t is the obseve s pope time. The obseve stats at x= with zeo velocity. Howeve, such an obseve could also be descibed with Rindle coodinates. With the coodinate tansfomation ct,x) τ,ρ) : ct = ρ sinhτ, x= coshτ,..35) ρ whee ρ = α/c, the Rindle metic eads ds = ρ dτ + ρ 4 dρ + dy + dz...36) Chistoffel symbols: Γ ρ ττ = ρ, Γ τ τρ = ρ, Γρ ρρ = ρ...37) Patial deivatives Γττ,ρ ρ =, Γτρ,ρ τ = ρ, Γρ ρρ,ρ = ρ...38) The Riemann and Ricci tensos as well as the Ricci and Ketschmann scala vanish identically. Local tetad: e τ) = ρ τ, e ρ) = ρ ρ, e y) = y, e z) = z...39) Ricci otation coefficients: γ τ)ρ)τ) = ρ, and γ ρ) = ρ...40)

22 8 CHAPTER. SPACETIMES. Schwazschild spacetime.. Schwazschild coodinates In Schwazschild coodinates{t Ê, Ê +,ϑ 0,π),ϕ [0,π)}, the Schwazschild metic eads ds = s ) c dt + s / d + dϑ + sin ϑdϕ ),..) whee s = GM/c is the Schwazschild adius, G is Newton s constant, c is the speed of light, and M is the mass of the black hole. The citical point = 0 is a eal cuvatue singulaity while the event hoizon, = s, is only a coodinate singulaity, see e.g. the Ketschmann scala. Chistoffel symbols: Γ tt = c s s ) 3, Γ t t = s s ), Γ = s ), s..a) Γ ϑ ϑ =, Γϕ ϕ =, Γ ϑϑ = s),..b) Γ ϕ ϑϕ = cotϑ, Γ ϕϕ = s)sin ϑ, Γϕϕ ϑ = sinϑ cosϑ...c) Patial deivatives Γ tt, = 3 s)c s 4, Γ t t, = s) s s ), Γ, = s) s s ),..3a) Γϑ, ϑ =, Γϕ ϕ, =, Γ ϑϑ, =,..3b) Γ ϕ ϑϕ,ϑ = sin ϑ, Γ ϕϕ, = sin ϑ, Γϕϕ,ϑ ϑ = cosϑ),..3c) Γ ϕϕ,ϑ = s)sinϑ)...3d) Riemann-Tenso: R tt = c s 3, R tϑtϑ = c s ) s, R tϕtϕ = s R ϑϑ =, R ϕϕ = s c s ) s sin ϑ,..4a) s sin ϑ s, R ϑϕϑϕ = s sin ϑ...4b) As aspected, the Ricci tenso as well as the Ricci scala vanish identically because the Schwazschild spacetime is a vacuum solution of the field equations. Hence, the Weyl tenso is identical to the Riemann tenso. The Ketschmann scala eads K = s 6. Hee, it becomes clea that at = s thee is no eal singulaity. Local tetad: e t) = Dual tetad: θ t) = c c s / t, e ) = s dt, θ) =..5) s, e ϑ) = ϑ, e ϕ) = sinϑ ϕ...6) d s /, θϑ) = dϑ, θ ϕ) = sinϑ dϕ...7) Ricci otation coefficients: s γ )t)t) = s /, γ ϑ))ϑ) = γ ϕ))ϕ) = s, γ ϕ)ϑ)ϕ) = cotϑ...8)

23 .. SCHWARZSCHILD SPACETIME 9 The contactions of the Ricci otation coefficients ead γ ) = 4 3 s s /, γ ϑ) = cotϑ...9) Stuctue coefficients: c t) t)) = s s /, cϑ) )ϑ) = cϕ) )ϕ) = s, ϑ)ϕ) = cotϑ...0) cϕ) Riemann-Tenso with espect to local tetad: R t))t)) = R ϑ)ϕ)ϑ)ϕ) = s 3, R t)ϑ)t)ϑ) = R t)ϕ)t)ϕ) = R )ϑ))ϑ) = R )ϕ))ϕ) = s 3. The covaiant deivatives of the Riemann tenso ead..a)..b) R t))t));) = R ϑ)ϕ)ϑ)ϕ);) = 3 s s 5 ), R t)))ϑ);ϑ) = R t))t)ϕ);ϕ) = R t)ϑ)t)ϑ);) = R t)ϕ)t)ϕ);) = = R )ϕ)ϑ)ϕ);ϑ) = 3 s 5 s ), R )ϑ))ϑ);) = R )ϑ)ϑ)ϕ);ϕ) = R )ϕ))ϕ);) = 3 s 5 s ). Newman-Penose tetad:..a)..b)..c) l= et) + e ) ), n= et) e ) ), m= eϑ) + ie ϕ) )...3) Non-vanishing spin coefficients: ρ = µ = s, γ = ε = s 4 cotϑ, α = β = s /...4) Embedding: The embedding function eads z= s s. Eule-Lagange: The Eule-Lagangian fomalism, Sec..8.4, fo geodesics in the ϑ = π/ hypeplane yields ṙ +V eff = k c, V eff = ) s h ) κc..5)..6) with the constants of motion k= s /)c ṫ, h= ϕ, and κ as in Eq..8.). Fo timelike geodesics, the effective potential has the extemal points ± = h ± h h 3c s c s,..7) whee + is a maximum and is a minimum. The innemost timelike cicula geodesic follows fom h = 3c s and eads itcg = 3 s. Null geodesics, howeve, have only a maximum at po = 3 s. The coesponding cicula obit is called photon obit. Futhe eading: Schwazschild[Sch6, Sch03], MTW[MTW73], Rindle[Rin0], Wald[Wal84], Chandasekha[Cha06], Mülle[Mül08b, Mül09].

24 0 CHAPTER. SPACETIMES.. Schwazschild in pseudo-catesian coodinates The Schwazschild spacetime in pseudo-catesian coodinatest,x,y,z) eads ds = + ) x c dt + s x + y + z s / ) ) dx s / + y + z x + + y dy s / + z ) dz +..8) s s ) xydxdy+xzdxdz+yzdydz), whee = x + y + z. Fo a natual local tetad that is adapted to the x-axis, we make the following ansatz: e 0) = c s / t, e ) = A x, e ) = B x +C y, e 3) = D x + E y + F z...9) A= gxx, B= g xy g xx g xy /g xx+ g yy, C= g xy /g xx+ g yy,..0a) D= g xyg yz g xz g yy, E = g xzg xy g xx g yz N, F =, NW NW W..0b) with N = g xx g yy g xy, W = g xx g yy g zz g xzg yy + g xz g xy g yz g xyg zz g xx g yz...a)..b)..3 Isotopic coodinates Spheical isotopic coodinates The Schwazschild metic..) in spheical isotopic coodinates t,ρ,ϑ,ϕ) eads ds = ) ρs /ρ c dt + + ρ ) 4 s [ dρ + ρ dϑ + sin ϑdϕ )],..) +ρ s /ρ ρ whee =ρ + ρ s ρ ) o ρ = s ± ) s ) 4..3) is the coodinate tansfomation between the Schwazschild adial coodinate and the isotopic adial coodinate ρ, see e.g. MTW[MTW73] page 840. The event hoizon is given by ρ s = s /4. The photon obit and the innemost timelike cicula geodesic ead ρ po = + ) 3 ρ s and ρ itcg = 5+ ) 6 ρ s...4) Chistoffel symbols: Γtt ρ = ρ ρ s)ρ 4 ρ s c ρ+ ρ s ) 7, Γ t tρ = ρ s ρ ρs, Γρρ ρ = ρ s ρ+ ρ s )ρ, Γ ϑ ρϑ = ρ ρ s ρ+ ρ s )ρ, Γ ϕ ϑϕ = cotϑ, Γϕ ρϕ = ρ ρ s ρ+ ρ s )ρ,..5a) Γρ ϑϑ = ρ ρ ρ s ρ+ ρ s,..5b) Γρ ϕϕ = ρ ρ s)ρ sin ϑ ρ+ ρ s, Γ ϑ ϕϕ = sinϑ cosϑ...5c)

25 .. SCHWARZSCHILD SPACETIME Riemann-Tenso: R tρtρ = 4 ρ ρ s) ρ s c ρ+ ρ s ) 4 ρ, R tϑtϑ = ρ ρ s) ρρ s c ρ+ ρ s ) 4,..6a) R tϕtϕ = ρ ρ s) ρc ρ s sin ϑ ρ+ ρ s ) 4, R ρϑρϑ = ρ+ ρ s) ρ s ρ 3,..6b) R ρϕρϕ = ρ+ ρ s) ρ s sin ϑ ρ 3, R ϑϕϑϕ = 4ρ+ ρ s) ρ s sin ϑ...6c) ρ The Ricci tenso and the Ricci scala vanish identically. Ketschmann scala: s s K = 9 = ρ 6 +ρ s /ρ) ρ) 6...7) Local tetad: e t) = +ρ s/ρ t ρ s /ρ c, e ) = [+ρ s /ρ] ρ, e ϑ) = ρ[+ρ s /ρ] ϑ, e ϕ) = ρ[+ρ s /ρ] sin ϑ ϕ...8a)..8b) Ricci otation coefficients: γ ρ)t)t) = γ ϕ)ϑ)ϕ) = ρ cotϑ ρ+ ρ s ). ρ s ρ ρ+ ρ s ) 3 ρ ρ s ), γ ϑ)ρ)ϑ) = γ ϕ)ρ)ϕ) = ρρ ρ s) ρ+ ρ s ) 3,..9a)..9b) The contactions of the Ricci otation coefficients ead γ ρ) = ρρ ρρ s + ρ s) ρ+ ρ s ) 3 ρ ρ s ), γ ϑ) = ρ cotϑ ρ+ ρ s )...30) Riemann-Tenso with espect to local tetad: R t)ρ)t)ρ) = R ϑ)ϕ)ϑ)ϕ) = s ρ) 3, R t)ϑ)t)ϑ) = R t)ϕ)t)ϕ) = R ρ)ϑ)ρ)ϑ) = R ρ)ϕ)ρ)ϕ) = s ρ) 3...3a)..3b) Futhe eading: Buchdahl[Buc85]. Catesian isotopic coodinates The Schwazschild metic..) in Catesian isotopic coodinatest,x,y,z) eads, ds = ) ρs /ρ c dt + + ρ ) 4 s [ dx + dy + dz ],..3) +ρ s /ρ ρ whee ρ = x + y + z and, as befoe, =ρ + ρ ) s. ρ..33)

26 CHAPTER. SPACETIMES Chistoffel symbols: Γtt x = c ρ 3 ρ s ρ ρ s )x ρ+ ρ s ) 7, Γtt y = c ρ 3 ρ s ρ ρ s )y ρ+ ρ s ) 7, Γtt z = c ρ 3 ρ s ρ ρ s )z ρ+ ρ s ) 7,..34a) Γ t tx = ρ s x ρ 3 [ ρs /ρ ], Γt ty = ρ s y ρ 3 [ ρs /ρ ], Γt tz = ρ s z ρ 3 [ ρs /ρ ],..34b) Γ x xx = Γy xy = Γz xz = Γx yy = Γx zz = ρ s ρ 3 Γ y xx = Γx xy = Γy yy = Γz yz = Γy zz = ρ s ρ 3 Γ z xx = Γ x xz = Γ z yy = Γ y yz = Γ z zz = ρ s ρ 3..4 Eddington-Finkelstein x +ρ s /ρ,..34c) y +ρ s /ρ,..34d) z +ρ s /ρ...34e) The tansfomation of the Schwazschild metic..) fom the usual Schwazschild time coodinate t to the advanced null coodinate v with cv=ct+ + s ln s )..35) leads to the ingoing Eddington-Finkelstein[Edd4, Fin58] metic with coodinatesv,,ϑ,ϕ), ds = Metic-Tenso: s g vv = c s Chistoffel symbols: ) c dv + cdvd+ dϑ + sin ϑdϕ )...36) ), g v = c, g ϑϑ =, g ϕϕ = sin ϑ...37) Γ v vv = c s, Γ vv = c s s ) 3, Γ v = c s, Γϑ ϑ =,..38a) Γ ϕ ϕ =, Γv ϑϑ = c, Γ ϑϑ = s), Γ ϕ ϑϕ = cotϑ,..38b) Γ v ϕϕ = sin ϑ, Γ ϕϕ = s )sin ϑ, Γϕϕ ϑ = sinϑ cosϑ...38c) c Patial deivatives Γ v vv, = c s 3, Γ vv, = 3 s)c s 4, Γ v, = c s 3,..39a) Γ ϑ ϑ, =, Γϕ ϕ, =, Γv ϑϑ, = c,..39b) Γ ϑϑ, =, Γϕ ϑϕ,ϑ = sin ϑ, Γv ϕϕ, = sin ϑ,..39c) c Γ v sinϑ) ϕϕ,ϑ =, c Γ ϕϕ, = sin ϑ, Γ ϑ ϕϕ,ϑ = cosϑ),..39d) Γ ϕϕ,ϑ = s)sinϑ)...39e) Riemann-Tenso: R vv = c s 3, R vϑvϑ = c s s ), R vϑϑ = c s,..40a) R vϕvϕ = c s s )sin ϑ, R vϕϕ = c s sin ϑ, R ϑϕϑϕ = s sin ϑ...40b)

27 .. SCHWARZSCHILD SPACETIME 3 While the Ricci tenso and the Ricci scala vanish identically, the Ketschmann scala is K = s /6. Static local tetad: e v) = c s / v, e ) = c s / v+ s, e ϑ) = ϑ, e ϕ) = sinϑ ϕ...4) Dual tetad: θ v) = c s dv d s /, d θ) = s /, θϑ) = dϑ, θ ϕ) = sinϑdϕ...4) Ricci otation coefficients: s γ )v)v) = s /, γ ϑ))ϑ) = γ ϕ))ϕ) = s, γ ϕ)ϑ)ϕ) = cotϑ...43) The contactions of the Ricci otation coefficients ead γ ) = 4 3 s s /, γ ϑ) = cotϑ...44) Riemann-Tenso with espect to local tetad: R v))v)) = R ϑ)ϕ)ϑ)ϕ) = s 3, R v)ϑ)v)ϑ) = R v)ϕ)v)ϕ) = R )ϑ))ϑ) = R )ϕ))ϕ) = s a)..45b)..5 Kuskal-Szekees The Schwazschild metic in Kuskal-Szekees[Ku60, Wal84] coodinatest,x,ϑ,ϕ) eads ds = 43 s e / s dt + dx ) + dω,..46) whee R + \{0} is given by means of the LambetW-function W, [ )e / s = X T o = s W s X T e ) ] ) The Schwazschild coodinate time t in tems of the Kuskal coodinates T and X eads t = s actanh T X, > s,..48a) t = s actanh X T, < s,..48b) t =, = s...48c) The tansfomations between Kuskal- and Schwazschild coodinates ead X = e /s) sinh ct e /s) cosh ct s s, T = s X = e /s) cosh ct, T = s s, 0<<,..49a) s e /s) sinh ct, s...49b) s s

28 4 CHAPTER. SPACETIMES Chistoffel symbols: Γ T T T = Γ X T X = Γ T XX = T s+ s ) e / s,..50a) Γ X T T = ΓT T X = ΓX XX = X s+ s ) e / s,..50b) Γ ϑ T ϑ = s T e / s, Γ ϑ Xϑ = s X e / s,..50c) Γ T ϑϑ = s T, Γ T ϑϑ = s T sin ϑ, Γ X ϑϑ = s X, Γ X ϑϑ = s X sin ϑ,..50d)..50e) Γ ϕ ϑϕ = cotϑ, Γϑ ϕϕ = sinϑ cosϑ...50f) Riemann-Tenso: R T XT X = 6 7 s 5 e / s, R T ϕt ϕ = 4 s e / s sin ϑ, R T ϑtϑ = 4 s e / s, R XϑXϑ = 4 s e / s,..5a)..5b) R XϕXϕ = 4 s e / s sin ϑ, R ϑϕϑϕ = s sin ϑ...5c) The Ricci-Tenso as well as the Ricci-scala vanish identically. Ketschmann scala: K = s 6...5) Local tetad: e T) = e /s) T, e X) = e /s) X, e s s ϑ) = s s ϑ, e ϕ) = sin ϑ ϕ..53) Riemann-Tenso with espect to local tetad: R T)X)T)X) = R X)ϑ)X)ϑ) = R X)ϕ)X)ϕ) = R ϑ)ϕ)ϑ)ϕ) = s 3, R T)ϑ)T)ϑ) = R T)ϕ)T)ϕ) = s a)..54b)..6 Totoise coodinates The Schwazschild metic epesented by totoise coodinates t,ρ,ϑ,ϕ) eads ds = ) s c dt + ) s dρ + ρ) dϑ + sin ϑdϕ ),..55) ρ) ρ) whee s = GM/c is the Schwazschild adius, G is Newton s constant, c is the speed of light, and M is the mass of the black hole. The totoise adial coodinate ρ and the Schwazschild adial coodinate ae elated by ) ρ = + s ln s o = s {+W [ )]} ρ exp...56) s

29 .. SCHWARZSCHILD SPACETIME 5 Chistoffel symbols: Γ ρ tt = c s ρ), Γt tρ = s ρ), Γρ ρρ = s ρ),..57a) Γρϑ ϑ = ρ), Γ ϕ ρϕ = s ρ), Γ ρ ϑϑ = ρ),..57b) s Γ ϕ ϑϕ = cotϑ, Γρ ϕϕ = ρ)sin ϑ, Γ ϑ ϕϕ = sinϑ cosϑ...57c) Riemann-Tenso: R tρtρ = c s ρ) 3 R tϕtϕ = c sin ϑ R ρϕρϕ = sin ϑ s ρ) s ρ) s ρ) ), R tϑtϑ = c s ρ) ) s ρ),..58a) ) s ρ), R ρϑρϑ = ) s s..58b) ρ) ρ) ) s ρ), R ϑϕϑϕ = ρ) s sin ϑ...58c) The Ricci tenso as well as the Ricci scala vanish identically because the Schwazschild spacetime is a vacuum solution of the field equations. Hence, the Weyl tenso is identical to the Riemann tenso. The Ketschmann scala eads s K = ρ) ) Local tetad: e t) = Dual tetad: θ t) = c c s /ρ) t, e ρ) = s ρ) dt, θρ) = Riemann-Tenso with espect to local tetad: s /ρ) ρ, e ϑ) = ρ) ϑ, e ϕ) = ρ)sinϑ ϕ...60) s ρ) dρ, θϑ) = ρ)dϑ, θ ϕ) = ρ)sin ϑ dϕ...6) R t)ρ)t)ρ) = R ϑ)ϕ)ϑ)ϕ) = s ρ) 3, R t)ϑ)t)ϑ) = R t)ϕ)t)ϕ) = R ρ)ϑ)ρ)ϑ) = R ρ)ϕ)ρ)ϕ) = s ρ) 3...6a)..6b) Futhe eading: MTW[MTW73]..7 Painlevé-Gullstand The Schwazschild metic expessed in Painlevé-Gullstand coodinates[mp0] eads ) ds = c dt s + d+ cdt + dϑ + sin ϑdϕ ),..63) whee the new time coodinate T follows fom the Schwazschild time t in the following way: ct = ct+ s + ) s ln /s /s )

30 6 CHAPTER. SPACETIMES Metic-Tenso: g TT = c s Chistoffel symbols: ) s, g T = c, g =, g ϑϑ =, g ϕϕ = sin ϑ...65) Γ T T T = c s Γ T = c s s, Γ T T = c s s ) 3, Γ T T = s, s, ΓT = s c, Γ = s s,..66a)..66b) s Γ ϑ ϑ =, Γϕ ϕ =, ΓT ϑϑ = c, Γ ϑϑ = s), Γ ϕ ϑϕ = cotϑ, ΓT ϕϕ = s c sin ϑ,..66c)..66d) Γ ϕϕ = s)sin ϑ, Γϕϕ ϑ = sinϑ cosϑ...66e) Riemann-Tenso: R TT = c s 3, R TϑTϑ = c s s ), R Tϑϑ = c s R TϕT ϕ = c s s )sin ϑ, R Tϕϕ = c s s sin ϑ, R ϑϑ = s, s,..67a)..67b) R ϕϕ = s sin ϑ, R ϑϕϑϕ = s sin ϑ...67c) The Ricci tenso and the Ricci scala vanish identically. Ketschmann scala: K = s /6...68) Fo the Painlevé-Gullstand coodinates, we can define two natual local tetads. Static local tetad: ê T) = c s / s T, ê ) = c T + s s, ê ϑ) = ϑ, ê ϕ) = sinϑ ϕ,..69) Dual tetad: ˆθ T) = c s dt d /s, ˆθ ) = d s /, ˆθ ϑ) = dϑ, ˆθ ϕ) = sin ϑ dϕ...70) Feely falling local tetad: e T) = c s T, e ) =, e ϑ) = ϑ, e ϕ) = sinϑ ϕ...7) Dual tetad: θ T) = cdt, θ ) s = c dt + d, θϑ) = dϑ, θ ϕ) = sin ϑdϕ...7) Riemann-Tenso with espect to local tetad: R T))T)) = R ϑ)ϕ)ϑ)ϕ) = s 3, R T)ϑ)T)ϑ) = R T)ϕ)T)ϕ) = R )ϑ))ϑ) = R )ϕ))ϕ) = s a)..73b)

31 .. SCHWARZSCHILD SPACETIME 7..8 Isael coodinates The Schwazschild metic in Isael coodinatesx,y,ϑ,ϕ) eads[skm + 03] ds = s [ ) 4dx dy+ y dx ++xy) dϑ + sin ϑdϕ )],..74) +xy whee the coodinates x and y follow fom the Schwazschild coodinates via t = s +xy+ln y ) and = s +xy)...75) x Chistoffel symbols: Γ x xx = y+xy) +xy), Γ ϑ xϑ = y +xy, Γy xx = y3 3+xy) +xy) 3, Γϕ xϕ = y +xy, Γy xy = y+xy) +xy), Γϑ yϑ = x +xy,..76a)..76b) Γ ϕ xϕ = x +xy, Γx ϑϑ = x +xy), Γy ϑϑ = y xy),..76c) Γ ϕ ϑϕ = cotϑ, Γx ϕϕ = x +xy)sin ϑ, Γ y ϕϕ = y xy)sin ϑ,..76d) Γ ϑ ϕϕ = sinϑ cosϑ...76e) Riemann-Tenso: s R xyxy = 4 +xy) 3, R y s xϑxϑ = +xy), R xϑyϑ = s +xy, R xϕxϕ = s y sin ϑ +xy), R xϕyϕ = s sin ϑ +xy, R ϑϕϑϕ =+xy) s sin ϑ...77a)..77b) The Ricci tenso as well as the Ricci scala vanish identically. Hence, the Weyl tenso is identical to the Riemann tenso. The Ketschmann scala eads K = Local tetad: 4 s +xy)6...78) +xy y +xy e 0) = x + y, e s y s +xy ) = x, s y e ) = s +xy) ϑ, e 3) = s +xy)sinϑ ϕ...79a)..79b) Dual tetad: θ 0) = s +xy dy, y θ ) = sy dx+ s +xy dy, +xy y..80a) θ ) = s +xy)dϑ, θ 3) = s +xy)sinϑ dϕ...80b)

32 8 CHAPTER. SPACETIMES.3 Alcubiee Wap The Wap metic given by Miguel Alcubiee[Alc94] eads whee ds = c dt +dx v s f s )dt) + dy + dz.3.) v s = dx st), dt s t)= x x s t)) + y + z,.3.a).3.b) f s )= tanhσ s+ R)) tanhσ s R))..3.c) tanhσr) The paamete R>0 defines the adius of the wap bubble and the paamete σ > 0 its thickness. Metic-Tenso: g tt = c + v s f s ), g tx = v s f s ), g xx = g yy = g zz =..3.3) Chistoffel symbols: Γ t tt = f f x v 3 s c, Γ z tt = f f zv s, Γy tt = f f y v s,.3.4a) Γ x tt = f 3 f x v 4 s c f f x v s c f t v s c, Γ t tx = f f xv s c, Γ x tx = f f x v 3 s c,.3.4b) Γ y tx = f yv s, Γz tx = f zv s, Γt ty = f f yv s c,.3.4c) Γ x ty = f f y v 3 s + c f y v s c, Γ t tz = f f zv s c, Γx tz = f f z v 3 s + c f z v s c,.3.4d) Γ t xx = f xv s c, Γ x xy = f f yv s c, Γt xz = f zv s c, with deivatives f t = d f s) = v sσx x s t)) dt s tanhσr) f x = d f s) = σx x st)) dx s tanhσr) f y = d f s) dy f z = d f s) dz σy = s tanhσr) σz = s tanhσr) Γx xx = f f xv s c, Γ t xy = f yv s c, [ ] sech σ s + R)) sech σ s R)) [ sech σ s + R)) sech σ s R)) [ ] sech σ s + R)) sech σ s R)) [ ] sech σ s + R)) sech σ s R)).3.4e) Γx xz = f f zv s c,.3.4f) ].3.5a).3.5b).3.5c).3.5d) Riemann- and Ricci-tenso as well as Ricci- and Ketschman-scala ae shown only in the Maple woksheet. Comoving local tetad: e 0) = c t + v s f x ), e ) = x, e ) = y, e 3) = z..3.6) Static local tetad: e 0) = c v s f v s f c t, e ) = c c v s f t + v s f x, e c ) = y, e 3) = z..3.7) Futhe eading: Pfenning[PF97], Clak[CHL99], Van Den Boeck[Bo99]

33 .4. BARRIOLA-VILENKIN MONOPOL 9.4 Baiola-Vilenkin monopol The Baiola-Vilenkin metic descibes the gavitational field of a global monopole[bv89]. In spheical coodinatest,, ϑ, ϕ), the metic eads ds = c dt + d + k dϑ + sin ϑ dϕ ),.4.) whee k is the scaling facto esponsible fo the deficit/suplus angle. Chistoffel symbols: Γ ϑϑ = k, Γ ϕϕ = k sin ϑ, Γ ϑ ϑ =,.4.a) Γϕϕ ϑ = sinϑ cosϑ, Γϕ ϕ =, Γϕ ϑϕ = cotϑ..4.b) Patial deivatives Γ ϑ ϑ, =, Γϕ ϕ, =, Γ ϑϑ, = k,.4.3a) Γ ϕ ϑϕ,ϑ = sin ϑ, Γ ϕϕ, = k sin ϑ, Γϕϕ,ϑ ϑ = cosϑ),.4.3b) Γ ϕϕ,ϑ = k sinϑ)..4.3c) Riemann-Tenso: R ϑϕϑϕ = k )k sin ϑ..4.4) Ricci tenso, Ricci and Ketschmann scala: R ϑϑ = k ), R ϕϕ = k )sin ϑ, R = k k, K = 4 k ) k ) Weyl-Tenso: C tt = c k ) 3k, C tϑtϑ = c 6 k ), C tϕtϕ = c 6 k )sin ϑ,.4.6a) C ϑϑ = 6 k ), C ϕϕ = 6 k )sin ϑ, C ϑϕϑϕ = k 3 k )sin ϑ..4.6b) Local tetad: e t) = c t, e ) =, e ϑ) = k ϑ, e ϕ) = Dual tetad: k sinϑ ϕ..4.7) θ t) = cdt, θ ) = d, θ ϑ) = k dϑ, θ ϕ) = k sin ϑ dϕ..4.8) Ricci otation coefficients: γ ϑ))ϑ) = γ ϕ))ϕ) =, The contactions of the Ricci otation coefficients ead γ ) =, γ ϕ)ϑ)ϕ) = cotϑ k..4.9) γ ϑ) = cotϑ k..4.0)

34 30 CHAPTER. SPACETIMES Riemann-Tenso with espect to local tetad: R ϑ)ϕ)ϑ)ϕ) = k k..4.) Ricci-Tenso with espect to local tetad: R ϑ)ϑ) = R ϕ)ϕ) = k k..4.) Weyl-Tenso with espect to local tetad: C t))t)) = C ϑ)ϕ)ϑ)ϕ) = k 3k, C t)ϑ)t)ϑ) = C t)ϕ)t)ϕ) = C )ϑ))ϑ) = C )ϕ))ϕ) = k 6k..4.3a).4.3b) Embedding: The embedding function, see Sec..7, fo k< eads z= k..4.4) Eule-Lagange: The Eule-Lagangian fomalism, Sec..8.4, fo geodesics in the ϑ = π/ hypeplane yields ṙ +V eff = h c, V eff = ) h k κc,.4.5) with the constants of motion h = c ṫ and h = k ϕ. The point of closest appoach pca fo a null geodesic that stats at = i with y=±e t) +cosξ e ) +sinξ e ϕ) is given by = i sinξ. Hence, the pca is independent of k. The same is also tue fo timelike geodesics. Futhe eading: Baiola and Vilenkin[BV89], Pelick[Pe04].

35 .5. BERTOTTI-KASNER 3.5 Betotti-Kasne The Betotti-Kasne spacetime in spheical coodinatest,,ϑ,ϕ) eads[rin98] ds = c dt + e Λct d + Λ dϑ + sin ϑdϕ ),.5.) whee the cosmological constant Λ must be positive. Chistoffel symbols: Γ t = c Λ, Γ t = Patial deivatives Λ Λct c e, Γ ϕ ϑϕ = cotϑ, Γϑ ϕϕ = sinϑ cosϑ..5.) Γ t,t = Λe Λct, Γ ϕ ϑϕ,ϑ = sin ϑ, Γϑ ϕϕ,ϑ = cosϑ)..5.3) Riemann-Tenso: R tt = Λc e Λct, R ϑϕϑϕ = sin ϑ Λ..5.4) Ricci-Tenso: R tt = Λc, R = Λe Λct, R ϑϑ =, R ϕϕ = sin ϑ..5.5) The Ricci and Ketschmann scalas ead R = 4Λ, K = 8Λ..5.6) Weyl-Tenso: C tt = 3 Λc e Λct, C tϑtϑ = c 3, C tϕtϕ = Λct 3 e, C ϑϑ = 3 e Λct, C ϕϕ = 3 e Λct sin ϑ, C ϑϕϑϕ = 3 Local tetad: e t) = c t, e ) = e Λct, e ϑ) = Λ ϑ, e ϕ) = Dual tetad: θ t) = cdt, θ ) = e Λct d, θ ϑ) = dϑ, Λ.5.7a) sin ϑ Λ..5.7b) Λ sinϑ ϕ..5.8) θ ϕ) = sinϑ Λ dϕ..5.9) Ricci otation coefficients: γ t))) = Λ, γ ϑ)ϕ)ϕ) = Λcotϑ..5.0) The contactions of the Ricci otation coefficients ead γ t) = Λ, γ ϑ) = Λcotϑ..5.) Riemann-Tenso with espect to local tetad: R t))t)) = R ϑ)ϕ)ϑ)ϕ) = Λ..5.)

36 3 CHAPTER. SPACETIMES Ricci-Tenso with espect to local tetad: R t)t) = R )) = R ϑ)ϑ) = R ϕ)ϕ) = Λ..5.3) Weyl-Tenso with espect to local tetad: C t))t)) = C ϑ)ϕ)ϑ)ϕ) = Λ 3, C t)ϑ)t)ϑ) = C t)ϕ)t)ϕ) = C )ϑ))ϑ) = C )ϕ))ϕ) = Λ a).5.4b) Eule-Lagange: The Eule-Lagangian fomalism, Sec..8.4, fo geodesics in the ϑ = π/ hypeplane yields c ṫ = h e Λct + Λh κ.5.5) with the constants of motion h = ṙe Λct and h = ϕ/λ. Thus, +qt) λ = c qt i ) ln Λ Λh κ qt) +qt i ) Λct ), qt)= h e whee t i is the initial time. We can also solve the obital equation: t)=wt) wt i )+ i, wt)= whee i is the initial adial position. h e Λct + Λh κ Λh +,.5.6) κ h Λ,.5.7) Futhe eading: Rindle[Rin98]: Evey spheically symmetic solution of the genealized vacuum field equations R i j = Λg i j is eithe equivalent to Kottle s genealization of Schwazschild space o to the [...] Betotti-Kasne space fo which Λ must be necessaily be positive).