the metric acts on (contravariant) vectors (tensor indices) symmetric means g(x,y) = g(y,x) R for X,Y T P bilinear means for X,Y,Z T P and a,b,c R

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1 2. General Relativity the metric mathematical definition: a metric in M is a symmetric, bilinear, non-degenerate function g P : T P T P R the metric acts on (contravariant) vectors (tensor indices) symmetric means g(x,y) = g(y,x) R for X,Y T P bilinear means for X,Y,Z T P and a,b,c R g(ax +by,cz) = acg(x,z)+bcg(y,z) R non-degenerate: g(x,y) = 0 for all Y M only if X = 0 acting on the basis vectors e (µ) gives the metric tensor g µν (P) = g( e (µ), e (ν) ) the metric allows length and angle measurements X := g(x,x) and cosϕ = g(x,y) X Y Thomas Gajdosik Concepts of Modern Cosmology P

2 2. General Relativity the metric examples for the metric in Euclidean space the metric is the normal dot-product: a b = g jk a j b k = δ jk a j b k so g j=k = 1 and g j k = 0 we can use this metric of the embedding to get the induced metric in S 2 g jk = g( e (j), e (k) ) = e (j) e (k) so g 11 = s 2 ϑ 0 g 12 = g 21 = 0 g 22 = 1 as we can see, the metric tensor depends on the position in Minkovsky space we had g 00 = 1, g ii = 1, and g µ ν = 0 we can generalize the line element s 2 = (c t) 2 ( x) 2 ( y) 2 ( z) 2 = g µν x µ x ν the differentials x here are not multiplied with their natural wedge product one can define the inverse metric g µν by g µν g µρ = δρ ν the inverse metric is the metric of the cotangent space Thomas Gajdosik Concepts of Modern Cosmology

3 2. General Relativity the metric connection and the metric the metric gives angles so we can relate angles between vector fields all over the manifold but we have not specified the vectorfield, that defines parallel transport that is done with the connection / gauge field the metric has to be compatible with this choice as seen with the Lie bracket, parallel transport has to do with derivation the Lie derivative was a map L(T P ) : T P T P leaving the differentiated object in its own representation the directional derivative was a map D(T P ) : R T P this is, what we want but it does not connect the different points in M including an affine connection Γ to make a covariant derivative like in gauge theories: = +Γ = d+ω Thomas Gajdosik Concepts of Modern Cosmology

4 2. General Relativity Levi Civita connection definition for a covariant derivative: the covariant derivative U in the direction of U should map tensors to tensors be linear in the direction: fu+gv T = f U T +g V T be linear in the argument: U (T +S) = U T + U S follow the Leibniz rule: U (T S) = ( U T) S +T ( U S) commute with contractions: U δ λ ρ = 0 this means in index notation µ (T λ λν) = µ (δ λ ρt ρ λν) = δ λ ρ µ (T ρ λν) = ( T) λ λν;µ be the partial derivative on scalars: U φ = U(φ) = (U) µ µ φ definition of the Levi Civita connection: the covariant derivative U in the direction of U is metric compatible: U g(x,y) = 0 is torsion free: ( X Y Y X) = [X,Y] this is also easier explained in index notation... Thomas Gajdosik Concepts of Modern Cosmology

5 2. General Relativity Levi Civita connection using the natural basis the covariant derivative µ goes into the direction of (µ) the covariant derivative in the direction of U is U = U µ µ linearity in the direction is obvious from U T P (M) writing the Levi Civita connection as: µ A ν = µ A ν +Γ ν µρa ρ and µ a ν = µ a ν +Γ ρ µνa ρ commutation with contractions means: with B = A µ a µ = δ λ ρa ρ a λ a scalar: µ B = δ λ ρ[( µ A ρ +Γ ρ µνa ν )a λ +A ρ ( µ a λ +Γ ν µλ a ν)] = ( µ A ρ )a ρ +A ρ ( µ a ρ )+[Γ ρ µνa ν a ρ +A λ Γ ν µλ a ν] = µ (A ν a ν )+A ν a ρ [Γ ρ µν +Γ ρ µν ] Γ ρ µν = Γ ρ µν covariant indices transform opposite from contravariant ones Thomas Gajdosik Concepts of Modern Cosmology

6 2. General Relativity Levi Civita connection writing the Levi Civita connection as: µ A ν = µ A ν +Γ ν µρ Aρ and µ a ν = µ a ν Γ ρ µν a ρ torsion free means: 0 = X µ µ Y ν Y µ µ X ν [X µ µ Y ν Y µ µ X ν ] = X µ Γ ν µρ Y ρ Y µ Γ ν µρ Xρ = X µ Y ρ (Γ ν µρ Γν ρµ ) the Christoffel symbols Γ ρ µν = Γ ρ νµ are symmetric metric compatible means: 0 = ρ g µν = ρ g µν Γ λ ρµg λν Γ λ ρνg λµ adding µ g νρ + ν g µρ ρ g µν we get 0 = µ g νρ Γ λ µνg λρ Γ λ µρg λν + ν g µρ Γ λ νµg λρ Γ λ νρg λµ ρ g µν +Γ λ ρµg λν +Γ λ ρνg λµ = µ g νρ + ν g µρ ρ g µν 2Γ λ µν g λρ the Christoffel symbols can be calculated from the metric: Γ λ µν = 1 2 gλρ ( µ g νρ + ν g µρ ρ g µν ) Thomas Gajdosik Concepts of Modern Cosmology

7 2. General Relativity Riemann curvature tensor The difference between a Riemannian manifold and flat space parallel transport of a vector V along a curve C(t) is defined by ddt C(t) V = 0 taking the curve as a Levi Civita parallelogramoid along the vectors A and B P 00 A( s) P 10 B ( t) P 11 A ( s) P 01 B( t) P 00 with B parallel transported along A, so A B = 0, and B = B we can compare the vector at P 11 between the two different paths P 00 A( s) P 10 B( t) P 11 and P 00 B( t) P 01 A( s) P 11 and the difference of the parallel transported vector at P 11 is B A V A B V Thomas Gajdosik Concepts of Modern Cosmology

8 2. General Relativity Riemann curvature tensor The difference between a Riemannian manifold and flat space evaluating the difference in the natural basis µ we get B A V A B V = B σ σ A ρ ρ V µ µ A ρ ρ B σ σ V µ µ = A ρ B σ ( σ ρ ρ σ )V µ µ due to the condition A B = B A = 0 using the affine connection we get ( σ ρ ρ σ )V µ = σ ( ρ V µ +Γ µ ρνv ν ) ρ ( σ V µ +Γ µ σνv ν ) = σ ( ρ V µ +Γ µ ρν V ν ) Γ λ σρ ( λv µ +Γ µ λν V ν )+Γ µ σλ ( ρv λ +Γ λ ρν V ν ) ρ ( σ V µ +Γ µ σν V ν )+Γ λ ρσ ( λv µ +Γ µ λν V ν ) Γ µ ρλ ( σv λ +Γ λ σν V ν ) = ( σ Γ µ ρν ργ µ σν Γµ ρλ Γλ σν +Γµ σλ Γλ ρν )V ν (Γ λ σρ Γλ ρσ )( λv µ +Γ µ λν V ν ) =: R µ νσρv ν Tσρ( ν ν V µ ) the Riemann curvature tensor R µ νσρ and the Torsion tensor T ν σρ T ν ρσ vanishes for the Christoffel symbols Γ λ µν = Γ λ νµ Thomas Gajdosik Concepts of Modern Cosmology

9 2. General Relativity Riemann curvature tensor The difference between a Riemannian manifold and flat space in Euclidean (or Minkovsky) flat space one can use Cartesian coordinates, where the metric is constant the connection and the Riemann tensor vanish identically curvlinear coordinates, where the metric is not constant the connection is non zero but the Riemann tensor still vanishes identically example: spherical coordinates in R 3 with unit vectors e (r) = (s ϑ c ϕ,s ϑ s ϕ,c ϑ ), e (ϑ) = (c ϑ c ϕ,c ϑ s ϕ, s ϑ ), and e (ϕ) = ( s ϕ,c ϕ,0) the metric is ds 2 = dx 2 +dy 2 +dz 2 = dr 2 +r 2 dϑ 2 +r 2 sin 2 ϑdϕ 2 with non-zero coefficients g rr = 1, g ϑϑ = r 2, and g ϕϕ = r 2 sin 2 ϑ the Christoffel symbols are Γ r µν = 1 2 grr ( µ g νr + ν g µr r g µν ) Γ r ϑϑ = r Γr ϕϕ = rsin 2 ϑ Γ ϑ µν = 1 2 gϑϑ ( µ g νϑ + ν g µϑ ϑ g µν ) Γ ϑ rϑ = r 1 Γ ϑ ϕϕ = sinϑcosϑ Γ ϕ µν = 1 2 gϕϕ ( µ g νϕ + ν g µϕ ϕ g µν ) Γ ϕ rϕ = r 1 Γ ϕ ϑϕ = cotϑ Thomas Gajdosik Concepts of Modern Cosmology

10 2. General Relativity Riemann curvature tensor The difference between a Riemannian manifold and flat space example: spherical coordinates in R 3 the Riemann tensor R µ νσρ = ( σ Γ µ ρν Γ µ ρλ Γλ σν) (ρ σ) in components R µ νσρ = ( σ Γ µ ρν Γ µ ρrγ r σν Γ µ ρϑ Γϑ σν Γ µ ρϕγ ϕ σν) (ρ σ) R r rσρ = ( σ Γ r ρr Γ r ρrγ r σr Γ r ρϑγ ϑ σr Γ r ρϕγ ϕ σr) (ρ σ) = ( σ 0 00 ( r)δ ϑ ρ(r 1 )δ ϑ σ ( rsin 2 ϑ)δ ϕ ρ(r 1 )δ ϕ σ) (ρ σ) = (δρ ϑ δϑ σ +sin2 ϑδρ ϕ δϕ σ ) (ρ σ) = 0 R r ϑσρ = ( σ Γ r ρϑ 0Γr σϑ Γr ρϑ Γϑ σϑ Γr ρϕ Γϕ σϑ ) (ρ σ) = ( σ ( r)δρ ϑ ( r)δϑ ρ (r 1 )δσ r ( rsin2 ϑ)δρ ϕ (cotϑ)δϕ σ ) (ρ σ) = ( δσ r δϑ ρ +δϑ ρ δr σ ) (ρ σ) = 0 R r ϕσρ = ( σ Γ r ρϕ 0Γ r σϕ Γ r ρϑ Γϑ σϕ Γ r ρϕγ ϕ σϕ) (ρ σ) = ( σ ( rs 2 ϑ )δϕ ρ ( r)δϑ ρ ( s ϑc ϑ )δσ ϕ ( rs2 ϑ )δϕ ρ [(r 1 )δσ r +(c ϑ/s ϑ )δσ ϑ ]) (ρ σ) = ( [δ r σs 2 ϑ +r2s ϑc ϑ δ ϑ σ]δ ϕ ρ rs ϑ c ϑ δ ϑ ρδ ϕ σ +s 2 ϑ δϕ ρδ r σ +rs ϑ c ϑ δ ϕ ρδ ϑ σ) (ρ σ) = rs ϑ c ϑ (2δ ϕ ρδ ϑ σ +δ ϕ σδ ϑ ρ δ ϕ ρδ ϑ σ) (ρ σ) = 0 and similar for R ϑ νσρ and R ϕ νσρ R 3 is flat also in spherical coordinates Thomas Gajdosik Concepts of Modern Cosmology

11 2. General Relativity Riemann curvature tensor The difference between a Riemannian manifold and flat space example: S 2 with coordinates (x 1 = ϕ,x 2 = ϑ) and basis vectors e (1), e (2) embedded in R 3 with (x,y,z) = r(s ϑ c ϕ,s ϑ s ϕ,c ϑ ) and r constant the metric is ds 2 = (dx 1 ) 2 +(dx 2 ) 2 = r 2 dϑ 2 +r 2 sin 2 ϑdϕ 2 with non-zero coefficients g ϑϑ = r 2 and g ϕϕ = r 2 sin 2 ϑ the Christoffel symbols are Γ ϑ µν = 1 2 gϑϑ ( µ g νϑ + ν g µϑ ϑ g µν ) Γ ϑ ϕϕ = sinϑcosϑ Γ ϕ µν = 1 2 gϕϕ ( µ g νϕ + ν g µϕ ϕ g µν ) Γ ϕ ϑϕ = cotϑ and the Riemann tensor R µ νσρ = ( σ Γ µ ρν Γ µ ρλ Γλ σν ) (ρ σ) in components R µ νσρ = ( σ Γ µ ρν Γ µ ρϑ Γϑ σν Γµ ρϕγ ϕ σν) (ρ σ) R ϑ ϑσρ = ( σ Γ ϑ ρϑ Γϑ ρϑ Γϑ σϑ Γϑ ρϕγ ϕ σϑ ) (ρ σ) = ( ( sinϑcosϑ)δϕ ρ(cotϑ)δ ϕ σ) (ρ σ) = 0 R ϑ ϕσρ = ( σ Γ ϑ ρϕ Γϑ ρϑ Γϑ σϕ Γϑ ρϕ Γϕ σϕ ) (ρ σ) = ( σ( sinϑcosϑ)δρ ϕ ( sinϑcosϑ)δϕ ρ (cotϑ)δϑ σ ) (ρ σ) = ([ cos 2 ϑ+sin 2 ϑ]+cos 2 ϑ)δ ϕ ρ δϑ σ (ρ σ) = sin2 ϑ(δ ϕ ρ δϑ σ δϕ σ δϑ ρ ) 0 R ϕ ϑσρ = ( σ Γ ϕ ρϑ Γϕ ρϑ Γϑ σϑ Γϕ ρϕ Γϕ σϑ ) (ρ σ) = ( σ(cotϑ)δρ ϕ (cotϑ)δϑ ρ (cotϑ)δϕ σ ) (ρ σ) = ([ sin 2 ϑ]δ ϕ ρ δϑ σ cot2 ϑδ ϑ ρ δϕ σ ) (ρ σ) = (δϕ ρ δϑ σ δϕ σ δϑ ρ ) 0 R ϕ ϕσρ = ( σ Γ ϕ ρϕ Γϕ ρϑ Γϑ σϕ Γϕ ρϕ Γϕ σϕ ) (ρ σ) = (δϑ ρ ϑ(cotϑ)δρ ϑ (cotϑ)δϕ ρ ( sinϑcosϑ)δϕ ρ ) (ρ σ) = 0 S 2 is not flat Thomas Gajdosik Concepts of Modern Cosmology

12 2. General Relativity Riemann curvature tensor Properties of the Riemann curvature tensor the first Bianchi identity R µ νρσ +Rµ ρσν +Rµ σνρ = 0 Rµ [νρσ] = 0 with the abbreviation λ X =: X ;λ, the second Bianchi identity R µ νρσ;λ +Rµ νλρ;σ +Rµ νσλ;ρ = 0 Rµ ν[ρσ;λ] = 0 with lowering the first index R µνρσ = g µλ R λ νρσ R µνρσ = R µνσρ = R νµρσ = R ρσµν contraction gives the symmetric Ricci tensor and the Ricci scalar R λ µλν = gλρ R λµρν = R µν = R νµ g µν R µν = R this allows the decomposition R µνρσ = S µνσρ +E νµρσ +C ρσµν into a scalar part S µνσρ = R n(n 1) (g µσg νρ g µρ g νσ ) using the traceless Ricci tensor S µν = R µν R n g µν into a semi traceless part E µνσρ = 1 n 2 (g µσs νρ g µρ S νσ +g νρ S µσ g νσ S µρ ) and into the fully traceless Weyl tensor C µνσρ Thomas Gajdosik Concepts of Modern Cosmology

13 2. General Relativity Riemann curvature tensor Curvature in general for a curve we can imagine, how it bends on the plane or in space this is the extrinsic curvature the intrinsic curvature of a curve is zero it is similar to a straight line for a two dimensional surface we can imagine its bending in space its intrinsic curvature is defined independently from the embedding in our example of the spere S 2 we calculated the Riemann curvature tensor R ϑ ϕϑϕ = Rϑ ϕϕϑ = sin2 ϑ R ϕ ϑϑϕ = Rϕ ϑϕϑ = 1 this gives the Ricci tensor R ϑϑ = R ϕ ϑϕϑ = 1, R ϕϕ = R ϑ ϕϑϕ = sin2 ϑ and the Ricci scalar R = g ϑϑ R ϑϑ +g ϕϕ R ϕϕ = r 2 1+r 2 sin 2 ϑ sin 2 ϑ = 2 r 2 which gives the radius of the sphere... for higher dimensional (hyper)surfaces the Ricci scalar is the generalisation of the curvature radius Thomas Gajdosik Concepts of Modern Cosmology

14 2. General Relativity orthonormal coordinates using the metric we can change our basis to make it orthonormal: defining a new set of local coordinates e (m) (x) = e µ m (x) (µ) the greek indices µ indicate the natural basis (µ) obtained from the coordinate functions X(x) on M the latin indices m indicate the orthonormal basis e (m) (x) defined by the relation g(e (m) (x),e (n) (x))(x) = η mn the flat Minkovsky metric η 00 = 1, η ii = 1, and η m n = 0 e (m) (x) is an orthonormal coordinate system e µ m(x) are called the vierbein or tetrad its inverse e m µ (x) is defined by e µ me m ν = δ µ ν and e µ me n µ = δ n m we can write the metric tensor in natural coordinates as g µν (x) = e m µ (x)e n ν(x)η mn Thomas Gajdosik Concepts of Modern Cosmology

15 2. General Relativity orthonormal coordinates using the tetrad similar like for the natural coordinate basis (µ) we can introduce a dual basis in T P by requiring em (e n ) = δ m n this dual basis can also be written as e m = e m µ dx(µ) with the tetrad we can change any index into the orthonormal frame the vectors are just represented in a different basis tensors can even have mixed components: T µ ρσ = e µ mt m ρσ = e µ me r ρt m rσ = e µ me r ρe s σt m rs changing the coordinates x x changes components of a tensor by T µ ρ σ (x ) = xµ x µ x ρ x ρ x σ x σ T µ ρσ(x) this is called a general coordinate transformation (GCT) changing the orthonormal basis e n e n to another orthonormal one the metric does not change η mn = η mn the transformation is a local Lorentz transformation (LLT) : e n (x) e n(x) = Λ m n(x)e m (x) Thomas Gajdosik Concepts of Modern Cosmology

16 2. General Relativity orthonormal coordinates a connection is not a tensor a transformation of its indices might not lead to a connection parametrizing the connection in the orthonormal indices as µ T a b = µt a b +ω µ a c Tc b ω µ c b Ta c studying the covariant derivative of a vector helps: dx V = dx µ ( µ V) c e c = ( µ V c a +ω µ b V b )dx µ (e σ c (σ) ) = e σ c( µ (e c νv ν c )+ω µ b e b ρv ρ )dx µ (σ) = e σ ce c ν(( µ V ν )+e ν a( µ e a ρ)v ρ +e ν ae b a ρω µ b V ρ )dx µ (σ) = ( µ V ν +e ν a ( µe a ρ )V ρ +e ν a eb ρ ω µ a b V ρ )dx µ (ν) = ( µ V ν +Γ ν µρ V ρ )dx µ (ν) so Γ ν µρ = e ν a( µ e a ρ)+e ν ae b ρω µ a b or ω µ a b = e a λ eρ b Γλ µρ e ρ b ( µe a ρ) multiplying with e b ν this can be written as the tetrad posulate µ e a ν = µ e a ν Γ λ µνe a λ +ω µ a b eb ν = 0 ω a µ b is called the spin connection since it transports an index that transforms under Lorentz transformations and Lorentz transformations act also on spinors Thomas Gajdosik Concepts of Modern Cosmology

17 2. General Relativity orthonormal coordinates the spin connection transforms as a one-form under GCTs but inhomogenously under LLTs ω a µ b ω µ a b = Λa c Λd b ω µ c d Λc b µλ a c allows the implementation of a covariant exterior derivative: by writing forms we can suppress the tensor index of the natural basis: EM potential A = A µ dx µ, field strength F = 1 2 F µνdx µ dx ν = 1 2 (da) µνdx µ dx ν having a vector valued (or group valued) one-form A a = A a µ dxµ, we can extend the exterior derivative to a covariant exterior derivative 2( A) a = ( A) a µν dxµ dx ν = (da) a µν dxµ dx ν +(ω A) a µν dxµ dx ν = ( µ A a a ν +ω µ b A b ν)dx µ dx ν writing the torsion as a vector valued two form: T a = 1 2 Ta µνdx µ dx ν = 1 2 ea ρ(γ ρ µν Γ ρ νµ)dx µ dx ν = ( µ e a a ν +ω µ b e b ν)dx µ dx ν = ( µ e a ν)dx µ dx ν = e a Thomas Gajdosik Concepts of Modern Cosmology

18 2. General Relativity orthonormal coordinates the spin connection calculating the curvature tensor as a tensor valued two form 1 2 ( µ ν ν µ )V a e ρ a dxµ dx ν (ρ) = e ρ a V a (ρ) = e ρ a (dv a +ω a b V b ) (ρ) = e ρ a(d(dv a +ω a b V b )+ω a c (dv c +ω c b V b )) (ρ) = (d 2 V a +(dω a b )V b ω a b dv b +ω a c dv c +ω a c ω c b V b ))e ρ a (ρ) = (dω a b +ωa c ω c b )V b e a = R a b V b e a gives the Maurer-Cartan structure equations T a = de a +ω a b eb and R a b = dωa b +ωa c ω c b this allows an easy generalisation of the Bianchi identities (R ρ [σµν] = 0): T a = d(de a +ω a b eb )+ω a b (deb +ω b c ec ) = (dω a b ) eb ω a b (deb )+ω a b (deb )+ω a b ωb c e c = R a b eb = 1 2 Ra bµν eb dx µ dx ν and (R ρ σ[µν;λ] = 0) R a b = d(dω a b +ωa c ωc b )+ωa c (dωc b +ωc d ωd b ) (dωa d +ωa c ωc d ) ωd b = (dω a c ) ωc b ωa c (dωc b )+ωa c (dωc b )+ωa c (ωc d ωd b ) (dωa d ) ωd b (ωa c ωc d ) ωd b = 0 metric compatibility 0 = µ η ab = µ η ab ω µ c a η cb ω µ c b η ac = ω µba ω µab gives an antisymmetric spin connection ω µab = ω µba Thomas Gajdosik Concepts of Modern Cosmology