Introduction Singular General Relativity Our approach is to explore singularities in General Relativity, by constructing and using canonical geometric objects. As it turned out, singularities are much nicer than is usually thought. The equations can be expressed, even at singularities, using finite and smooth fundamental geometric objects. We can do covariant calculus, write down field equations. The FLRW Big Bang singularity is of this type. Isotropic singularities are of this type. Degenerate warped products are of this type. The stationary black holes turn out to be of this type. Non-stationary black holes are compatible with global hyperbolicity. The information is not necessarily lost. Implications to the Weyl Curvature Hypothesis of Penrose. Implications to dimensional reduction regularization in QFT and QG. 18 / 142
Introduction Two types of singularities 19 / 142
Introduction Two types of singularities 1 Malign singularities : some of the components g ab → ∞ . 20 / 142
Introduction Two types of singularities 1 Malign singularities : some of the components g ab → ∞ . 2 Benign singularities : g ab are smooth and finite, but det g → 0. 21 / 142
Introduction What is wrong with singularities 22 / 142
Introduction What is wrong with singularities 1 For PDE on curved spacetimes: the covariant derivatives blow up: Γ cab = 1 2 g cs ( ∂ a g bs + ∂ b g sa − ∂ s g ab ) (1) 23 / 142
Introduction What is wrong with singularities 1 For PDE on curved spacetimes: the covariant derivatives blow up: Γ cab = 1 2 g cs ( ∂ a g bs + ∂ b g sa − ∂ s g ab ) (1) 2 For Einstein’s equation blows up in addition because it is expressed in terms of the curvature, which is defined in terms of the covariant derivative: R d abc = Γ d ac , b − Γ d ab , c + Γ d bs Γ sac − Γ d cs Γ sab (2) 24 / 142
Introduction What is wrong with singularities 1 For PDE on curved spacetimes: the covariant derivatives blow up: Γ cab = 1 2 g cs ( ∂ a g bs + ∂ b g sa − ∂ s g ab ) (1) 2 For Einstein’s equation blows up in addition because it is expressed in terms of the curvature, which is defined in terms of the covariant derivative: R d abc = Γ d ac , b − Γ d ab , c + Γ d bs Γ sac − Γ d cs Γ sab (2) G ab = R ab − 1 (3) 2 Rg ab 25 / 142
Introduction What is wrong with singularities 1 For PDE on curved spacetimes: the covariant derivatives blow up: Γ cab = 1 2 g cs ( ∂ a g bs + ∂ b g sa − ∂ s g ab ) (1) 2 For Einstein’s equation blows up in addition because it is expressed in terms of the curvature, which is defined in terms of the covariant derivative: R d abc = Γ d ac , b − Γ d ab , c + Γ d bs Γ sac − Γ d cs Γ sab (2) G ab = R ab − 1 (3) 2 Rg ab R ab = R sasb , R = g pq R pq (4) 26 / 142
Introduction What is wrong with singularities 1 For PDE on curved spacetimes: the covariant derivatives blow up: Γ cab = 1 2 g cs ( ∂ a g bs + ∂ b g sa − ∂ s g ab ) (1) 2 For Einstein’s equation blows up in addition because it is expressed in terms of the curvature, which is defined in terms of the covariant derivative: R d abc = Γ d ac , b − Γ d ab , c + Γ d bs Γ sac − Γ d cs Γ sab (2) G ab = R ab − 1 (3) 2 Rg ab R ab = R sasb , R = g pq R pq (4) Even if g ab are all finite, these equations are also in terms of g ab , and g ab → ∞ when det g → 0. 27 / 142
Introduction What are the non-singular objects? 1 Some quantities which are part of the equations are indeed singular, but this is not a problem if we use instead other quantities, equivalent to them when the metric is non-degenerate. Singular Non-Singular When g is... Γ cab (2-nd) Γ abc (1-st) smooth R d abc R abcd semi-regular � W , W ≤ 2 R ab R ab | det g | semi-regular � W , W ≤ 2 R R | det g | semi-regular Ric Ric ◦ g quasi-regular R Rg ◦ g quasi-regular 1 (Stoica, 2011b; Stoica, 2012b) 28 / 142
Examples of singularities Examples of singularities 29 / 142
The mathematics of singularities Degenerate inner product - algebraic properties Degenerate inner product Definition An inner product on a vector space V is a symmetric bilinear form g ∈ V ∗ ⊗ V ∗ . The pair ( V , g ) is named inner product space . We use alternatively the notation � u , v � := g ( u , v ), for u , v ∈ V . The inner product g is degenerate if there is a vector v ∈ V , v � = 0, so that � u , v � = 0 for all u ∈ V , otherwise g is non-degenerate . There is always a basis, named orthonormal basis , in which g takes a diagonal form: O r . g = − I s (5) + I t where O r is the zero operator on R r , and I q , q ∈ { s , t } is the identity operator in R q . The signature of g is defined as the triple ( r , s , t ). 30 / 142
The mathematics of singularities Degenerate inner product - algebraic properties (V,g) V* u+w (V ● ,g ● ) w u V ● =V/V ○ (V ● ,g ● ) u ● ( V , g ) is an inner product vector space. 31 / 142
The mathematics of singularities Degenerate inner product - algebraic properties (V,g) V* u+w (V ● ,g ● ) w u V ● =V/V ○ (V ● ,g ● ) u ● The morphism ♭ : V → V ∗ is defined by ( V , g ) is an inner product vector space. u �→ u • := ♭ ( u ) = u ♭ = g ( u , ). 32 / 142
The mathematics of singularities Degenerate inner product - algebraic properties (V,g) V* u+w (V ● ,g ● ) w u V ● =V/V ○ (V ● ,g ● ) u ● The morphism ♭ : V → V ∗ is defined by ( V , g ) is an inner product vector space. u �→ u • := ♭ ( u ) = u ♭ = g ( u , ). The radical V ◦ := ker ♭ = V ⊥ is the set of isotropic vectors in V . 33 / 142
The mathematics of singularities Degenerate inner product - algebraic properties (V,g) V* u+w (V ● ,g ● ) w u V ● =V/V ○ (V ● ,g ● ) u ● The morphism ♭ : V → V ∗ is defined by ( V , g ) is an inner product vector space. u �→ u • := ♭ ( u ) = u ♭ = g ( u , ). The radical V ◦ := ker ♭ = V ⊥ is the set of isotropic vectors in V . V • := im ♭ ≤ V ∗ is the image of ♭ . 34 / 142
The mathematics of singularities Degenerate inner product - algebraic properties (V,g) V* u+w (V ● ,g ● ) w u V ● =V/V ○ (V ● ,g ● ) u ● The morphism ♭ : V → V ∗ is defined by ( V , g ) is an inner product vector space. u �→ u • := ♭ ( u ) = u ♭ = g ( u , ). The radical V ◦ := ker ♭ = V ⊥ is the set of isotropic vectors in V . V • := im ♭ ≤ V ∗ is the image of ♭ . The inner product g induces on V • an inner product defined by g • ( u ♭ 1 , u ♭ 1 ) := g ( u 1 , u 2 ) 35 / 142
The mathematics of singularities Degenerate inner product - algebraic properties (V,g) V* u+w (V ● ,g ● ) w u V ● =V/V ○ (V ● ,g ● ) u ● The morphism ♭ : V → V ∗ is defined by ( V , g ) is an inner product vector space. u �→ u • := ♭ ( u ) = u ♭ = g ( u , ). The radical V ◦ := ker ♭ = V ⊥ is the set of isotropic vectors in V . V • := im ♭ ≤ V ∗ is the image of ♭ . The inner product g induces on V • an inner product defined by g • ( u ♭ 1 , u ♭ 1 ) := g ( u 1 , u 2 ), which is the inverse of g iff det g � = 0. 36 / 142
The mathematics of singularities Degenerate inner product - algebraic properties (V,g) V* u+w (V ● ,g ● ) w u V ● =V/V ○ (V ● ,g ● ) u ● The morphism ♭ : V → V ∗ is defined by ( V , g ) is an inner product vector space. u �→ u • := ♭ ( u ) = u ♭ = g ( u , ). The radical V ◦ := ker ♭ = V ⊥ is the set of isotropic vectors in V . V • := im ♭ ≤ V ∗ is the image of ♭ . The inner product g induces on V • an inner product defined by g • ( u ♭ 1 , u ♭ 1 ) := g ( u 1 , u 2 ), which is the inverse of g iff det g � = 0. The quotient V • := V / V ◦ consists in the equivalence classes of the form u + V ◦ . 37 / 142
The mathematics of singularities Degenerate inner product - algebraic properties (V,g) V* u+w (V ● ,g ● ) w u V ● =V/V ○ (V ● ,g ● ) u ● The morphism ♭ : V → V ∗ is defined by ( V , g ) is an inner product vector space. u �→ u • := ♭ ( u ) = u ♭ = g ( u , ). The radical V ◦ := ker ♭ = V ⊥ is the set of isotropic vectors in V . V • := im ♭ ≤ V ∗ is the image of ♭ . The inner product g induces on V • an inner product defined by g • ( u ♭ 1 , u ♭ 1 ) := g ( u 1 , u 2 ), which is the inverse of g iff det g � = 0. The quotient V • := V / V ◦ consists in the equivalence classes of the form u + V ◦ . On V • , g induces an inner product g • ( u 1 + V ◦ , u 2 + V ◦ ) := g ( u 1 , u 2 ). (Stoica, 2011c) 38 / 142
The mathematics of singularities Degenerate inner product - algebraic properties Relations between the various spaces 2 The relations between the radical, the radical annihilator and the factor spaces can be collected in the diagram: π • i ◦ ( V • , g • ) 0 V ◦ ( V , g ) 0 ♭ V ♯ ♭ π ◦ i • V ◦ V ∗ ( V • , g • ) 0 0 where V • = V •∗ = V and V ◦ = V ◦∗ = V ∗ V • . V ◦ 2 (Stoica, 2011c) 39 / 142
The mathematics of singularities Degenerate inner product - algebraic properties Netric contraction between covariant indices 1 We define it first on tensors T ∈ V • ⊗ V • , by C 12 T = g • ab T ab . 40 / 142
The mathematics of singularities Degenerate inner product - algebraic properties Netric contraction between covariant indices 1 We define it first on tensors T ∈ V • ⊗ V • , by C 12 T = g • ab T ab . 2 Let T ∈ T r s V be a tensor with r ≥ 0 and s ≥ 2, which satisfies T ∈ V ⊗ r ⊗ V ∗⊗ s − 2 ⊗ V • ⊗ V • . Then, we define s V ⊗ V • ⊗ V • → T r s − 2 V ⊗ g • : T r C s − 1 s := 1 T r s − 2 V , (6) 41 / 142
The mathematics of singularities Degenerate inner product - algebraic properties Netric contraction between covariant indices 1 We define it first on tensors T ∈ V • ⊗ V • , by C 12 T = g • ab T ab . 2 Let T ∈ T r s V be a tensor with r ≥ 0 and s ≥ 2, which satisfies T ∈ V ⊗ r ⊗ V ∗⊗ s − 2 ⊗ V • ⊗ V • . Then, we define s V ⊗ V • ⊗ V • → T r s − 2 V ⊗ g • : T r C s − 1 s := 1 T r s − 2 V , (6) 3 Let T ∈ T r s V be a tensor with r ≥ 0 and s ≥ 2, which satisfies T ∈ V ⊗ r ⊗ V ∗⊗ k − 1 ⊗ V • ⊗ V ∗⊗ l − k − 1 ⊗ V • ⊗ V ∗⊗ s − l , 1 ≤ k < l ≤ s . We define the contraction C kl : V ⊗ r ⊗ V ∗⊗ k − 1 ⊗ V • ⊗ V ∗⊗ l − k − 1 ⊗ V • ⊗ V ∗⊗ s − l → V ⊗ r ⊗ V ∗⊗ s − 2 , (7) by C kl := C s − 1 s ◦ P k , s − 1; l , s , where P k , s − 1; l , s : T ∈ T r s V → T ∈ T r s V is the permutation isomorphisms which moves the k -th and l -th slots in the last two positions. 42 / 142
The mathematics of singularities Definition of singular semi-Riemannian manifolds Singular semi-Riemannian manifolds Definition A singular semi-Riemannian manifold is a pair ( M , g ), where M is a differentiable manifold, and g is a symmetric bilinear form on M , named metric tensor or metric . 43 / 142
The mathematics of singularities Definition of singular semi-Riemannian manifolds Singular semi-Riemannian manifolds Definition A singular semi-Riemannian manifold is a pair ( M , g ), where M is a differentiable manifold, and g is a symmetric bilinear form on M , named metric tensor or metric . Constant signature : the signature of g is fixed. 44 / 142
The mathematics of singularities Definition of singular semi-Riemannian manifolds Singular semi-Riemannian manifolds Definition A singular semi-Riemannian manifold is a pair ( M , g ), where M is a differentiable manifold, and g is a symmetric bilinear form on M , named metric tensor or metric . Constant signature : the signature of g is fixed. Variable signature : the signature of g varies from point to point. 45 / 142
The mathematics of singularities Definition of singular semi-Riemannian manifolds Singular semi-Riemannian manifolds Definition A singular semi-Riemannian manifold is a pair ( M , g ), where M is a differentiable manifold, and g is a symmetric bilinear form on M , named metric tensor or metric . Constant signature : the signature of g is fixed. Variable signature : the signature of g varies from point to point. If g is non-degenerate, then ( M , g ) is a semi-Riemannian manifold . 46 / 142
The mathematics of singularities Definition of singular semi-Riemannian manifolds Singular semi-Riemannian manifolds Definition A singular semi-Riemannian manifold is a pair ( M , g ), where M is a differentiable manifold, and g is a symmetric bilinear form on M , named metric tensor or metric . Constant signature : the signature of g is fixed. Variable signature : the signature of g varies from point to point. If g is non-degenerate, then ( M , g ) is a semi-Riemannian manifold . If g is positive definite, ( M , g ) is a Riemannian manifold . 47 / 142
The mathematics of singularities Definition of singular semi-Riemannian manifolds Degenerate metric - algebraic properties For the tangent bundle T p M at a point p ∈ M , the spaces and associated metrics are defined as usual: i ◦ π • ( V • , g • ) 0 ( T p M , g ) 0 T ◦ p M ♭ T p M ♯ ♭ π ◦ i • T ◦ p M T ∗ ( T • p M , g • ) 0 p M 0 p M = T p M T ∗ T ◦ p M and T ◦ p M = ( T ◦ p M ) ∗ = p M where T • p M = T •∗ T • p M . 48 / 142
The mathematics of singularities Covariant derivative The Koszul object The Koszul object is defined as K : X ( M ) 3 → R , 1 K ( X , Y , Z ) := 2 { X � Y , Z � + Y � Z , X � − Z � X , Y � (8) −� X , [ Y , Z ] � + � Y , [ Z , X ] � + � Z , [ X , Y ] �} . 49 / 142
The mathematics of singularities Covariant derivative The Koszul object The Koszul object is defined as K : X ( M ) 3 → R , 1 K ( X , Y , Z ) := 2 { X � Y , Z � + Y � Z , X � − Z � X , Y � (8) −� X , [ Y , Z ] � + � Y , [ Z , X ] � + � Z , [ X , Y ] �} . In local coordinates it is the Christoffel’s symbols of the first kind: K abc = K ( ∂ a , ∂ b , ∂ c ) = 1 2( ∂ a g bc + ∂ b g ca − ∂ c g ab ) = Γ abc , (9) 50 / 142
The mathematics of singularities Covariant derivative The Koszul object The Koszul object is defined as K : X ( M ) 3 → R , 1 K ( X , Y , Z ) := 2 { X � Y , Z � + Y � Z , X � − Z � X , Y � (8) −� X , [ Y , Z ] � + � Y , [ Z , X ] � + � Z , [ X , Y ] �} . In local coordinates it is the Christoffel’s symbols of the first kind: K abc = K ( ∂ a , ∂ b , ∂ c ) = 1 2( ∂ a g bc + ∂ b g ca − ∂ c g ab ) = Γ abc , (9) For non-degenerate metrics, the Levi-Civita connection is obtained uniquely: ∇ X Y = K ( X , Y , ) ♯ . (10) 51 / 142
The mathematics of singularities Covariant derivative The covariant derivatives 3 The lower covariant derivative of a vector field Y in the direction of a vector field X : ( ∇ ♭ X Y )( Z ) := K ( X , Y , Z ) (11) 3 (Stoica, 2011b) 52 / 142
The mathematics of singularities Covariant derivative The covariant derivatives 3 The lower covariant derivative of a vector field Y in the direction of a vector field X : ( ∇ ♭ X Y )( Z ) := K ( X , Y , Z ) (11) The covariant derivative of differential forms : ( ∇ X ω ) ( Y ) := X ( ω ( Y )) − g • ( ∇ ♭ X Y , ω ) , 3 (Stoica, 2011b) 53 / 142
The mathematics of singularities Covariant derivative The covariant derivatives 3 The lower covariant derivative of a vector field Y in the direction of a vector field X : ( ∇ ♭ X Y )( Z ) := K ( X , Y , Z ) (11) The covariant derivative of differential forms : ( ∇ X ω ) ( Y ) := X ( ω ( Y )) − g • ( ∇ ♭ X Y , ω ) , ∇ X ( ω 1 ⊗ . . . ⊗ ω s ) := ∇ X ( ω 1 ) ⊗ . . . ⊗ ω s + . . . + ω 1 ⊗ . . . ⊗ ∇ X ( ω s ) 3 (Stoica, 2011b) 54 / 142
The mathematics of singularities Covariant derivative The covariant derivatives 3 The lower covariant derivative of a vector field Y in the direction of a vector field X : ( ∇ ♭ X Y )( Z ) := K ( X , Y , Z ) (11) The covariant derivative of differential forms : ( ∇ X ω ) ( Y ) := X ( ω ( Y )) − g • ( ∇ ♭ X Y , ω ) , ∇ X ( ω 1 ⊗ . . . ⊗ ω s ) := ∇ X ( ω 1 ) ⊗ . . . ⊗ ω s + . . . + ω 1 ⊗ . . . ⊗ ∇ X ( ω s ) ( ∇ X T ) ( Y 1 , . . . , Y k ) = X ( T ( Y 1 , . . . , Y k )) − � k i =1 K ( X , Y i , • ) T ( Y 1 , , . . . , • , . . . , Y k ) 3 (Stoica, 2011b) 55 / 142
The mathematics of singularities Covariant derivative Semi-regular manifolds. Riemann curvature tensor 4 A semi-regular semi-Riemannian manifold is defined by the condition ∇ X ∇ ♭ Y Z ∈ A • ( M ) . (12) 4 (Stoica, 2011b) 56 / 142
The mathematics of singularities Covariant derivative Semi-regular manifolds. Riemann curvature tensor 4 A semi-regular semi-Riemannian manifold is defined by the condition ∇ X ∇ ♭ Y Z ∈ A • ( M ) . (12) Equivalently, K ( X , Y , • ) K ( Z , T , • ) ∈ F ( M ) . (13) 4 (Stoica, 2011b) 57 / 142
The mathematics of singularities Covariant derivative Semi-regular manifolds. Riemann curvature tensor 4 A semi-regular semi-Riemannian manifold is defined by the condition ∇ X ∇ ♭ Y Z ∈ A • ( M ) . (12) Equivalently, K ( X , Y , • ) K ( Z , T , • ) ∈ F ( M ) . (13) Riemann curvature tensor: R ( X , Y , Z , T ) = ( ∇ X ∇ ♭ Y Z )( T ) − ( ∇ Y ∇ ♭ X Z )( T ) − ( ∇ ♭ [ X , Y ] Z )( T ) (14) 4 (Stoica, 2011b) 58 / 142
The mathematics of singularities Covariant derivative Semi-regular manifolds. Riemann curvature tensor 4 A semi-regular semi-Riemannian manifold is defined by the condition ∇ X ∇ ♭ Y Z ∈ A • ( M ) . (12) Equivalently, K ( X , Y , • ) K ( Z , T , • ) ∈ F ( M ) . (13) Riemann curvature tensor: R ( X , Y , Z , T ) = ( ∇ X ∇ ♭ Y Z )( T ) − ( ∇ Y ∇ ♭ X Z )( T ) − ( ∇ ♭ [ X , Y ] Z )( T ) (14) R abcd = ∂ a K bcd − ∂ b K acd + ( K ac • K bd • − K bc • K ad • ) (15) 4 (Stoica, 2011b) 59 / 142
The mathematics of singularities Covariant derivative Semi-regular manifolds. Riemann curvature tensor 4 A semi-regular semi-Riemannian manifold is defined by the condition ∇ X ∇ ♭ Y Z ∈ A • ( M ) . (12) Equivalently, K ( X , Y , • ) K ( Z , T , • ) ∈ F ( M ) . (13) Riemann curvature tensor: R ( X , Y , Z , T ) = ( ∇ X ∇ ♭ Y Z )( T ) − ( ∇ Y ∇ ♭ X Z )( T ) − ( ∇ ♭ [ X , Y ] Z )( T ) (14) R abcd = ∂ a K bcd − ∂ b K acd + ( K ac • K bd • − K bc • K ad • ) (15) Is a tensor field. 4 (Stoica, 2011b) 60 / 142
The mathematics of singularities Covariant derivative Semi-regular manifolds. Riemann curvature tensor 4 A semi-regular semi-Riemannian manifold is defined by the condition ∇ X ∇ ♭ Y Z ∈ A • ( M ) . (12) Equivalently, K ( X , Y , • ) K ( Z , T , • ) ∈ F ( M ) . (13) Riemann curvature tensor: R ( X , Y , Z , T ) = ( ∇ X ∇ ♭ Y Z )( T ) − ( ∇ Y ∇ ♭ X Z )( T ) − ( ∇ ♭ [ X , Y ] Z )( T ) (14) R abcd = ∂ a K bcd − ∂ b K acd + ( K ac • K bd • − K bc • K ad • ) (15) Is a tensor field. Has the same symmetry properties as for det g � = 0. 4 (Stoica, 2011b) 61 / 142
The mathematics of singularities Covariant derivative Semi-regular manifolds. Riemann curvature tensor 4 A semi-regular semi-Riemannian manifold is defined by the condition ∇ X ∇ ♭ Y Z ∈ A • ( M ) . (12) Equivalently, K ( X , Y , • ) K ( Z , T , • ) ∈ F ( M ) . (13) Riemann curvature tensor: R ( X , Y , Z , T ) = ( ∇ X ∇ ♭ Y Z )( T ) − ( ∇ Y ∇ ♭ X Z )( T ) − ( ∇ ♭ [ X , Y ] Z )( T ) (14) R abcd = ∂ a K bcd − ∂ b K acd + ( K ac • K bd • − K bc • K ad • ) (15) Is a tensor field. Has the same symmetry properties as for det g � = 0. It is radical-annihilator. 4 (Stoica, 2011b) 62 / 142
The mathematics of singularities Covariant derivative Semi-regular manifolds. Riemann curvature tensor 4 A semi-regular semi-Riemannian manifold is defined by the condition ∇ X ∇ ♭ Y Z ∈ A • ( M ) . (12) Equivalently, K ( X , Y , • ) K ( Z , T , • ) ∈ F ( M ) . (13) Riemann curvature tensor: R ( X , Y , Z , T ) = ( ∇ X ∇ ♭ Y Z )( T ) − ( ∇ Y ∇ ♭ X Z )( T ) − ( ∇ ♭ [ X , Y ] Z )( T ) (14) R abcd = ∂ a K bcd − ∂ b K acd + ( K ac • K bd • − K bc • K ad • ) (15) Is a tensor field. Has the same symmetry properties as for det g � = 0. It is radical-annihilator. It is smooth for semi-regular metrics. 4 (Stoica, 2011b) 63 / 142
The mathematics of singularities Examples of semi-regular semi-Riemannian manifolds Examples of semi-regular semi-Riemannian manifolds 5 Isotropic singularities: g = Ω 2 ˜ g . 5 (Stoica, 2011b; Stoica, 2011d) 64 / 142
The mathematics of singularities Examples of semi-regular semi-Riemannian manifolds Examples of semi-regular semi-Riemannian manifolds 5 Isotropic singularities: g = Ω 2 ˜ g . Degenerate warped products ( f allowed to vanish): d s 2 = d s 2 B + f 2 ( p )d s 2 F . (16) 5 (Stoica, 2011b; Stoica, 2011d) 65 / 142
The mathematics of singularities Examples of semi-regular semi-Riemannian manifolds Examples of semi-regular semi-Riemannian manifolds 5 Isotropic singularities: g = Ω 2 ˜ g . Degenerate warped products ( f allowed to vanish): d s 2 = d s 2 B + f 2 ( p )d s 2 F . (16) FLRW spacetimes are degenerate warped products: d s 2 = − d t 2 + a 2 ( t )dΣ 2 (17) d r 2 1 − kr 2 + r 2 � d θ 2 + sin 2 θ d φ 2 � dΣ 2 = , (18) where k = 1 for S 3 , k = 0 for R 3 , and k = − 1 for H 3 . 5 (Stoica, 2011b; Stoica, 2011d) 66 / 142
Einstein’s equation on semi-regular spacetimes Einstein’s equation on semi-regular spacetimes 6 On 4 D semi-regular spacetimes Einstein tensor density G det g is smooth. At the points p where the metric is non-degenerate, the Einstein tensor can be expressed by: G ab det g = g kl ǫ akst ǫ blpq R stpq , (19) where ǫ abcd is the Levi-Civita symbol. 6 (Stoica, 2011b) 67 / 142
Einstein’s equation on semi-regular spacetimes Einstein’s equation on semi-regular spacetimes 6 On 4 D semi-regular spacetimes Einstein tensor density G det g is smooth. At the points p where the metric is non-degenerate, the Einstein tensor can be expressed by: G ab det g = g kl ǫ akst ǫ blpq R stpq , (19) where ǫ abcd is the Levi-Civita symbol. Therefore, G ab det g is smooth too, and it makes sense to write a densitized version of Einstein’s equation G ab det g + Λ g ab det g = κ T ab det g , (20) where κ := 8 π G c 4 , G and c being Newton’s constant and the speed of light. 6 (Stoica, 2011b) 68 / 142
Einstein’s equation on semi-regular spacetimes Einstein’s equation on semi-regular spacetimes 6 On 4 D semi-regular spacetimes Einstein tensor density G det g is smooth. At the points p where the metric is non-degenerate, the Einstein tensor can be expressed by: G ab det g = g kl ǫ akst ǫ blpq R stpq , (19) where ǫ abcd is the Levi-Civita symbol. Therefore, G ab det g is smooth too, and it makes sense to write a densitized version of Einstein’s equation G ab det g + Λ g ab det g = κ T ab det g , (20) where κ := 8 π G c 4 , G and c being Newton’s constant and the speed of light. √ det g . In many cases, the densitized Einstein equation works even with G ab 6 (Stoica, 2011b) 69 / 142
Einstein’s equation on semi-regular spacetimes Einstein’s equation on semi-regular spacetimes 6 On 4 D semi-regular spacetimes Einstein tensor density G det g is smooth. At the points p where the metric is non-degenerate, the Einstein tensor can be expressed by: G ab det g = g kl ǫ akst ǫ blpq R stpq , (19) where ǫ abcd is the Levi-Civita symbol. Therefore, G ab det g is smooth too, and it makes sense to write a densitized version of Einstein’s equation G ab det g + Λ g ab det g = κ T ab det g , (20) where κ := 8 π G c 4 , G and c being Newton’s constant and the speed of light. √ det g . In many cases, the densitized Einstein equation works even with G ab It is not allowed to divide by det g , when det g = 0. 6 (Stoica, 2011b) 70 / 142
Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime If S is a connected three-dimensional Riemannian manifold of constant cur- vature k ∈ {− 1 , 0 , 1 } ( i.e. H 3 , R 3 or S 3 ) and a ∈ ( A , B ), −∞ ≤ A < B ≤ ∞ , a ≥ 0, then the warped product I × a S is called a Friedmann-Lemaˆ ıtre- Robertson-Walker spacetime. d s 2 = − d t 2 + a 2 ( t )dΣ 2 (21) d r 2 1 − kr 2 + r 2 � d θ 2 + sin 2 θ d φ 2 � dΣ 2 = , (22) where k = 1 for S 3 , k = 0 for R 3 , and k = − 1 for H 3 . 71 / 142
Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime If S is a connected three-dimensional Riemannian manifold of constant cur- vature k ∈ {− 1 , 0 , 1 } ( i.e. H 3 , R 3 or S 3 ) and a ∈ ( A , B ), −∞ ≤ A < B ≤ ∞ , a ≥ 0, then the warped product I × a S is called a Friedmann-Lemaˆ ıtre- Robertson-Walker spacetime. d s 2 = − d t 2 + a 2 ( t )dΣ 2 (21) d r 2 1 − kr 2 + r 2 � d θ 2 + sin 2 θ d φ 2 � dΣ 2 = , (22) where k = 1 for S 3 , k = 0 for R 3 , and k = − 1 for H 3 . In general the warping function is taken a ∈ F ( I ) is a > 0. Here we allow it to be a ≥ 0, including possible singularities. 72 / 142
Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime If S is a connected three-dimensional Riemannian manifold of constant cur- vature k ∈ {− 1 , 0 , 1 } ( i.e. H 3 , R 3 or S 3 ) and a ∈ ( A , B ), −∞ ≤ A < B ≤ ∞ , a ≥ 0, then the warped product I × a S is called a Friedmann-Lemaˆ ıtre- Robertson-Walker spacetime. d s 2 = − d t 2 + a 2 ( t )dΣ 2 (21) d r 2 1 − kr 2 + r 2 � d θ 2 + sin 2 θ d φ 2 � dΣ 2 = , (22) where k = 1 for S 3 , k = 0 for R 3 , and k = − 1 for H 3 . In general the warping function is taken a ∈ F ( I ) is a > 0. Here we allow it to be a ≥ 0, including possible singularities. The resulting singularities are semi-regular. 73 / 142
Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime Distance separation vs. topological separation The old method of resolution of singularities shows how we can “untie” the singularity of a cone and obtain a cylinder. Similarly, it is not necessary to assume that, at the Big Bang singularity, the entire space was a point, but only that the space metric was degenerate. 74 / 142
Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime Friedman equations The stress-energy tensor is T ab = ( ρ + p ) u a u b + pg ab , (23) where u a is the timelike vector field ∂ t , normalized. 75 / 142
Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime Friedman equations The stress-energy tensor is T ab = ( ρ + p ) u a u b + pg ab , (23) where u a is the timelike vector field ∂ t , normalized. The Friedmann equation a 2 + k ρ = 3 ˙ , (24) a 2 κ 76 / 142
Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime Friedman equations The stress-energy tensor is T ab = ( ρ + p ) u a u b + pg ab , (23) where u a is the timelike vector field ∂ t , normalized. The Friedmann equation a 2 + k ρ = 3 ˙ , (24) a 2 κ The acceleration equation ρ + 3 p = − 6 ¨ a a . (25) κ 77 / 142
Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime Friedman equations The stress-energy tensor is T ab = ( ρ + p ) u a u b + pg ab , (23) where u a is the timelike vector field ∂ t , normalized. The Friedmann equation a 2 + k ρ = 3 ˙ , (24) a 2 κ The acceleration equation ρ + 3 p = − 6 ¨ a a . (25) κ The fluid equation , expressing the conservation of mass-energy: ρ = − 3 ˙ a ˙ a ( ρ + p ) . (26) 78 / 142
Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime Friedman equations The stress-energy tensor is T ab = ( ρ + p ) u a u b + pg ab , (23) where u a is the timelike vector field ∂ t , normalized. The Friedmann equation a 2 + k ρ = 3 ˙ , (24) a 2 κ The acceleration equation ρ + 3 p = − 6 ¨ a a . (25) κ The fluid equation , expressing the conservation of mass-energy: ρ = − 3 ˙ a ˙ a ( ρ + p ) . (26) They are singular for a = 0. 79 / 142
Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime Friedman equations, densitized 7 The actual densities contain in fact √− g 7 (Stoica, 2011a) 80 / 142
Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime Friedman equations, densitized 7 The actual densities contain in fact √− g (= a 3 √ g Σ ): 7 (Stoica, 2011a) 81 / 142
Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime Friedman equations, densitized 7 The actual densities contain in fact √− g (= a 3 √ g Σ ): � � ρ = ρ √− g = ρ a 3 √ g Σ p = p √− g = pa 3 √ g Σ (27) � 7 (Stoica, 2011a) 82 / 142
Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime Friedman equations, densitized 7 The actual densities contain in fact √− g (= a 3 √ g Σ ): � � ρ = ρ √− g = ρ a 3 √ g Σ p = p √− g = pa 3 √ g Σ (27) � The Friedmann equation (24) becomes � � √ g Σ , ρ = 3 a 2 + k � κ a ˙ (28) 7 (Stoica, 2011a) 83 / 142
Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime Friedman equations, densitized 7 The actual densities contain in fact √− g (= a 3 √ g Σ ): � � ρ = ρ √− g = ρ a 3 √ g Σ p = p √− g = pa 3 √ g Σ (27) � The Friedmann equation (24) becomes � � √ g Σ , ρ = 3 a 2 + k � κ a ˙ (28) The acceleration equation (25) becomes p = − 6 a √ g Σ , κ a 2 ¨ � ρ + 3 � (29) 7 (Stoica, 2011a) 84 / 142
Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime Friedman equations, densitized 7 The actual densities contain in fact √− g (= a 3 √ g Σ ): � � ρ = ρ √− g = ρ a 3 √ g Σ p = p √− g = pa 3 √ g Σ (27) � The Friedmann equation (24) becomes � � √ g Σ , ρ = 3 a 2 + k � κ a ˙ (28) The acceleration equation (25) becomes p = − 6 a √ g Σ , κ a 2 ¨ � ρ + 3 � (29) Hence, � ρ and � p are smooth 7 (Stoica, 2011a) 85 / 142
Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime Friedman equations, densitized 7 The actual densities contain in fact √− g (= a 3 √ g Σ ): � � ρ = ρ √− g = ρ a 3 √ g Σ p = p √− g = pa 3 √ g Σ (27) � The Friedmann equation (24) becomes � � √ g Σ , ρ = 3 a 2 + k � κ a ˙ (28) The acceleration equation (25) becomes p = − 6 a √ g Σ , κ a 2 ¨ � ρ + 3 � (29) Hence, � ρ and � p are smooth, as it is the densitized stress-energy tensor √− g = ( � T ab ρ + � p ) u a u b + � pg ab . (30) 7 (Stoica, 2011a) 86 / 142
Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime FLRW Big Bang 8 Big Bang singularity, corresponding to a (0) = 0, ˙ a (0) > 0. 8 (Stoica, 2011a) 87 / 142
Friedmann-Lemaˆ ıtre-Robertson-Walker spacetime FLRW Big Bounce 9 Big Bounce, corresponding to a (0) = 0, ˙ a (0) = 0, ¨ a (0) > 0. 9 (Stoica, 2011a) 88 / 142
Black hole singularities Schwarzschild black holes Schwarzschild singularity is semi-regular 10 � � � � − 1 1 − 2 m 1 − 2 m d s 2 = − d t 2 + d r 2 + r 2 d σ 2 , (31) r r where d σ 2 = d θ 2 + sin 2 θ d φ 2 (32) 10 (Stoica, 2012e) 89 / 142
Black hole singularities Schwarzschild black holes Schwarzschild singularity is semi-regular 10 � � � � − 1 1 − 2 m 1 − 2 m d s 2 = − d t 2 + d r 2 + r 2 d σ 2 , (31) r r where d σ 2 = d θ 2 + sin 2 θ d φ 2 (32) Let’s change the coordinates to � r = τ 2 (33) = ξτ 4 t 10 (Stoica, 2012e) 90 / 142
Black hole singularities Schwarzschild black holes Schwarzschild singularity is semi-regular 10 � � � � − 1 1 − 2 m 1 − 2 m d s 2 = − d t 2 + d r 2 + r 2 d σ 2 , (31) r r where d σ 2 = d θ 2 + sin 2 θ d φ 2 (32) Let’s change the coordinates to � r = τ 2 (33) = ξτ 4 t The four-metric becomes: 4 τ 4 d s 2 = − 2 m − τ 2 d τ 2 + (2 m − τ 2 ) τ 4 (4 ξ d τ + τ d ξ ) 2 + τ 4 d σ 2 (34) which is analytic and semi-regular at r = 0. 10 (Stoica, 2012e) 91 / 142
Black hole singularities Schwarzschild black holes Evaporating Schwarzschild black hole and information 11 A. Standard evaporating black hole, whose singularity destroys the information. B. Evaporating black hole extended through the singularity preserves information. 11 (Stoica, 2012e) 92 / 142
Black hole singularities Reissner-Nordstr¨ om black holes om singularity is analytic 12 Reissner-Nordstr¨ � � � � − 1 + q 2 + q 2 1 − 2 m 1 − 2 m d s 2 = − d t 2 + d r 2 + r 2 d σ 2 , (35) r 2 r 2 r r 12 (Stoica, 2012a) 93 / 142
Black hole singularities Reissner-Nordstr¨ om black holes om singularity is analytic 12 Reissner-Nordstr¨ � � � � − 1 + q 2 + q 2 1 − 2 m 1 − 2 m d s 2 = − d t 2 + d r 2 + r 2 d σ 2 , (35) r 2 r 2 r r � t = τρ T We choose the coordinates ρ and τ , so that = ρ S r 12 (Stoica, 2012a) 94 / 142
Black hole singularities Reissner-Nordstr¨ om black holes om singularity is analytic 12 Reissner-Nordstr¨ � � � � − 1 + q 2 + q 2 1 − 2 m 1 − 2 m d s 2 = − d t 2 + d r 2 + r 2 d σ 2 , (35) r 2 r 2 r r � t = τρ T We choose the coordinates ρ and τ , so that = ρ S r The metric has, in the new coordinates, the following form d s 2 = − ∆ ρ 2 T − 2 S − 2 ( ρ d τ + T τ d ρ ) 2 + S 2 ∆ ρ 4 S − 2 d ρ 2 + ρ 2 S d σ 2 , (36) where ∆ := ρ 2 S − 2 m ρ S + q 2 . (37) 12 (Stoica, 2012a) 95 / 142
Black hole singularities Reissner-Nordstr¨ om black holes om singularity is analytic 12 Reissner-Nordstr¨ � � � � − 1 + q 2 + q 2 1 − 2 m 1 − 2 m d s 2 = − d t 2 + d r 2 + r 2 d σ 2 , (35) r 2 r 2 r r � t = τρ T We choose the coordinates ρ and τ , so that = ρ S r The metric has, in the new coordinates, the following form d s 2 = − ∆ ρ 2 T − 2 S − 2 ( ρ d τ + T τ d ρ ) 2 + S 2 ∆ ρ 4 S − 2 d ρ 2 + ρ 2 S d σ 2 , (36) where ∆ := ρ 2 S − 2 m ρ S + q 2 . (37) � S ≥ 1 To remove the infinity of the metric at r = 0, take T ≥ S + 1 which also ensure that the metric is analytic at r = 0. 12 (Stoica, 2012a) 96 / 142
Black hole singularities Reissner-Nordstr¨ om black holes Non-singular electromagnetic field 13 The electromagnetic potential in the coordinates ( t , r , φ, θ ) is singular at r = 0: A = − q r d t , (38) 13 (Stoica, 2012a) 97 / 142
Black hole singularities Reissner-Nordstr¨ om black holes Non-singular electromagnetic field 13 The electromagnetic potential in the coordinates ( t , r , φ, θ ) is singular at r = 0: A = − q r d t , (38) In the new coordinates ( τ, ρ, φ, θ ), the electromagnetic potential is A = − q ρ T − S − 1 ( ρ d τ + T τ d ρ ) , (39) 13 (Stoica, 2012a) 98 / 142
Black hole singularities Reissner-Nordstr¨ om black holes Non-singular electromagnetic field 13 The electromagnetic potential in the coordinates ( t , r , φ, θ ) is singular at r = 0: A = − q r d t , (38) In the new coordinates ( τ, ρ, φ, θ ), the electromagnetic potential is A = − q ρ T − S − 1 ( ρ d τ + T τ d ρ ) , (39) the electromagnetic field is F = q (2 T − S ) ρ T − S − 1 d τ ∧ d ρ, (40) 13 (Stoica, 2012a) 99 / 142
Black hole singularities Reissner-Nordstr¨ om black holes Non-singular electromagnetic field 13 The electromagnetic potential in the coordinates ( t , r , φ, θ ) is singular at r = 0: A = − q r d t , (38) In the new coordinates ( τ, ρ, φ, θ ), the electromagnetic potential is A = − q ρ T − S − 1 ( ρ d τ + T τ d ρ ) , (39) the electromagnetic field is F = q (2 T − S ) ρ T − S − 1 d τ ∧ d ρ, (40) and they are analytic everywhere, including at the singularity ρ = 0. 13 (Stoica, 2012a) 100 / 142
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