(The Singularity Theorems of) Lorentzian geometry Melanie Graf University of Vienna 19.10.2016 Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 1 / 17
Outline An introduction to Lorentzian geometry 1 Elementary Lorentzian geometry Spacetimes Examples Singularities 2 Examples of singular spacetimes The singularity theorems My thesis project Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 2 / 17
Lorentzian vector spaces Definition (Lorentz vector space.) A Lorentz vector space is a vector space V (real, 2 ≤ dim ( V ) < ∞ ) with a scalar product g : V × V → R of index 1, i.e., there exists a one dimensional subspace W ⊂ V such that g | W × W is negative definite, but g | Z × Z is not negative definite for any subspace Z with dim ( Z ) ≥ 2. Example V = R n , g ( v , w ) = − v 1 w 1 + � n i = 2 v i w i Fact To each scalar product one can associate a symmetric invertible Matrix ( g ij ) n i , j = 1 such that g ( v , w ) = g ij v i w j . The scalar product is Lorentzian if and only if ( g ij ) n i , j = 1 has exactly one negative and n − 1 positive eigenvalues. Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 3 / 17
Lorentzian vector spaces Definition (Lorentz vector space.) A Lorentz vector space is a vector space V (real, 2 ≤ dim ( V ) < ∞ ) with a scalar product g : V × V → R of index 1, i.e., there exists a one dimensional subspace W ⊂ V such that g | W × W is negative definite, but g | Z × Z is not negative definite for any subspace Z with dim ( Z ) ≥ 2. Example V = R n , g ( v , w ) = − v 1 w 1 + � n i = 2 v i w i Fact To each scalar product one can associate a symmetric invertible Matrix ( g ij ) n i , j = 1 such that g ( v , w ) = g ij v i w j . The scalar product is Lorentzian if and only if ( g ij ) n i , j = 1 has exactly one negative and n − 1 positive eigenvalues. Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 3 / 17
Lorentzian vector spaces Definition (Lorentz vector space.) A Lorentz vector space is a vector space V (real, 2 ≤ dim ( V ) < ∞ ) with a scalar product g : V × V → R of index 1, i.e., there exists a one dimensional subspace W ⊂ V such that g | W × W is negative definite, but g | Z × Z is not negative definite for any subspace Z with dim ( Z ) ≥ 2. Example V = R n , g ( v , w ) = − v 1 w 1 + � n i = 2 v i w i Fact To each scalar product one can associate a symmetric invertible Matrix ( g ij ) n i , j = 1 such that g ( v , w ) = g ij v i w j . The scalar product is Lorentzian if and only if ( g ij ) n i , j = 1 has exactly one negative and n − 1 positive eigenvalues. Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 3 / 17
Classification of vectors A vector v ∈ V is called spacelike if g ( v , v ) > 0 or v = 0 causal if g ( v , v ) ≤ 0, null if g ( v , v ) = 0, timelike if g ( v , v ) < 0. Orthonormal Basis. Given any scalar product, there exists an othonormal basis { e 1 , . . . , e n } . Use this basis to identify V = R n . In this basis a Lorentzian metric has the form − 1 0 0 0 0 1 0 0 g = ... 0 0 0 0 0 0 1 ⇒ | v 1 | ≥ | ( 0 , v 2 , . . . , v n ) | e v causal ⇐ Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 4 / 17
Classification of vectors A vector v ∈ V is called spacelike if g ( v , v ) > 0 or v = 0 causal if g ( v , v ) ≤ 0, null if g ( v , v ) = 0, timelike if g ( v , v ) < 0. Orthonormal Basis. Given any scalar product, there exists an othonormal basis { e 1 , . . . , e n } . Use this basis to identify V = R n . In this basis a Lorentzian metric has the form − 1 0 0 0 0 1 0 0 g = ... 0 0 0 0 0 0 1 ⇒ | v 1 | ≥ | ( 0 , v 2 , . . . , v n ) | e v causal ⇐ Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 4 / 17
Classification of vectors A vector v ∈ V is called spacelike if g ( v , v ) > 0 or v = 0 causal if g ( v , v ) ≤ 0, null if g ( v , v ) = 0, timelike if g ( v , v ) < 0. Orthonormal Basis. Given any scalar product, there exists an othonormal basis { e 1 , . . . , e n } . Use this basis to identify V = R n . In this basis a Lorentzian metric has the form − 1 0 0 0 0 1 0 0 g = ... 0 0 0 0 0 0 1 ⇒ | v 1 | ≥ | ( 0 , v 2 , . . . , v n ) | e v causal ⇐ Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 4 / 17
The reverse triangle inequality For timelike vectors one can derive substitutes for some of the tools of inner product spaces, e.g., a concept of length, a reverse Cachy-Schwarz inequality and a reverse triangle inequality. Definition (Length of causal vectors.) The (Lorentzian) length of a causal vector v is given by � | v | g := − g ( v , v ) . Proposition (Reverse Cauchy-Schwarz and triangle inequality.) Let v , w ∈ V be timelike. Then | g ( v , w ) | ≥ | v | g | w | g with equality if and only if x and y are collinear. If furthermore g ( v , w ) < 0, then | v | g + | w | g ≤ | v + w | g . Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 5 / 17
The reverse triangle inequality For timelike vectors one can derive substitutes for some of the tools of inner product spaces, e.g., a concept of length, a reverse Cachy-Schwarz inequality and a reverse triangle inequality. Definition (Length of causal vectors.) The (Lorentzian) length of a causal vector v is given by � | v | g := − g ( v , v ) . Proposition (Reverse Cauchy-Schwarz and triangle inequality.) Let v , w ∈ V be timelike. Then | g ( v , w ) | ≥ | v | g | w | g with equality if and only if x and y are collinear. If furthermore g ( v , w ) < 0, then | v | g + | w | g ≤ | v + w | g . Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 5 / 17
The reverse triangle inequality For timelike vectors one can derive substitutes for some of the tools of inner product spaces, e.g., a concept of length, a reverse Cachy-Schwarz inequality and a reverse triangle inequality. Definition (Length of causal vectors.) The (Lorentzian) length of a causal vector v is given by � | v | g := − g ( v , v ) . Proposition (Reverse Cauchy-Schwarz and triangle inequality.) Let v , w ∈ V be timelike. Then | g ( v , w ) | ≥ | v | g | w | g with equality if and only if x and y are collinear. If furthermore g ( v , w ) < 0, then | v | g + | w | g ≤ | v + w | g . Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 5 / 17
Lorentzian manifolds (Smooth) Manifold: (Nice) topological space M , such that each p ∈ M has a neighborhood U that is homeomorphic to a subset of R n via a chart ψ . Transition functions between charts have to be diffeomorphisms! E.g., R n , (2-dimensional) surfaces in R 3 , n -dimensional spheres,... Tangent space: To each p ∈ M one associates an n -dimensional vector space T p M , the tangent space. E.g., the usual tangent space for surfaces in R 3 or S n ⊂ R n + 1 . Lorentzian manifold: Each T p M is a Lorentz vector space with Lorentzian scalar product g p . Usually: g : p �→ g p smooth, C 2 . Low regularity: g only C 1 , 1 , C 0 ,α , C E.g., for M = R n : smooth matrix valued map g : R n → R n × n Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 6 / 17
Lorentzian manifolds (Smooth) Manifold: (Nice) topological space M , such that each p ∈ M has a neighborhood U that is homeomorphic to a subset of R n via a chart ψ . Transition functions between charts have to be diffeomorphisms! E.g., R n , (2-dimensional) surfaces in R 3 , n -dimensional spheres,... Tangent space: To each p ∈ M one associates an n -dimensional vector space T p M , the tangent space. E.g., the usual tangent space for surfaces in R 3 or S n ⊂ R n + 1 . Lorentzian manifold: Each T p M is a Lorentz vector space with Lorentzian scalar product g p . Usually: g : p �→ g p smooth, C 2 . Low regularity: g only C 1 , 1 , C 0 ,α , C E.g., for M = R n : smooth matrix valued map g : R n → R n × n Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 6 / 17
Lorentzian manifolds (Smooth) Manifold: (Nice) topological space M , such that each p ∈ M has a neighborhood U that is homeomorphic to a subset of R n via a chart ψ . Transition functions between charts have to be diffeomorphisms! E.g., R n , (2-dimensional) surfaces in R 3 , n -dimensional spheres,... Tangent space: To each p ∈ M one associates an n -dimensional vector space T p M , the tangent space. E.g., the usual tangent space for surfaces in R 3 or S n ⊂ R n + 1 . Lorentzian manifold: Each T p M is a Lorentz vector space with Lorentzian scalar product g p . Usually: g : p �→ g p smooth, C 2 . Low regularity: g only C 1 , 1 , C 0 ,α , C E.g., for M = R n : smooth matrix valued map g : R n → R n × n Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 6 / 17
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