(The Singularity Theorems of) Lorentzian geometry Melanie Graf - - PowerPoint PPT Presentation

(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

1

An introduction to Lorentzian geometry Elementary Lorentzian geometry Spacetimes Examples

2

Singularities 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 = Rn, g(v, w) = −v1w1 + n

i=2 viwi

Fact

To each scalar product one can associate a symmetric invertible Matrix (gij)n

i,j=1 such that

g(v, w) = gijviwj. The scalar product is Lorentzian if and only if (gij)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 = Rn, g(v, w) = −v1w1 + n

i=2 viwi

Fact

To each scalar product one can associate a symmetric invertible Matrix (gij)n

i,j=1 such that

g(v, w) = gijviwj. The scalar product is Lorentzian if and only if (gij)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 = Rn, g(v, w) = −v1w1 + n

i=2 viwi

Fact

To each scalar product one can associate a symmetric invertible Matrix (gij)n

i,j=1 such that

g(v, w) = gijviwj. The scalar product is Lorentzian if and only if (gij)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 {e1, . . . , en}. Use this basis to identify V = Rn. In this basis a Lorentzian metric has the form g =

     

−1 1 ... 1

     

v causal ⇐ ⇒ |v1| ≥ |(0, v2, . . . , vn)|e

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 {e1, . . . , en}. Use this basis to identify V = Rn. In this basis a Lorentzian metric has the form g =

     

−1 1 ... 1

     

v causal ⇐ ⇒ |v1| ≥ |(0, v2, . . . , vn)|e

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 {e1, . . . , en}. Use this basis to identify V = Rn. In this basis a Lorentzian metric has the form g =

     

−1 1 ... 1

     

v causal ⇐ ⇒ |v1| ≥ |(0, v2, . . . , vn)|e

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 Rn via a chart ψ. Transition functions between charts have to be diffeomorphisms!

E.g., Rn, (2-dimensional) surfaces in R3, n-dimensional spheres,...

Tangent space: To each p ∈ M one associates an n-dimensional vector space TpM, the tangent space.

E.g., the usual tangent space for surfaces in R3 or Sn ⊂ Rn+1.

Lorentzian manifold: Each TpM is a Lorentz vector space with Lorentzian scalar product gp. Usually: g : p → gp smooth, C2. Low regularity: g only C1,1, C0,α, C

E.g., for M = Rn: smooth matrix valued map g : Rn → Rn×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 Rn via a chart ψ. Transition functions between charts have to be diffeomorphisms!

E.g., Rn, (2-dimensional) surfaces in R3, n-dimensional spheres,...

Tangent space: To each p ∈ M one associates an n-dimensional vector space TpM, the tangent space.

E.g., the usual tangent space for surfaces in R3 or Sn ⊂ Rn+1.

Lorentzian manifold: Each TpM is a Lorentz vector space with Lorentzian scalar product gp. Usually: g : p → gp smooth, C2. Low regularity: g only C1,1, C0,α, C

E.g., for M = Rn: smooth matrix valued map g : Rn → Rn×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 Rn via a chart ψ. Transition functions between charts have to be diffeomorphisms!

E.g., Rn, (2-dimensional) surfaces in R3, n-dimensional spheres,...

Tangent space: To each p ∈ M one associates an n-dimensional vector space TpM, the tangent space.

E.g., the usual tangent space for surfaces in R3 or Sn ⊂ Rn+1.

Lorentzian manifold: Each TpM is a Lorentz vector space with Lorentzian scalar product gp. Usually: g : p → gp smooth, C2. Low regularity: g only C1,1, C0,α, C

E.g., for M = Rn: smooth matrix valued map g : Rn → Rn×n

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 6 / 17

Time orientation

In a Lorentz vector space the set of all timelike vectors has two connected components: Fixing any timelike vector u, they can be described by C+ = {v : g(u, v) < 0} and C− = {v : g(u, v) > 0} On manifolds: Smoothly assign each p ∈ M a future light cone C+(p) ⊂ TpM. A timelike vector v ∈ TpM is called future pointing iff v ∈ C+(p) Equivalently: Choose a smooth vector field X ∈ X(M) with g(X, X) < 0. A timelike vector v ∈ TpM is future pointing iff gp(X(p), v) < 0.

Definition (Spacetime.)

A spacetime (M, g) is a Lorentzian manifold with a time orientation.

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 7 / 17

Time orientation

In a Lorentz vector space the set of all timelike vectors has two connected components: Fixing any timelike vector u, they can be described by C+ = {v : g(u, v) < 0} and C− = {v : g(u, v) > 0} On manifolds: Smoothly assign each p ∈ M a future light cone C+(p) ⊂ TpM. A timelike vector v ∈ TpM is called future pointing iff v ∈ C+(p) Equivalently: Choose a smooth vector field X ∈ X(M) with g(X, X) < 0. A timelike vector v ∈ TpM is future pointing iff gp(X(p), v) < 0.

Definition (Spacetime.)

A spacetime (M, g) is a Lorentzian manifold with a time orientation.

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 7 / 17

Time orientation

In a Lorentz vector space the set of all timelike vectors has two connected components: Fixing any timelike vector u, they can be described by C+ = {v : g(u, v) < 0} and C− = {v : g(u, v) > 0} On manifolds: Smoothly assign each p ∈ M a future light cone C+(p) ⊂ TpM. A timelike vector v ∈ TpM is called future pointing iff v ∈ C+(p) Equivalently: Choose a smooth vector field X ∈ X(M) with g(X, X) < 0. A timelike vector v ∈ TpM is future pointing iff gp(X(p), v) < 0.

Definition (Spacetime.)

A spacetime (M, g) is a Lorentzian manifold with a time orientation.

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 7 / 17

Time orientation

In a Lorentz vector space the set of all timelike vectors has two connected components: Fixing any timelike vector u, they can be described by C+ = {v : g(u, v) < 0} and C− = {v : g(u, v) > 0} On manifolds: Smoothly assign each p ∈ M a future light cone C+(p) ⊂ TpM. A timelike vector v ∈ TpM is called future pointing iff v ∈ C+(p) Equivalently: Choose a smooth vector field X ∈ X(M) with g(X, X) < 0. A timelike vector v ∈ TpM is future pointing iff gp(X(p), v) < 0.

Definition (Spacetime.)

A spacetime (M, g) is a Lorentzian manifold with a time orientation.

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 7 / 17

Causal curves

Definition (Causal curve.)

A (smooth) curve γ : I → M is called causal (timelike, null, future directed, ...) if ˙ γ(t) ∈ Tγ(t)M is causal (timelike, ...) for all t ∈ M. Given p, q ∈ M we write

p ≤ q if there exists a future directed causal curve from p to q (or p = q) p ≪ q if there exists a future directed timelike curve from p to q

Define J+(p) := {q ∈ M : p ≤ q} and I+(p) := {q ∈ M : p ≪ q} “Push-up” principle: If p ≤ q and q ≪ r, then p ≪ r (Note: No longer true for metrics of to little regularity!)

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 8 / 17

Causal curves

Definition (Causal curve.)

A (smooth) curve γ : I → M is called causal (timelike, null, future directed, ...) if ˙ γ(t) ∈ Tγ(t)M is causal (timelike, ...) for all t ∈ M. Given p, q ∈ M we write

p ≤ q if there exists a future directed causal curve from p to q (or p = q) p ≪ q if there exists a future directed timelike curve from p to q

Define J+(p) := {q ∈ M : p ≤ q} and I+(p) := {q ∈ M : p ≪ q} “Push-up” principle: If p ≤ q and q ≪ r, then p ≪ r (Note: No longer true for metrics of to little regularity!)

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 8 / 17

Causal curves

Definition (Causal curve.)

A (smooth) curve γ : I → M is called causal (timelike, null, future directed, ...) if ˙ γ(t) ∈ Tγ(t)M is causal (timelike, ...) for all t ∈ M. Given p, q ∈ M we write

p ≤ q if there exists a future directed causal curve from p to q (or p = q) p ≪ q if there exists a future directed timelike curve from p to q

Define J+(p) := {q ∈ M : p ≤ q} and I+(p) := {q ∈ M : p ≪ q} “Push-up” principle: If p ≤ q and q ≪ r, then p ≪ r (Note: No longer true for metrics of to little regularity!)

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 8 / 17

Causal curves

Definition (Causal curve.)

A (smooth) curve γ : I → M is called causal (timelike, null, future directed, ...) if ˙ γ(t) ∈ Tγ(t)M is causal (timelike, ...) for all t ∈ M. Given p, q ∈ M we write

p ≤ q if there exists a future directed causal curve from p to q (or p = q) p ≪ q if there exists a future directed timelike curve from p to q

Define J+(p) := {q ∈ M : p ≤ q} and I+(p) := {q ∈ M : p ≪ q} “Push-up” principle: If p ≤ q and q ≪ r, then p ≪ r (Note: No longer true for metrics of to little regularity!)

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 8 / 17

Simple examples

Minkowski space: M = Rn, gp = diag(−1, 1, . . . , 1). Physically: Special relativity Minkowski space without a point: Geodesically incomplete, causal diamonds J+(p) ∩ J−(q) can be non-compact, J+(p) not necessarily closed Lorentz cylinder: M = R × S1, g = −dt2 + ds2 (where ds2 is the standard metric on S1), boundary of I+(p) is compact A different cylinder: M = S1 × R, g = −ds2 + dx2, closed timelike curves, I+(p) = M The cylinder from before with two rays removed: no closed timelike curves, but “almost” closed timelike curves ... and many more

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 9 / 17

Simple examples

Minkowski space: M = Rn, gp = diag(−1, 1, . . . , 1). Physically: Special relativity Minkowski space without a point: Geodesically incomplete, causal diamonds J+(p) ∩ J−(q) can be non-compact, J+(p) not necessarily closed Lorentz cylinder: M = R × S1, g = −dt2 + ds2 (where ds2 is the standard metric on S1), boundary of I+(p) is compact A different cylinder: M = S1 × R, g = −ds2 + dx2, closed timelike curves, I+(p) = M The cylinder from before with two rays removed: no closed timelike curves, but “almost” closed timelike curves ... and many more

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 9 / 17

Simple examples

Minkowski space: M = Rn, gp = diag(−1, 1, . . . , 1). Physically: Special relativity Minkowski space without a point: Geodesically incomplete, causal diamonds J+(p) ∩ J−(q) can be non-compact, J+(p) not necessarily closed Lorentz cylinder: M = R × S1, g = −dt2 + ds2 (where ds2 is the standard metric on S1), boundary of I+(p) is compact A different cylinder: M = S1 × R, g = −ds2 + dx2, closed timelike curves, I+(p) = M The cylinder from before with two rays removed: no closed timelike curves, but “almost” closed timelike curves ... and many more

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 9 / 17

Simple examples

Minkowski space: M = Rn, gp = diag(−1, 1, . . . , 1). Physically: Special relativity Minkowski space without a point: Geodesically incomplete, causal diamonds J+(p) ∩ J−(q) can be non-compact, J+(p) not necessarily closed Lorentz cylinder: M = R × S1, g = −dt2 + ds2 (where ds2 is the standard metric on S1), boundary of I+(p) is compact A different cylinder: M = S1 × R, g = −ds2 + dx2, closed timelike curves, I+(p) = M The cylinder from before with two rays removed: no closed timelike curves, but “almost” closed timelike curves ... and many more

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 9 / 17

Simple examples

Minkowski space: M = Rn, gp = diag(−1, 1, . . . , 1). Physically: Special relativity Minkowski space without a point: Geodesically incomplete, causal diamonds J+(p) ∩ J−(q) can be non-compact, J+(p) not necessarily closed Lorentz cylinder: M = R × S1, g = −dt2 + ds2 (where ds2 is the standard metric on S1), boundary of I+(p) is compact A different cylinder: M = S1 × R, g = −ds2 + dx2, closed timelike curves, I+(p) = M The cylinder from before with two rays removed: no closed timelike curves, but “almost” closed timelike curves ... and many more

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 9 / 17

Simple examples

Minkowski space: M = Rn, gp = diag(−1, 1, . . . , 1). Physically: Special relativity Minkowski space without a point: Geodesically incomplete, causal diamonds J+(p) ∩ J−(q) can be non-compact, J+(p) not necessarily closed Lorentz cylinder: M = R × S1, g = −dt2 + ds2 (where ds2 is the standard metric on S1), boundary of I+(p) is compact A different cylinder: M = S1 × R, g = −ds2 + dx2, closed timelike curves, I+(p) = M The cylinder from before with two rays removed: no closed timelike curves, but “almost” closed timelike curves ... and many more

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 9 / 17

Geodesics

Define the length of a causal curve γ : (a, b) → M by L(γ) = ˆ b

a

• −g(˙

γ(τ), ˙ γ(τ))dτ Look for curves of maximal length = ⇒ geodesic equation ∇ ˙

γ ˙

γ = 0

• r

¨ γm + Γm

ij ˙

γi ˙ γj = 0 Geodesic = solution of the geodesic equation (unique for any given γ(0) ∈ M, ˙ γ(0) ∈ Tγ(0)M, denote maximal intervall of existence by Iγ) Morally: causal geodesics in a Lorentzian manifold — geodesics in Riemannian manifolds

Causal geodesics are always length maximizing for short times but they stop being maximizing after encountering a conjugate point

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 10 / 17

Geodesics

Define the length of a causal curve γ : (a, b) → M by L(γ) = ˆ b

a

• −g(˙

γ(τ), ˙ γ(τ))dτ Look for curves of maximal length = ⇒ geodesic equation ∇ ˙

γ ˙

γ = 0

• r

¨ γm + Γm

ij ˙

γi ˙ γj = 0 Geodesic = solution of the geodesic equation (unique for any given γ(0) ∈ M, ˙ γ(0) ∈ Tγ(0)M, denote maximal intervall of existence by Iγ) Morally: causal geodesics in a Lorentzian manifold — geodesics in Riemannian manifolds

Causal geodesics are always length maximizing for short times but they stop being maximizing after encountering a conjugate point

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 10 / 17

Geodesics

Define the length of a causal curve γ : (a, b) → M by L(γ) = ˆ b

a

• −g(˙

γ(τ), ˙ γ(τ))dτ Look for curves of maximal length = ⇒ geodesic equation ∇ ˙

γ ˙

γ = 0

• r

¨ γm + Γm

ij ˙

γi ˙ γj = 0 Geodesic = solution of the geodesic equation (unique for any given γ(0) ∈ M, ˙ γ(0) ∈ Tγ(0)M, denote maximal intervall of existence by Iγ) Morally: causal geodesics in a Lorentzian manifold — geodesics in Riemannian manifolds

Causal geodesics are always length maximizing for short times but they stop being maximizing after encountering a conjugate point

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 10 / 17

Geodesics

Define the length of a causal curve γ : (a, b) → M by L(γ) = ˆ b

a

• −g(˙

γ(τ), ˙ γ(τ))dτ Look for curves of maximal length = ⇒ geodesic equation ∇ ˙

γ ˙

γ = 0

• r

¨ γm + Γm

ij ˙

γi ˙ γj = 0 Geodesic = solution of the geodesic equation (unique for any given γ(0) ∈ M, ˙ γ(0) ∈ Tγ(0)M, denote maximal intervall of existence by Iγ) Morally: causal geodesics in a Lorentzian manifold — geodesics in Riemannian manifolds

Causal geodesics are always length maximizing for short times but they stop being maximizing after encountering a conjugate point

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 10 / 17

Robertson-Walker spacetimes

Let S be H3, R3, or S3 and let f : I → R+ be a positive function defined on an open interval. Set M(S, f ) := I ×f S. The line element of M(S, f ) is −dt2 + f 2(t)dσ2, where dσ2 is the standard Riemannian line element for S. Relativistic model of the flow of a perfect fluid Big bang and big crunch: Given t∗ ∈ R the spacetime has a big bang (resp. big crunch) at t∗ if f → 0 and f ′ → ∞ (resp. f ′ → −∞) as t → t∗.

In this case the energy density diverges as t → t∗ = ⇒ physical singularities.

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 11 / 17

Robertson-Walker spacetimes

Let S be H3, R3, or S3 and let f : I → R+ be a positive function defined on an open interval. Set M(S, f ) := I ×f S. The line element of M(S, f ) is −dt2 + f 2(t)dσ2, where dσ2 is the standard Riemannian line element for S. Relativistic model of the flow of a perfect fluid Big bang and big crunch: Given t∗ ∈ R the spacetime has a big bang (resp. big crunch) at t∗ if f → 0 and f ′ → ∞ (resp. f ′ → −∞) as t → t∗.

In this case the energy density diverges as t → t∗ = ⇒ physical singularities.

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 11 / 17

Robertson-Walker spacetimes

Let S be H3, R3, or S3 and let f : I → R+ be a positive function defined on an open interval. Set M(S, f ) := I ×f S. The line element of M(S, f ) is −dt2 + f 2(t)dσ2, where dσ2 is the standard Riemannian line element for S. Relativistic model of the flow of a perfect fluid Big bang and big crunch: Given t∗ ∈ R the spacetime has a big bang (resp. big crunch) at t∗ if f → 0 and f ′ → ∞ (resp. f ′ → −∞) as t → t∗.

In this case the energy density diverges as t → t∗ = ⇒ physical singularities.

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 11 / 17

Robertson-Walker spacetimes

Let S be H3, R3, or S3 and let f : I → R+ be a positive function defined on an open interval. Set M(S, f ) := I ×f S. The line element of M(S, f ) is −dt2 + f 2(t)dσ2, where dσ2 is the standard Riemannian line element for S. Relativistic model of the flow of a perfect fluid Big bang and big crunch: Given t∗ ∈ R the spacetime has a big bang (resp. big crunch) at t∗ if f → 0 and f ′ → ∞ (resp. f ′ → −∞) as t → t∗.

In this case the energy density diverges as t → t∗ = ⇒ physical singularities.

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 11 / 17

The Schwarzschild metric

Exterior and interior region:

Mext = R × (2m, ∞) × S2, gext = −

• 1 − 2m

r

• dt2 +
• 1 − 2m

r

−1 dr 2 + r 2dΩ2 Mint = R × (0, 2m) × S2 ⊂ R4, gint = −

• 1 − 2m

r

• dt2 +
• 1 − 2m

r

−1 dr 2 + r 2dΩ2

Models a (static) black hole (of mass m sitting at r = 0) Vacuum spacetime (Ricci flat) In the exterior: ∂t is timelike and future pointing, ∂r spacelike In the interior: ∂r is timelike and past pointing, ∂t spacelike gext → gMinkowski for r → ∞

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 12 / 17

The Schwarzschild metric

Exterior and interior region:

Mext = R × (2m, ∞) × S2, gext = −

• 1 − 2m

r

• dt2 +
• 1 − 2m

r

−1 dr 2 + r 2dΩ2 Mint = R × (0, 2m) × S2 ⊂ R4, gint = −

• 1 − 2m

r

• dt2 +
• 1 − 2m

r

−1 dr 2 + r 2dΩ2

Models a (static) black hole (of mass m sitting at r = 0) Vacuum spacetime (Ricci flat) In the exterior: ∂t is timelike and future pointing, ∂r spacelike In the interior: ∂r is timelike and past pointing, ∂t spacelike gext → gMinkowski for r → ∞

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 12 / 17

The Schwarzschild metric

Exterior and interior region:

Mext = R × (2m, ∞) × S2, gext = −

• 1 − 2m

r

• dt2 +
• 1 − 2m

r

−1 dr 2 + r 2dΩ2 Mint = R × (0, 2m) × S2 ⊂ R4, gint = −

• 1 − 2m

r

• dt2 +
• 1 − 2m

r

−1 dr 2 + r 2dΩ2

Models a (static) black hole (of mass m sitting at r = 0) Vacuum spacetime (Ricci flat) In the exterior: ∂t is timelike and future pointing, ∂r spacelike In the interior: ∂r is timelike and past pointing, ∂t spacelike gext → gMinkowski for r → ∞

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 12 / 17

The Schwarzschild metric

Exterior and interior region:

Mext = R × (2m, ∞) × S2, gext = −

• 1 − 2m

r

• dt2 +
• 1 − 2m

r

−1 dr 2 + r 2dΩ2 Mint = R × (0, 2m) × S2 ⊂ R4, gint = −

• 1 − 2m

r

• dt2 +
• 1 − 2m

r

−1 dr 2 + r 2dΩ2

Models a (static) black hole (of mass m sitting at r = 0) Vacuum spacetime (Ricci flat) In the exterior: ∂t is timelike and future pointing, ∂r spacelike In the interior: ∂r is timelike and past pointing, ∂t spacelike gext → gMinkowski for r → ∞

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 12 / 17

The Schwarzschild metric

Exterior and interior region:

Mext = R × (2m, ∞) × S2, gext = −

• 1 − 2m

r

• dt2 +
• 1 − 2m

r

−1 dr 2 + r 2dΩ2 Mint = R × (0, 2m) × S2 ⊂ R4, gint = −

• 1 − 2m

r

• dt2 +
• 1 − 2m

r

−1 dr 2 + r 2dΩ2

Models a (static) black hole (of mass m sitting at r = 0) Vacuum spacetime (Ricci flat) In the exterior: ∂t is timelike and future pointing, ∂r spacelike In the interior: ∂r is timelike and past pointing, ∂t spacelike gext → gMinkowski for r → ∞

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 12 / 17

The Schwarzschild metric

Exterior and interior region:

Mext = R × (2m, ∞) × S2, gext = −

• 1 − 2m

r

• dt2 +
• 1 − 2m

r

−1 dr 2 + r 2dΩ2 Mint = R × (0, 2m) × S2 ⊂ R4, gint = −

• 1 − 2m

r

• dt2 +
• 1 − 2m

r

−1 dr 2 + r 2dΩ2

Models a (static) black hole (of mass m sitting at r = 0) Vacuum spacetime (Ricci flat) In the exterior: ∂t is timelike and future pointing, ∂r spacelike In the interior: ∂r is timelike and past pointing, ∂t spacelike gext → gMinkowski for r → ∞

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 12 / 17

The Schwarzschild metric

Exterior and interior region:

Mext = R × (2m, ∞) × S2, gext = −

• 1 − 2m

r

• dt2 +
• 1 − 2m

r

−1 dr 2 + r 2dΩ2 Mint = R × (0, 2m) × S2 ⊂ R4, gint = −

• 1 − 2m

r

• dt2 +
• 1 − 2m

r

−1 dr 2 + r 2dΩ2

Models a (static) black hole (of mass m sitting at r = 0) Vacuum spacetime (Ricci flat) In the exterior: ∂t is timelike and future pointing, ∂r spacelike In the interior: ∂r is timelike and past pointing, ∂t spacelike gext → gMinkowski for r → ∞

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 12 / 17

The Schwarzschild metric

Exterior and interior region:

Mext = R × (2m, ∞) × S2, gext = −

• 1 − 2m

r

• dt2 +
• 1 − 2m

r

−1 dr 2 + r 2dΩ2 Mint = R × (0, 2m) × S2 ⊂ R4, gint = −

• 1 − 2m

r

• dt2 +
• 1 − 2m

r

−1 dr 2 + r 2dΩ2

Models a (static) black hole (of mass m sitting at r = 0) Vacuum spacetime (Ricci flat) In the exterior: ∂t is timelike and future pointing, ∂r spacelike In the interior: ∂r is timelike and past pointing, ∂t spacelike gext → gMinkowski for r → ∞

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 12 / 17

The Schwarzschild metric, cont.

What happens at r = 2m? gint and gext become singular, but (t, r, ω) are not the only coordinates for Mint, Mext! The maximal Schwarzschild spacetime Mmax is a smooth extension of Mint, Mmax to include r = 2m. In new coordinates u, v (Kruskal coordinates): gmax = −16m3 r e− r

2m (du ⊗ dv + dv ⊗ du) + r 2dΩ2

∂u and ∂v are both future pointing null vectors What happens at r = 0? Physical singularity Curvature diverges as r → 0 There exists no continuous extension of Mmax (no C2 extension because of curvature, no continuous extension by Sbierski, 2015).

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 13 / 17

The Schwarzschild metric, cont.

What happens at r = 2m? gint and gext become singular, but (t, r, ω) are not the only coordinates for Mint, Mext! The maximal Schwarzschild spacetime Mmax is a smooth extension of Mint, Mmax to include r = 2m. In new coordinates u, v (Kruskal coordinates): gmax = −16m3 r e− r

2m (du ⊗ dv + dv ⊗ du) + r 2dΩ2

∂u and ∂v are both future pointing null vectors What happens at r = 0? Physical singularity Curvature diverges as r → 0 There exists no continuous extension of Mmax (no C2 extension because of curvature, no continuous extension by Sbierski, 2015).

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 13 / 17

The Schwarzschild metric, cont.

What happens at r = 2m? gint and gext become singular, but (t, r, ω) are not the only coordinates for Mint, Mext! The maximal Schwarzschild spacetime Mmax is a smooth extension of Mint, Mmax to include r = 2m. In new coordinates u, v (Kruskal coordinates): gmax = −16m3 r e− r

2m (du ⊗ dv + dv ⊗ du) + r 2dΩ2

∂u and ∂v are both future pointing null vectors What happens at r = 0? Physical singularity Curvature diverges as r → 0 There exists no continuous extension of Mmax (no C2 extension because of curvature, no continuous extension by Sbierski, 2015).

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 13 / 17

The Schwarzschild metric, cont.

What happens at r = 2m? gint and gext become singular, but (t, r, ω) are not the only coordinates for Mint, Mext! The maximal Schwarzschild spacetime Mmax is a smooth extension of Mint, Mmax to include r = 2m. In new coordinates u, v (Kruskal coordinates): gmax = −16m3 r e− r

2m (du ⊗ dv + dv ⊗ du) + r 2dΩ2

∂u and ∂v are both future pointing null vectors What happens at r = 0? Physical singularity Curvature diverges as r → 0 There exists no continuous extension of Mmax (no C2 extension because of curvature, no continuous extension by Sbierski, 2015).

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 13 / 17

The Schwarzschild metric, cont.

What happens at r = 2m? gint and gext become singular, but (t, r, ω) are not the only coordinates for Mint, Mext! The maximal Schwarzschild spacetime Mmax is a smooth extension of Mint, Mmax to include r = 2m. In new coordinates u, v (Kruskal coordinates): gmax = −16m3 r e− r

2m (du ⊗ dv + dv ⊗ du) + r 2dΩ2

∂u and ∂v are both future pointing null vectors What happens at r = 0? Physical singularity Curvature diverges as r → 0 There exists no continuous extension of Mmax (no C2 extension because of curvature, no continuous extension by Sbierski, 2015).

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 13 / 17

The Schwarzschild metric, cont.

What happens at r = 2m? gint and gext become singular, but (t, r, ω) are not the only coordinates for Mint, Mext! The maximal Schwarzschild spacetime Mmax is a smooth extension of Mint, Mmax to include r = 2m. In new coordinates u, v (Kruskal coordinates): gmax = −16m3 r e− r

2m (du ⊗ dv + dv ⊗ du) + r 2dΩ2

∂u and ∂v are both future pointing null vectors What happens at r = 0? Physical singularity Curvature diverges as r → 0 There exists no continuous extension of Mmax (no C2 extension because of curvature, no continuous extension by Sbierski, 2015).

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 13 / 17

The Schwarzschild metric, cont.

What happens at r = 2m? gint and gext become singular, but (t, r, ω) are not the only coordinates for Mint, Mext! The maximal Schwarzschild spacetime Mmax is a smooth extension of Mint, Mmax to include r = 2m. In new coordinates u, v (Kruskal coordinates): gmax = −16m3 r e− r

2m (du ⊗ dv + dv ⊗ du) + r 2dΩ2

∂u and ∂v are both future pointing null vectors What happens at r = 0? Physical singularity Curvature diverges as r → 0 There exists no continuous extension of Mmax (no C2 extension because of curvature, no continuous extension by Sbierski, 2015).

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 13 / 17

Mathematical definition of singularities

Definition

A spacetime is said to be singular if it contains an incomplete causal geodesic. An incomplete causal geodesic is a causal geodesic (i.e., solution of the geodesic equation) with Imax = (−∞, ∞). Examples for singular spacetimes:

Rn \ {p} with gMinkowski Robertson-Walker spacetimes with I = R Schwarzschild

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 14 / 17

Mathematical definition of singularities

Definition

A spacetime is said to be singular if it contains an incomplete causal geodesic. An incomplete causal geodesic is a causal geodesic (i.e., solution of the geodesic equation) with Imax = (−∞, ∞). Examples for singular spacetimes:

Rn \ {p} with gMinkowski Robertson-Walker spacetimes with I = R Schwarzschild

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 14 / 17

Mathematical definition of singularities

Definition

A spacetime is said to be singular if it contains an incomplete causal geodesic. An incomplete causal geodesic is a causal geodesic (i.e., solution of the geodesic equation) with Imax = (−∞, ∞). Examples for singular spacetimes:

Rn \ {p} with gMinkowski Robertson-Walker spacetimes with I = R Schwarzschild

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 14 / 17

What are “singularity theorems”...

The first singularity theorem was published by R. Penrose in 1965 (followed by one of Hawking and Hawking-Penrose) On a very basic level they have the following form:

Theorem (Pattern Singularity Theorem.)

If a spacetime satisfies

1 an energy (i.e., curvature) condition, 2 a causality condition and 3 a boundary or initial condition

then it contains an incomplete causal geodesic.

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 15 / 17

What are “singularity theorems”...

The first singularity theorem was published by R. Penrose in 1965 (followed by one of Hawking and Hawking-Penrose) On a very basic level they have the following form:

Theorem (Pattern Singularity Theorem.)

If a spacetime satisfies

1 an energy (i.e., curvature) condition, 2 a causality condition and 3 a boundary or initial condition

then it contains an incomplete causal geodesic.

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 15 / 17

...and why are they interesting?

Physically singularities are, e.g., things like black holes or the big bang The conditions of the singularity theorems are (mostly) physically reasonable A mathematical singularity does not necessarily imply a physical one (“bad” things happen, curvature blows up) Singularity theorems can generally only predict the former Discontinuities in the matter/energy-density give rise to C1,1-metrics for which classical singularity theorems no longer hold I like the math involved

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 16 / 17

...and why are they interesting?

Physically singularities are, e.g., things like black holes or the big bang The conditions of the singularity theorems are (mostly) physically reasonable A mathematical singularity does not necessarily imply a physical one (“bad” things happen, curvature blows up) Singularity theorems can generally only predict the former Discontinuities in the matter/energy-density give rise to C1,1-metrics for which classical singularity theorems no longer hold I like the math involved

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 16 / 17

...and why are they interesting?

Physically singularities are, e.g., things like black holes or the big bang The conditions of the singularity theorems are (mostly) physically reasonable A mathematical singularity does not necessarily imply a physical one (“bad” things happen, curvature blows up) Singularity theorems can generally only predict the former Discontinuities in the matter/energy-density give rise to C1,1-metrics for which classical singularity theorems no longer hold I like the math involved

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 16 / 17

...and why are they interesting?

Physically singularities are, e.g., things like black holes or the big bang The conditions of the singularity theorems are (mostly) physically reasonable A mathematical singularity does not necessarily imply a physical one (“bad” things happen, curvature blows up) Singularity theorems can generally only predict the former Discontinuities in the matter/energy-density give rise to C1,1-metrics for which classical singularity theorems no longer hold I like the math involved

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 16 / 17

...and why are they interesting?

Physically singularities are, e.g., things like black holes or the big bang The conditions of the singularity theorems are (mostly) physically reasonable A mathematical singularity does not necessarily imply a physical one (“bad” things happen, curvature blows up) Singularity theorems can generally only predict the former Discontinuities in the matter/energy-density give rise to C1,1-metrics for which classical singularity theorems no longer hold I like the math involved

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 16 / 17

...and why are they interesting?

Physically singularities are, e.g., things like black holes or the big bang The conditions of the singularity theorems are (mostly) physically reasonable A mathematical singularity does not necessarily imply a physical one (“bad” things happen, curvature blows up) Singularity theorems can generally only predict the former Discontinuities in the matter/energy-density give rise to C1,1-metrics for which classical singularity theorems no longer hold I like the math involved

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 16 / 17

My thesis project

Long term project on Lorentzian geometry and general relativity with metrics of low regularity

with M.G., James Grant, Michael Kunzinger, Alexander Lecke, Clemens Sämann, Milena Stojkovic, Roland Steinbauer, James Vickers

Try to generalize the Hawking-Penrose singularity theorem to C1,1-metrics Develop comparison techniques in the Lorentzian framework and apply them to singularity theorems Study rigidity properties, analogous to Riemannian geometry (the simplest example being the maximal diameter theorem), resulting from the typical assumptions of the singularity theorems

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 17 / 17

My thesis project

Long term project on Lorentzian geometry and general relativity with metrics of low regularity

with M.G., James Grant, Michael Kunzinger, Alexander Lecke, Clemens Sämann, Milena Stojkovic, Roland Steinbauer, James Vickers

Try to generalize the Hawking-Penrose singularity theorem to C1,1-metrics Develop comparison techniques in the Lorentzian framework and apply them to singularity theorems Study rigidity properties, analogous to Riemannian geometry (the simplest example being the maximal diameter theorem), resulting from the typical assumptions of the singularity theorems

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 17 / 17

My thesis project

Long term project on Lorentzian geometry and general relativity with metrics of low regularity

with M.G., James Grant, Michael Kunzinger, Alexander Lecke, Clemens Sämann, Milena Stojkovic, Roland Steinbauer, James Vickers

Try to generalize the Hawking-Penrose singularity theorem to C1,1-metrics Develop comparison techniques in the Lorentzian framework and apply them to singularity theorems Study rigidity properties, analogous to Riemannian geometry (the simplest example being the maximal diameter theorem), resulting from the typical assumptions of the singularity theorems

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 17 / 17

My thesis project

Long term project on Lorentzian geometry and general relativity with metrics of low regularity

with M.G., James Grant, Michael Kunzinger, Alexander Lecke, Clemens Sämann, Milena Stojkovic, Roland Steinbauer, James Vickers

Try to generalize the Hawking-Penrose singularity theorem to C1,1-metrics Develop comparison techniques in the Lorentzian framework and apply them to singularity theorems Study rigidity properties, analogous to Riemannian geometry (the simplest example being the maximal diameter theorem), resulting from the typical assumptions of the singularity theorems

Melanie Graf (University of Vienna) Singularity Thms in Lorentzian geometry 19.10.2016 17 / 17