The globally hyperbolic splitting in non-smoothLorentzian manifolds

Clemens Sämann

joint work withGünther Hörmann

University of Vienna, AustriaFaculty of Mathematics

DIANA group

ISAAC, Krakow, August 5, 2013

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 1 / 10

Introduction

Global hyperbolicity: strongest causality condition, well-posedness of theCauchy problem

(M , g) a space-time, i.e., a time-oriented, connected Lorentzian manifold(M , g) is globally hyperbolic if one of the following equivalent conditions issatisfied:

(M , g) is causal and ∀p, q ∈ M with q ∈ J +(p): J +(p) ∩ J−(q) iscompactThere is a Cauchy hypersurface S in (M , g)(M , g) is isometric to R× S , where S is a Cauchy hypersurface in M

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 2 / 10

Introduction

Global hyperbolicity: strongest causality condition, well-posedness of theCauchy problem

(M , g) a space-time, i.e., a time-oriented, connected Lorentzian manifold(M , g) is globally hyperbolic if one of the following equivalent conditions issatisfied:

(M , g) is causal and ∀p, q ∈ M with q ∈ J +(p): J +(p) ∩ J−(q) iscompactThere is a Cauchy hypersurface S in (M , g)(M , g) is isometric to R× S , where S is a Cauchy hypersurface in M

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 2 / 10

Introduction

Global hyperbolicity: strongest causality condition, well-posedness of theCauchy problem

(M , g) a space-time, i.e., a time-oriented, connected Lorentzian manifold(M , g) is globally hyperbolic if one of the following equivalent conditions issatisfied:

(M , g) is causal and ∀p, q ∈ M with q ∈ J +(p): J +(p) ∩ J−(q) iscompactThere is a Cauchy hypersurface S in (M , g)(M , g) is isometric to R× S , where S is a Cauchy hypersurface in M

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 2 / 10

Introduction

Global hyperbolicity: strongest causality condition, well-posedness of theCauchy problem

(M , g) a space-time, i.e., a time-oriented, connected Lorentzian manifold(M , g) is globally hyperbolic if one of the following equivalent conditions issatisfied:

(M , g) is causal and ∀p, q ∈ M with q ∈ J +(p): J +(p) ∩ J−(q) iscompactThere is a Cauchy hypersurface S in (M , g)(M , g) is isometric to R× S , where S is a Cauchy hypersurface in M

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 2 / 10

Introduction

Global hyperbolicity: strongest causality condition, well-posedness of theCauchy problem

(M , g) a space-time, i.e., a time-oriented, connected Lorentzian manifold(M , g) is globally hyperbolic if one of the following equivalent conditions issatisfied:

(M , g) is causal and ∀p, q ∈ M with q ∈ J +(p): J +(p) ∩ J−(q) iscompactThere is a Cauchy hypersurface S in (M , g)(M , g) is isometric to R× S , where S is a Cauchy hypersurface in M

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 2 / 10

The smooth metric splitting

want: generalization of global hyperbolicity to non-smooth Lorentzianmanifoldsapproach: via the globally hyperbolic metric splitting

Theorem (Bernal, Sánchez 2005)Let (M , g) be a globally hyperbolic Lorentzian manifold, then (M , g) isisometric to (R× S ,−βdt2 + ht), where S is a smooth spacelike Cauchyhypersurface, t : R× S → R is the projection on the first factor,β : R× S → (0,∞) a smooth function, and ht is a Riemannian metric oneach St := {t} × S , which varies smoothly with t.

strengthening of Geroch’s topological splitting (1970)

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 3 / 10

The smooth metric splitting

want: generalization of global hyperbolicity to non-smooth Lorentzianmanifoldsapproach: via the globally hyperbolic metric splitting

Theorem (Bernal, Sánchez 2005)Let (M , g) be a globally hyperbolic Lorentzian manifold, then (M , g) isisometric to (R× S ,−βdt2 + ht), where S is a smooth spacelike Cauchyhypersurface, t : R× S → R is the projection on the first factor,β : R× S → (0,∞) a smooth function, and ht is a Riemannian metric oneach St := {t} × S , which varies smoothly with t.

strengthening of Geroch’s topological splitting (1970)

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 3 / 10

The smooth metric splitting

want: generalization of global hyperbolicity to non-smooth Lorentzianmanifoldsapproach: via the globally hyperbolic metric splitting

Theorem (Bernal, Sánchez 2005)Let (M , g) be a globally hyperbolic Lorentzian manifold, then (M , g) isisometric to (R× S ,−βdt2 + ht), where S is a smooth spacelike Cauchyhypersurface, t : R× S → R is the projection on the first factor,β : R× S → (0,∞) a smooth function, and ht is a Riemannian metric oneach St := {t} × S , which varies smoothly with t.

strengthening of Geroch’s topological splitting (1970)

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 3 / 10

The smooth metric splitting

want: generalization of global hyperbolicity to non-smooth Lorentzianmanifoldsapproach: via the globally hyperbolic metric splitting

Theorem (Bernal, Sánchez 2005)Let (M , g) be a globally hyperbolic Lorentzian manifold, then (M , g) isisometric to (R× S ,−βdt2 + ht), where S is a smooth spacelike Cauchyhypersurface, t : R× S → R is the projection on the first factor,β : R× S → (0,∞) a smooth function, and ht is a Riemannian metric oneach St := {t} × S , which varies smoothly with t.

strengthening of Geroch’s topological splitting (1970)

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 3 / 10

N -fronted waves with parallel rays (NPWs) (1/2)

Definition (NPWs)N-fronted waves with parallel rays (NPWs), are space-times with Lorentzmetric

π∗(h) + 2dudv − a(., u)du2

on M := N × R2, where (N , h) is a connected Riemannian manifold.

Profile function a : N × R→ R, π : N × R→ N projectionN as wave surfacean NPW is globally hyperbolic if (N , h) is complete and a behavessubquadraticallya(x, u) = f (x)δ(u) ; impulsive gravitational waves

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 4 / 10

N -fronted waves with parallel rays (NPWs) (1/2)

Definition (NPWs)N-fronted waves with parallel rays (NPWs), are space-times with Lorentzmetric

π∗(h) + 2dudv − a(., u)du2

on M := N × R2, where (N , h) is a connected Riemannian manifold.

Profile function a : N × R→ R, π : N × R→ N projectionN as wave surfacean NPW is globally hyperbolic if (N , h) is complete and a behavessubquadraticallya(x, u) = f (x)δ(u) ; impulsive gravitational waves

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 4 / 10

N -fronted waves with parallel rays (NPWs) (1/2)

Definition (NPWs)N-fronted waves with parallel rays (NPWs), are space-times with Lorentzmetric

π∗(h) + 2dudv − a(., u)du2

on M := N × R2, where (N , h) is a connected Riemannian manifold.

Profile function a : N × R→ R, π : N × R→ N projectionN as wave surfacean NPW is globally hyperbolic if (N , h) is complete and a behavessubquadraticallya(x, u) = f (x)δ(u) ; impulsive gravitational waves

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 4 / 10

N -fronted waves with parallel rays (NPWs) (1/2)

Definition (NPWs)N-fronted waves with parallel rays (NPWs), are space-times with Lorentzmetric

π∗(h) + 2dudv − a(., u)du2

on M := N × R2, where (N , h) is a connected Riemannian manifold.

Profile function a : N × R→ R, π : N × R→ N projectionN as wave surfacean NPW is globally hyperbolic if (N , h) is complete and a behavessubquadraticallya(x, u) = f (x)δ(u) ; impulsive gravitational waves

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 4 / 10

N -fronted waves with parallel rays (NPWs) (2/2)

Special case M = R2, 0 ≤ a < λ(u0, v0) ∈ R2

a(u0) gives the slope of the lightlike linev = a(u0)2 uspacelike line S = {(u, λu) : u ∈ R}indicates the “maximal” slope

Behaviour of lightcones

v

u

∂v

space

space

future

past

slope a(u0)2

S

(M , g) is globally hyperbolic if a ∈ C∞(N ×R) with 0 ≤ a < λ and (N , h)is complete

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 5 / 10

N -fronted waves with parallel rays (NPWs) (2/2)

Special case M = R2, 0 ≤ a < λ(u0, v0) ∈ R2

a(u0) gives the slope of the lightlike linev = a(u0)2 uspacelike line S = {(u, λu) : u ∈ R}indicates the “maximal” slope

Behaviour of lightcones

v

u

∂v

space

space

future

past

slope a(u0)2

S

(M , g) is globally hyperbolic if a ∈ C∞(N ×R) with 0 ≤ a < λ and (N , h)is complete

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 5 / 10

N -fronted waves with parallel rays (NPWs) (2/2)

Special case M = R2, 0 ≤ a < λ(u0, v0) ∈ R2

a(u0) gives the slope of the lightlike linev = a(u0)2 uspacelike line S = {(u, λu) : u ∈ R}indicates the “maximal” slope

Behaviour of lightcones

v

u

∂v

space

space

future

past

slope a(u0)2

S

(M , g) is globally hyperbolic if a ∈ C∞(N ×R) with 0 ≤ a < λ and (N , h)is complete

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 5 / 10

N -fronted waves with parallel rays (NPWs) (2/2)

Special case M = R2, 0 ≤ a < λ(u0, v0) ∈ R2

a(u0) gives the slope of the lightlike linev = a(u0)2 uspacelike line S = {(u, λu) : u ∈ R}indicates the “maximal” slope

Behaviour of lightcones

v

u

∂v

space

space

future

past

slope a(u0)2

S

(M , g) is globally hyperbolic if a ∈ C∞(N ×R) with 0 ≤ a < λ and (N , h)is complete

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 5 / 10

N -fronted waves with parallel rays (NPWs) (2/2)

Special case M = R2, 0 ≤ a < λ(u0, v0) ∈ R2

a(u0) gives the slope of the lightlike linev = a(u0)2 uspacelike line S = {(u, λu) : u ∈ R}indicates the “maximal” slope

Behaviour of lightcones

v

u

∂v

space

space

future

past

slope a(u0)2

S

(M , g) is globally hyperbolic if a ∈ C∞(N ×R) with 0 ≤ a < λ and (N , h)is complete

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 5 / 10

N -fronted waves with parallel rays (NPWs) (2/2)

Special case M = R2, 0 ≤ a < λ(u0, v0) ∈ R2

a(u0) gives the slope of the lightlike linev = a(u0)2 uspacelike line S = {(u, λu) : u ∈ R}indicates the “maximal” slope

Behaviour of lightcones

v

u

∂v

space

space

future

past

slope a(u0)2

S

(M , g) is globally hyperbolic if a ∈ C∞(N ×R) with 0 ≤ a < λ and (N , h)is complete

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 5 / 10

N -fronted waves with parallel rays (NPWs) (2/2)

Special case M = R2, 0 ≤ a < λ(u0, v0) ∈ R2

a(u0) gives the slope of the lightlike linev = a(u0)2 uspacelike line S = {(u, λu) : u ∈ R}indicates the “maximal” slope

Behaviour of lightcones

v

u

∂v

space

space

future

past

slope a(u0)2

S

(M , g) is globally hyperbolic if a ∈ C∞(N ×R) with 0 ≤ a < λ and (N , h)is complete

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 5 / 10

N -fronted waves with parallel rays (NPWs) (2/2)

Special case M = R2, 0 ≤ a < λ(u0, v0) ∈ R2

a(u0) gives the slope of the lightlike linev = a(u0)2 uspacelike line S = {(u, λu) : u ∈ R}indicates the “maximal” slope

Behaviour of lightcones

v

u

∂v

space

space

future

past

slope a(u0)2

S

(M , g) is globally hyperbolic if a ∈ C∞(N ×R) with 0 ≤ a < λ and (N , h)is complete

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 5 / 10

The non-smooth metric splitting for NPWs (1/3)

First explicit calculations in the smooth case, then use these �-wise

M = N × R2 an NPW with a ∈ C∞(N × R) and 0 ≤ a < λ

τλ : M → R, τλ(x, u, v) := λu − v is a time function for (M , g)if DN

ξ̇ξ̇ = 12∇xa(ξ, α) has complete trajectories or M = R

2 thenSkλ := τ

−1λ ({k}) = {(x, u, λu − k) : x ∈ N , u ∈ R} is a Cauchy

hypersurface for every k ∈ Runder these hypotheses we obtain the metric splitting explicitly

Theorem (G. Hörmann, C.S. 2013)(M , g) is isometric to (R× S , l), where S = S0λ and

l = −12λ−Adt2 + Ht .

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 6 / 10

The non-smooth metric splitting for NPWs (1/3)

First explicit calculations in the smooth case, then use these �-wise

M = N × R2 an NPW with a ∈ C∞(N × R) and 0 ≤ a < λ

τλ : M → R, τλ(x, u, v) := λu − v is a time function for (M , g)if DN

ξ̇ξ̇ = 12∇xa(ξ, α) has complete trajectories or M = R

2 thenSkλ := τ

−1λ ({k}) = {(x, u, λu − k) : x ∈ N , u ∈ R} is a Cauchy

hypersurface for every k ∈ Runder these hypotheses we obtain the metric splitting explicitly

Theorem (G. Hörmann, C.S. 2013)(M , g) is isometric to (R× S , l), where S = S0λ and

l = −12λ−Adt2 + Ht .

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 6 / 10

The non-smooth metric splitting for NPWs (1/3)

First explicit calculations in the smooth case, then use these �-wise

M = N × R2 an NPW with a ∈ C∞(N × R) and 0 ≤ a < λ

τλ : M → R, τλ(x, u, v) := λu − v is a time function for (M , g)if DN

ξ̇ξ̇ = 12∇xa(ξ, α) has complete trajectories or M = R

2 thenSkλ := τ

−1λ ({k}) = {(x, u, λu − k) : x ∈ N , u ∈ R} is a Cauchy

hypersurface for every k ∈ Runder these hypotheses we obtain the metric splitting explicitly

Theorem (G. Hörmann, C.S. 2013)(M , g) is isometric to (R× S , l), where S = S0λ and

l = −12λ−Adt2 + Ht .

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 6 / 10

The non-smooth metric splitting for NPWs (1/3)

First explicit calculations in the smooth case, then use these �-wise

M = N × R2 an NPW with a ∈ C∞(N × R) and 0 ≤ a < λ

τλ : M → R, τλ(x, u, v) := λu − v is a time function for (M , g)if DN

ξ̇ξ̇ = 12∇xa(ξ, α) has complete trajectories or M = R

2 thenSkλ := τ

−1λ ({k}) = {(x, u, λu − k) : x ∈ N , u ∈ R} is a Cauchy

hypersurface for every k ∈ Runder these hypotheses we obtain the metric splitting explicitly

Theorem (G. Hörmann, C.S. 2013)(M , g) is isometric to (R× S , l), where S = S0λ and

l = −12λ−Adt2 + Ht .

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 6 / 10

The non-smooth metric splitting for NPWs (1/3)

First explicit calculations in the smooth case, then use these �-wise

M = N × R2 an NPW with a ∈ C∞(N × R) and 0 ≤ a < λ

τλ : M → R, τλ(x, u, v) := λu − v is a time function for (M , g)if DN

ξ̇ξ̇ = 12∇xa(ξ, α) has complete trajectories or M = R

2 thenSkλ := τ

−1λ ({k}) = {(x, u, λu − k) : x ∈ N , u ∈ R} is a Cauchy

hypersurface for every k ∈ Runder these hypotheses we obtain the metric splitting explicitly

Theorem (G. Hörmann, C.S. 2013)(M , g) is isometric to (R× S , l), where S = S0λ and

l = −12λ−Adt2 + Ht .

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 6 / 10

The non-smooth metric splitting for NPWs (1/3)

First explicit calculations in the smooth case, then use these �-wise

M = N × R2 an NPW with a ∈ C∞(N × R) and 0 ≤ a < λ

τλ : M → R, τλ(x, u, v) := λu − v is a time function for (M , g)if DN

ξ̇ξ̇ = 12∇xa(ξ, α) has complete trajectories or M = R

2 thenSkλ := τ

−1λ ({k}) = {(x, u, λu − k) : x ∈ N , u ∈ R} is a Cauchy

hypersurface for every k ∈ Runder these hypotheses we obtain the metric splitting explicitly

Theorem (G. Hörmann, C.S. 2013)(M , g) is isometric to (R× S , l), where S = S0λ and

l = −12λ−Adt2 + Ht .

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 6 / 10

The non-smooth metric splitting for NPWs (2/3)

non-smooth profile function a : N × R→ R, Riemannian metric h onN still smooth(a�)� a net of smooth functions N × R→ R with 0 ≤ a� < λ for all� ∈ (0, 1], depending smoothly on � and moderate, hence defines aclass in G(N × R)g := [(g�)�] ∈ G02(M ), where g has representative

g� := π∗(h) + 2dudv − a�(., u)du2

so (M , g) is a generalized Lorentzian manifoldunder the assumptions before we obtain �-wise the metric splitting

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 7 / 10

The non-smooth metric splitting for NPWs (2/3)

non-smooth profile function a : N × R→ R, Riemannian metric h onN still smooth(a�)� a net of smooth functions N × R→ R with 0 ≤ a� < λ for all� ∈ (0, 1], depending smoothly on � and moderate, hence defines aclass in G(N × R)g := [(g�)�] ∈ G02(M ), where g has representative

g� := π∗(h) + 2dudv − a�(., u)du2

so (M , g) is a generalized Lorentzian manifoldunder the assumptions before we obtain �-wise the metric splitting

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 7 / 10

The non-smooth metric splitting for NPWs (2/3)

non-smooth profile function a : N × R→ R, Riemannian metric h onN still smooth(a�)� a net of smooth functions N × R→ R with 0 ≤ a� < λ for all� ∈ (0, 1], depending smoothly on � and moderate, hence defines aclass in G(N × R)g := [(g�)�] ∈ G02(M ), where g has representative

g� := π∗(h) + 2dudv − a�(., u)du2

so (M , g) is a generalized Lorentzian manifoldunder the assumptions before we obtain �-wise the metric splitting

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 7 / 10

The non-smooth metric splitting for NPWs (2/3)

non-smooth profile function a : N × R→ R, Riemannian metric h onN still smooth(a�)� a net of smooth functions N × R→ R with 0 ≤ a� < λ for all� ∈ (0, 1], depending smoothly on � and moderate, hence defines aclass in G(N × R)g := [(g�)�] ∈ G02(M ), where g has representative

g� := π∗(h) + 2dudv − a�(., u)du2

so (M , g) is a generalized Lorentzian manifoldunder the assumptions before we obtain �-wise the metric splitting

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 7 / 10

The non-smooth metric splitting for NPWs (3/3)

; generalized diffeomorphism between (M , g) and (R× S , l), whichis c-bounded and its inverse alsothis yields the globally hyperbolic metric splitting (Hörmann,Kunzinger, Steinbauer 2012), i.e., the generalized metric l satisfies(a) ∃ representative (l�) of l such that every l� is a Lorentzian metric and

each slice {t} × S is a Cauchy hypersurface for every l�(b) ∃ metric splitting of l in the form l = −βdt2 + ht

ht is a t-dependent generalized Riemannian metric on S ,β ∈ G(R× S) is globally bounded, locally uniformly positive

(c) ∀T > 0 ∃ representative (hε) of h and a smooth complete Riemannianmetric ρ on S such that

hεt (X ,X) ≥ ρ(X ,X) ∀t ∈ [−T ,T ] ∀X ∈ X(M )

uniformly in ε

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 8 / 10

The non-smooth metric splitting for NPWs (3/3)

; generalized diffeomorphism between (M , g) and (R× S , l), whichis c-bounded and its inverse alsothis yields the globally hyperbolic metric splitting (Hörmann,Kunzinger, Steinbauer 2012), i.e., the generalized metric l satisfies(a) ∃ representative (l�) of l such that every l� is a Lorentzian metric and

each slice {t} × S is a Cauchy hypersurface for every l�(b) ∃ metric splitting of l in the form l = −βdt2 + ht

ht is a t-dependent generalized Riemannian metric on S ,β ∈ G(R× S) is globally bounded, locally uniformly positive

(c) ∀T > 0 ∃ representative (hε) of h and a smooth complete Riemannianmetric ρ on S such that

hεt (X ,X) ≥ ρ(X ,X) ∀t ∈ [−T ,T ] ∀X ∈ X(M )

uniformly in ε

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 8 / 10

The non-smooth metric splitting for NPWs (3/3)

; generalized diffeomorphism between (M , g) and (R× S , l), whichis c-bounded and its inverse alsothis yields the globally hyperbolic metric splitting (Hörmann,Kunzinger, Steinbauer 2012), i.e., the generalized metric l satisfies(a) ∃ representative (l�) of l such that every l� is a Lorentzian metric and

each slice {t} × S is a Cauchy hypersurface for every l�(b) ∃ metric splitting of l in the form l = −βdt2 + ht

ht is a t-dependent generalized Riemannian metric on S ,β ∈ G(R× S) is globally bounded, locally uniformly positive

(c) ∀T > 0 ∃ representative (hε) of h and a smooth complete Riemannianmetric ρ on S such that

hεt (X ,X) ≥ ρ(X ,X) ∀t ∈ [−T ,T ] ∀X ∈ X(M )

uniformly in ε

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 8 / 10

The non-smooth metric splitting for NPWs (3/3)

; generalized diffeomorphism between (M , g) and (R× S , l), whichis c-bounded and its inverse alsothis yields the globally hyperbolic metric splitting (Hörmann,Kunzinger, Steinbauer 2012), i.e., the generalized metric l satisfies(a) ∃ representative (l�) of l such that every l� is a Lorentzian metric and

each slice {t} × S is a Cauchy hypersurface for every l�(b) ∃ metric splitting of l in the form l = −βdt2 + ht

ht is a t-dependent generalized Riemannian metric on S ,β ∈ G(R× S) is globally bounded, locally uniformly positive

(c) ∀T > 0 ∃ representative (hε) of h and a smooth complete Riemannianmetric ρ on S such that

hεt (X ,X) ≥ ρ(X ,X) ∀t ∈ [−T ,T ] ∀X ∈ X(M )

uniformly in ε

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 8 / 10

The non-smooth metric splitting for NPWs (3/3)

; generalized diffeomorphism between (M , g) and (R× S , l), whichis c-bounded and its inverse alsothis yields the globally hyperbolic metric splitting (Hörmann,Kunzinger, Steinbauer 2012), i.e., the generalized metric l satisfies(a) ∃ representative (l�) of l such that every l� is a Lorentzian metric and

each slice {t} × S is a Cauchy hypersurface for every l�(b) ∃ metric splitting of l in the form l = −βdt2 + ht

ht is a t-dependent generalized Riemannian metric on S ,β ∈ G(R× S) is globally bounded, locally uniformly positive

(c) ∀T > 0 ∃ representative (hε) of h and a smooth complete Riemannianmetric ρ on S such that

hεt (X ,X) ≥ ρ(X ,X) ∀t ∈ [−T ,T ] ∀X ∈ X(M )

uniformly in ε

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 8 / 10

Applications and Outlook

a0 ∈ L∞(N × R) with a0 ≥ 0 almost everywhere, ‖a0‖∞ < λ andthere is a function C : R→ (0,∞) that is locally integrable satisfying

|a0(x, s)− a0(y, s)| ≤ C (s) dh(x, y) (∀x, y ∈ N , s ∈ R) ,

then M ∼= R× S via a local bi-Lipschitz homeomorphisma0 as above, not necessarily with the Lipschitz condition and (a�)� asbefore with a� → a0 in L1loc(N × R), then the correspondingdiffeomorphisms converge in L1loc(R× S)approximation via appropriate convolution “a� := a0 ∗ ρ�”apply existence and uniqueness result of the Cauchy problem(Hörmann, Kunzinger, Steinbauer 2012) in this case

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 9 / 10

Applications and Outlook

a0 ∈ L∞(N × R) with a0 ≥ 0 almost everywhere, ‖a0‖∞ < λ andthere is a function C : R→ (0,∞) that is locally integrable satisfying

|a0(x, s)− a0(y, s)| ≤ C (s) dh(x, y) (∀x, y ∈ N , s ∈ R) ,

then M ∼= R× S via a local bi-Lipschitz homeomorphisma0 as above, not necessarily with the Lipschitz condition and (a�)� asbefore with a� → a0 in L1loc(N × R), then the correspondingdiffeomorphisms converge in L1loc(R× S)approximation via appropriate convolution “a� := a0 ∗ ρ�”apply existence and uniqueness result of the Cauchy problem(Hörmann, Kunzinger, Steinbauer 2012) in this case

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 9 / 10

Applications and Outlook

a0 ∈ L∞(N × R) with a0 ≥ 0 almost everywhere, ‖a0‖∞ < λ andthere is a function C : R→ (0,∞) that is locally integrable satisfying

|a0(x, s)− a0(y, s)| ≤ C (s) dh(x, y) (∀x, y ∈ N , s ∈ R) ,

then M ∼= R× S via a local bi-Lipschitz homeomorphisma0 as above, not necessarily with the Lipschitz condition and (a�)� asbefore with a� → a0 in L1loc(N × R), then the correspondingdiffeomorphisms converge in L1loc(R× S)approximation via appropriate convolution “a� := a0 ∗ ρ�”apply existence and uniqueness result of the Cauchy problem(Hörmann, Kunzinger, Steinbauer 2012) in this case

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 9 / 10

Applications and Outlook

a0 ∈ L∞(N × R) with a0 ≥ 0 almost everywhere, ‖a0‖∞ < λ andthere is a function C : R→ (0,∞) that is locally integrable satisfying

|a0(x, s)− a0(y, s)| ≤ C (s) dh(x, y) (∀x, y ∈ N , s ∈ R) ,

then M ∼= R× S via a local bi-Lipschitz homeomorphisma0 as above, not necessarily with the Lipschitz condition and (a�)� asbefore with a� → a0 in L1loc(N × R), then the correspondingdiffeomorphisms converge in L1loc(R× S)approximation via appropriate convolution “a� := a0 ∗ ρ�”apply existence and uniqueness result of the Cauchy problem(Hörmann, Kunzinger, Steinbauer 2012) in this case

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 9 / 10

References

A. N. Bernal, M. Sánchez. On smooth Cauchy hypersurfaces and Geroch’ssplitting theorem. Commun. Math. Phys., 243(3):461–470, 2003.

A. N. Bernal, M. Sánchez. Smoothness of time functions and the metricsplitting of globally hyperbolic spacetimes. Commun. Math. Phys.,257(1):43-50, 2005.

J. L. Flores, M. Sánchez. On the geometry of pp-wave type spacetimes. InAnalytical and numerical approaches to mathematical relativity, volume 692of Lecture Notes in Phys., pages 79–98. Springer, Berlin, 2006.

G. Hörmann, M. Kunzinger, R. Steinbauer. Wave equations on non-smoothspace-times. In Evolution Equations of Hyperbolic and Schrödinger Type,volume 301 of Progress in Mathematics, pages 163–186. Birkhäuser, 2012

G. Hörmann, C. S. The globally hyperbolic metric splitting for non-smoothwave-type space-times. In preparation

Clemens Sämann, University of Vienna ISAAC, August 5, 2013 10 / 10