Wave equations with initial data on compact Cauchy horizons
Oliver Lindblad Petersen

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We study the following problem: Given initial data on a compact Cauchy horizon, does there exist a unique solution to wave equations on the globally hyperbolic region? 
Our main results apply to any spacetime satisfying the null energy condition and containing a compact Cauchy horizon with surface gravity that can be normalised to a non-zero constant.
Examples include the Misner spacetime and the Taub-NUT spacetime.
We prove an energy estimate close to the Cauchy horizon for wave equations acting on sections of vector bundles.
Using this estimate we prove that if a linear wave equation can be solved up to any order at the Cauchy horizon, then there exists a unique solution on the globally hyperbolic region.
As a consequence, we prove several existence and uniqueness results for linear and non-linear wave equations without assuming analyticity or symmetry of the spacetime and without assuming that the generators close.
We overcome in particular the essential remaining difficulty in proving that vacuum spacetimes with a compact Cauchy horizon with constant non-zero  surface gravity necessarily admits a Killing vector field.
This work is therefore related to the strong cosmic censorship conjecture.



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§ INTRODUCTION


The purpose of this paper is to present new methods to treat the characteristic Cauchy problem for wave equations with initial data on a smooth, compact, totally geodesic Cauchy horizon with surface gravity that can be normalised to a non-zero constant.
To the best of our knowledge, all known examples of compact Cauchy horizons in (electro-)vacuum spacetimes satisfy these conditions.
In fact, it was recently proven that any compact Cauchy horizon in a spacetime satisfiying the null energy condition is necessarily smooth and totally geodesic (see <cit.> and  <cit.>).

Since Cauchy horizons are lightlike hypersurfaces, the metric degenerates at the Cauchy horizon.
This paper therefore concerns a geometric singular initial value problem for wave equations.
We prove:

	
  * If a linear wave equation can be solved up to any order at the Cauchy horizon, it can be solved uniquely on the globally hyperbolic region (Theorem <ref>).
	In particular, if a solution to a linear wave equation vanishes up to any order at the Cauchy horizon, then it vanishes on the globally hyperbolic region.
	
  * If a solution to what we call an admissible linear wave equation vanishes at the Cauchy horizon, then it vanishes on the globally hyperbolic region (Theorem <ref>).
	
  * Given any initial data to an admissible linear wave equation for scalar valued functions, there exists a unique solution on the globally hyperbolic region (Theorem <ref>).
	
  * Given any initial data to an admissible semi-linear wave equation for scalar valued functions, there exists a unique solution on a neighbourhood of the Cauchy horizon (Theorem <ref>).

The statements (3) and (4) require a condition on the Ricci curvature at the Cauchy horizon, implied for example by the dominant energy condition. 
All four statements hold in any spacetime dimension greater or equal to 2. 
Rather surprisingly, simple counter examples show that the statements (2-4) were false for general wave operators. 
We also give counter examples implying that all four statements were false if we allowed for Cauchy horizons with vanishing surface gravity.

Our results have a natural application in general relativity. 
The strong cosmic censorship conjecture says that the maximal globally hyperbolic vacuum (or suitable matter) developments of generic relativistic initial data is inextendible. 
In other words, the maximally globally hyperbolic hyperbolic vacuum spacetimes which are extendible over a Cauchy horizon are expected to be non-generic. 
Moncrief and Isenberg conjectured in <cit.> that a vacuum spacetime with a compact Cauchy horizon necessarily admits a Killing vector field in the globally hyperbolic region.
Together with István Rácz we show in <cit.> that the Killing equation can be solved up to any order at a compact Cauchy horizon, if the spacetime is vacuum and the surface gravity can be normalised to a non-zero constant.
Applying the main result of this paper, Theorem <ref>, then proves the Moncrief-Isenberg conjecture under the assumption that the surface gravity of the Cauchy horizon can be normalised to a non-zero constant.
This generalises classical results by Moncrief-Isenberg and Friedrich-Rácz-Wald, who assume that the generators are closed or densely fill a 2-torus (<cit.> <cit.> <cit.>).
In particular, generic maximally globally hyperbolic vacuum developments of asymptotically flat or compact initial data cannot be extended over a compact Cauchy horizon with surface gravity that can be normalised to a non-zero constant.
This is a natural step towards the strong cosmic censorship conjecture without symmetry assumptions.
We postpone a more detailed discussion of this until we have presented the precise formulation of our main results.

Let (M, g) denote a spacetime, i.e. a connected time-oriented Lorentzian manifold. 
Assume that  is an acausal topological hypersurface of M, such that ⊂ M is a closed subset.
Let n+1 denote the dimension of M, we only require that n+1 ≥ 2.
The domain of dependence D() ⊂ M (also called Cauchy development) is a globally hyperbolic submanifold of M and  is a Cauchy hypersurface for D() (see <cit.>*Lem. 43). 
The classical well-posedness statement for the Cauchy problem says that if  is smooth and spacelike, then linear wave equations can be solved uniquely on D() if one specifies initial data on , see e.g. <cit.>*Thm. 3.2.11. 
In this paper, we instead specify the initial data on a part of the boundary of D(). 
By <cit.>*Prop. 14.53, the boundary of D() can be divided into disjoint sets

    Ḍ() = _̋+ ⊔_̋-,
 
where _̋± := D_±()\ D_±() are called the future and the past Cauchy horizon of  respectively. 
_̋+ and _̋- are closed achronal lightlike Lipschitz hypersurfaces (if non-empty) of M. 
Let from now on $̋ denote the future or the past Cauchy horizon of.
If$̋ is a smooth hypersurface, then ⊔̋D() is a smooth manifold with boundary. 
In our case, $̋ will be smooth and compact.
Therefore there will be closed or almost closed lightlike curves in$̋, which implies that M is not globally hyperbolic.
Our results therefore concern wave equations on non-globally hyperbolic spacetimes.
A first example is given by the classical Misner spacetime, where M = × S^1 with coordinates t and x and g = 2dtdx + tdx^2. 
As shown in Figure <ref>, the Misner spacetime admits a compact past Cauchy horizon :̋= {t = 0} of any level set := {t = C}, where C is a positive constant (see Example <ref> for more details).


Let us explain what we mean by normalising the surface gravity of $̋.
Any lightlike vector fieldVtangent to$̋ is pre-geodesic, i.e. there is a smooth function κ such that

    _V V = κ V

on $̋.
Motivated by the corresponding equation on black hole horizons, we callκthe surface gravity associated withV. 
It is important, however, to note that there is no canonical way to normalise the surface gravity on a compact Cauchy horizon.
Any vector fieldfVfor a functionfwill give a different surface gravity.



Assume from now on that $̋ is a (non-empty) smooth, compact, totally geodesic future or past Cauchy horizon ofand that there is a nowhere vanishing lightlike vector fieldVtangent to$̋ such that

    _V V = κ V

for a non-zero constantκ.
 
By substitutingVwith1/κV, we may from now on assume thatκ= 1.
The integral curves ofV(or their reparametrisations as geodesics) are called generators of the Cauchy horizon.