Globally Hyperbolic Spacetimes and the Loc Categories

Locally covariant quantum field theory first fixes which spacetimes may carry a theory and which embeddings may transport observables. The standard source category, Loc, uses oriented and time-oriented globally hyperbolic Lorentzian spacetimes and causally convex isometric embeddings. Those restrictions are not cosmetic: they are precisely what lets hyperbolic propagation and causal separation survive an embedding.

Required background. QFT Frameworks: Object Classes and Maps supplies the discipline of typing objects and arrows. Smooth Manifolds, Tangent Spaces, and Tensors supplies embeddings and pulled-back metrics. Hyperbolic Equations, Green Operators, and Causal Propagators supplies the propagation theorem used below.

Helpful background. Globally Hyperbolic Spacetimes and Cauchy Surfaces reviews the geometric equivalences and examples.

The source category Loc

Fix a spacetime dimension $n\geq2$. An object

$$\boldsymbol M=(M,g,o,t)$$

consists of a smooth $n$-manifold, a Lorentzian metric of signature $(+---)$, an orientation $o$, and a time-orientation $t$, such that $(M,g)$ is globally hyperbolic. It is common to require finitely many connected components and to treat the connected category separately; every component then has a Cauchy surface.

A morphism $\psi:\boldsymbol M\to\boldsymbol N$ is a smooth embedding satisfying four conditions:

1. $\psi^*g_N=g_M$;
2. $\psi$ preserves orientation and time-orientation;
3. $\psi(M)$ is open in $N$;
4. $\psi(M)$ is causally convex: every causal curve in $N$ with endpoints in $\psi(M)$ lies wholly in $\psi(M)$.

Because source and target have the same dimension, an isometric embedding is locally a diffeomorphism and therefore has open image. It is nevertheless useful to display openness because it identifies $M$ with the induced spacetime on its image. Identities and composites again satisfy all four conditions; causal convexity of a composite follows by applying it first in $N$ and then in $M$. This is the category used in Brunetti, Fredenhagen, and Verch 2003, § 2.2, pp. 5–9.

A morphism is Cauchy when its image contains a Cauchy surface of the target. That condition is stronger than admissibility. It is the geometric input for the time-slice axiom, which will turn the induced algebra map into an isomorphism. If disjoint unions are admitted, one often adds a monoidal structure and permits a map from $M_1\sqcup M_2$ only when the two images are causally disjoint; this packages Einstein causality but is not needed for the basic definition.

Examples and a first application

Minkowski spacetime, an ultrastatic cylinder $(\mathbb R\times\Sigma,dt^2-h)$ with $(\Sigma,h)$ complete, and the domain of dependence of a suitable open ball in a Cauchy surface are Loc objects. A causally convex diamond $D(B)\hookrightarrow M$ gives a Loc morphism. It need not be Cauchy in $M$: its image generally contains a Cauchy surface for the diamond, not for all of $M$. By contrast, a causally convex neighborhood of a full Cauchy surface can define a Cauchy morphism.

This classification is useful in curved-spacetime calculations. Treat Minkowski space, an ultrastatic cylinder, and a causally convex diamond as Loc objects and decide which inclusions are admissible or Cauchy before transporting a field algebra; the necessary geometry is developed in Globally Hyperbolic Spacetimes and Cauchy Surfaces. The check is geometric, not merely topological: for every proposed inclusion, test causal convexity in the target and ask whether a target Cauchy surface lies in the image.

Why causal convexity is indispensable

Suppose an isometric open embedding has an image that is not causally convex. A causal curve may leave the image and later re-enter. The retarded or advanced solution computed in the target can then sample points that are absent from the source. Consequently one cannot generally expect

$$\psi_*E_M^{\pm}f=E_N^{\pm}\psi_*f$$

on the image, and an algebra map defined from the causal propagator can fail to preserve commutators. This is an explicit failure boundary, not evidence that global hyperbolicity itself failed.

Three independent checks catch most mistakes: verify the pulled-back metric and both orientations; verify causal convexity using target causal curves; and distinguish a Cauchy surface of the subspacetime from one of the target. The third check prevents a small diamond from being misclassified as a Cauchy morphism.

Exercise

Let $O\subset M$ be an open causally convex globally hyperbolic subset, with its induced structures. Show that the inclusion $\iota:O\hookrightarrow M$ is a Loc morphism. Give the additional condition that makes it Cauchy.

Solution

The inclusion is smooth, isometric, orientation- and time-orientation-preserving, and has the causally convex open image $O$, so it is a Loc morphism. It is Cauchy exactly when $O$ contains a Cauchy surface of $M$. Having a Cauchy surface of its own is insufficient; every globally hyperbolic $O$ has one.

References

• Bär, Christian, Nicolas Ginoux, and Frank Pfäffle. Wave Equations on Lorentzian Manifolds and Quantization. Zürich: European Mathematical Society, 2007. DOI.
• Brunetti, Romeo, Klaus Fredenhagen, and Rainer Verch. “The Generally Covariant Locality Principle—A New Paradigm for Local Quantum Physics.” Communications in Mathematical Physics 237 (2003): 31–68. DOI; Open PDF.
• Fewster, Christopher J., and Rainer Verch. “Dynamical Locality and Covariance: What Makes a Physical Theory the Same in All Spacetimes?” Annales Henri Poincaré 13 (2012): 1613–1674. DOI; Open PDF.