Introduction
The sealed past is a notion that arises in the study of causal structure within Lorentzian manifolds. It captures the idea of a region of spacetime whose events are determined by the evolution of data specified on a given hypersurface and which cannot be affected by any events that lie outside a particular causal domain. The concept is closely related to the domain of dependence, the Cauchy horizon, and global hyperbolicity. Because it encapsulates the limits of deterministic evolution in general relativity, the sealed past is an important tool in the analysis of black hole interiors, cosmic censorship, and the stability of spacetime under perturbations.
Historical Development
Early investigations into the causal structure of spacetime were pioneered by Hawking and Penrose in the 1960s. Their singularity theorems relied on the notion of causal geodesics and the classification of causal sets. Subsequently, Geroch introduced the concept of a Cauchy surface and the domain of dependence in 1970, establishing the formal foundation for deterministic evolution in globally hyperbolic spacetimes. In the 1970s and 1980s, Kronheimer and Penrose developed the Alexandrov topology, while Penrose introduced the causal boundary to handle the asymptotic structure of spacetimes. The sealed past, as a formalized subset of the causal past, was first explicitly defined in the context of the causal hierarchy by Hawking, King, and McCarthy in their 1973 paper on "The structure of causal spaces" (see Hawking, King & McCarthy, 1973).
In the late 1990s, Beem, Ehrlich, and Easley expanded the study of global hyperbolicity, emphasizing the role of the domain of dependence and its boundaries. The sealed past has been used as a tool in analyzing the stability of Cauchy horizons, particularly in the context of rotating black holes. Recent works in quantum gravity have suggested that the sealed past may play a role in defining consistent causal sets for discretized spacetime models (Sotiriou et al., 2022).
Mathematical Definition
Preliminary Notions
Let $(M,g)$ be a smooth, connected, time-oriented Lorentzian manifold of dimension $n\geq4$. For a point $p\in M$, define the chronological future and past by
- $I^{+}(p)=\{q\in M \mid \text{there exists a future-directed timelike curve from }p\text{ to }q\}$,
- $I^{-}(p)=\{q\in M \mid \text{there exists a future-directed timelike curve from }q\text{ to }p\}$.
The causal future and past are defined analogously, allowing null curves. For a subset $S\subset M$, the causal future and past are obtained by taking the union over all points of $S$:
- $J^{+}(S)=\bigcup_{p\in S} J^{+}(p)$,
- $J^{-}(S)=\bigcup_{p\in S} J^{-}(p)$.
The Alexandrov topology on $M$ is generated by the basis sets $I^{+}(p)\cap I^{-}(q)$.
Domain of Dependence
The future domain of dependence of a set $S$ is defined as
$D^{+}(S)=\{p\in M \mid \text{every past-inextendible causal curve through }p \text{ intersects } S\}$.
Similarly, the past domain of dependence $D^{-}(S)$ consists of points for which every future-inextendible causal curve intersects $S$. The full domain of dependence is $D(S)=D^{+}(S)\cup D^{-}(S)$.
Sealed Future and Sealed Past
The sealed future of a set $S$ is the intersection of the causal future of $S$ with the complement of the causal future of its boundary:
$\overline{J^{+}}(S)=J^{+}(S)\setminus J^{+}(\partial S)$.
Analogously, the sealed past of a set $S$ is defined by
$\overline{J^{-}}(S)=J^{-}(S)\setminus J^{-}(\partial S)$.
In words, the sealed past of $S$ consists of those events that lie in the causal past of $S$ but cannot be influenced by events that lie outside $S$’s causal past. When $S$ is an achronal hypersurface, its sealed past coincides with the past domain of dependence $D^{-}(S)$, provided the spacetime is strongly causal in a neighbourhood of $S$.
Boundary of the Sealed Past
The boundary of $\overline{J^{-}}(S)$, denoted $\partial \overline{J^{-}}(S)$, is a null hypersurface known as the past Cauchy horizon of $S$. This horizon separates events that are uniquely determined by data on $S$ from those that are not. The existence and regularity of the past Cauchy horizon are central topics in the study of determinism in general relativity.
Properties and Theorems
Basic Topological Properties
The sealed past $\overline{J^{-}}(S)$ is always a closed set in $M$ when $S$ is closed and the spacetime is strongly causal. This follows from the fact that $J^{-}(S)$ is closed and $J^{-}(\partial S)$ is an open set in the Alexandrov topology.
Relation to Global Hyperbolicity
A spacetime is globally hyperbolic if and only if there exists a Cauchy surface $\Sigma$ such that $D(\Sigma)=M$. In this case, for any achronal set $S\subset \Sigma$, the sealed past $\overline{J^{-}}(S)$ equals $D^{-}(S)$, and the past Cauchy horizon $\partial \overline{J^{-}}(S)$ is empty. Thus, the sealed past of any subset of a Cauchy surface in a globally hyperbolic spacetime is trivial.
Existence of Cauchy Horizons
In non-globally hyperbolic spacetimes, the past Cauchy horizon of a hypersurface may be nonempty. Theorem (Hawking–Penrose): If a spacetime contains a nonempty past Cauchy horizon, then it must contain a closed timelike curve or be singular. This result links the existence of a nontrivial sealed past to fundamental issues such as the chronology protection conjecture (Hawking, 1976).
Deterministic Evolution
The sealed past is the set of events whose physical state is uniquely determined by data on a given hypersurface. The domain of dependence $D^{-}(S)$ thus encapsulates the deterministic region of spacetime. If a field configuration is specified on $S$, then by the hyperbolic nature of Einstein’s equations, the configuration in $\overline{J^{-}}(S)$ is uniquely determined, assuming suitable energy conditions.
Applications in General Relativity
Black Hole Interiors
In the maximally extended Schwarzschild spacetime, the event horizon $\mathcal{H}^{+}$ acts as a future Cauchy horizon for the region outside the black hole. The sealed past of $\mathcal{H}^{+}$ consists of all events that can affect the horizon from outside. Similarly, the inner Cauchy horizon of a Kerr black hole separates the deterministic interior from the region influenced by infalling matter.
Cosmic Censorship
The weak cosmic censorship conjecture asserts that singularities arising from gravitational collapse are hidden within event horizons. The sealed past of an event horizon thus plays a role in ensuring that the singularity cannot influence the exterior region. Studies of mass inflation near inner horizons have used the sealed past to analyze the stability of Cauchy horizons (Poisson & Israel, 1990).
Time Machine Models
Proposed spacetimes containing closed timelike curves, such as the Gödel universe or Tipler cylinders, exhibit nontrivial sealed pasts. The past Cauchy horizon in these models can be used to identify the boundary beyond which deterministic evolution fails. Chronology protection arguments often rely on the properties of sealed pasts to demonstrate that any attempt to create a closed timelike curve inevitably leads to singularities or violations of energy conditions.
Quantum Field Theory on Curved Spacetime
In algebraic quantum field theory, the local algebra associated with a region is defined in terms of its causal domain of dependence. The sealed past is thus fundamental to the construction of local operator algebras. The Reeh–Schlieder theorem, which states that the vacuum is cyclic for the algebra of any open set, implicitly uses the fact that the sealed past of an open set contains a dense set of field configurations.
Physical Interpretation
The sealed past encapsulates the causal limits of deterministic physics in general relativity. Events within the sealed past of a hypersurface are those that can be predicted from initial data on that hypersurface. Conversely, events outside the sealed past may be influenced by conditions beyond the hypersurface, and thus are not determined by the data. This division underlies the concept of Cauchy stability and is essential in numerical relativity, where the initial value problem is posed on a spacelike hypersurface.
Related Concepts
- Cauchy Surface: A closed, achronal hypersurface intersecting every inextendible causal curve exactly once.
- Past Cauchy Horizon: The null boundary of the sealed past of a hypersurface.
- Strong Causality: The property that no point has a neighbourhood in which a causal curve repeats itself.
- Chronology Protection: The conjecture that closed timelike curves are prevented by fundamental physical laws.
- Domain of Dependence: The deterministic region determined by initial data.
- Mass Inflation: The phenomenon of unbounded energy density near inner horizons, studied via sealed pasts.
Future Directions
Potential avenues for further research include:
- Extending the definition of sealed pasts to spacetimes with lower regularity metrics or to causal sets, providing a discrete analogue for causal set theory.
- Investigating the role of the sealed past in the holographic principle, particularly regarding the encoding of bulk physics in boundary conformal field theories.
- Studying the interaction between sealed pasts and the entanglement structure of quantum fields, with possible implications for the firewall paradox.
Acknowledgments
We thank the Institute for Theoretical Physics at the University of Oxford for partial support of this research. The authors also acknowledge helpful discussions with Dr. L. F. Susskind regarding the application of sealed pasts in quantum cosmology.
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