Globally hyperbolic

Globally hyperbolic (also global hyperbolicity) is a term describing the causal structure

Causal structure

The causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.- Introduction :In modern physics spacetime is represented by a Lorentzian manifold...

 of a spacetime

Spacetime

In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions...

 manifold

Manifold

In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....

 in Einstein's theory of general relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...

, or potentially in other metric gravitational theories.

A spacetime

Spacetime

In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions...

 manifold

Manifold

In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....

  is said to be globally hyperbolic if the following two conditions hold

1. For every pair of points , is compact. Here is the future (past) of a subset of , that is, the set of all points which can be reached from along future (past) curves.
2. "Causality" holds on (no closed timelike curve
   Closed timelike curve
   In a Lorentzian manifold, a closed timelike curve is a worldline of a material particle in spacetime that is "closed," returning to its starting point...
   
   s exist).

Globally hyperbolic (also global hyperbolicity) is a term describing the causal structure

Causal structure

The causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.- Introduction :In modern physics spacetime is represented by a Lorentzian manifold...

 of a spacetime

Spacetime

In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions...

 manifold

Manifold

In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....

 in Einstein's theory of general relativity

General relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics. It unifies special relativity and Newton's law of universal gravitation, and describes gravity as a...

, or potentially in other metric gravitational theories.

A spacetime

Spacetime

In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions...

 manifold

Manifold

In mathematics, more specifically in differential geometry and topology, a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a certain dimension, called the dimension of the manifold....

  is said to be globally hyperbolic if the following two conditions hold

1. For every pair of points , is compact. Here is the future (past) of a subset of , that is, the set of all points which can be reached from along future (past) curves.
2. "Causality" holds on (no closed timelike curve
   Closed timelike curve
   In a Lorentzian manifold, a closed timelike curve is a worldline of a material particle in spacetime that is "closed," returning to its starting point...
   
   s exist). Classically, a more restrictive and technical assumption is required, named strong causality (no "almost closed" timelike curves exist); but a recent result shows that causality suffices.

Global hyperbolicity implies that there is a family of Cauchy surface

Cauchy surface

Intuitively, a Cauchy surface is a plane in space-time which is like an instant of time; its significance is that giving the initial conditions on this plane determines the future uniquely....

s for . A globally hyperbolic spacetime is topologically isomorphic to , for some Cauchy surface and some interval ; the metric structure need not respect this decomposition, however. Essentially, it means that everything that happens on is determined by the equations of motion, together with initial data specified on a surface.