In the study of Lorentzian manifold
spacetimeIn 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...
s there exists a hierarchy of
causality conditions which are important in proving mathematical theorems about the global structure of such manifolds. These conditions were collected during the late 1970s.
The weaker the causality condition on a spacetime, the more
unphysical the spacetime is. Spacetimes with
closed timelike curveIn 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, for example, present severe interpretational difficulties.
In the study of Lorentzian manifold
spacetimeIn 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...
s there exists a hierarchy of
causality conditions which are important in proving mathematical theorems about the global structure of such manifolds. These conditions were collected during the late 1970s.
The weaker the causality condition on a spacetime, the more
unphysical the spacetime is. Spacetimes with
closed timelike curveIn 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, for example, present severe interpretational difficulties. See the
grandfather paradoxThe grandfather paradox is a proposed paradox of time travel first described by the science fiction writer René Barjavel in his 1943 book Le Voyageur Imprudent . Nevertheless, similar paradoxes had already been described, for instance by Robert A. Heinlein in "By His Bootstraps"...
.
It is reasonable to believe that any physical spacetime will satisfy the strongest causality condtion: global hyperbolicity. For such spacetimes the equations in
general relativityGeneral 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...
can be posed as an
initial value problemIn mathematics, in the field of differential equations, an initial value problem is an ordinary differential equation together with specified value, called the initial condition, of the unknown function at a given point in the domain of the solution...
on a
Cauchy surfaceIntuitively, 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....
.
The hierarchy
There is a hierarchy of causality conditions, each one of which is strictly stronger than the previous. This is sometimes called the
causal ladder. The conditions, from weakest to strongest, are:
- Non-totally vicious
- Chronological
- Causal
- Distinguishing
- Strongly causal
- Stably causal
- Causally continuous
- Causally simple
- Globally hyperbolic
We now give definitions of these causality conditions for a Lorentzian manifold . Where two or more are given they are equivalent.
Notation:
- denotes the chronological relation.
- denotes the causal relation.
(See causal structure for definitions.)
Chronological
- There are no closed chronological (timelike) curves.
- The chronological relation is irreflexive: for all .
Past-distinguishing
- Two points which share the same chronological past are the same point:
- For any neighborhood of there exists a neighborhood such that no past-directed non-spacelike curve from intersects more than once.
Future-distinguishing
- Two points which share the same chronological future are the same point:
- For any neighborhood of there exists a neighborhood such that no future-directed non-spacelike curve from intersects more than once.
Strongly causal
- For any neighborhood of there exists no timelike curve that passes through more than once.
- For any neighborhood of there exists a neighborhood such that is causally convex in (and thus in ).
- The Alexandrov topology agrees with the manifold topology.
Stably causal
A manifold satisfying any of the weaker causality conditions defined above may fail to do so if the metric is given a small
perturbationPerturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem...
. A spacetime is stably causal if it cannot be made to contain closed causal curves by arbitrarily small perturbations of the metric.
Stephen HawkingStephen William Hawking, CH, CBE, FRS, FRSA is a British theoretical physicist. He is known for his contributions to the fields of cosmology and quantum gravity, especially in the context of black holes...
showed that this is equivalent to:
- There exists a global time function on . This is a scalar
In physics, a scalar is a simple physical quantity that is not changed by coordinate system rotations or translations , or by Lorentz transformations or space-time translations . -Physical quantity:...
field on whose gradient is everywhere timelike and future-directed. This global time function gives us a stable way to distinguish between future and past for each point of the spacetime (and so we have no causal violations).
Globally hyperbolic
- is strongly causal and every set (for points ) is compact.
Robert GerochRobert Geroch is a theoretical physicist and professor at the University of Chicago. He has worked prominently on general relativity and mathematical physics and has promoted the use of category theory in mathematics and physics. He was the Ph.D. supervisor for Abhay Ashtekar.Geroch obtained his Ph.D...
showed that a spacetime is globally hyperbolic
if and only ifIn logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements. In that it is biconditional, the connective can be likened to the standard material conditional combined with its reverse ; hence the name...
there exists a
Cauchy surfaceIntuitively, 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....
for . This means that:
- is topologically equivalent to for some 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....
(Here denotes the real lineIn mathematics, the real line is the line whose points correspond to the real numbers. That is, the real line is the set R of all real numbers, viewed as a geometric space...
).
See also
- 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...
- Lorentzian manifold
- 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...
- Globally hyperbolic
Globally hyperbolic is a term describing the causal structure of a spacetime manifold in Einstein's theory of general relativity, or potentially in other metric gravitational theories....
- 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...