In a Lorentzian manifold, a closed timelike curve (CTC) is a worldline of a material particle in spacetime that is closed. This possibility was raised by Willem Jacob van Stockum in 1937 and by Kurt Gödel in 1949. If CTCs exist, their existence would seem to imply at least the theoretical possibility of making a time machine, as well as raising the spectre of the grandfather paradox. In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... A world line of an object or person is the sequence of events labeled with time and place, that marks the history of the object or person. ... In physics, spacetime is a mathematical model that combines space and time into a single construct called the space-time continuum. ... Willem Jacob van Stockum (November 20, 1910-June 10, 1944) was a physicist who made an important contribution to the early development of general relativity. ... [...]I dont believe in natural science. ... Time travel is a concept that has long fascinated humanity—whether it is Merlin experiencing time backwards, or religious traditions like Mohammeds trip to Jerusalem and ascent to heaven, returning before a glass knocked over had spilt its contents. ... The grandfather paradox is a paradox of time travel, first conceived by the science fiction writer René Barjavel in his 1943 book Le Voyageur Imprudent (The Imprudent Traveller) [1]. The paradox, stated in the second person, is this: Suppose you traveled back in time and killed your biological grandfather before...

Light cones

The lower light cone is characteristic of light cones in flat space - all spacetime coordinates included in the light cone have later times. The upper light cone not only includes other spatial locations at the same time, it doesn't include x=0 at future times, and includes earlier times.

When discussing the evolution of a system in general relativity, or more specifically Minkowski space, physicists often refer to a "light cone". A light cone represents any possible future evolution of an object given its current state, or every possible location given its current location. An object's possible future locations are limited by the speed that the object can move, which is at best the speed of light. For instance, an object located at position p at time t₀ can only move to locations within c(t₁ − t₀) in time t₁. Image File history File links No higher resolution available. ... General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... In special relativity, a light cone is the pattern describing the temporal evolution of a flash of light in Minkowski spacetime. ... A line showing the speed of light on a scale model of Earth and the Moon The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. It is the speed of all electromagnetic radiation...

This is commonly represented on a graph with physical locations along the horizontal axis and time running vertically, with units of t for time and ct for space. Light cones in this representation appear as lines at 45 degrees centered on the object, as light travels at ct per t. On such a diagram, every possible future location of the object lies within the cone. Additionally, every space location has a future time, implying that an object may stay at any location in space indefinitely.

Any single point on such a diagram is known as an event. Separate events are considered to be timelike if they are separated across the time axis, or spacelike if they differ along the space axis. If the object were in free fall it would travel up the t axis, if it accelerates it moves across the x axis as well. The actual path an object takes through spacetime, as opposed to the ones it could take, is known as the worldline. Another definition is that the light cone represents all possible worldlines. Free fall in its strictest sense is the condition of acceleration which is due only to gravity. ... In physics, the world line of an object is the singular path of that object as it travels through 4-dimensional spacetime. ...

In "simple" examples of spacetime metrics the light cone is directed forward in time. This corresponds to the common case that an object cannot be in two places at once, or alternately that it cannot moved instantly to another location. In these spacetimes, the worldlines of physical objects are, by definition, timelike. However this orientation is only true of "locally flat" spacetimes. In curved spacetimes the light cone will be "tilted" along the spacetime's geodesic. For instance, while moving in the vicinity a star, the star's gravity will "pull" on the object, affecting its worldline, so its possible future positions lie closer to the star. This appears as a slightly tilted lightcone on the corresponding spacetime diagram. An object in free fall in this circumstance continues to move along its local t axis, but to an external observer it appears it is accelerating in space as well – a common situation if the object is in orbit, for instance. In general relativity, the metric tensor (or simply the metric) is the fundamental object of study. ... In mathematics, a geodesic is a generalization of the notion of a straight line to curved spaces. In presence of a metric, geodesics are defined to be (locally) the shortest path between points on the space. ...

In extreme examples, in spacetimes with suitably high-curvature metrics, the light cone can be tilted beyond 45 degrees. That means there are potential "future" positions, from the object's frame of reference, that are spacelike separated to observers in an external rest frame. From this outside viewpoint, the object can move instantaneously through space. In these situations the object would have to move, since its present spacial location would not be in its own future light cone. Additionally, with enough of a tilt, there are event locations that lie in the "past" as seen from the outside. With a suitable movement of what appears to it its own space axis, the object appears to travel though time as seen externally. In special relativity the rest frame of a particle is the coordinate system (frame of reference) in which the particle is at rest. ...

A closed timelike curve can be created if a series of such light cones are set up so as to loop back on themselves, so it would be possible for an object to move around this loop and return to the same place and time that it started. Orbits around high-density objects with extreme gravitational forces are an example of such a closed loop. An object in such an orbit would repeatedly return to the same point in spacetime if it stays in free fall. Returning to the original spacetime location would be only one possibility; the object's future light cone would include spacetime points both forwards and backwards in time, and so it should be possible for the object to engage in time travel under these conditions. This is the mechanism the Tipler Cylinder would use to be a time machine. Time travel is a concept that has long fascinated humanity—whether it is Merlin experiencing time backwards, or religious traditions like Mohammeds trip to Jerusalem and ascent to heaven, returning before a glass knocked over had spilt its contents. ... This article is in need of attention from an expert on the subject. ... Time travel is a concept that has long fascinated humanity—whether it is Merlin experiencing time backwards, or religious traditions like Mohammeds trip to Jerusalem and ascent to heaven, returning before a glass knocked over had spilt its contents. ...

General relativity

CTCs have an unnerving habit of appearing in locally unobjectionable exact solutions to the Einstein field equation of general relativity, including some of the most important solutions. These include: // In general relativity, an exact solution is a Lorentzian manifold equipped with certain tensor fields which are taken to model states of ordinary matter, such as a fluid, or classical nongravitational fields such as the electromagnetic field. ... For other topics related to Einstein see Einstein (disambig) In physics, the Einstein field equation or the Einstein equation is a tensor equation in the theory of gravitation. ... General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...

• the Kerr vacuum (which models a rotating uncharged black hole)
• the van Stockum dust (which models a cylindrically symmetric configuration of dust),
• the Gödel lambdadust (which models a dust with a carefully chosen cosmological constant term).
• J. Richard Gott has proposed a mechanism for creating CTCs using cosmic strings.

Some of these examples are, like the Tipler cylinder, rather artificial, but the exterior part of the Kerr solution is thought to be in some sense generic, so it is rather unnerving to learn that its interior contains CTCs. Most physicists feel that CTCs in such solutions are artifacts. In general relativity, the Kerr metric (or Kerr vacuum) describes the geometry of spacetime around a rotating massive body, such as a rotating black hole. ... In general relativity, the van Stockum dust is an exact solution of the Einstein field equation in which the gravitational field is generated by dust particles which are rotating about an axis of cylindrical symmetry. ... In general relativity, a dust solution is an exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a perfect fluid which has positive mass density but vanishing pressure. ... The Gödel solution is an exact solution of the Einstein field equation in which the stress-energy tensor contains two terms, the first representing the matter density of a homogeneous distribution of swirling dust particles, and the second associated with a nonzero cosmological constant (see lambdavacuum solution). ... J. Richard Gott is a professor of astrophysical sciences at Princeton University John Richard Gott III is especially well known for developing and advocating two cosmological theories with the flavour of science fiction: Time travel, and the Doomsday argument. ... A cosmic string is a hypothetical 1-dimensional topological defect in the fabric of spacetime. ...

Consequences

One feature of a CTC is that it opens the possibility of a worldline which is not connected to earlier times, and so the existence of events that cannot be traced to an earlier cause. Ordinarily, causality demands that each event in spacetime is preceded by its cause in every rest frame. This principle is critical in determinism, which in the language of general relativity states complete knowledge of the universe on a spacelike Cauchy surface can be used to calculate the complete state of the rest of spacetime. However, in a CTC, causality breaks down, because an event can be "simultaneous" with its cause – in some sense an event may be able to cause itself. It is impossible to determine based only on knowledge of the past whether or not something exists in the CTC that can interfere with other objects in spacetime. A CTC therefore results in a Cauchy horizon, and a region of spacetime that cannot be predicted from perfect knowledge of some past time. It has been suggested that this article be split into multiple articles accessible from a disambiguation page. ... Determinism is the philosophical proposition that every event, including human cognition, decision and action, is causally determined by an unbroken chain of prior occurrences. ... General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ... This article is in need of attention from an expert on the subject. ... In physics, a Cauchy horizon is a light_like boundary of the domain of validity of a Cauchy problem. ...

No CTC can be continuously deformed as a CTC (is timelike homotopic) to a point, as that point would not be causally well behaved. The topological feature which prevents the CTC from being deformed to a point is known as a timelike topological feature. On a Lorentzian manifold, certain curves are distinguished as timelike. ... No closed timelike curve (CTC) on a Lorentzian manifold can be continuously deformed as a CTC to a point, because Lorentzian manifolds are locally causally well-behaved. ...

Existence of CTCs places restrictions on physically allowable states of matter-energy fields in the universe. Propagating a field configuration along the family of closed timelike wordlines must eventually result in the state that is identical to the original one. This has been explored by some scientists as a possible approach towards disproving the existence of CTCs.

See also

• Timelike
• General relativity

In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... General relativity (GR) [also called the general theory of relativity (GTR) and general relativity theory (GRT)] is the geometrical theory of gravitation published by Albert Einstein in 1915/16. ...

References

• S. Carroll (2004). Spacetime and Geometry. Addison Wesley. ISBN 0-8053-8732-3. 
• Kurt Gödel (1949). "An Example of a New Type of Cosmological Solution of Einstein's Field Equations of Gravitation". Rev. Mod. Phys. 21: 447. 

External links

• A Primer on Time Travel -(backup in the Internet Archive)