mathematical physics Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...

, a closed timelike curve (CTC) is a

world line The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics. The concept of a "world line" is distinguished from c ...

in a

Lorentzian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...

, of a material particle in

spacetime In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...

, that is "closed", returning to its starting point. This possibility was first discovered by

Willem Jacob van Stockum Willem Jacob van Stockum (20 November 1910 – 10 June 1944) was a Dutch mathematician who made an important contribution to the early development of general relativity. Biography Van Stockum was born in Hattem in the Netherlands. His father wa ...

in 1937 and later confirmed by

Kurt Gödel Kurt Friedrich Gödel ( , ; April 28, 1906 – January 14, 1978) was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel had an imme ...

in 1949,Stephen Hawking, '' My Brief History'', chapter 11 who discovered a solution to the equations of

general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...

(GR) allowing CTCs known as the

Gödel metric The Gödel metric, also known as the Gödel solution or Gödel universe, is an exact solution of the Einstein field equations in which the stress–energy tensor contains two terms, the first representing the matter density of a homogeneous d ...

; and since then other GR solutions containing CTCs have been found, such as the

Tipler cylinder A Tipler cylinder, also called a Tipler time machine, is a hypothetical object Theory, theorized to be a potential mode of time travel—although results have shown that a Tipler cylinder could only allow time travel if its length were infinite ...

and traversable wormholes. If CTCs exist, their existence would seem to imply at least the theoretical possibility of

time travel Time travel is the concept of movement between certain points in time, analogous to movement between different points in space by an object or a person, typically with the use of a hypothetical device known as a time machine. Time travel is a w ...

backwards in time, raising the spectre of the

grandfather paradox A temporal paradox, time paradox, or time travel paradox is a paradox, an apparent contradiction, or logical contradiction associated with the idea of time and time travel. The notion of time travel to the future complies with current understanding ...

, although the

Novikov self-consistency principle The Novikov self-consistency principle, also known as the Novikov self-consistency conjecture and Larry Niven's law of conservation of history, is a principle developed by Russian physicist Igor Dmitriyevich Novikov in the mid-1980s. Novikov int ...

seems to show that such paradoxes could be avoided. Some physicists speculate that the CTCs which appear in certain GR solutions might be ruled out by a future theory of

quantum gravity Quantum gravity (QG) is a field of theoretical physics that seeks to describe gravity according to the principles of quantum mechanics; it deals with environments in which neither gravitational nor quantum effects can be ignored, such as in the vi ...

which would replace GR, an idea which Stephen Hawking labeled the

chronology protection conjecture The chronology protection conjecture is a hypothesis first proposed by Stephen Hawking that laws of physics beyond those of standard general relativity prevent time travel on all but microscopic scales - even when the latter theory states that it ...

. Others note that if every closed timelike curve in a given space-time passes through an

event horizon In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s. In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact obj ...

, a property which can be called chronological censorship, then that space-time with event horizons excised would still be causally well behaved and an observer might not be able to detect the causal violation.

Light cones

When discussing the evolution of a system in

, or more specifically

Minkowski space In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inerti ...

, physicists often refer to a "

light cone In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take thro ...

". 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 The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...

. For instance, an object located at position ''p'' at time ''t''₀ can only move to locations within ''p'' + ''c''(''t''₁ − ''t''₀) by time ''t''₁. 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 ''timewise separated'' if they differ along the time axis, or ''spacewise separated'' if they differ along the space axis. If the object were in

free fall In Newtonian physics, free fall is any motion of a body where gravity is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on ...

, 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 The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics. The concept of a "world line" is distinguished from con ...

''. Another definition is that the light cone represents all possible worldlines. 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 move instantly to another location. In these spacetimes, the worldlines of physical objects are, by definition, timewise. However this orientation is only true of "locally flat" spacetimes. In curved spacetimes the light cone will be "tilted" along the spacetime's

geodesic In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...

. For instance, while moving in the vicinity of 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 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 In special relativity, the rest frame of a particle is the frame of reference (a coordinate system attached to physical markers) in which the particle is at rest. The rest frame of compound objects (such as a fluid, or a solid made of many vibratin ...

. From this outside viewpoint, the object can move instantaneously through space. In these situations the object would ''have'' to move, since its present spatial 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 through time as seen externally. 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. 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

under these conditions.

General relativity

CTCs appear in ''locally'' unobjectionable

exact solutions In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first ...

to the

Einstein field equation In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form ...

, including some of the most important solutions. These include: *the

Misner space Misner space is an abstract mathematical spacetime, first described by Charles W. Misner. It is also known as the Lorentzian orbifold \mathbb^/\text. It is a simplified, two-dimensional version of the Taub–NUT spacetime. It contains a non-curvat ...

(which is

orbifold In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space. D ...

ed by a discrete boost) *the Kerr vacuum (which models a rotating uncharged

black hole A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts t ...

) *the interior of a rotating

BTZ black hole The BTZ black hole, named after Máximo Bañados, Claudio Teitelboim, and Jorge Zanelli, is a black hole solution for (2+1)-dimensional topological gravity with a negative cosmological constant. History In 1992 Bañados, Teitelboim and Zanel ...

*the

van Stockum dust In general relativity, the van Stockum dust is an exact solution of the Einstein field equations in which the gravitational field is generated by dust rotating about an axis of cylindrical symmetry. Since the density of the dust is ''increasing' ...

(which models a cylindrically symmetric configuration of

dust Dust is made of fine particles of solid matter. On Earth, it generally consists of particles in the atmosphere that come from various sources such as soil lifted by wind (an aeolian process), volcanic eruptions, and pollution. Dust in homes ...

) *the Gödel lambdadust (which models a dust with a carefully chosen cosmological constant term) *the

(a cylindrically symmetric metric with CTCs) * Bonnor-Steadman solutions describing laboratory situations such as two spinning balls *

J. Richard Gott John Richard Gott III (born February 8, 1947) is a professor of astrophysics, astrophysical sciences at Princeton University. He is known for his work on time travel and the Doomsday argument. Exotic matter time travel theories Paul Davies's ...

has proposed a mechanism for creating CTCs using

cosmic strings Cosmic strings are hypothetical 1-dimensional topological defects which may have formed during a symmetry-breaking phase transition in the early universe when the topology of the vacuum manifold associated to this symmetry breaking was not simp ...

. 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.Roy Kerr (Crafoord Prize Symposium in Astronomy):
Spinning Black Holes
'. (YouTube, Timestamp 26m)

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 Causality (also referred to as causation, or cause and effect) is influence by which one event, process, state, or object (''a'' ''cause'') contributes to the production of another event, process, state, or object (an ''effect'') where the cau ...

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

states complete knowledge of the universe on a spacelike

Cauchy surface In the mathematical field of Lorentzian geometry, a Cauchy surface is a certain kind of submanifold of a Lorentzian manifold. In the application of Lorentzian geometry to the physics of general relativity, a Cauchy surface is usually interpreted as ...

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 In physics, a Cauchy horizon is a light-like boundary of the domain of validity of a Cauchy problem (a particular boundary value problem of the theory of partial differential equations). One side of the horizon contains closed space-like geod ...

, and a region of spacetime that cannot be predicted from perfect knowledge of some past time. No CTC can be continuously deformed as a CTC to a point (that is, a CTC and a point are not

timelike homotopic On a Lorentzian manifold, certain curves are distinguished as timelike. A timelike homotopy between two timelike curves is a homotopy such that each intermediate curve is timelike. No closed timelike curve (CTC) on a Lorentzian manifold is timelike ...

), as the manifold would not be causally well behaved at that point. The topological feature which prevents the CTC from being deformed to a point is known as a

timelike topological feature In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime, that is "closed", returning to its starting point. This possibility was first discovered by Willem Jacob van Sto ...

. The existence of CTCs would arguably place restrictions on physically allowable states of matter-energy fields in the universe. Propagating a field configuration along the family of closed timelike worldlines must, according to such arguments, eventually result in the state that is identical to the original one. This idea has been explored by some scientists as a possible approach towards disproving the existence of CTCs. While quantum formulations of CTCs have been proposed, a strong challenge to them is their ability to freely create entanglement, which quantum theory predicts is impossible. If Deutsch's prescription holds, the existence of these CTCs implies also equivalence of quantum and classical computation (both in

PSPACE In computational complexity theory, PSPACE is the set of all decision problems that can be solved by a Turing machine using a polynomial amount of space. Formal definition If we denote by SPACE(''t''(''n'')), the set of all problems that can b ...

). If Lloyd's prescription holds, quantum computations would be PP-complete.

Contractible versus noncontractible

There are two classes of CTCs. We have CTCs contractible to a point (if we no longer insist it has to be future-directed timelike everywhere), and we have CTCs which are not contractible. For the latter, we can always go to the

universal covering space A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...

, and reestablish causality. For the former, such a procedure is not possible. No closed timelike curve is contractible to a point by a timelike homotopy among timelike curves, as that point would not be causally well behaved.

Cauchy horizon

The

chronology violating set In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold. Introduction In modern physics (especially general relativity) spacetime is represented by a Lorentzian m ...

is the set of points through which CTCs pass. The boundary of this set is the

. The Cauchy horizon is generated by closed null geodesics. Associated with each closed null geodesic is a redshift factor describing the rescaling of the rate of change of the affine parameter around a loop. Because of this redshift factor, the affine parameter terminates at a finite value after infinitely many revolutions because the geometric series converges.

Notes

References

* * * *

External links

A Primer on Time Travel
(backup in the

Internet Archive The Internet Archive is an American digital library with the stated mission of "universal access to all knowledge". It provides free public access to collections of digitized materials, including websites, software applications/games, music, ...

) {{time travel Time travel Lorentzian manifolds