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| 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 raised{{citation needed|date=August 2013}} by [[Kurt Gödel]] in 1949, who discovered a solution to the equations of [[general relativity]] (GR) allowing CTCs known as the [[Gödel metric]]; and since then other GR solutions containing CTCs have been found, such as the [[Tipler cylinder]] and [[Wormhole#Traversable wormholes|traversable wormholes]]. If CTCs exist, their existence would seem to imply at least the theoretical possibility of [[time travel]] backwards in time, raising the spectre of the [[grandfather paradox]], although the [[Novikov self-consistency principle]] 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]] which would replace GR, an idea which [[Stephen Hawking]] has labeled the [[chronology protection conjecture]]. Others note that if every closed timelike curve in a given space-time passes through an event horizon, 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.<ref name=monroe>{{cite journal | doi= 10.1007/s10701-008-9254-9 | author= H. Monroe | title = Are Causality Violations Undesirable? | year = 2008 | journal = Foundations of Physics | volume = 38 | pages = 1065–1069 |arxiv = gr-qc/0609054 | issue= 11 |bibcode = 2008FoPh...38.1065M }}</ref>
| | We frequently become emotionally disturbed in the course of a disagreement, or an argument, or when a person criticizes us or disagrees with us. In such situations our personality generally feels hurt, demeaned and in danger.<br><br>When we really feel this way, we destroy our personal happiness, clarity and overall health and frequently behave in approaches which we later regret.<br><br>A simple strategy for gaining clarity is to:<br><br>1. Get further on the affiliated web site - Hit this link: [http://www.newswire.net/newsroom/pr/00083003-adya-clarity.html newswire.net/newsroom/pr/00083003-adya-clarity.html/ reviews]. [http://www.britannica.com/search?query=Remember Remember] that concerning criticism, there are two possibilities:<br><br>a. The...<br><br>Why this is critical<br><br>We typically turn out to be emotionally disturbed during a disagreement, or an argument, or when a person criticizes us or disagrees with us. In such instances our personality normally feels hurt, demeaned and in danger.<br><br>When we really feel this way, we destroy our own happiness, clarity and health and frequently behave in techniques which we later regret.<br><br>A basic strategy for gaining clarity is to:<br><br>1. Don"t forget that regarding criticism, there are two possibilities:<br><br>a. If you have an opinion about jewelry, you will maybe fancy to study about [https://www.facebook.com/AdyaWater adya clarity profile]. The other particular person might be right in his or her observations and criticism. In this case we would advantage by admitting it and generating the correct adjustment in our behavior. We have every thing to acquire by listening and evolving by way of others comments.<br><br>What prevents us is the belief that we are not lovable if we are not perfect. As a result, we do not want to see or admit our faults. When we comprehend that we are worthy of love and respect even when we are not excellent or right, then we will be in a position to appear at our faults.<br><br>b. The other individual may be wrong. In this case, it is his or her projection, and we need not be affected by these misconceptions or projections. We have in this case the lesson of loving ourselves and also the other folks even when they perceive us in distorted and damaging approaches.<br><br>2. One resolution would be simply not to react 1 way or the other at very first, but to [http://photobucket.com/images/reflect reflect] upon what has been stated for some time so as to evaluate whether or not or not it is correct.<br><br>We can establish a space in our minds exactly where we can shop such queries about our character structure or actions so as to observe objectively for ourselves if they are true. If they are not, then we just continue on in the way we have been.<br><br>We want not really feel hurt, angry, defend ourselves, prove ourselves, or attack. To get another viewpoint, please gaze at: [http://www.blackmicaextract.biz/ clicky]. When we feel inner security and self-worth, we do not need to have to react in these approaches.<br><br>3. We can just thank the other for this feedback and tell him or her that we will consider about this observation and will progressively come to our conclusions, and if needed, make changes.<br><br>We do not want to live our lives in accordance with others perceptions. We can listen to all, but follow ourselves.<br><br>Note:<br><br>A "thank you" is sufficient.<br><br>(Adapted from the "The Psychology of Happiness" by Robert Najemy).<br><br>If you have any thoughts pertaining to wherever and how to use [http://ovalranch1562.soup.io health insurance rates], you can call us at the website. |
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| == Light cones == | |
| [[Image:Tilted light cone.png|frame|right|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 <math>x=0</math> at future times, and includes earlier times.]]
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| 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 ''t''<sub>0</sub> can only move to locations within ''c''(''t''<sub>1</sub> − ''t''<sub>0</sub>) by time ''t''<sub>1</sub>.
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| This is commonly represented on a graph with physical locations along the horizontal axis and time running vertically, with units of <math>t</math> 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 <math>ct</math> per <math>t</math>. 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.
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| 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 ''[[world line|worldline]]''. Another definition is that the light cone represents all possible worldlines.
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| In "simple" examples of [[Metric tensor (general relativity)|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, 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 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 <math>t</math> 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.
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| 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 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 though time as seen externally.
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| 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 [[time travel]] under these conditions.
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| == General relativity ==
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| CTCs appear in ''locally'' unobjectionable [[exact solutions in general relativity|exact solutions]] to the [[Einstein field equation]] of [[general relativity]], including some of the most important solutions. These include:
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| *the [[Misner space]] (which is [[Minkowski space]] [[orbifold]]ed by a discrete boost)
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| *the [[Kerr metric|Kerr vacuum]] (which models a rotating uncharged [[black hole]])
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| *the interior of a rotating [[BTZ black hole]]
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| *the [[van Stockum dust]] (which models a cylindrically symmetric configuration of [[dust solution|dust]])
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| *the [[Gödel metric|Gödel lambdadust]] (which models a dust with a carefully chosen cosmological constant term)
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| *the [[Tipler cylinder]] (a cylindrically symmetric metric with CTCs)
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| *[[Bonnor Steadman]] solutions describing laboratory situations such as two spinning balls
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| *[[J. Richard Gott]] has proposed a mechanism for creating CTCs using [[cosmic strings]].
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| 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.
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| == Consequences ==
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| 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.
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| No CTC can be continuously deformed as a CTC to a point (that is, a CTC and a point are not [[timelike homotopic]]), 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]].
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| 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 worldlines 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.
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| Existence of CTCs implies also equivalence of quantum and classical computation (both in [[PSPACE]]).<ref name=aaronson>{{cite journal | doi=10.1098/rspa.2008.0350|bibcode = 2009RSPSA.465..631A | title=Closed timelike curves make quantum and classical computing equivalent | year=2009 | last1=Watrous | first1=John | last2=Aaronson | first2=Scott | journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences | volume=465 | issue=2102 | pages=631 |arxiv = 0808.2669 }}</ref>
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| == Contractible versus noncontractible ==
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| 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]], 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.<ref name="monroe"/>
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| == Cauchy horizon ==
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| The [[chronology violating set]] is the set of points through which CTCs pass. The boundary of this set is the [[Cauchy horizon]]. 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.
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| == See also ==
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| * [[Timelike]]
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| * [[Causal structure]]
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| * [[Causality conditions]]
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| * [[Roman ring]]
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| == References ==
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| {{reflist}}
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| ===General References===
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| {{refbegin}}
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| *{{cite book | author=S. Carroll | title=Spacetime and Geometry | publisher=Addison Wesley | year=2004 | isbn=0-8053-8732-3}}
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| *{{cite journal | author=Kurt Gödel | title=An Example of a New Type of Cosmological Solution of Einstein's Field Equations of Gravitation | journal=Rev. Mod. Phys. | year=1949 | volume=21 | pages=447 | doi=10.1103/RevModPhys.21.447 | issue=3|bibcode = 1949RvMP...21..447G }}
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| *{{cite journal | doi= 10.1007/s10714-005-0163-3 | author= W. Bonnor | author2= B.R. Steadman | title = Exact solutions of the Einstein-Maxwell equations with closed timelike curves | year = 2005 | journal = Gen. Rel. Grav. | volume = 37 | pages = 1833 | issue= 11 |bibcode = 2005GReGr..37.1833B }}
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| {{refend}}
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| == External links ==
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| * [http://web.archive.org/web/20041015234901sh_re_/www.readmag.com/Columns/timetravel.htm A Primer on Time Travel] – (backup in the [[Internet Archive]])
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| <references/>
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| {{time travel}}
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| [[Category:Time travel]]
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| [[Category:Lorentzian manifolds]]
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We frequently become emotionally disturbed in the course of a disagreement, or an argument, or when a person criticizes us or disagrees with us. In such situations our personality generally feels hurt, demeaned and in danger.
When we really feel this way, we destroy our personal happiness, clarity and overall health and frequently behave in approaches which we later regret.
A simple strategy for gaining clarity is to:
1. Get further on the affiliated web site - Hit this link: newswire.net/newsroom/pr/00083003-adya-clarity.html/ reviews. Remember that concerning criticism, there are two possibilities:
a. The...
Why this is critical
We typically turn out to be emotionally disturbed during a disagreement, or an argument, or when a person criticizes us or disagrees with us. In such instances our personality normally feels hurt, demeaned and in danger.
When we really feel this way, we destroy our own happiness, clarity and health and frequently behave in techniques which we later regret.
A basic strategy for gaining clarity is to:
1. Don"t forget that regarding criticism, there are two possibilities:
a. If you have an opinion about jewelry, you will maybe fancy to study about adya clarity profile. The other particular person might be right in his or her observations and criticism. In this case we would advantage by admitting it and generating the correct adjustment in our behavior. We have every thing to acquire by listening and evolving by way of others comments.
What prevents us is the belief that we are not lovable if we are not perfect. As a result, we do not want to see or admit our faults. When we comprehend that we are worthy of love and respect even when we are not excellent or right, then we will be in a position to appear at our faults.
b. The other individual may be wrong. In this case, it is his or her projection, and we need not be affected by these misconceptions or projections. We have in this case the lesson of loving ourselves and also the other folks even when they perceive us in distorted and damaging approaches.
2. One resolution would be simply not to react 1 way or the other at very first, but to reflect upon what has been stated for some time so as to evaluate whether or not or not it is correct.
We can establish a space in our minds exactly where we can shop such queries about our character structure or actions so as to observe objectively for ourselves if they are true. If they are not, then we just continue on in the way we have been.
We want not really feel hurt, angry, defend ourselves, prove ourselves, or attack. To get another viewpoint, please gaze at: clicky. When we feel inner security and self-worth, we do not need to have to react in these approaches.
3. We can just thank the other for this feedback and tell him or her that we will consider about this observation and will progressively come to our conclusions, and if needed, make changes.
We do not want to live our lives in accordance with others perceptions. We can listen to all, but follow ourselves.
Note:
A "thank you" is sufficient.
(Adapted from the "The Psychology of Happiness" by Robert Najemy).
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