Time
TIME
TIME, CLOSED TIMELIKE CURVES AND CAUSALITYF Introduction It seems to be extremely difficult to give a precise definition of Time, this mysterious ingredient of the Universe. Intuitively, we have the notion of timeas something that flows. Ancient religions have registered it as somethingunusual, and many myths are built into their dogmas.The ancient Greeks conveyed the image of Chronos, or Father Time.Plato assumed that time had a beginning, looping back into itself. Thisnotion of circular time, was probably inspired by phenomena observed inNature, namely the alternation of day and night, the repetition of the sea-sons, etc. But, it was in the Christian theological doctrine that the uniquecharacter of historical events gave rise to a linear notion of time. Aristotle,a keen natural philosopher, stated that time was related to motion, i.e.,to change. An idea reflected in his famous metaphor: Time is the movingimage of Eternity.Reflections on time can be encountered in many philosophical consid-erations and works over the ages, culminating in Newton�s Absolute Time.Newton stated that time flowed at the same rate for all observers in theUniverse. But in 1905, Einstein changed altogether our notion of time. Timeflowed at different rates for different observers, and Minkowski, three yearslater, formally united the parameters of time and space, giving rise to thenotion of a four-dimensional entity, spacetime.Later, Einstein influenced by Mach�s Principle, was motivated to seeka theory in which the structure of spacetime was influenced by the pres-ence of matter, and presented the field equations of the General Theory ofRelativity in 1915. Adopting a pragmatic point of view, to measure time achanging configuration of matter is needed, i.e., a swinging pendulum, etc.Change seems to be imperative to have an emergent notion of time. Therefore, time is empirically related to change. But change can beconsidered as a variation or sequence of occurrences. Thus, intuitively, asequence of successive occurrences, provides us with a notion of somethingthat flows, i.e., it provides us with the notion of Time. Time flows andeverything relentlessly moves along this stream.In Relativity, we can substitute the above empirical notion of a sequenceof occurrences by a sequence of events. We idealize the concept of an eventto become a point in space and an instant in time.Following this reasoning, a sequence of events has a determined temporalorder. We experimentally verify that specific events occur before others andnot vice-versa. Certain events (effects) are triggered off by others (causes),providing us with the notion of causality.2. Closed Timelike Curves and Associated Paradoxes of TimeTravelThe conceptual definition and understanding of Time, both quantitativelyand qualitatively is of the utmost difficulty and importance. Special Relativ-ity provides us with important quantitative elucidations of the fundamentalprocesses related to time dilation effects. The General Theory of Relativity(GTR) provides a deep analysis to effects of time flow in the presence ofstrong and weak gravitational fields.As time is incorporated into the proper structure of the fabric of space-time, it is interesting to note that GTR is contaminated with non-trivialgeometries which generate closed timelike curves [1]. A closed timelike curve(CTC) allows time travel, in the sense that an observer which travels on atrajectory in spacetime along this curve, returns to an event which coincideswith the departure. The arrow of time leads forward, as measured locallyby the observer, but globally he/she may return to an event in the past.This fact apparently violates causality, opening Pandora�s box and produc-ing time travel paradoxes [2], throwing a veil over our understanding of thefundamental nature of Time. The notion of causality is fundamental in theconstruction of physical theories, therefore time travel and its associatedparadoxes have to be treated with great caution. The paradoxes fall intotwo broad groups, namely the consistency paradoxes and the causal loops.The consistency paradoxes include the classical grandfather paradox.Imagine travelling into the past and meeting one�s grandfather. Nurturinghomicidal tendencies, the time traveller murders his grandfather, prevent-ing the birth of his father, therefore making his own birth impossible. Infact, there are many versions of the grandfather paradox, limited only byone�s imagination. The consistency paradoxes occur whenever possibilitiesof changing events in the past arise. TIME, CLOSED TIMELIKE CURVES AND CAUSALIT The paradoxes associated with causal loops are related to self-existinginformation or objects, trapped in spacetime. Imagine a time traveller go-ing back to his past, handing his younger self a manual for the constructionof a time machine. The younger version then constructs the time machineover the years, and eventually goes back to the past to give the manualto his younger self. The time machine exists in the future because it wasconstructed in the past by the younger version of the time traveller. Theconstruction of the time machine was possible because the manual was re-ceived from the future. Both parts considered by themselves are consistent,and the paradox appears when considered as a whole. One might inquireas to the origin of the manual, since its worldline is a closed loop. There isa manual never created, nevertheless existing in spacetime, although thereare no causality violations.3. Solutions of the EFEs Generating CTCsA great variety of solutions to the Einstein Field Equations (EFEs) con-taining CTCs exist, but two particularly notorious features seem to standout. Solutions with a tipping over of the light cones due to a rotation abouta cylindrically symmetric axis; and solutions that violate the Energy Con-ditions of GTR, which are fundamental in the singularity theorems andtheorems of classical black hole thermodynamics [3].3.1. STATIONARY, AXISYMMETRIC SOLUTIONSThe tipping over of light cones seem to be a generic feature of some solutionswith a rotating cylindrical symmetry. The general metric for a stationary,axisymmetric solution with rotation is given by [1, 4]:ds2= -A(r)dt2+ 2B(r)df dt + C(r)df2+ D(r)(dr2+ dz2)(1)The range of the coordinates is: t ? (-8,+8), r ? (0,+8), f ? [0,2p],and z ? (-8,+8), respectively. The metric components are functions ofr alone. It is clear that the determinant, g = det(g�?) = -(AC + B2)D2isLorentzian, provided that (AC + B2) > 0.Due to the periodic nature of the angular coordinate, f, an azimuthalcurve with ? = {t = const,r = const,z = const} is a closed curve ofinvariant length s2?= C(r)(2p)2. If C(r) is negative then the integral curvewith (t, r, z) fixed is a CTC.The present work is far from making an exhaustive search of all theEFE solutions generating CTCs with these features, but the best knownspacetimes will be briefly analyzed, namely, the van Stockum spacetime,the G�odel universe and spinning cosmic strings. Spacetime The earliest solution to the EFEs containing CTCs, is probably that of thevan Stockum spacetime [1, 5]. It is a stationary, cylindrically symmetricsolution describing a rapidly rotating, infinitely long cylinder of dust, sur-rounded by vacuum. The centrifugal forces of the dust are balanced by thegravitational attraction. Consider R the surface of the cylinder.The metric for the interior solution r < R, is given by:ds2= -dt2+ 2?r2df dt + r2(1 - ?2r2)df2+ exp(-?2r2)(dr2+ dz2) (2)where ? is the angular velocity of the cylinder. It is trivial to verify thatCTCs arise if ?r > 1. Causality violation can also be verified for ?R > 1/2,in the exterior region.3.1.2. Spinning Cosmic StringConsider an infinitely long straight string that lies along and spins aroundthe z-axis. The symmetries are analogous to the van Stockum spacetime,but the asymptotic behavior is different [1].We restrict the analysis to an infinitely long straight string, with a delta-function source confined to the z-axis. It is characterized by a mass per unitlength, �; a tension, t , and an angular momentum per unit length, J.In cylindrical coordinates the metric takes the following form:ds2= -[d(t + 4GJf)]2+ dr2+ (1 - 4G�)2r2df2+ dz2(3)Consider an azimuthal curve, i.e., an integral curve of f. Closed timelikecurves appear whenever r < 4GJ/(1 - 4G�).3.1.3. The G�odel UniverseKurt G�odel in 1949 discovered an exact solution to the EFEs of a uni-formly rotating universe containing dust and a nonzero cosmological con-stant. Writing the metric in a form in which the rotational symmetry of thesolution, around the axis r = 0, is manifest and suppressing the irrelevantz coordinate, we have [3, 6]:ds2= 2w2(-dt2+ dr2-(sinh4r -sinh2r) df2+ 2(v2) sinh2r df dt) (4)Moving away from the axis, the light cones open out and tilt in the f-direction. The azimuthal curves with ? = {t = const,r = const,z = const}are CTCs if the condition r > ln(1 + v2) is satisfied.3.2. SOLUTIONS VIOLATING THE ENERGY CONDITIONSThe traditional manner of solving the EFEs, G�?= 8pGT�?, consists inconsidering a plausible stress-energy tensor, T�?, and finding the geomet-rical structure, G�?. But one can run the EFE in the reverse direction by TIME, CLOSED TIMELIKE CURVES AND CAUSALITY imposing an exotic metric g�?, and eventually finding the matter source forthe respective geometry.In this fashion, solutions violating the energy conditions have been ob-tained. One of the simplest energy conditions is the weak energy condition(WEC), which states: T�?U�U?= 0, in which U�is a timelike vector.This condition is equivalent to the assumption that any timelike observermeasures a local positive energy density. Although classical forms of matterobey these energy conditions, violations have been encountered in quantumfield theory, the Casimir effect being a well-known example.Adopting the reverse philosophy, solutions such as traversable worm-holes, the warp drive, the Krasnikov tube and the Ori-Soen spacetime havebeen obtained. These solutions violate the energy conditions and with sim-ple manipulations generate CTCs.3.2.1. Traversable Wormholes, the Gott Spacetime and the Ori-SoenSolutionMuch interest in traversable wormholes had been aroused since the classicalarticle by Morris and Thorne [7]. A wormhole is a hypothetical tunnelwhich connects different regions in spacetime. These solutions are multiply-connected and probably involve a topology change, which by itself is aproblematic issue. One of the most fascinating aspects of wormholes is theirapparent ease in generating CTCs. There are several ways to generate atime machine using multiple wormholes [1], but a manipulation of a singlewormhole seems to be the simplest way [8].An extremely elegant model of a time-machine was constructed by Gott[9]. It is an exact solution to the EFE for the general case of two movingstraight cosmic strings that do not intersect. This solution produces CTCseven though they do not violate the WEC, have no singularities and eventhorizons, and are not topologically multiply-connected as the wormholesolution. The appearance of CTCs relies solely on the gravitational lenseffect and the relativity of simultaneity.A time-machine model was also proposed by Amos Ori and Yoav Soenwhich significantly ameliorates the conditions of the EFE�s solutions whichgenerate CTCs [10]. The Ori-Soen model presents some notable features.It was verified that CTCs evolve from a well-defined initial slice, a partialCauchy surface, which does not display causality violation. The partialCauchy surface and spacetime are asymptotically flat, contrary to the Gottspacetime, and topologically trivial, contrary to the wormhole solutions.The causality violation region is constrained within a bounded region ofspace, and not at infinity as in the Gott solution. The WEC is satisfieduntil and beyond a time slice t = 1/a, on which the CTCs appear. The Alcubierre Warp Drive and the Krasnikov Solution Within the framework of general relativity, it is possible to warp spacetime in a small bubblelike region [11], in such a way that the bubble may attainarbitrarily large velocities, v(t). Inspired in the inflationary phase of theearly Universe, the enormous speed of separation arises from the expansionof spacetime itself. The model for hyperfast travel is to create a local dis-tortion of spacetime, producing an expansion behind the bubble, and anopposite contraction ahead of it.One may consider a spaceship immersed within the bubble, movingalong a timelike curve, regardless of the value of v(t). Due to the arbitraryvalue of the warp bubble velocity, the metric of the warp drive permitssuperluminal travel. This possibility raises the question of the existenceof CTCs. Although the solution deduced by Alcubierre by itself does notpossess CTCs, Everett demonstrated that these are created by a simplemodification of the Alcubierre metric [12], by applying a similar analysisas in tachyons.Krasnikov discovered an interesting feature of the warp drive, in whichan observer in the center of the bubble is causally separated from the frontedge of the bubble. Therefore he/she cannot control the Alcubierre bubbleon demand. Krasnikov proposed a two-dimensional metric [13], which waslater extended to a four-dimensional model [14]. Using two such tubes it isa simple matter, in principle, to generate CTCs.4. ConclusionGTR has been an extremely successful theory, with a well established ex-perimental footing, at least for weak gravitational fields. Its predictionsrange from the existence of black holes and gravitational radiation to thecosmological models, which predict a primordial beginning, namely the big-bang.However, it was seen that it is possible to find solutions to the EFEs,with certain ease, which generate CTCs. This implies that if we considerGTR valid, we need to include the possibility of time travel in the form ofCTCs. A typical reaction is to exclude time travel due to the associatedparadoxes. But the paradoxes do not prove that time travel is mathemat-ically or physically impossible. Consistent mathematical solutions to theEFEs have been found, based on plausible physical processes. What theydo seem to indicate is that local information in spacetimes containing CTCsare restricted in unfamiliar ways.The grandfather paradox, without doubt, does indicate some strangeaspects of spacetimes that contain CTCs. It is logically inconsistent thatthe time traveller murders his grandfather. But, one can ask, what exactly TIME, CLOSED TIMELIKE CURVES AND CAUSALITY prevented him from accomplishing his murderous act if he had ample op-portunities and the free will to do so. It seems that certain conditions inlocal events are to be fulfilled for the solution to be globally self-consistent.These conditions are denoted consistency constraints [15]. To eliminate theproblem of free will, mechanical systems were developed, such as the self-collision of billiard balls in the presence of CTCs [16]. These do not conveythe associated philosophical speculations on free will related to human be-ings. Much has been written on two possible remedies to the paradoxes,namely the Principle of Self-Consistency and the Chronology ProtectionConjecture.Igor Novikov is a leading advocate for the Principle of Self-Consistency,which stipulates that events on a CTC are self-consistent, i.e., events influ-ence one another along the curve in a cyclic and self-consistent way. In thepresence of CTCs the distinction between past and future events is ambigu-ous, and the definitions considered in the causal structure of well-behavedspacetimes break down. What is important to note is that events in thefuture can influence, but cannot change, events in the past.The Principle of Self-Consistency permits one to construct local solu-tions of the laws of physics, only if these can be prolonged to a unique globalsolution, defined throughout non-singular regions of spacetime. Therefore,according to this principle, the only solutions of the laws of physics that areallowed locally, reinforced by the consistency constraints, are those whichare globally self-consistent.Hawking�s Chronology Protection Conjecture is a more conservativeway of dealing with the paradoxes. Hawking notes the strong experimentalevidence in favour of the conjecture from the fact that �we have not beeninvaded by hordes of tourists from the future� [17].An analysis reveals that the renormalized expectation value of the quan-tum stress-energy tensor diverges as one gets close to CTC formation. Thisconjecture permits the existence of traversable wormoles, but prohibits theappearance of CTCs. The transformation of a wormhole into a time machineresults in enormous effects of the vacuum polarization, which destroys itsinternal structure. There is no convincing demonstration of the ChronologyProtection Conjecture, but the hope exists that a future theory of quantumgravity may prohibit CTCs.In addition to these remedies, Visser considers two other conjectures[1]. The first is the radical reformulation of physics conjecture, in which oneabandons the causal structure of the laws of physics and allows, withoutrestriction, time travel, reformulating physics from the ground up. The sec-ond is the boring physics conjecture, in which one simply ceases to considerthe solutions to the EFEs generating CTCs.Perhaps an eventual quantum gravity theory will provide us with the But, as stated by Thorne [18], it is by extending the theory to its extreme predictions that one can get important insights to its limitations,and probably ways to overcome them. Therefore, time travel in the formof CTCs is more than a justification for theoretical speculation, it is aconceptual tool and an epistemological instrument to probe the deepestlevels of GTR and extract clarifying views.
2 Responses to “Time”
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February 4th, 2008 at 11:02 pm
I do believe you are trying to reach to far from your point of origin. Since 1 day has been stretched out into 7 ages, counting down to 1 day. Time travel is done by the use of Static Electricity. Traveling in straight lines from point A to point B by setting Cosmic Coordinates in your timeship you travel in a tunnel of space, not thru space.
June 30th, 2008 at 3:35 am
If time travel exists in the future and enables either travelers or objects to travel to the past, we should either see evidence of it now or else traveling to the past is limited only to the timeline of the person attempting it. In addition, if one goes at a speed that slows time down significantly, so that when you return to Earth you visit the planet hundreds of years after you left, did you really travel in time? We know how to and do speed time up and slow it down. However, it seems somewhat pointless unless one can travel in all times and dimensions. At that point, we will have to argue about the definition of eternal life.