PHYSICAL REVIEW D 72, 043532 (2005) Cosmic string lensing and closed timelike curves
Benjamin Shlaer* and S.-H. Henry Tye† Laboratory for Elementary Particle Physics, Cornell University, Ithaca, New York 14853, USA (Received 3 March 2005; revised manuscript received 21 July 2005; published 31 August 2005) In an analysis of the gravitational lensing by two relativistic cosmic strings, we argue that the formation of closed timelike curves proposed by Gott is unstable in the presence of particles (e.g. the cosmic microwave background radiation). Because of the attractorlike behavior of the closed timelike curve, we argue that this instability is very generic. A single graviton or photon in the vicinity, no matter how soft, is sufficient to bend the strings and prevent the formation of closed timelike curves. We also show that the gravitational lensing due to a moving cosmic string is enhanced by its motion, not suppressed.
DOI: 10.1103/PhysRevD.72.043532 PACS numbers: 98.80.Cq, 04.20.Gz, 04.60.Kz, 11.27.+d
I. INTRODUCTION strings in the sense opposite to their motion becomes a CTC. This is sometimes called the Gott spacetime or Gott Although cosmic strings as the seed of structure forma- time machine. tion [1] has been ruled out by observations, their presence Although there is no proof that a time machine cannot at a lower level is still possible. Indeed, cosmic strings are exist in our world [13], their puzzling causal nature leads generically present in brane inflation in superstring theory, many physicists to believe that CTCs cannot be formed. and their properties are close to, but within all observatio- This skepticism has been encoded in Hawking’s chronol- nal bounds [2–7]. This is exciting for many reasons since ogy protection conjecture (CPC) [14]. However, CPC as current and near future cosmological experiments/obser- proposed is not very precise; even if we assume CPC is vations will be able to confirm or rule out this explicit correct, it is not clear exactly what law of physics will stringy prediction. If detected, the rich properties of cosmic prevent the specific Gott spacetime. There are a number of strings as well as their inflationary signatures provide a interesting and insightful studies attempting to apply CPC window to both the superstring theory and our preinfla- against the Gott’s spacetime: tionary universe. The cosmic strings are expected to evolve (a) Recall that the original Chronology Protection to a scaling string network with a spectrum of tensions. Conjecture [14] is motivated by two results. The Roughly speaking, the physics and the cosmological im- more general of the two is the semiclassical diver- plications are entirely dictated by the ground state cosmic gence of the renormalized stress-energy tensor near string tension , or the dimensionless number G , where the ‘‘chronology horizon,’’ or Cauchy surface sepa- G is the Newton’s constant [1]. The present observational rating the regions of spacetime containing CTCs bound is around G & 6 10 7. Recently, it was shown from those without. These (vacuum)‘‘polarized’’ [7] that the cosmic string tension can easily saturate this hypersurfaces led Hawking to conjecture that any bound in the simplest scenario in string theory, namely, the classical spacetime containing a chronology horizon realistic D3 D3-brane inflationary scenario [4]. This will be excluded from the quantum theory of grav- means gravitational lensing by such cosmic strings, pro- ity. However, a recent paper [15] found an example viding image separation of order of an arc second or less, is where this chronology horizon is well defined in the an excellent way to search for cosmic strings. Generic background of a string theory (to all orders in 0). features of cosmic strings include a conical ‘‘deficit angle’’ So, in superstring theory, the CPC is not generically geometry, so a straight string provides the very distinct true: the Taub-NUT geometry receives corrections signature of an undistorted double image. Cosmic string that preserve the travsersibility of the chronology lensing has been extensively studied [8–11]. horizon. Unlike classical Taub-NUT, stringy Taub- In the string network, some segments of cosmic strings Nut contains timelike singularities, although they will move at relativistic speeds. It is therefore reasonable to are far from the regions containing CTCs. The sec- consider the gravitational lensing by highly relativistic ond hint that CPC is true is the theorem proved by cosmic strings. We also confront another interesting fea- Hawking and Tipler [14,16] that spacetimes obeying ture of cosmic strings, first realized by Gott [12]: the the weak energy condition with regular initial data possible appearance of closed timelike curves (CTCs) and whose chronology horizon is compactly gener- from two parallel cosmic strings moving relativistically ated cannot exist. These theorem depends crucially past each other. As the strings approach each other fast on the absence of a singularity, and so Hawking’s enough in Minkowski spacetime, the path encircling the claim that finite lengths of cosmic string cannot produce CTCs is only true if one rejects the possi- *Electronic address: [email protected] bility of a singularity being present somewhere on †Electronic address: [email protected] the chronology horizon [17].
1550-7998=2005=72(4)=043532(11)$23.00 043532-1 2005 The American Physical Society BENJAMIN SHLAER AND S.-H. HENRY TYE PHYSICAL REVIEW D 72, 043532 (2005) (b) When we consider possible formation of CTCs and its sources need not encounter the singularity. In coming from cosmic string loops (though this is fact Tipler’s physical argument against the creation not necessary for our general argument), one may of CTCs is the unfeasibility of creating singularities. make use of the null energy condition and smooth- However, it is well-known that singularities such as ness to argue against CTCs. The null energy condi- orbifold fixed points and conifolds are perfectly fine tion can be satisfied even when one smooths out the in superstring theory, where Einstein gravity is re- conical singularity (with a field theory model) at the covered as a low energy effective theory. Fur- core of the cosmic strings. However, in superstring thermore, under the appropriate circumstances, to- theory, such a procedure is not permitted. The cos- pology changes are perfectly sensible. This consis- mic strings are either D1-strings or fundamental tency is due to the extended nature of string modes. strings. Classically, the core is a -function with (f) One may consider the Gott spacetime in 2 1 no internal structure (e.g., energy distribution) in dimensions. The 2 1 dimensional gravity relevant the string cross-sections, so the string has only for the problem has been studied by Deser, Jackiw transverse excitation modes. Suppose we smooth and ’t Hooft [19]. For a closed universe, ’t Hooft out the -function. Then one can rotate the string [20] argues that the universe will shrink to zero around its axis and endow it with longitudinal volume before any CTCs can be formed. For an modes. The presence of such longitudinal modes open universe, Carroll, Farhi, Guth, and Olum violate the unitarity property of the superstring the- (CFGO) [17] show that it will take infinite energy ory. (In fact, in the presence of such longitudinal to reach Gott’s two-particle system from strings at modes, general relativity is no longer assured in rest. Holonomy calculations show that this system superstring theory.) In this sense, the string geome- has spacelike total momentum, much like a tachyon. try is not differentiable, and one must generalize the For this reason one might consider Gott spacetime to appropriate theorems before they may be applied. be unphysical, although there are differences be- (c) Cutler [18] has shown that the Gott spacetime con- tween the global properties (including holonomy) tains regions free of CTCs, that the chronology of tachyons and Gott’s cosmic strings. In 3+1 di- horizon is classically well defined, and that Gott mensions, the physicality of Gott spacetime is less spacetime contains no closed (as opposed to just clear. self-intersecting) null geodesics. Hawking points If none of the above arguments against CTCs are fully out that this last feature must be discarded for applicable to the Gott 3 1 spacetime, does this mean bounded versions of Gott (and similar) spacetimes. CPC fails and Gott spacetime can be realized in the real This means that Gott spacetime undergoes qualita- universe? Or are there other mechanisms which prevent the tive changes when embedded in asymptotically flat formation of CTCs? spacetime. Cutler found a global picture of the Gott In this paper, we use a lensing framework to demonstrate spacetime very much in agreement with general the classical instability near the Cauchy horizon which we arguments made by Hawking regarding the instabil- argue will prevent the formation of cosmic string CTCs in ity of Cauchy horizons, specifically the blueshifting any realistic situation. To be specific, our argument goes as of particles in CTCs. Here, we find a concrete follows: example of this phenomenon using a lensing (a) A particle or a photon gets a positive kick in its perspective. momentum in the plane orthogonal to the strings (d) As parallel strings move relativistically past each each time it goes around a CTC [17,21,22]. other to create CTCs, a black hole may be formed by (b) Once inside the chronology horizon, such a particle the strings before the CTC appears, thus preventing is generically attracted to a CTC; that is, a worldline CTCs. If this happens, one can consider the forma- in the vicinity of the CTC will coalesce with the tion of a black hole as a realization of CPC. CTC. This is our main observation. However, it is easy to see that, using Thorne’s (c) The particle will go through the CTC numerous hoop conjecture, there is a range of string speed times (actually an arbitrarily large number of times) where the CTCs appear, but no black hole is formed. instantaneously; that is, the particle will be instanta- The results of Tipler and Hawking suggest that the neously blueshifted by an amount unbounded from strings are slowed to prevent CTCs, or a singularity above. forms somewhere else in the geometry. This singu- (d) It follows that the backreaction must be important; larity does not need to disrupt or isolate the region conservation of angular momentum and energy im- containing CTCs. plies that the cosmic strings will slow down, or, (e) Tipler [16] proved that whenever a CTC is produced more likely, bend; this in turn prevents the formation in a finite region of spacetime, a singularity must of CTCs. Note that this backreaction must disrupt necessarily accompany the CTC. This singularity the closed timelike curve, otherwise the infinite does not represent a no-go theorem, since the CTC blueshift can not be prevented. So, a single particle,
043532-2 COSMIC STRING LENSING AND CLOSED TIMELIKE CURVES PHYSICAL REVIEW D 72, 043532 (2005) say a graviton or photon, no matter how soft, will collapsing scenario occurs. Here, we agree with the chro- bend the cosmic strings so that CTC cannot be nology protection principle and ’t Hooft that a CTC is formed. The following picture seems reasonable: not formed. However, we believe that, due to the energy- as the two segments of cosmic strings move toward momentum-angular momentum conservations, the back- each other, they are bent and so radiate gravitation- reaction will bend the cosmic strings and induce gravita- ally. This slows them down to below the critical tional radiation so that the CTC is never formed. Neither value for CTC formation. We expect no singular- the curvature nor the energy of the photon blows up. Note ity/divergence to appear. that the bending of the strings can not happen in 2 1 (e) Since there is a cosmic microwave background ra- dimensions. diation in our universe, these photons preclude the In this paper, the motion of a photon/graviton around the existence of CTCs. Of course, the cosmic micro- cosmic strings is a crucial ingredient of the analysis. We wave background radiation is not the only wrench in shall start with a review of the gravitational lensing by a the machine. Gravitons or some other particles can straight moving cosmic string. Here we correct a mistake be emitted by the moving strings, either classically lensing formula mistake in the literature (see Appendix A). or via quantum fluctuation. In particular, gravitons Next we review the Gott spacetime. Finally we show that must be present in spacetimes of dimensions 3 1 the CTCs encircling the two strings are attractors for or greater. A single graviton, no matter how soft, particles. Our analysis ends here. Supplemented with plau- will lead to the above effect. We argue this is how sible reasonings, we argue that the above mechanism is a the chronology protection conjecture works in the way to prevent CTCs in the real universe. Throughout, we Gott spacetime. shall assume 0 8G to be very small (say, less than This result is not too surprising in light of the likely 10 5). (blueshift) instability of Cauchy horizons discussed by Hawking and others, although a counter example was II. COSMIC STRING LENSING found by Li and Gott [23] while analyzing possible vacua of Misner space, whose Cauchy horizon can be free of One may calculate the observational signatures of rap- instability. As in our example, a divergence occurs only in idly moving cosmic strings (straight and loops), in particu- the presence of particles, although we find that blueshifting lar, their lensing effects on distant galaxies and the CMB. (and not particle number) is the cause. The simplest case of a straight, nearly static cosmic string The blushift instability is well studied in the literature has the distinctive signature of producing two identical [24]. The strong cosmic censorship conjecture predicts that images, each being undistorted and equidistant from the a Cauchy horizon is, in general unstable (e.g. that of a observer. Reisner-Nordstro¨m black hole in asymptotically flat space- In Fig. 1 below is pictured a cosmic string moving to the time), and that this instability is the result of the infinite right across a distant galaxy. We call the angular separation blueshift of in falling perturbations. This must be similar to of the images ’, and the photon deflection angle ,asin the instability we describe, but our picture is resolved Fig. 2 below. Although the double images on the left differently. We find that the CTC never forms because picture of Fig. 1 may be due to two almost identical surrounding particles scatter off the cosmic strings, bend- galaxies (a rare but not impossible scenario), the picture ing and slowing them. Hence no Cauchy horizon (stable or on the right will be a much cleaner signature of cosmic not) ever forms. string lensing. If one sees a double image candidate [26], ’t Hooft [25] argues that, since the local equations of one expects to see other candidates nearby. Searching for motion for a cosmic string are well-defined, one should be incomplete images will be important. able to list the Cauchy data at any particular time, and This leads to the well-known result ’ Ds;cs . The Ds;O demand the Laws of Nature to be applied in a strictly spacetime around a static cosmic string is Minkowski causal order. If one phrases the logic this way, there are space with the identification of two semi-infinite hyper- no CTCs by construction, in agreement with the chronol- planes whose intersection is the cosmic string world sheet. ogy protection conjecture and strong cosmic censorship. This is equivalent to identifying every event s in space time So the question is: what is wrong with Gott spacetime? His with a dual s0 where the relation between s and s0 with a answer to this question is that the Cauchy planes become static cosmic string located at r is unstable: in terms of these, the Universe shrinks to a line in cs 3 1 dimensions. The moment a disturbance from any 0 s R0 s rcs rcs: (1) tiny particle is added somewhere in the past, it generates so much curvature that the inhabitants of this universe are Where R0 is a pure rotation (counter clockwise). (Notice killed by it. In our scenario, we give a specific mechanism rcs can be any point on the cosmic string world sheet.) It with more details: a single graviton or photon, even a very should be noted that s is visible only when it appears to the soft one, will suffice. An infinitely blueshifted photon (or right of the cosmic string, and s0 is visible only when to the any particle) will cause so much curvature that ’t Hooft’s left.
043532-3 BENJAMIN SHLAER AND S.-H. HENRY TYE PHYSICAL REVIEW D 72, 043532 (2005) 0.03 0.03
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-0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 -0.04-0.03-0.02-0.01 0 0.01 0.02 0.03
FIG. 1. Gravitational lensing by a straight cosmic string. The two images are identical. Note that they are not mirror reflections of each other, but one is a translational displacement of the other. As the source moves to the left (or equivalently, the cosmic string moves to the right), part of the right image is cut off, eventually leaving only a single image.
The general case involves a cosmic string moving at at a distance r from the string vertex. The photon is some four-velocity vcs. We will always take vcs to be redirected by an angle and makes a spacial and temporal perpendicular to the cosmic string world sheet, since any jump. This jump (s0 s) is found using the coordinate parallel component is unphysical (assuming a pure tension identification in Eq. (2), where s A is the photon striking string). Interestingly, this velocity is only well defined in the deficit angle and s0 B is its emergence from the combination with a ‘‘branch cut.’’ This is related to the fact deficit angle. If we choose the deficit angle to be perpen- that a passing cosmic string will induce a relative velocity dicular to vcs, the spacial jump is parallel to vcs, and is between originally static points in space, so a constant given by velocity field will not be everywhere single valued. More physically, parallel geodesics moving past a cosmic string x 2r tan 0=2 cs t 2r tan 0=2 csvcs: will be bent toward each other, provided they pass the (3) string on opposite sides. Specifying a branch cut enables the conical geometry to be mapped to Minkowski space p 2 (minus a wedge), where things are simpler. The pure where cs 1= 1 vcs. In Fig. 3, events A and B are rotation identification is valid only in the center-of-mass identified, while events A and C are simultaneous. If the frame of the cosmic string, so in general photon strikes the leading edge of the deficit angle, the jump is behind the cosmic string and backward in time. If 0 s v R v s rcsrcs: (2) the photon strikes the trailing edge, the jump is in front of cs 0 cs the cosmic string and forward in time. For ultrarelativistic cosmic strings, only photons traveling almost exactly par- Where vcs is a pure boost such that vcs vcs 1; 0; 0; 0. We have simply boosted into the strings refer- ence frame and then performed the rotation-identification. Then we boost back. δ Here we investigate the lensing due to nearly straight ∆x ∆t segments moving at arbitrarily relativistic speeds. We con- A B sider the interaction of a photon with a cosmic string’s n2 deficit angle. In Fig. 3, a photon is crossing the deficit angle C n1 r
Ds,O α
s’ * vcs δϕ cs δ * O s FIG. 3. The path of a photon coming from the source at the n * lower left and reaching the observer at the right. The straight Ds,cs cosmic string is moving with speed vcs to the left. The photon’s initial velocity n^ 1 makes an angle with respect to vcs.AtA FIG. 2. A straight cosmic string introduces a deficit angle . the photon strikes the leading edge of the deficit wedge (a Here, the sources s and s0 are to be identified. distance r from the position of the cosmic string).
043532-4 COSMIC STRING LENSING AND CLOSED TIMELIKE CURVES PHYSICAL REVIEW D 72, 043532 (2005) allel to the string’s velocity will strike the trailing edge of where the string is moving at velocity vcs. Then the deficit angle. We will calculate the change in momentum of a particle kfinal vcs R0 vcs kinitial (5) interacting with the cosmic string. In the string rest frame, we know that To calculate the directional change in a photon’s veloc- ity, we take the above formula with k2 m2 0 and for kfinal R0 kinitial (4) simplicity we take the photon to travel in a plane perpen- dicular to the cosmic string. Then we find (dropping the cs so we simply boost the above equation into the frame subscript)
A B cos sin v sin cos sin = cos p 0 0 ; (6) 2 2 sin 0 v cos cos 0 sin = B A sin sin 0
where The above formula only applies to cosmic strings per- pendicular to the line of sight. For the most general lensing 2 A cos v cos 0; due to straight cosmic strings, see Ref. [28].
B v cos 0 1: III. THE POSSIBILITY OF RAPID COSMIC The blueshift can be calculated as well, yielding STRING FORMATION
!final 2 Naively, ultrarelativistic straight strings are rather un- sin sin 0 1 v cos !initial likely, and two parallel ultrarelativistic straight strings v cos cos v2 cos : (7) passing each other with such a large kinetic energy density 0 0 must take some arrangement. It is along this line of rea- Straight portions of a cosmic string moving at ultrarelativ- soning that CFGO [17] argues against the formation of istic speeds produce photon deflection of order in con- Gott spacetime in 2 1 dimensions. The situation is no junction with severe blueshift. Slower moving cosmic more likely in 3 1 dimensions, although one cannot rule strings will obey the Kaiser-Stebbins formula [21] and out the possibility of ultrarelativistic cosmic string forma- cause the sky behind the moving string to be blueshifted tion. Recent brane inflation models have proposed having relative to the sky in front of the string. the inflationary branes located in the tip of a Klebanov- With the exception of loops, we expect cosmic strings to Strassler throat of a Calabi-Yau 3-fold [4]. In some scenar- move moderately relativistically, but with 0 <<1.In ios, the standard model branes are in a different throat. One this limit, the above formula reduces to feature of this construction is that cosmic strings produced after inflation are meta stable, and will not be able to decay 0 1 v cos ! via open string interaction with the standard model branes D 8G (8) [5]. It should be pointed out that extra dimensions have no ’ s;cs p 1 n^ v: 2 effect on Gott’s construction of CTCs, provided that the Ds;O 1 v radii of the extra dimensions are small compared to the The first factor Ds;cs=Ds;O is a plane-geometric coefficient radius of the CTC. This can be seen as follows. A cosmic for 8G , which is the relativistic energy of the string. string in our universe is an object extended in one non- The third term is the result of the finite travel time of light, compact direction, and zero or more compact directions. and does not represent the coordinate locations of the The four dimensional effective theory will always have a images in the observer’s frame. Notice that a moving string conical singularity on the location of the string, and any (except one moving toward the observer) has a stronger corrections to this will be due to massive axion and (KK) lensing effect. This result disagrees with that given in modes of the metric—particles whose range is limited to Ref. [10], which has the factor in the denominator. To sizes of order the radii of the extra dimensions. Thus as see this difference more clearly, we give in Appendix A the long as we do not probe distances so near the string that simple derivation of Ref. [10] and point out where the error these corrections are significant, the conical geometry is occurs. Recall that the typical speed of the cosmic strings valid and Gott’s construction is meaningful. In fact, the in the network is rather large, v 2=3 [27]. There will be CTCs in Gott’s construction exist at large radii from the segments of strings that have 1 and they have the best cosmic strings, and so one never needs to probe the near- chance to be detected via lensing. string geometry.
043532-5 BENJAMIN SHLAER AND S.-H. HENRY TYE PHYSICAL REVIEW D 72, 043532 (2005) IV. GOTT’S CONSTRUCTION OF CTCS (In our analysis, we focus on lightlike motion since time- like motion is, in a sense bounded by this case.) For large Here we give a brief review of Gott’s original construc- enough x, this velocity is greater than that of light and so tion. The key feature is the conical deficit angle ( 0 we may boost to a frame where the departure and arrival 8G ) around a cosmic string. This results in a ‘‘cosmic events are simultaneous. This is true for any d. In the shortcut,’’ since two geodesics passing on opposite sides of above picture, we will sever the spacetime at the y the string will differ in length. This shortcut, like a worm- 0-hyperplane and boost such that the top string is moving hole, leads to apparent superluminal travel. Under boost, to the left at speed v > cos =2 and the bottom string is ‘‘superluminal’’ travel becomes ‘‘instantaneous’’ travel. cs 0 moving to the right at the same speed. This means that we Gott realized that this instantaneous travel could be per- can take A ! B on the upper path and B ! A on the lower formed in one direction, and then back again when two path, and in both directions the elapsed time is zero. This is cosmic strings approach each other at very high speed. The possible when actual trajectory is timelike, i.e. performed by a massive body, and resembles an orbit around the center-of-mass of 2 < 2 sin =2 : (11) the cosmic string system (in a direction with opposite 0 0 angular momentum as the cosmic strings). Below we illus- It is sufficient that the y 0 hyperplane has vanishing trate the geometry with two strings at rest. intrinsic and extrinsic curvature for us to consistently sew In Fig. 4, there are three (geodesic) paths from A to B. the two halves together. Notice that the limiting case of The central path is not necessarily the shortest, in fact it can 0 2 corresponds to closed lightlike curves. (We will be seen that d assume that x ! 0 for simplicity.) The (two-fold) boosted version of the above setup is 0 0 w x cos d sin : (9) pictured in Fig. 5 below. The events are numbered in 2 2 ‘‘proper’’ chronological order (that is, the order in which Thus although A ! B is traversed by a particle on a time- they occur on the particle worldline), but the center-of- like trajectory above or below the cosmic strings, the mass coordinate frame chronological order needs to be departure and arrival events may have spacelike separation explained. Event 1 is the light ray initially traveling up to in the y 0 hyperplane which extends between the two meet the rapidly moving deficit angle, which happens at strings. In this hyperplane, the average velocity of this event 2. This meeting is identified (under Eq. (2)) with particle can be calculated as event 3, although in center-of-mass coordinates event 3 happens before the previous events occur. Events 1, 4, and 2x 1 7 are cm-simultaneous at t 0 while events 2 and 5 occur v d : (10) 2w cos 0=2x sin 0=2
3 2
IDENTIFY
δ 0
w w v v 1 δ 0 δ0 2 d 2 4 7 A B v v x x
δ 0
6 5 IDENTIFY
FIG. 5. This is the critical case where 0 2. The deflection angle is calculated using Eq. (6) and the discontinuity in the FIG. 4. The Gott spacetime, before the strings are moving. world line using Eq. (3). We have drawn d>0 for clarity. The This spacetime will be severed at the y 0 hyperplane (line particle travels along the path labeled 1 to 7 and back to 1, AB), boosted, and then smoothly glued together again. A and B arriving at the same point in space and time, i.e., a closed are separated by a distance of 2x, while the cosmic strings are timelike curve. In this case, the particle is neither blueshifted separated by a distance of 2d. nor redshifted.
043532-6 COSMIC STRING LENSING AND CLOSED TIMELIKE CURVES PHYSICAL REVIEW D 72, 043532 (2005) at tcm 1, and events 3 and 6 at tcm 1. Notice that cause a slight increase in . A slight increase in . will then decrease the next , and return the system We would like to apply our understanding of ultrarela- to equilibrium. (This assumes 0 < 1, which is tivistic cosmic strings to Gott spacetime. We can use the always the case.) Fig. 7 illustrates the attractor in jumps in location, time and direction (t, x, and )to action. construct all photon paths around a cosmic string system (b) Photon momentum in the plane perpendicular to the and determine the complete lensing behavior. Below is a strings will be blueshifted as given by Eq. (7). This graph of for several values of 0, as given by Eq. (6). formula applies to any relativistic particle. The blue- Of importance is where the graphs take the value . shift occurs twice for each revolution, once from Because the two cosmic string velocities differ in direction each string; by , is a fixed point of photon direction. As Cutler (c) Since momenta along the string lengths are not showed, a crossing with positive slope is stable and blue- blueshifted, nonrelativistic particles and particles shifted, while the one with negative slope is unstable and with velocities along the length of the cosmic strings redshifted. The stable blueshifted fixed point will always will be attracted to relativistic trajectories perpen- exist for 0 > 2 (supercritical case) and thus represents a dicular to the string length; catastrophic divergence. This is because a particle in the (d) In the limit of small d the average distance to the presence of this geometry will fall into a stable orbit with core of the CTC will not vary. This means that the exponentially diverging energy. The above graph is accu- orbits will close, rather than shrink; rate for massless particles, but massive particles (equiva- (e) The energy of each particle in the CTC will diverge lently: particles with nonzero momentum along the strings) exponentially as a function of number of cycles will behave similarly once they become blueshifted. In total, we find γδ = (a) Particle trajectories in the vicinity of a 0 > 2 pair 0 2.5 of cosmic strings will be attracted to the stable 200 131197153 10121004826 orientation , 0 > 0 (see Fig. 6). This 5 6 is because the two cosmic strings velocities are 100 equal and opposite, that is, when a particle incident at angle n is deflected by an angle n, n1 1 2 n n. Thus not only is a ‘‘fixed cs2 cs1 point’’ of incidence angle, but it is a stable one if -300 -200 -100 100 200 300 0 > 0, since then a slight increase in will 4 3 -100
8 7 δ (α) 101260482 131115937 4 -200 γδ = 3.5 0 3 3 -300 2.5 2 2 1 1.5 -400 1 0.1 0.5 FIG. 7. The supercritical case. Cosmic string #1 is moving to 123456 the left and cosmic string #2 is moving to the right. The particle enters from the bottom right and reaches cs1’s upper deficit angle FIG. 6. This plot derives entirely from Eq. (6). Here, is at 1. Its world line appears to jump to the right to 2, now directed the deflection angle of the photon as a function of the angle down and slightly to the right. This process continues indef- between the photon direction and (minus) the cosmic string initely, 3, 4, ...1, the world line spiraling clockwise outward as velocity (see Fig. 3). 0 2 is the critical case, when the it falls into a stable orbit (the two long, outermost parallel peak of touches the value (the solid horizontal line). A segments). The coordinate discontinuity seen here does not 0 > 2 curve crosses at 2 points. Notable is where reflect any actual discontinuity. The numbers on the world line crosses (see e.g. the 0 3 curve) with positive slope satisfy 2n 1 2n 2 where the equivalence is identification at s. A positive slope crossing implies a stable fixed point. via Eq. (2). The coordinate discontinuities plotted here are Photons with different initial not far from s will follow calculated with Eq. (3), and the trajectory angle is determined closed timelike paths and approach s. Photons are blue- by Eq. (6). The initial conditions of the above trajectory are not shifted at all positive slope crossings. The amount of blueshift tuned, but rather generic. Approximately half of all initial each time is given by Eq. (7). particle trajectories will end up in a CTC.
043532-7 BENJAMIN SHLAER AND S.-H. HENRY TYE PHYSICAL REVIEW D 72, 043532 (2005) taken; since it takes no time to travel any number of reaches that limit, a black hole is formed. An elliptic or cycles, unbounded blueshift takes place; rectangular loop can be made to collapse to parallel, rela- (f) Therefore, 0 > 2 implies a catastrophic diver- tivistic segments without forming a black hole. Theorems gence in the presence of even a single particle. by Tippler and Hawking suggest that such a loop cannot It should be noted that the exponential divergence in form CTCs without creating a singularity, although we energy is kinematic, and has nothing to do with particle argue that potential CTCs will be disrupted by the presence number. Below is pictured the trajectory of a photon in a of photons (or any other mode). It is also possible that the supercritical Gott space. The photon enters from the lower huge blueshift will cause the formation of a black hole, right, and then spirals clockwise outward into a stable, resulting in a black hole with cosmic string loops protrud- blueshifted orbit. The upper and lower deficit angle wedges ing. In this case, the presence of CTCs inside the black hole are moving to the left and right, respectively. is acceptable, since they are not observable. More analysis We thus may conclude that although closed lightlike is needed to fully address this issue. curves may exist ( 0 2), a purely classical divergence destroys the Gott solution of closed timelike curves B. Specific discussions on 2 1 dimensions ( 0 > 2) in the presence of a dynamical field (e.g. grav- itons or photons). Cosmic strings cannot produce closed A simpler scenario with CTCs was found by van timelike curves. This is in agreement with Li and Gott [23] Stockum [31] and Deser, Jackiw, and ’t Hooft (DJtH) which finds Misner space (and by implication Grant space [19] whereby a single stationary cosmic string is given and Gott space) to suffer from a classical instability similar angular momentum about its axis. The resulting back- to the one found here. It should be noted that the hoop ground is given by conjecture is not involved in this breakdown. Even the 2 ds2 dt Jd2 dz2 dr2 1 4G 2r2d2: 1 dimensional Gott space, where the hoop conjecture is not (12) applicable, is unstable in the presence of dynamical fields. It seems unlikely for such a cosmic string to exist in string V. COMMENTS theory (at least as fundamental objects[32]), since the cosmic strings in superstring theory lack the internal de- A. General discussions gree of freedom ‘‘spin.’’ DJtH noticed an unusual feature of Semiclassical gravity raises an objection to the existence the Gott spacetime. They classified the energy momentum of CTCs, or at least to spacetimes that contain both regions of Gott’s solution in terms of the Lorentz transformations with CTCs and regions out of causal contact with them. encountered under parallel transport (PT) around the The boundary between such regions is called the ‘‘chro- strings (holonomy). It is well known that the PT trans- nology horizon,’’ and in known examples this horizon formations around a single cosmic string is rotationlike, i.e. coincides with a divergence of the renormalized energy- can be expressed as momentum tensor. This led Hawking to pose the ‘‘chro- nology protection conjecture,’’ in which he proposes that R0 ; (13) any classical examples of spacetimes containing CTCs will where is a pure boost with rapidity and R is a pure be excluded by a quantum theory of gravity. Thorne and 0 Kim disagreed [29] on the grounds that the divergence of rotation about the angle 0. One may calculate the holon- omy of a Gott pair via
043532-8 COSMIC STRING LENSING AND CLOSED TIMELIKE CURVES PHYSICAL REVIEW D 72, 043532 (2005) pair geometries. A more complete description of geometry strings. In this paper, we find that nearby photons and would include not just the PT transformation, but the gravitons will be attracted to the closed timelike curves, homotopy class of the PT transformation as well. resulting in an instantaneous infinite blueshift. This implies Equivalently, one should extend SO 2; 1 to its universal backreaction must be large enough to disrupt the formation covering group. of such closed timelike curves. We interpret this as a One may interpret the ‘‘boostlike’’ holonomy as boost realization of the chronology protection conjecture in the identification, as was done by Grant [34]. This makes the case of the Gott spacetime. Gott spacetime akin to a generalized Misner space. Grant was able to show that the Gott/Misner spacetime suffers ACKNOWLEDGMENTS from large quantum mechanical divergences on an infinite We thank Robert Bradenberger, Eanna Flanagan, Gerard family of polarized lightlike hypersurfaces. This diver- ’t Hooft, Mark Jackson, Ken Olum, Geoff Potvin, Alex gence is stronger than that at the chronology horizon [34]. Vilenkin, Ira Wasserman, and Mark Wyman for discus- Regardless of whether the Gott spacetime is physical or sions. This work is supported by the National Science not, it can be shown that the Gott spacetime in 2 1 Foundation under Grant No. PHY-009831. dimensions cannot evolve from cosmic strings initially at rest [17]. This is quite different from the 3 1 dimensional case. APPENDIX A: LENSING BY A MOVING COSMIC STRING C. The instability in 3 1 dimensions In the text, the lensing due to a moving cosmic string is Recent realization of the inflationary scenario in super- too general for normal application. Here we reproduce a string theory strongly suggests that cosmic superstrings simple elegant argument due to Vilenkin [10] to calculate were indeed produced toward the end of the inflationary the angular separation of images produced by a moving epoch. With this possibility, the issue has renewed urgency. cosmic string in the case when the cosmic string tension as In the above discussions, we argue that the reasoning well as the lensing effect are small. We point out where the against the Gott spacetime in 2 1 dimensions does not (small) error in Ref. [10] is. Vilenkin argues that the necessarily apply to the 3 1 dimensional case. In short, angular deflection of light by a cosmic string may be the Gott spacetime is entirely possible in an ideal classical calculated by appealing to Lorentz invariance. For a string situation. However, we argue that instability is manifest if at rest, the angular separation is given by there is a quanta/particle nearby. The particle will be Ds;cs Ds;cs attracted to the closed timelike curve and is arbitrarily ’ 0 sin 0; (A1) blueshifted instantly. Of course, backreaction takes place DO;cs Ds;cs Ds;O before this happens. This backreaction must disrupt the where is the relative angle between the cosmic string and closed timelike curve, otherwise the infinite blueshift will the line of sight, and Dx;cs is the normal distance between x not be prevented. In an ideal situation where there is no and the cosmic string. We may consider two light waves, quanta nearby, one still expects particles like (very soft) one from each image, k and k0. Their scalar product is gravitons/photons can emerge due to quantum fluctuation. given by In fact, quantum fluctuation of the cosmic strings them- selves as they move rapidly toward each other will produce 1 k k0 !!0 1 cos ’ !!0 ’2: (A2) graviton radiations. One graviton, no matter how soft, is 2 sufficient to cause the instability. So we believe that the We may assume that the two light waves have the same Gott spacetime is unstable under tiny perturbations and so frequency: ! !0. (This can be true in all reference cannot be formed in any realistic situation. frames since we are expanding to first order in 0.) We can then relate the angular separation in any two reference VI. CONCLUSION frames by the frequency of the light waves: Recent implementation of the inflationary scenario into ! ’ !’ (A3) superstring theory led to the possibility that cosmic strings 0 0 were produced toward the end of brane inflation in the i.e., the higher the observed frequency, the lower the brane world. This possibility leads us to reexamine the Gott observed angular separation. The frequency in a reference spacetime, where closed timelike curves appear as two frame where the string moves at velocity v relative to the cosmic strings move ultrarelativistically pass each other. observer is given by In an ideal situation, the Gott spacetime is an exact solution !0 to Einstein equation, with a well-defined chronology hori- ! ; (A4) 1 n^ v zon. It does not collapse into a black hole and can be readily reached. So it seems perfectly sensible that such a where n^ is the direction from the observer to the source spacetime can be present in a universe that contains cosmic (and thus string), and hence
043532-9 BENJAMIN SHLAER AND S.-H. HENRY TYE PHYSICAL REVIEW D 72, 043532 (2005)
’ 1 n^ v’0: (A5) The effective potential energy is plotted below. Notable features are the existence of stable orbits for generic initial The roles of ! and !0 are erroneously swapped in conditions, and exponentially long orbital periods as a Ref. [10], which thus agrees with ours only for n^ / v. function of energy. Physically, a traveler moving transverse to the light we Notice that the total energy of the system contains an observe should measure a higher frequency than we do (i.e. arbitrary additive constant. This should be fixed by the !
[1] A. Vilenkin and E. P.S. Shellard, Cosmic Strings and L. McAllister, and S. P. Trivedi, J. Cosmol. Astropart. Other Topological Defects, (Cambridge University Press, Phys. 10 (2003) 013. Cambridge, England, 2000). [5] E. J. Copeland, R. C. Myers, and J. Polchinski, J. High [2] N. Jones, H. Stoica, and S.-H. H. Tye, J. High Energy Energy Phys. 06 (2004) 013. Phys. 07 (2002) 051; S. Sarangi and S.-H. H. Tye, Phys. [6] T. W. B. Kibble, astro-ph/0410073; T. Damour and A. Lett. B 536, 185 (2002). Vilenkin, Phys. Rev. D 71, 063510 (2005); J. Polchinski, [3] L. Pogosian, S.-H. H. Tye, I. Wasserman, and M. Wyman, Phys. Rev. D 71, 063510 (2005). Phys. Rev. D 68, 023506 (2003). [7] H. Firouzjahi and S.-H. H. Tye, J. Cosmol. Astropart. [4] S. Kachru, R. Kallosh, A. Linde, J. Maldacena, Phys. 03 (2005) 009.
043532-10 COSMIC STRING LENSING AND CLOSED TIMELIKE CURVES PHYSICAL REVIEW D 72, 043532 (2005) [8] A. Vilenkin, Astrophys. J. 282, L51 (1984). G. Covone, O. Khovanskaya, M. Pavlov, M. Pannella, M. [9] A. Vilenkin, Phys. Rep. 121, 263 (1985). Radovich, and V. Testa, Mon. Not. R. Astron. Soc. 343, [10] A. Vilenkin, Nature (London), 322, 613 (1986). 353 (2003); M. V. Sazhin, O. S. Khovanskaya, [11] A. A. de Laix and T. Vachaspati, Phys. Rev. D 54, 4780 M. Capaccioli, G. Longo, J. M. Alcala, R. Silvotti, and (1996). M. V. Pavlov, Mon. Not. R. Astron. Soc. 343, 353 (2003); [12] J. R. Gott, III Phys. Rev. Lett. 66, 1126 (1991). J. M. Alcala’, M. Pannella, E. Puddu, M. Radovich, [13] See e.g., K. Thorne, Black Holes and Time Warps, R. Silvotti, M. Arnaboldi, M. Capaccioli, G. Covone, M. (Norton, New York, 1994); B. Greene, The Fabric of the Dall’Ora, G. De Lucia, A. Grado, G. Longo, A. Mercurio, Cosmo, (Knopf, New York, 2004). I. Musella, N. Napolitano, M. Pavlov, A. Rifatto, V. [14] S. W. Hawking, Phys. Rev. D 46, 603 (1992). Ripepi, and R. Scaramella, astro-ph/0408220. [15] C. V. Johnson and H. G. Svendsen, Phys. Rev. D 70, [27] A. Albrecht and N. Turok, Phys. Rev. Lett. 54, 1868 126011 (2004). (1985); D. P. Bennett and F. R. Bouchet, Phys. Rev. Lett. [16] F. J. Tipler, Ann. Phys. (N.Y.) 108, 1 (1977). 60, 257 (1988); B. Allen and E. P.S. Shellard, Phys. Rev. [17] S. M. Carroll, E. Farhi, A. H. Guth, and K. D. Olum, Phys. Lett. 64, 119 (1990). Rev. D 50, 6190 (1994). [28] B. Shlaer and M. C. Wyman (to be published). [18] C. Cutler, Phys. Rev. D 45, 487 (1992); A. Ori, Phys. Rev. [29] S. W. Kim and K. S. Thorne, Phys. Rev. D 43, 3929 D 44, R2214 (1991). (1991). [19] S. Deser, R. Jackiw, and G. ’t Hooft, Ann. Phys. (N.Y.) [30] L. Jarv and C. V. Johnson, Phys. Rev. D 67, 066003 152, 220 (1984). (2003); E. G. Gimon and P. Horava, Phys. Rev. D 67, [20] G. ’t Hooft, Classical Quantum Gravity 9, 1335 (1992); 066003 (2003); N. Drukker, Phys. Rev. D 70, 084031 10, 1023 (1993); 10, 91993 (1992). (2004). [21] N. Kaiser and A. Stebbins, Nature (London) 310, 391 [31] W. J. van Stockum, Proc. R. Soc. Edinburgh, Sect. A: (1984). Math. 57, 135 (1937). [22] S. M. Carroll, E. Farhi, and A. H. Guth, hep-th/9207037. [32] P.O. Mazur, Phys. Rev. Lett. 57, 929 (1986). [23] L. X. Li and J. R. Gott III, Phys. Rev. Lett. 80, 2980 [33] M. P. Headrick and J. R. Gott III, Phys. Rev. D 50, 7244 (1998). (1994). [24] E. Poisson, gr-qc/9709022. [34] J. D. E. Grant, Phys. Rev. D 47, 2388 (1993). [25] G. ’t Hooft (private communication). [35] C. P. Burgess, P. Martineau, F. Quevedo, and R. Rabadan, [26] M. V. Sazhin, G. Longo, J. M. Alcala’, R. Silvotti, J. High Energy Phys. 06 (2003) 037.
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