MMass and spin for classical strings in dS Klaas Parmentier
Columbia University, Department of Physics
Abstract
We demonstrate that all rigidly rotating strings with center of mass at the originof the dS static patch satisfy the Higuchi bound. This extends the observation ofNoumi et al. for the open GKP-like string to all solutions of the Larsen-Sanchez class.We argue that strings violating the bound end up expanding towards the horizon andprovide a numerical example. Adding point masses to the open string only increases themass/spin ratio. For segmented strings, we write the conserved quantities, invariantunder Gubser’s algebraic evolution equation, in terms of discrete lightcone coordinatesdescribing kink collisions. Randomly generated strings are found to have a tendencyto escape through the horizon that is mostly determined by their energy. For rapidlyrotating segmented strings with mass/spin <
1, the kink collisions eventually becomecausally disconnected. Finally we consider the scenario of cosmic strings captured bya black hole in dS and find that horizon friction can make the strings longer. a r X i v : . [ h e p - t h ] F e b ontents dS dS . . . . . . . . . . . . . . . . . . . . M l < L . . . . . . . . . . . . . . . . . . . . . . . . dS Introduction
In de Sitter space, the flat space Regge relation M ∼ L/α (cid:48) (1.1) between the mass M , angular momentum L and Regge slope α (cid:48) , is expected to bemodified when the strings become large and start to feel the curvature of spacetime. Indeed,combining (1.1) with the Higuchi bound [1] for unitary representations of S O (1 , d + 1) atlarge L M l > L , (1.2) with l being the de Sitter scale, led [2, 3] to conclude that for L > l /α (cid:48) unitarity isviolated unless the spectrum is modified. That this happens seems clear, even withoutresorting to unitarity, because at this point, the string has a length comparable to l andnotices the curvature of dS . In this note, since it is not clear what unitarity imposes atthe level of classical strings, we set out to explore the bound (1.2) by other means.We first limit ourselves to rigidly rotating strings in dS , for which the analysis is analyt-ically tractable and for which we can show explicitly that (1.2) holds. In higher dimensionsone can consider strings that lie in the equatorial plane. Such solutions of the Polyakovaction with constraints are captured by an Ansatz of [4]. We begin section 2 by giving aconcise review of this solution class, which we call the Larsen-Sanchez class. We do thiswith a particular emphasis on the three different regimes that occur: strings with inwardcusps, outward cusps or no cusps at all. We find an analytic expression for the ratio oftheir energy and angular momentum and verify (1.2) . It turns out that M l/L is boundedbelow by the corresponding ratio for the rigidly rotating straight string, which was studiedby [2]. That specific solution was first obtained by [5], whose
AdS version is known asthe GKP string [6]. Implications of the Higuchi bound for towers of string states in thecontext of inflation were discussed in [2, 3, 7].The solutions with cusps are related via Wick rotation to Kruczenski’s famous spikystrings in AdS [8] as we discuss in appendix A. There is a vast literature on integrabilityresults in these cases, starting with the work of Pohlmeyer [9] who showed how a for-mulation of the string equations of motion as an O ( n ) σ -model with Virasoro constraintsreduces to sinh-Gordon, cosh-Gordon or Liouville type equations, as described in [10] aswell. In AdS at least, such strings with cusps that grow close to the boundary are dual to ingle-trace boundary operators whose anomalous dimension grows logarithmically withspin [8, 11, 12, 13]. In dS it was shown recently by [14] how the different regimes can bemapped to band structures of the Lam´e potential.After having discussed the Larsen-Sanchez class of solutions in conformal gauge, weapproach the problem more generally via the Nambu-Goto action. There, we can take acylindrical gauge to show that every rigidly rotating solution is indeed part of the Larsen-Sanchez class. It is moreover straightforward to prove that if (1.2) is to be violated, bysome other non-rigid solution, the string needs to extend partly beyond r > l/ √ This is not unexpected, since (1.3) is the stability bound for the closed circular string[15] whose (in)stability was analyzed in [16]. We discuss the behavior of the perturbationsaround such an unstable circular string when it expands towards the horizon, where fromthe point of view of the static patch observer the perturbations get exponentially frozen.The Nambu-Goto equations also lend themselves to numerical analysis, allowing us togive an example of a string that violates (1.2) and expands towards the horizon, stillconsistent with the idea that (1.2) applies only to those strings remaining within the staticpatch and do not reach the horizon in finite proper time. Looking at open strings, we findthat the only rigidly rotating open string that goes through the origin is the GKP-likestring. We can however imagine varying the setup by adding point masses to its ends,which turns out to always increase the value of
M l/L .After that, we will use the method, as applied by [17, 18] in
AdS , of approximating theclassical string worldsheet in a piecewise manner by totally geodesic dS ⊂ dS segments.Such strings are exact solutions, consisting of kinks moving around and colliding with eachother. This can be described in terms of discrete lightcone coordinates determining thecollision events. These events are subject to the algebraic evolution equation of Gubser[19]. We will quickly review this equation and the ideas leading to it. Afterwards, weshall derive expressions for the energy and angular of the segmented string that are indeedinvariant under this evolution. We will show how these discrete expressions can be obtainedfrom the Nambu-Goto conserved currents.The discretization makes it easier for the computer to handle the string evolution withoutnumerical errors and allows us to further investigate how the total energy and angularmomentum influence the general behavior of the string and its tendency to escape throughthe horizon. Typically, some time after parts of the string have left the static patch, the inks end up in different causal patches. At that point the algebraic evolution law breaksdown and the string keeps expanding without generating new kink collisions.We will first generate several random strings, see Figure 4.2, confirming that the generalstability is mostly determined by the total energy. After that, we choose specific initialconditions with a large number of kinks, resulting in a segmented string with M l > L .Whereas the numerical evolution of the Nambu-Goto solution quickly becomes imprecise,here we can very accurately follow how the string rotates rapidly and expands, as in Figure4.3. Working in embedding space, we see how at some point after leaving the static patchno further collisions are generated and the string expands indefinitely.Finally we apply our results to a string which is captured by a black hole in asymptot-ically dS -space. In recent work [20], based on [21, 22], a cosmic string [23] captured by amassive compact black hole is considered in flat space. Several effects such as the ejectionof daughter loops and shrinking due to horizon friction were identified. Here we note thatin dS , due to the shape of the spectrum of the GKP-like string, the horizon friction, whichlowers energy and angular momentum, can increase the length of the string. For a rotatingblack hole, [20] demonstrate curve lengthening as well as a black hole bomb mechanism [24]due to circularly polarized tension waves on the captured string loop. Given our previousobservations that large angular momentum makes the string expand towards the horizon,we expect this to provide a natural ending to the black hole bomb process in dS . Note added:
At the final stages of this project, we became aware of the recent work [25]which has overlap with our section 2. dS At this point we are interested in rigidly rotating strings in the static patch of dS , forwhich we will demonstrate (1.2) . Looking to solve the Polyakov equations of motion withconstraints, we follow [4] and review their Ansatz in static patch coordinates, where themetric g µν is given by d s = g µν d X µ d X ν = − (1 − r l )d t + d r − r l + r d ϕ . (2.1) We need to solve the Polyakov equations ∂ α ∂ α X µ = − Γ µνρ ∂ α X ν ∂ α X ρ . (2.2)4 here Γ µνρ are the Christoffel symbols. The conformal flatness of the induced metric onthe worldsheet in ensured by the constraints g µν ˙ X µ X (cid:48) ν = 0 = g µν ˙ X µ ˙ X ν + g µν X (cid:48) µ X (cid:48) ν . (2.3) The Ansatz of Larsen and Sanchez is to take [4] X µ = ( t, r, ϕ ) = ( c τ + f ( σ ) , r ( σ ) , c τ + g ( σ )) . (2.4) where the constants c , c are real and f , g are real functions of σ . Letting τ run whilekeeping σ fixed, we see that the string indeed does not change its shape, which is why wecall it rigidly rotating. The equations of motion (2.2) and constraints (2.3) are satisfiedwhen f (cid:48) = k − r l , g (cid:48) = k r , c k − c k = 0 (2.5) and r (cid:48) = − (1 − r l )( r c + k r ) + k + (1 − r l ) c , (2.6) where k , k are again real constants. By interchanging the direction of rotation theycan be taken positive. The above Ansatz can also be used with σ and τ interchanged.The resulting surface that is traced out in spacetime will be the same but the timelike andspacelike worldsheet coordinates are interchanged. The induced metric on the string is d s = ( c − ( c l + c ) r )( − d τ + d σ ) . (2.7) Special cases of (2.4) include c = k = 0 , which gives the oscillating closed string [15].The GKP-like open rotor [5] is obtained for k = k = 0 . For c = c = 0 the worldsheetdegenerates to a worldline and we get a point particle instead. We will first review theexplicit solutions, which did not appear in the original paper, but showed up in slightlydifferent notation in [14]. First note that both τ, σ can be rescaled such that c + c l = l .Then we call c = l sin a and c = cos a for some a which we can choose between and π/ , by picking a convention for the direction of rotation. Now we can find out at whichradii the string reaches an extremum, by equating (2.6) to zero. The result is, after using c k = c k , a = l sin a, r b = l √ − (cid:114) − k sin al cos a ) , r c = l √ (cid:114) − k k sin al cos a ) . (2.8) If r b and r c are real we can call them r b = l sin b and r c = l cos b for some b ∈ [0 , π/ sincetheir squares sum to 1. When they are complex conjugate one finds that the discriminant ofthe Weierstrass elliptic function that solves (2.6) is negative and that it is not satisfactoryas a solution. Then, again up to orientation reversal, we can call k = l sin 2 b cos a and k = l sin 2 b sin a . We then find that a bounded solution is given by r = l ( p [ σ + w ] + 16 (3 − cos 2 a )) (2.9) with p the Weierstrass elliptic function with imaginary half-period w and Weierstrassinvariants g = 16 (4 + cos 4 a + 3 cos 4 b ) , g = 154 cos 2 a (8 − cos 4 a + 9 cos 4 b ) . (2.10) Now we can integrate (2.5) using 5.141.5 in [26] f = l cos a sin 2 b p (cid:48) [ v ] log s [ σ + v + w ] s [ σ − v + w ] − σ + w ) ζ [ v ] (2.11) where v = p − [ (3 + cos 2 a )] , ζ is the Weierstrass ζ -function and by abuse of notation,since σ is already a worldsheet coordinate, we used s to denote the Weierstrass σ -function.Similarly g = − sin a sin 2 b p (cid:48) [ w ] log s [ σ + w + w ] s [ σ − w + w ] − σ + w ) ζ [ w ] (2.12) where w = p − [ (cos 2 a − . When using the above in Mathematica, one has to becareful about branch cuts , and in particular it is easier to plot the strings using numericalintegration. The Weierstrass elliptic function is periodic under a shift of σ equal to twicethe real half-period . Moreover, t and ϕ are quasiperiodic in the sense that after one periodthey shift by a constant ∆ t and ∆ ϕ that can be obtained explicitly using the integrals in[26]. We are interested now in closed strings, symmetric around the origin. To have astring that closes in on itself at a fixed time t , we need See [26] or the appendix of [14] for an exposition of the most relevant properties of the Weierstrasselliptic functions. The interpretation of the branched worldsheet as describing (in)finitely many strings at the same time,has been discussed in [15], there referred to as the multi-string property. In Mathematica this is either WeierstrassHalfPeriodW1 or -W1-W2. cot( a )∆ t + ∆ ϕ = 2 π mn (2.13) which means that after n periods the string will have gone around the origin m timesbefore closing in on itself. In the next subsection we discuss some numerically obtainedexamples with small m, n . First we observe that at r a we have r (cid:48) =0 but also − cot( a ) f (cid:48) + g (cid:48) = 0 , at r a . (2.14) This means that the angular derivative at fixed time also vanishes and that there is acusp. We can then distinguish three regimes, apart from the previously mentioned specialcases, using that (2.6) gets positive at large r and negative as r → . When r a < r b < r c thecusps are on the inside of the string. They move with the speed of light. The worldsheetis of course still Lorentzian, though from (2.7) we can see that in this case it is σ that istimelike. When r a = r b < r c we have a point-particle moving around at the speed of light.The second regime is when r b < r a < r c in which case the string extends between r b and r a and has cusps on the outside. A final regime occurs when r b < r c < r a . In this casethere are no cusps and the string extends between r b and r c . Note that r = l/ √ lies inbetween these two, where r = l/ √ is the radius at which a static circular string exists[15]. At smaller radii those circular strings oscillate through the origin and at larger radiithey expand towards the horizon. Associated with the Killing vectors ∂ t and ∂ ϕ are conserved worldsheet currents J ξ and L ξ for the energy and angular momentum respectively. They are obtained from the Noetherprocedure and satisfy ∂ ξ J ξ = ∂ ξ L ξ = 0 (2.15) with ξ ranging over τ and σ . When integrated over the worldsheet at fixed time t , orover any homologous cycle, they yield the energy and angular momentum. Using (2.4) , weobtain This regime does not seem to exist in
AdS since the equation for the extrema only has only 2 real rootsinstead of three and therefore only has the regimes with cusps on the inside or outside depending on theirorder. Related to this, the straightforward Wick rotation of Kruczenski’s solutions gives the first two regimesas we show in A. For the third regime an extra shift is needed. τ = − (1 − r l ) l sin a, J σ = 12 l sin 2 b cos a (2.16) and for the angular momentum L τ = − r cos a, L σ = 12 l sin 2 b sin a. (2.17) Now we integrate over a closed loop, which exists at fixed t , but not at fixed τ . In orderfor t to remain the same, when we integrate over the period ∆ σ , there is an accompanyingchange of ∆ τ = − ∆ tl sin a . We can also write ∆ t as an integral over σ since t (cid:48) = f (cid:48) = l sin 2 b cos a/ (1 − r /l ) , from (2.4) . This gives the following expressions M = (cid:90) ∆ σ d σ [ l sin a (1 − r l ) − l sin b cos a a (1 − r l ) ] (2.18) L = (cid:90) ∆ σ d σ [ r cos a − l sin b cos a − r l ) ] . (2.19) In the previously used notation we have ∆ σ = 2 n v , where v is the real Weierstrasshalf-period. From this, one finds a useful combination M − cot al L = l sin a (cid:90) ∆ σ d σ (1 − r l sin a ) . (2.20) It is easy to see, since the string never extends beyond l sin a , that this quantity willbe positive. In AdS there is a coth a on the LHS instead, which is often called ω andone typically looks at the ω → limit, in which case one finds an anomalous dimensionthat scales logarithmically with L [6, 8]. From this one can also observe, as before, thatif tan a < , or equivalently r < l/ √ , the stronger bound M l > L follows, since theintegrand itself is positive. In general, using section 5.14 of [26], one can find analyticexpressions for (2.18) and (2.19) . In particular one obtains the exact expression
M l − L = n ( αζ [ v ] + β v + γ v ζ [ v ] − vζ [ v ] p (cid:48) [ v ] ) (2.21) where v = p − [ (3 + cos 2 a )] and α = 2 l (sin a + cos a ) β = l a − a + sin 3 a + cos 3 a ) γ = l a (1 − sin 2 a + cos 2 a ) sin b. (2.22)8igure 2.1: We plot M l − L . This quantity is seen to be positive everywhere and becomeszero at the point where the parametrization degenerates and we have a point particle movingat the speed of light. Smaller a gives a string with cusps on the inside and larger a givescusps on the outside. The strange bumps occur when we go from outwards cusps to no cuspsat all, namely when a > π/ − b . Each graph corresponds to a value of b , starting with 0.1in red, 0.3 in purple, 0.5 in magenta and 0.7 in cyan. In Figure 2.1 one sees numerically that (1.2) is indeed always satisfied, for each regime.In going to regime 1, which corresponds to the smaller values of a on the left side of 2.1,the induced metric (2.7) flips sign, as its conformal factor is r − l sin a . Indeed at theboundary between the two regimes we get a particle at the speed of light with inducedmetric that vanishes. This is where the curve hits zero. In Figure 2.2 we plotted the moretelling normalized relation. As the examples in the next section will make clear, eachstring is composed of a number n of identical pieces. Each of these pieces has an angularmomentum no larger than that of the open straight string with highest angular momentum.This is in line with the ‘maximal spin’ noted in [2]. Now we can briefly give some examples of what the solutions look like at fixed time in staticpatch coordinates. For each of the regimes we give an example of a closed string, with a, b determined numerically such that (2.13) is true. In the corresponding plots we also give thenumerical results for their energy and angular momentum. Whether these solutions are ofsinh- or cosh-Gordon (or Liouville) type depends on the sign of cos 4 a − cos 4 b , according L/M l as a function of the parameter a and find that itis everywhere smaller than 1, demonstrating (1.2). The colors are as in figure 2.1. It is clearthat smaller b give higher maxima and for b = 0 we are back at the GKP-like string.. to the discussion in [4] translated to our notation.The first case has r a < r b and is characterized by inward cusps. Such solutions are ofsinh-Gordon type. A particular example with n = m = 1 is obtained for a = 0 . and b = 0 . as shown in Figure 2.3a.In the second case we have r b < r a < r c and hence there are cusps on the outside.These solutions are of cosh-Gordon type as cos 4 a < cos 4 b . An example with m = 1 and n = 4 is found by taking a = 0 . and b = 0 . and can be seen in Figure 2.3b.As discussed in A, these two cases with cusps are the ones that correspond to the Wickrotation of Kruczenski’s solutions in AdS [8].Finally, there is the case r b < r c < r a , which means there are no cusps. These solutionsare again of sinh-Gordon type. An example with m = 1 and n = 3 is given by a = 0 . and b = 0 . . It is shown in Figure 2.3c. Note that the parts of the string inside r = l/ √ balance off those parts outside of it, which would otherwise expand toward the horizon.This also helps to understand how the cusps have been smoothed out, which does notoccur in AdS . a) Inward cusps (b) Outward cusps (c) No cusps Figure 2.3: Examples from each of the three types of solutions in the Larsan-Sanchez classthat are shifted by an imaginary half-period. The origin of the plots is that of the staticpatch and the axes measure the dimensionless coordinate distance r/l . One type of Larsen-Sanchez solutions that so far we haven’t discussed are the ones wherethe Weierstrass elliptic functions have not been shifted by the imaginary half-period. Thosecan describe expanding strings that reach the horizon in finite proper time, such as cir-cular strings that start at r > l/ √ . The rigidly rotating solutions of this unshifted typemoreover extend to the horizon and consequently have infinite mass and angular momen-tum. Nonetheless, the general behaviour of such a solution near the horizon may still beof relevance. We can see from (2.5) that t (cid:48) diverges as / (1 − r l ) . Therefore, to find theshape at fixed t , we must let τ run over a compensating range. This will result in a change ∆ ϕ ∝ log 1 + rl − rl (2.23) which is similar to the logaritmic winding of a static string near the horizon of a Kerrblack-hole [27, 28]. In this case however the string itself rotates uniformly and the horizondoes not, resulting in unbounded energy and making the solution less physical. One couldhowever imagine a rotating string achieving a certain winding until it possibly reconnectsat one of its self-intersections. From the mathematical point of view, one can make a linkwith classification results of [29], in the specific case of Liouville type of solutions wherethe Gauss curvature is constant. We discuss this in appendix B. Of course, the energyand angular momentum integrals will diverge, but integrating over finite parts, one canfind examples where M l < L . Again this does not contradict (1.2) since these strings do a = arctan and b = arcsin 2. The string winds aroundthe horizon as in the case of a stationary string around a Kerr horizon. However, in this casethe horizon is static and the string itself rotates uniformly and therefore has infinite massand is rather unphysical. The string keeps winding around and has only been plotted partlysince Mathematica takes more and more time when getting near the horizon at r/l = 1. not remain strictly within the static patch, as they stretch ever closer to the horizon. Atypical solution of this unshifted kind is given in Figure 2.4. Instead of starting from the Polyakov action in conformal gauge, in some classical cases itis easier to start instead from the Nambu-Goto action S = (cid:90) d τ d σ L ng = (cid:90) d τ d σ (cid:113) ( ˙ X µ X (cid:48) µ ) − ˙ X µ ˙ X µ X (cid:48) ν X (cid:48) ν (3.1) where contractions are made with the static patch metric (2.1) . Here we can take a staticgauge t = τ to find solutions of interest. Simply calculating the equations of motion inthe static patch of dS would give three equations that are proportional to each other, dueto the reparametrization invariance, so we need to impose an extra condition apart fromstatic gauge. For the rest of this section we will also take l = 1 . .1 Closed strings near the horizon For closed strings that are single-valued as a function of ϕ , in particular they must goaround the origin only once, we can fix the invariance by taking a cylindrical static gauge,where t = τ and ϕ = σ . The equation of motion ∂ σ P σt + ∂ τ P τt = 0 (3.2) with P σt = ∂ L ng ∂t (cid:48) , P τt = ∂ L ng ∂ ˙ t (3.3) then becomes, with l = 1¨ r ( r − r − rr (cid:48) ) + r (cid:48)(cid:48) ( r (1 − ˙ r ) + r − r ) + 2 r ˙ rr (cid:48) ˙ r (cid:48) = r − r + 5 r − r + ˙ r (4 r − r ) + r (cid:48) (2 − r + 4 r ) . (3.4) We can use (3.4) to numerically solve for the string motion, given a certain initial shapeand velocity. For strings that rotate rigidly, i.e. don’t change shape but just increasetheir value of ϕ linearly in time, we have r ( τ, σ ) = r ( σ − ωτ ) . In that case, we end upwith a single non-linear ODE for r . This ODE determines the shape of rigidly rotatingstring segment, given a point on that segment and its tangent vector. It shows that theLarsen-Sanchez solutions are in fact the unique rigidly rotating solutions . Finding generalLarsen-Sanchez solutions in this gauge is more involved, but for a circular string oscillatingat frequency ω we have r ( τ ) satisfying ˙ r = (1 − r )(1 − ω r + ω r ) (3.5) describing the solution in [15]. For ω = 2 we have the special static string at r = 1 / √ .For smaller ω the string moves through the origin and then back to the horizon, whereasfor ω > the string either oscillates near the origin or it comes in from the near-horizonregion, reaches a minimum distance to the origin and expands again towards the horizon.We can now add a perturbation r = r ( t ) + εr ( τ, σ ) to this. To first order in (cid:15) , using that r is a circular solution Though of course for the ones with cusps the cylindrical parametrization breaks down at the cuspsthemselves. r − r (cid:48)(cid:48) (1 − r ) ω + 8 r − r (cid:113) − ω r (1 − r ) ˙ r − r (2+ ω − r + ω r (3 − r )) = 0 . (3.6) Later on we want to look at an example for which
M l > L , which means we will have totake a solution that is not close to the circular one. Still, as we noticed previously it meansthat the string will have to move near the horizon, so it is useful to look at what happensin that regime, even in the more tractable circular case. To understand perturbations inthis limit, we can take r → in (3.6) and find ¨ r + 6 ˙ r + 8 r = 0 (3.7) showing that irrespective of the precise shape of the perturbations, to a static patchobserver they will appear exponentially damped as r ( t ) ∼ e λt with λ = − or − . This isdue to the fact that the static patch time diverges exponentially as a function of the globaltime, in the neighborhood of the horizon. The above describes the perturbation in thecoordinate distance r . Near the horizon however, for the circular string, t itself behaves as t ≈ − log(1 − r ) , so the physical distance δs evolves as δs = δr (1 − r ) / ∼ e ( λ +1) t (3.8) which stills decreases exponentially. For more general perturbations, the above getsinvolved and perhaps a more technical approach along the lines of [30] could be beneficial.To find an example with M > L , we need to move away from the non-rotating circularcase and take as initial position r and velocity ˙ r something rapidly moving and necessarilyclose to the horizon, like r (0 , σ ) = 0 .
012 sin 8 σ + 0 . , ˙ r (0 , σ ) = 0 .
03 cos 8 σ. (3.9) The energy and angular momentum are determined by M = (cid:90) d σ P τt , L = (cid:90) d σ P τϕ (3.10) with P τt = r − r + r (cid:48) (cid:113) r (cid:48) + r (1 − ˙ r − r ) − r , P τϕ = r ˙ rr (cid:48) (1 − r ) (cid:113) r (cid:48) + r (1 − ˙ r − r ) − r . (3.11) While still having a Lorentzian worldsheet. he square roots are real whenever the worldsheet is indeed Lorentzian. For (3.9) , wefind numerically M = 1 . and L = 1 . . So this goes to show that, for strings with centerof mass at the origin, the bound M > L can be violated when the string will end up inthe horizon. For τ not too large, we can solve (3.4) numerically and the shape changes asin Figure 3.1.The string as a whole moves outwards and the amplitude of the perturbation growssmaller in the coordinate r . We see also that the tips of the perturbation get stuck on thehorizon, while the inner parts of the string continue to move by, resulting in a change inshape. By taking r (0 , σ ) = 1 n sin nσ + (1 − n ) , ˙ r (0 , σ ) = 1 √ n cos nσ. (3.12) with n ∈ N large, we can make L/M arbitrarily large, growing like √ n , while still havinga Lorentzian worldsheet.Finally, from (3.11) one can also see that any string, with L ≥ M will need to have apart outside of / √ . Previously, in the discussion around (2.20) , we only obtained this forrigid strings. Here we see it generally, noting that having a Lorentzian worldsheet imposes r ˙ r − r ≤ r − r + r (cid:48) . (3.13) This means that even locally P τt > P τϕ unless r (cid:48) > ˙ r . If that is the case, then P τϕ P τt < r r (cid:48) (1 − r )( r − r + r (cid:48) ) , (3.14) where the RHS will be less than if r < / √ . One could also wonder about the movement of open strings. In this case it is natural to take t = τ and impose the orthogonality condition of vanishing dS inner product g µν ˙ X µ X (cid:48) ν = 0 .Then we obtain as equation of motion ∂ τ P τt = 0 , P τt = (1 − r ) X (cid:48) (cid:112) − ˙ X X (cid:48) . (3.15) with X denoting the coordinates and taking as inner product the dS one. Solutions arefound by considering some P τt = K ( σ ) . Near the ends of the open string, where ˙ X = 0 ,due to the vanishing of P σϕ and P σr , we also necessarily have X (cid:48) = 0 whenever K remains a) string at τ = 0 (b) string at τ = 0 . τ = 0 (d) radius as at τ = 1 . Figure 3.1: Numerical example with
M l > L , showing that this bound need not holdfor strings that expand indefinitely towards the horizon. Initial conditions are r (0 , σ ) =0 .
012 sin 8 σ + 0 .
98 and ˙ r (0 , σ ) = 0 .
03 cos 8 σ . The axes give the dimensionless coordinatedistance r/l , with horizon at r/l = 1. One can see how the string moves towards the horizonand how in doing so the outer extrema get slowed down most, while the inner parts continueto move. 16 nite. In this way one can also numerically evolve strings with a given initial shape r ( σ ) and velocity profile determined by K ( σ ) by solving both the above and the orthogonalityconstraint. This simple-minded approach seems to run in to serious numerical errors ratherquickly. Nevertheless, at least at small times one can consider a straight string that hasbeen given a higher angular velocity for its inner parts than in the rigid rotor. The stringstarts to bulge and the endpoints move outward towards the horizon.One can however see that the only rigidly rotating solution through the origin is theGKP-like rigid rotor. This can be done by forgetting about the orthogonality constraintand instead parametrizing as r ( σ ) and ϕ = ωτ + ϕ ( σ ) . The equation of motion can berewritten as an equation for ϕ ( r ) alone, ( − ω ) r − ω ) r ) d ϕ d r + r (1 − r ) ( − r )( d ϕ d r ) = r (1 − r )(1 − (1 + ω ) r ) d ϕ d r . (3.16) The straight rotor is obtained by ϕ (cid:48) ( r ) = 0 everywhere. Apart from this solution, (3.16) has no other solutions that can be Taylor-expanded around the origin. By uniqueness, theother solutions that do not reach the origin again describe parts of the Larsen-Sanchezstrings.To change the setup and gain some more intuition one could also imagine modeling arotating string as having an inner part consisting of a straight, rigidly rotating open stringand outer parts that are described by some blobs of mass. In the extremely simplified case,we would be adding a point-mass to each end of the open GKP-like string string rotatingat angular velocity ω . This configuration, when having the masses M at radius r has totalenergy (still setting l = 1 ). E = 2 M − r (cid:112) − (1 + ω ) r + (cid:90) r − r d r √ − r − (1 + ω ) r . (3.17) The total angular momentum is L = 2 M r ω (cid:112) − (1 + ω ) r + (cid:90) r − r d r r ω √ − r (cid:112) − (1 + ω ) r . (3.18) In equilibrium one needs to solve the equation of motion for r , ∂ r L = 0 , for the equilib-rium length r . This tells us that the tension balances the centripetal acceleration M r (1 + ω ) (cid:112) ω ) r − (cid:114) − (1 + ω ) r − r = 0 . (3.19)17igure 3.2: In black L/M for the straight open string that rigidly rotates, as in [2]. Theother curves are for increasing masses added at the two ends of the string, ranging from 0.25in red to 1 in cyan.
We solve for r and plug into E and L . The result is that the ratio L/E is alwayssmaller than that of the straight rotor with same angular velocity as seen in Figure 3.2.If we imagine modeling the outer part of the string by some blob of mass attached to aninner straight part, it means that for high
L/E the configuration can only be stabilizedby letting the blob expand towards the horizon, in line with what we argued previously.For the straight string itself, the
L/E ratio is bounded by . at ω = 0 . , where E − L/ω = 0 . Both E and L reach a maximum at this point, as previously reported in [2]. In [17] and [18] it was found that the evolution of classical strings in
AdS could beapproximated by exact solutions given in terms of a finite number of kinks moving atthe speed of light, pairwise lying in a totally geodesic AdS subspace and colliding witheach other after some time, thereby changing the shape of the string. Soon after, itwas realized by Gubser in [19] that such solutions could be completely described by analgebraic evolution law for discrete lightcone coordinates on the worldsheet. Since this iseasily implemented on a computer, we intend to follow this procedure here for dS .We will begin with reviewing the evolution law of [19]. In the next section, we buildfurther on this work, by obtaining a simple expression for the conserved energy and angularmomentum of the string. For given initial configurations with 4, 6 and 8 collisions we enerate a random sample of initial velocities. The generic movement is chaotic and theenergy rather than angular moments in general determines whether the string will exit thestatic patch, after which the evolution law will break down. Finally, inspired by section 3.1,we give an explicit example of a segmented string with its angular momentum larger thanits mass. The discrete approach makes it easy to see how the string expands as it rotatesand leaves the static patch, until at some point the kinks become causally disconnectedand do not collide again. dS Here we briefly review the ideas leading to the Gubser evolution law [19]. Let us beginwith a string in flat space, where we can decouple the evolution in terms of left- andright-movers. A worldsheet of the form X ( τ, σ ) = Y L ( τ + σ ) + Y R ( τ − σ ) (4.1) with Y L and Y R lightlike trajectories will solve the Polyakov equations of motion withconstraints. Some neat examples are when these trajectories move around circles of dif-ferent radii. In that case the resulting curve is an epicycloid or hypocycloid depending onthe relative direction of motion of Y L and Y R . The strings rigidly rotate and are the flatspace analogs of the Larsen-Sanchez strings in regime 1 and 2 respectively, see section 2.Now one can discretize these lightlike trajectories, by making them piecewise linear andthe resulting string still solves the equations of motion. It can be given in terms of discretelightlike coordinates ( i, j ) on the worldsheet X i,j = Y L,i + Y R,j . (4.2) These events X i,j are the kink collisions at which two kinks collide and subsequentlymove apart. The entire string at fixed time is then the collection of these ordered kinks,connected by straight line segments. An example is given in Figure 4.1. For closed stringsthere is a periodicity X i + n,j − n = X i,j for some n .A very analogous approach can be now be considered in AdS and dS [17, 18, 19].In this case, we still have kinks moving at the speed of light, in straight line segmentsin embedding space. A collision happens at points X i,j , after which one kink moves bya lightlike amount ∆ L,i,j and similarly the other moves by ∆ R,i,j . As opposed to flatspace, these ∆ now generally depend on both i and j . The three quantities X , ∆ L and a) before collision (b) kink collision (c) after collision Figure 4.1: This segmented string in flat space is obtained by taking Y L of radius 4 and Y R or radius 3. There are 32 kinks and 16 collisions, which in this special case occur all at thesame time. This string is an exact solution and approximates a rigidly rotating hypocycloid,which is the flat space analog of the regime 2 Larsen-Sanchez solutions of section 2.∆ R determine a plane in embedding space and, upon taking the intersection with thehyperboloid, an AdS or dS subspace in which the two adjacent kinks move. The straightlines connecting the kinks in flat space are therefore replaced by fixed time sections oftotally geodesic AdS and dS subspaces. Given the causal wedge X i,j , X i +1 ,j = X i,j + ∆ L,i,j , X i,j +1 = X i,j + ∆ R,i,j (4.3)
Gubser’s evolution equation, stated here for dS , still taking the dS length l = 1 , uniquelydetermines X i +1 ,j +1 as X i +1 ,j +1 = X i,j + 2 ∆ L + ∆ R − (∆ L · ∆ R ) X i,j L · ∆ R (4.4) where the inner product is that of the embedding space R , and we omitted the indiceson ∆ L,R to avoid clutter. It was already noted in [19] that if ∆ L · ∆ R < − the kinks arecausally disconnected, which is possible due to the expansion of dS , and there will not bea future collision. The evolution (4.4) itself is not to be used there, since it would give acollision at an earlier time. Having reviewed the result of [19], we will now continue by obtaining expressions for theenergy and angular momentum of segmented strings in dS . We will first take a step back rom the discrete coordinates to continuous ones, and find the conserved quantities in theusual approach. The result will be entirely in terms of the discrete coordinates and iseasily checked to be invariant under (4.4) .First we start by extending the evolution equation to describe the dS subspace in whichthe kinks move. This is easily done by taking X ( α, β ) = X i,j + 2 α ∆ L + β ∆ R − αβ (∆ L · ∆ R ) X i,j αβ ∆ L · ∆ R . (4.5) Clearly X ( α, and X (0 , β ) are the kink trajectories emerging from the collision X i,j .Energy corresponds to the symmetry δX = X and δX = X , whereas angular momen-tum corresponds to δX = − X and δX = X The conserved currents obtained from theNambu-Goto action are messy, but along the piecewise linear contour consisting of the linesegments α = 0 and β = 0 , they simplify, since ∂ α X | β =0 = ∆ L , ∂ β X | α =0 = ∆ R . (4.6) The square root of the induced metric becomes | ∆ L · ∆ R | there. Using moreover that the ∆ are null and therefore sg (∆ L · ∆ R ) = − , we find that the contribution of either of thesetwo line segments to the energy is simply M = 12 (cid:90) d ξ ( X ∂ ξ X − X ∂ ξ X )= 12 ( X i,j + ∆ i,j ) X i,j − ( X i,j + ∆ i,j ) X i,j (4.7) with ξ = α ( β ) whenever β ( α ) is zero and correspondingly in the last line we have ∆ = ∆ L or ∆ R respectively. Similarly for the angular momentum, L = 12 (cid:90) d ξ ( X ∂ ξ X − X ∂ ξ X )= 12 ( X i,j + ∆ i,j ) X i,j − ( X i,j + ∆ i,j ) X i,j . (4.8) The total conserved quantities for the segmented string in dS are then found by takingany closed contour γ on the (discretized) worldsheet, consisting of such α and β pieces andsumming their contributions given above. For instance, in the next section we will evolvestrings from initial conditions X i, − i and X i +1 , − i , with i periodic modulo some integer n . Inthat case the total mass and angular momentum are obtained from the initial conditionsas = 12 n (cid:88) k =1 (( X k +1 , − k + X k, − k +1 ) X k, − k − ( X k +1 , − k + X k, − k +1 ) X k, − k ) (4.9) L = 12 n (cid:88) k =1 (( X k +1 , − k + X k, − k +1 ) X k, − k − ( X k +1 , − k + X k, − k +1 ) X k, − k ) (4.10) We made brief use of the continuous Nambu-Goto description only to arrive at the rightexpressions. These expressions however clearly make sense within the discrete descriptionalone and can be checked to be indeed invariant under (4.4) . Since they are easily treated in a numerically precise way, we can generate some randomsegmented strings and look at their energy and angular momentum, comparing it withhow long they are expected to stay within the static patch. To do this, we take a simpleapproach. For 4, 6 and 8 specified initial collisions X i, − i at embedding space time t = 0 wewill randomly generate future collisions X i +1 , − i . Through (4.9) and (4.10) we obtain theconserved quantities.We take the initial conditions to be point-symmetric around the origin of the X X -plane. This is a quick way to ensure that the string has center of mass at the origin.For the X i,i we choose a configuration that is close enough to the origin, so that it doesnot immediately escape the static patch, yet far enough from it so as to not reduce itsdynamics to that of a string in flat space. We also did not take completely symmetricconditions, such as a square, which would result in periodic motion as noticed by [18] andzero angular momentum.The initial configurations are shown, for 4 initial collisions X i, − i in Figure 4.2a, 6 collisionsin Figure 4.2b and 8 in Figure 4.2c. Different randomly generated X i +1 , − i result in stringswith different conserved quantities. We take a random sample of size 40 for each case anddisplay energy M versus angular momentum L in the scatterplots of Figure 4.2. The colorsof the points indicate how long the segmented string remains within the static patch. Weperformed ten iterations of the evolution equation so that the color is determined by aninteger between 1 and 10. Strings in blue are the ones that remain inside the static patchfor at least ten iterations and are likely to be stable.It is notable how the colors are clearly most sensitive to the total energy M , as expected.Highly energetic strings, even with low angular momentum are likely to escape to thehorizon. All of these randomly generated strings have a rather low angular momentum to a) 4 initial collisions (b) 6 initial collisions (c) 8 initial collisions Figure 4.2: Considering strings with the above initial collision data X i, − i , we generate random X i +1 , − i , lightlike separated from the initial ones and lying on the dS hyperbola. This thenfully determines the motion of the string. For each of the initial configurations we showthe angular momentum L versus energy M for a sample of 40 thus randomly generatedsegmented strings. The color scale shows for how many iterations the string remains withinthe static patch. We stopped after 10 iterations, so that if the string remains within r ≤ nergy ratio, but those with larger L/M mostly have both large M and L , in line with theexpectation that they will leave the static patch. M l < L
In this section we again wish to find, as in section 3.1, an explicit example of a stringthat has angular momentum higher than its energy, violating the bound (1.2) . Previouslythe numerical evolution of the solution became erratic rather quickly. The method ofsegmented strings should allow us to understand more exactly its behavior. We will startwith a configuration in embedding space at time t = 0 , point-symmetric around the originof the X X -plane. As before, we want a string that is close to the static patch horizonand has many wiggles. We therefore consider an initial configuration of 32 kink collisions,determined by X i, − i = (0 , √ . , . πi , . πi ) , i even mod 32(0 , √ . , . πi , . πi ) , i odd mod 32 . (4.11) For the first step we now want to generate 32 new collisions. To this end, we consider X , and X , − . We need to find a suitable new collision that lies on the dS hyperbola andlies on a null ray coming from each of these previous two collisions. We do the same thingfor X , and X − , . After generating some examples we can take one in which the two newcollisions have moved counterclockwise, resulting in high angular momentum, namely X , = (0 . , . , . , . X , = (0 . , . , . , . where we rounded the numbers just to be able to show them in a single line. Whenwe rotate this by π/ , we retain the Z symmetry and have determined all X i +1 , − i . Thisinitial data, as shown in Figure 4.3a, is all we need to evolve the string using (4.4) . From (4.9) and (4.10) we find M = 5 . , L = 5 .
97 (4.13) and therefore
M < L . We expect that this implies that the string must leave the staticpatch, and indeed it can be seen that already at the next iteration half of the collisions lieoutside the static patch. The string rotates counterclockwise and expands as it does so, asone can see in Figure 4.3. At the fourth step all collisions lie outside the static patch and nally, after the tenth step neighboring collisions are no longer causally connected. Theevolution law breaks down and the string keeps expanding. As mentioned, the discretecollision points are all that is needed to determine the evolution, even without mentioningthe string segments that connect them. It is pleasant however to see how the 2 kinks thatemerge from each collision are connected by a fixed time curve in a totally geodesic dS fragment and collide again with other kinks. This is why we show in Figure 4.3b the stringat times in between collisions, when one can distinguish 64 kinks. (a) Initial configuration at embeddingspace time t = 0 with the 32 dots indi-cating the kink collisions (b) Configurations at embedding spacetimes t = 0 .
6, 1 . .
8. There are 64kinks at each time
Figure 4.3: The segmented string with initial configuration determined by (4.11) and (4.12)has angular momentum larger than its energy and expands towards the horizon. After 3iterations the string has partially left the static patch. After 10 iterations the differentcollisions are causally disconnected and the evolution law breaks down. On the right we seethe configuration at intermediate times, when two kinks have come out of each collision andmove on to collide with the next neighbor. dS The possibility of cosmic strings is an old subject [23, 31, 32]. Recently though, interestingnew phenomena have been discussed in [20]. Their setup was to consider a string loopcaptured by a much more massive and compact black hole in four dimensions. At zerothorder it can be modelled, as a closed string in flat space, pinned to a specific point. The inearity of the equations of motion allows for a description using an auxiliary curve. Usingthis to their advantage, [20] gave a beautiful and intuitive picture of the main effects inthe evolution of such a string, being due to the black hole’s finite mass, torques acting onthe black hole and possible superradiance effects. In dS , as we have noticed, the equationsbecome non-linear and much less pleasant. However, we can think of a situation wherethere is a (non)-rotating black hole with event horizon much smaller than the cosmichorizon, such that there is a regime close to the black hole where the evolution is verywell described by [20], but needs to be complemented by curvature effects when it movesoutwards too much. As argued in [20], in flat space, one can expect the string of length s to develop self-intersections after a certain time. At reconnection, which for cosmic strings happens withprobability p ∼ [33], one expects the string to emit daughter loops, causing it to shrink.If R is the Schwarzschild radius and µ the string tension, then for p (cid:29) µs/Rt shrink = Rµ . (5.1)
The derivation starts from the periodicity of the auxiliary curve and therefore changesare expected. At intermediate radii however one expects the order of magnitude estimateto be correct. There is a second effect due to the torques of the moving string on the blackhole, which is called horizon friction. The result is that the string loses energy and angularmomentum to the black hole , which happens on a timescale t fr = s R . (5.2) The derivation of this was rather general and only assumes that the tangent vectors tothe string at the 2 points where it is attached to the black hole move on a timescale s .Again, non-linearity will become important, but at intermediate ranges such estimate isexpected to hold.In dS as in flat space, the torques act as to decrease the energy and angular momentumof the string. In flat space this inevitably makes the string shorter. In dS , we can consider This is not unlike the case of a string in
AdS , with one end moving at constant velocity on the boundary,and the other stretching down to the horizon. In such a configuration energy is also dissipated, resulting inthe drag force calculated by [34, 35]. he straight open string [5], for which the energy and angular momentum both attain amaximum at [2] K [ x ] = 2 E [ x ] , x = 1 ω + 1 , (5.3) with E and K the elliptic functions, corresponding to the previously mentioned ω ≈ . , see also Figure 3.2. When ω is smaller than this, the horizon friction will then tendto make the string longer until it stretches towards the horizon.For generic configurations, possible self-intersections and reconnections may quickly re-duce s until the flat space limit is reached. For s < µ − / R (5.4) the horizon friction will be the most relevant effect, so for small enough µ one couldexpect the friction to be the most important one, even closer to the cosmic horizon. In the rotating black hole there is, beyond the shrinking and friction, the additional inter-esting possibility of superradiance, where circularly polarized tension waves on the stringget amplified upon reflection from the black hole. The fastest growth is achieved for modesof frequency ω = 12 Ω BH cos θ min (5.5) with θ min being the minimum angle between the black hole spin axis and the string [20].In flat space, due to the periodicity of the string, after bouncing back, the tension wavewill move along the string and approach the black hole from the same direction once more.Modes are then amplified on a timescale of t bomb = s R Ω BH cos θ min ) . (5.6) This leads to a black hole bomb scenario [24], where the string itself plays the role of acavity [20]. Additionally the length of the string increases. For a straight string in the flatspace regime with Ω BH (cid:29) /s , this happens as s = (cid:113) s + 16 π R Ω BH t. (5.7)27 enerically (5.6) with be the smallest timescale in the system and the exponential growthwill manifest. For strings that are close enough to the dS equatorial plane it is possiblethat the lengthening process happens at a faster rate. When this happens and the stringexpands towards the horizon, the bomb will not go off. That excessive lengthening wouldspoil the linearity assumptions in deriving the black hole bomb process actually happensin flat space as well. The difference is that the dS horizon will have the additional effectsof initially accelerating the expansion of the string and, as it moves close to the horizon,exponentially damping perturbations in static patch time.In dS , even before reaching the horizon the non-linearities in the equations of motionfor the string will become important and the direction at which tension waves approachthe black hole will no longer be strictly periodic, which would also disable the bombmechanism, or at least reduce its efficiency.The importance of reconnections in the rotating case, even in flat space, is not veryclear, as [20] already noted. It may well make a generic string stay within the flat spaceregime such that the above scenario where the string moves towards the horizon does notgo through. In that case one could expect that already in flat space the bomb gets defusedthrough reconnections.As noted in [20] investigating this oscillatory behaviour and possible limit cycles throughsimulations deserves further attention. Considering the setup within the static patchthrough simulations is bound to make the phase space structure even more interesting,though arguably for realistic parameters its influence will be negligible. We showed that for rigidly rotating strings with center of mass at the origin of the staticpatch of dS , with length scale l , the Higuchi bound M l > L (1.2) on energy M versusangular momentum L holds, as seen in Figure 2.2, thereby extending the observation of[2] and providing evidence to the idea that all strings that remain within the static patchsatisfy (1.2) . These rotating strings are described by the Larsen-Sanchez class of solutionsof the Polyakov action with constraints [4]. We gave a brief review of these, with emphasison the three types of string in Figure 2.3. We argued that for strings with higher angularmomentum per mass would end up expanding towards the horizon.After having commented on the unshifted Larsen-Sanchez solutions, leaving their relationto the classification [29] for appendix B, we approached the problem via the Nambu-Goto ction. This proved useful in understanding uniqueness and perturbations, as well as fornumerical analysis. In Figure 3.1, we gave a numerical example of a string that falls intothe horizon and can therefore violate (1.2) , zooming in on how its shape changes as itmoves nearer to the horizon. We further discussed the case of the straight open rotor withpoint masses attached which always has a higher M l/L ratio than the corresponding openGKP-like string, as can be seen in Figure 3.2.We continued by approximating the classical string worldsheet in a piecewise manner bytotally geodesic dS ⊂ dS segments, following the work done in AdS by [17, 18]. Thosestrings are exact solutions and can be described in terms of discrete lightcone coordinates,subject to the algebraic evolution equation of Gubser [19]. We derived discrete expressionsfor the energy and angular of the segmented string that are indeed invariant under thisevolution. The algebraic evolution law breaks down when the expansion is sufficiently largefor the kinks to end up in different causal patches, which typically happens some time afterparts of the string left the static patch.For different initial conditions we generated random samples of segmented strings. Thescatterplots in Figure 4.2 suggest that the probability for the string to remain in thestatic patch is mostly determined by its total energy. After that, we choose specific initialconditions with 64 kinks, and 32 kink collisions, resulting in a segmented string with M l < L . Whereas the evolution of the Nambu-Goto solution quickly became numericallyimprecise, here we can very precisely follow how the string evolves in embedding spacetime, see Figure 4.3. The string leaves the static patch at the third iteration. The Gubserevolution law breaks down after the tenth iteration, when the collisions are no longercausally connected. After this the segmented string simply keeps expanding.Finally we pointed out possible changes to the phenomena described by [20] when con-sidering their cosmic strings captured by black holes in case the configuration takes placein dS instead of flat space. For the case of a non-rotating black hole we found that thehorizon friction can make the string longer. In this last section, particularly for rotatingblack holes, we were not able to provide much details, as the role of reconnections even inflat space is not well understood and calls for future research. It would be very interestingto figure out the complete phase space structure.Likewise it would be good to understand if (1.2) , which was referred to as the Higuchibound in [2, 3] has such a group theoretic interpretation at the level of classical strings. TheHiguchi bound [1], corresponds to positivity of the quadratic Casimir, for the principal andcomplementary series representations of S O (1 , d + 1) . At large quantum numbers, taking ll conserved charges zero, apart from the static patch energy and angular momentum, thequadratic Casimir reduces, at the classical level, to M l − L . For the rigidly rotatingstrings, which remain strictly within the static patch, this was shown to be positive. Forstrings that expand towards the horizon it can be negative, as we demonstrated in sections3.1 and 4.4. This could be related to the fact that the string exits the horizon in finiteproper time and that different parts of it lose causal contact after doing so.It would in general be interesting to understand better how strict the bound is for classicalstrings. For instance, is the value of M l/L for the open GKP-like string [2] a lower boundfor all strings that remain in the static patch?We did not pay much attention to decay effects, which are enhanced especially near thecusps [36], or backreaction, which we assumed to be negligible. We mostly discussed the3-dimensional case as we can always restrict to strings lying in a dS subspace of dS d .Additionally this is also technically the more straightforward case to analyze. It could befruitful to take a more general approach. Perhaps in this respect the many integrabilityresults mentioned in the Introduction, such as the relation to the sinh-Gordon model viaPohlmeyer reduction [9, 10], or the band structure of the Lam´e potential [14], could givevaluable insights. Acknowledgements:
I would like to thank Nikolay Bobev, Frederik Denef, KristofDekimpe, Fri ð rik Freyr Gautason, Yuri Levin, Jef Pauwels and Marco Scalisi for theiruseful suggestions and comments on the draft. A Wick rotation of Kruczenski solutions
The Kruczenski strings [8] in conformal gauge are given in [13]. By their Ansatz, they arerigidly rotating and upon Wick rotation from
AdS to dS should therefore fit into theLarsen-Sanchez class. In AdS global coordinates we have r = l √ a cn [ u, k ] + cosh 2 b sn [ u, k ] − (A.1) with u = ( cosh 2 a + cosh 2 b cosh 2 a − σ, k = cosh 2 a − cosh 2 b cosh 2 a + cosh 2 b . (A.2) In AdS there are only 2 cases, with inward or outward cusps respectively, depending onwhether a < b or a ≥ b . The well-studied case of spiky strings in AdS [8] corresponds o the regime with outward cusps close to the boundary. These are dual to particularsingle-trace operators of the form O = Tr D k + F D k + F · · · D k + F (A.3) with k large. The strings extend from l sinh b to l sinh a . Shifting by a half-period givescomplex results. We can rotate the above result by l (cid:55)→ i l into dS , correspondingly rotating a and b , and find r = l √ − cos 2 a cn [ u, k ] − cos 2 b sn [ u, k ] ) (A.4) with u = σl ( cos 2 a + cos 2 b − cos 2 a ) , k = cos 2 a − cos 2 b cos 2 a + cos 2 b . (A.5) For the regime with cusps, the above is real. With the previously used conventions for c , c , k , k and making use of relation 8.169 of [26] between the Weierstrass and Jacobielliptic functions, we retrieve the previous expression for the Larsen-Sanchez solution uponidentifying w = i √ K [1 − k ] √ cos 2 a + cos 2 b . (A.6) B Classification results of Chen
In [4] it is noted that the conformal factor e α of the worldsheet metric satisfies a sinh-Gordon, cosh-Gordon or Liouville type equation depending, in our notation, on the signof cos 4 a − cos 4 b (positive, negative or zero respectively). If a = b or π/ − b it is zero andthere is the Liouville type equation in lightcone coordinates α + − − e α = 0 . (B.1) This occurs precisely in the cases of constant Gauss curvature. Compact solutions withinthe static patch then degenerate to point particles moving at the speed of light. Thereare however the unshifted solutions that extend to the horizon describing more interestingminimal surfaces from the mathematical point of view. We can relate these to the ones weobtained via a classification result of [29]. here, the question is asked which Lorentzian minimal surfaces of constant Gauss cur-vature can occur in dS d +1 . In other words, Lorentzian minimal surfaces for which theinduced metric can be taken to have conformal factor α = − u + v ) + log 2 . It is proventhat one of three possibilities occurs. Either i) the surface is totally geodesic, or ii) it canbe written as S ( u, v ) = z ( u ) u + v − z (cid:48) ( u )2 (B.2) for a certain curve z in embedding space R ,d +1 such that ( z, z ) = 0 , ( z (cid:48) , z (cid:48) ) = 4 , ( z (cid:48)(cid:48) , z (cid:48)(cid:48) ) = 0 , z (cid:48)(cid:48)(cid:48) (cid:54) = 0 (B.3) with the embedding space inner product, or iii) it can be written as S ( u, v ) = z ( u ) + w ( v ) u + v − z (cid:48) ( u ) + w (cid:48) ( v )2 (B.4) with more involved conditions on the curves z and w , see [29]. Here we will just look atsome examples that can be obtained from i) and ii).An example that gives i) is to take z ( u ) = ( − − u , u, − u , , (B.5) z (cid:48)(cid:48)(cid:48) = 0 and therefore we have a totally geodesic dS ⊂ dS . Indeed S ( u, v ) = ( − uv u + v ) , − uu + v , − uv u + v ) ,
0) (B.6) parametrizes this. Another simple example, of ii), is when we take as a seed curve z ( u ) = √ u, sinh u, cos u, sin u ) . (B.7) The surface becomes S ( u, v ) = ( √ uu + v − sinh u √ , √ uu + v − cosh u √ , √ uu + v + sin u √ , √ uu + v − cos u √ . (B.8) Already here one sees that this surface is unbounded due to the denominator u + v . Onecan convert to static patch coordinates S ( u, v ) = r ( u, v )(cosh t ( u, v ) , sinh t ( u, v ) , cos ϕ ( u, v ) , sin ϕ ( u, v )) . (B.9)32 otably, this solution turns out to belong to the Larsen-Sanchez class, where the Weier-strass function is not shifted by the half-period, i.e. the ones that expand towards thehorizon and indeed have infinite mass and angular momentum. The above example hasLarsen-Sanchez parameters a = b = π/ and its simple expression allows to calculate theintegrals for M, L explicitly. Converting u, v to τ = ( u + v ) / √ and σ = ( u − v ) / √ wehave r = (cid:114)
12 + 1 τ . (B.10) We reach the horizon as τ → √ and, when integrated up to τ we find LM = τ log |√ τ | − τ log |√ − τ | − √ τ log |√ τ | − τ log |√ − τ | + 2 √ both M and L diverge, but their ratio approaches one.A seemingly different example can be found by taking as a seed curve √ u (cosh 1 /u, sinh 1 /u, cos 1 /u, sin 1 /u ) . (B.12) However, it is the same solution as before, with coordinates /u and /v .Other solutions can be found for instance by taking a static patch parametrization anda specific Ansatz for θ, ϕ . Satisfying ( z (cid:48) , z (cid:48) ) = 4 is done by taking r ( u ) = 2( θ (cid:48) + ϕ (cid:48) ) − / (B.13) and then the condition on ( z (cid:48)(cid:48) , z (cid:48)(cid:48) ) gives for each choice of ϕ a nasty non-linear ODEfor θ . Generally, tractable Ans¨atze include the cases where ϕ is a multiple of θ . Theseturn out to give Larsen-Sanchez solutions once again, through conservation of misery. Forexample ϕ = 2 θ results in a solution θ = 2 √
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