Planetoid string solutions in axisymmetric spacetimes
Abstract
The string propagation equations in axisymmetric spacetimes are exactly solved by quadratures for a planetoid Ansatz. This is a straight nonoscillating string, radially disposed, which rotates uniformly around the symmetry axis of the spacetime. In Schwarzschild black holes, the string stays outside the horizon pointing towards the origin. In de Sitter spacetime the planetoid rotates around its center. We quantize semiclassically these solutions and analyze the spin/(mass) (Regge) relation for the planetoids, which turns out to be nonlinear.
pacs:
11.25.w, 04.70.s, 98.80.CqI Introduction and Motivations
The systematic investigation of strings in curved spacetimes started in [3] has uncovered a variety of new physical phenomena (see [4, 5] for a general review). These results are relevant both for fundamental (quantum) strings and for cosmic strings, which behave in an essentially classical way.
The study of classical and semiclassical strings in curved backgrounds will provide and is indeed providing us with a better comprehension of what a consistent string theory and gravity theory entail. In this context we place the present paper, which continues the line of research set by [3].
Among the heretofore existing analysis of the motion of classical strings in gravitational backgrounds a special place is to be granted to exact solutions, usually obtained by means of separable ansätze (nonseparable exact solutions were systematically constructed for de Sitter spacetime [6]). Such are the circular string ansatz [7, 8], which for stationary axially symmetric spacetimes reduces the nonlinear equations of string motion to an equivalent onedimensional dynamical system [9], or the stationary string ansatz [10].
In this paper we examine a different ansatz, which we have called the planetoid ansatz, in stationary axisymmetric spacetime backgrounds. The planetoid solutions are straight nonoscillating string solutions that rotate uniformly around the symmetry axis of the spacetime. In Schwarzschild black holes, they are permanently pointing towards while they rotate outside the horizon. In de Sitter spacetime the planetoid rotates around its center.
We call our ansatz planetoid since it generalizes to strings the bounded circular orbits of point particles in such spacetimes. In the case of the Schwarzschild geometry the planetoid string solutions presented here are generalization of the circular orbits of planets.
We will show how our planetoid ansatz produces either orbiting strings with bounded worldsheet and length or strings of unbounded length. The main competing physical forces in the context of this ansatz are the attraction of gravity, the centrifugal force, and the string tension. The combination of these three causes in different proportions produce different effects, as we will now see.
It should be noted that the effects mentioned take place even when the gravitational field acting on the string is not strong. They are due to the non local character of the string.
We quantize semiclassically the planetoid string solutions using the WKB method adapted to periodic string solutions [11]. We obtain in this way their masses as a function of the angular momentum. Such relations are nonlinear and can be considered as a (generalized) Regge trajectory [See figs. 1 and 2].
Ii Equatorial planetoid ansatz
ii.1 The ansatz and the string equations of motion
We consider our classical strings propagating in dimensional stationary axisymmetric spacetime. For simplicity, we restrict in this paper to strings propagating in the equatorial plane . We can thus restrict ourselves to the metric with line element of the form
(1) 
Let and be the timelike and spacelike worldsheet coordinate respectively in the conformal gauge. Under the ansatz
(2) 
the equations of motion for a string in this background are given by the following onedimensional equivalent system:
(3) 
The function will then be given by the zero energy motion in “time” of under the potential . , with
Quite obviously, the movement of the string will be periodic. The physical period in coordinate time relates to through
It will prove useful to introduce the ‘physical’ potential
since it only depends on the physical parameter .
The boundary conditions for open strings, namely, at the ends of the string, are naturally fulfilled by this ansatz.
Following [12], we see that this is the only ansatz that separates variables, lets strings be dynamical, and respects the open string boundary conditions, when is chosen.
Note also that this ansatz differs from the circular string ansatz (i.e., , , ) in the dependence of in the spacelike worldsheet (conformal) coordinate and in the form of the equivalent onedimensional energy equation, which for this later case reads , where the dot stands for the derivative with respect to .
The invariant size of the planetoid string is given by the substitution of the ansatz in the line element:
(4) 
ii.2 Energy and angular momentum
It is well known that the definition of a stressenergy tensor for an extended object in general relativity is no mean task [13]. In the case at hand, however, there exists a favored time coordinate, for which a Killing vector exists (). This allows us to define clearly what is meant as energy, [13]: .
Similarly, the existence of the Killing vector , associated with the rotational symmetry, allows for the definition of an angular momentum about the axis. In particular, this is performed as follows: the function appears in the string Lagrangian only through its derivatives, whence the conserved worldsheet current is obtained by Noether’s theorem
The integration of this current provides us with the string angular momentum ,
where we used eq.(3) and and denote the minimum and maximum radius reached by the string, respectively.
ii.3 General expressions and quantization condition
We will collect here the expressions for the physical string magnitudes: angular momentum , classical action for solutions , mass , and reduced action . The mass will be defined as , with the period. The reduced action [11] is thus obtained as . The quantization condition will read (in units with ).
For the case at hand, closed expressions in terms of quadratures can be obtained for all these quantities, as follows:
(6)  
(8)  
(10)  
(11) 
As is immediately obvious from these expressions, it is not necessary to have the solution in a closed form for the quantities indicated to be evaluated, and in what follows we will not use the explicit expressions for , which, after all, is dependent on the parametrization of the worldsheet. It should be noted that the previously mentioned quantization condition [] is equivalent for this class of solutions to . This should be interpreted as a consistency check of the semiclassical quantization being performed.
The invariant string length at a fixed time follows from eq.(4)
(12) 
Iii Explicit solutions and their analysis
iii.1 Minkowski spacetime
In order to improve our understanding of the physical meaning of the solutions being examined, let us take the easy Minkowski case, for which , , , and . Equation (3) then becomes
The solution is immediate:
where . In cartesian coordinates,
It is easy to see that this is a string of length rotating around its middle point which coincides with the origin of coordinates.
The action, reduced action, mass and angular momentum are therefore
(14)  
(16)  
(17) 
from which the relation follows
(18) 
It should be noted that this relation differs from the standard one by a factor 4. This is due to the different normalization of the string tension parameter .
iii.2 Static RobertsonWalker spacetimes
As a first curved spacetime, we examine the static RobertsonWalker universe, with line element
(19) 
If , the potential
is smaller than zero if , as in Minkowski spacetime. On the other hand, were we to take , the number of possible types of solutions increases. Consider first . Let , and .
Our computations result in
(21)  
(23)  
(24) 
where and are complete elliptic integrals of the first and second kind respectively, with the elliptic modulus as their argument.
Let us now pass to the situation. There are two classes of solutions: those that extend from to , and those from to infinity, where and . The second class of solutions lead to infinite reduced action. As to the first class, computations yield
(26)  
(28)  
(29) 
for the case , and
(31)  
(33)  
(34) 
for the case .
We see that the string angular momentum is not proportional to yielding a nonlinear Regge trajectory.
In the limit, we have
(35) 
and, consequently,
In the limit we find a linear Regge trajectory, recovering the previous results for Minkowski spacetime.
iii.3 Cosmological and black hole spacetimes
Let us consider spacetimes with the generic form , . The potential is then given by . Since the “motion” of in can only take place when , we have to determine the zeroes of and of , together with the asymptotics in the different physical regions.
iii.3.1 de Sitter spacetime
Included within this set of metrics we find the de Sitter metric, for which and . The radius of the horizon, , is given by . Thus,
The zeroes of the potential in this case are and .
There are two types of planetoid strings: those of infinite length that are to be found outside the horizon, and those completely within the horizon, that are of finite length. Let us concentrate on the later. The maximum radius is and . This is a string rotating around its middle point located precisely at .
The integrals to be performed are complete elliptic integrals, with elliptic modulus
Let . Our computations result in the following:
(37)  
(39)  
(40) 
It is here obvious that is not proportional to .
For small , the quantization condition reads , as in flat spacetime, and the mass of the string is in this case (compare with [11])
(41) 
It follows from eqs.(37) that is a twovalued function of and hence of . Therefore, there are two values of for each . This is easy to see from the behaviour of for and for . vanishes in both cases.
There is a maximum on the values can take, given by
(42) 
This correspond to the maximal planetoid mass.
The first branch yields masses in the range
and the second branch in the range
[Notice that ].
We find from eq.(12) for the invariant string length
takes its maximum value for the lightest states in the second branch
With respect to the infinite length planetoid solutions (that is to say, those restricted to be outside the horizon), the corresponding action, reduced action and mass are all infinite.
iii.3.2 Antide Sitter spacetime
In this case, and . Only for a restricted set of values of will there be a change of sign in , since only if will there be a zero of , namely at . Therefore, lower values of correspond to strings of infinite length, whereas those strings for which will be of finite length. They will rotate around its middle point located precisely at with period .
The results for this spacetime are as follows, where :
(44)  
(46)  
(47) 
In this case is a monotonous function of , and so is , so the doubling of mass eigenvalues found in de Sitter spacetime is not present here.
For the lowlying mass states we find
There is no upper bound in the mass spectrum for antide Sitter spacetime. For large masses we find
The heavy states spacing is given by whereas the small mass spacing is determined by .
iii.3.3 Schwarzschild black hole
For the Schwarzschild black hole and , where stands for the Schwarzschild radius.
There will be positive zeroes of other than that at if and only if . Of the two additional zeroes in this case, one will be placed between and , and the other will be larger than in units of . In the extreme case the two will coalesce onto , which is the minimal (unstable) radius for a circular null geodesic [14]. As we turn to larger values, one of the zeroes runs to , and the other out to infinity, these extreme values being reached for , thus corresponding to an infinite static string from the horizon to infinity [15].
Let us choose the following parametrization for , and consequently for the roots of , with :
(49)  
(51)  
(53)  
whence
(54)  
(55) 
The parameter is a function of as defined by eq.(49). runs from to , and the roots are ordered as . The planetoid string extends from to . Its invariant length follows from eq.(4)
where
The classical, reduced action and mass are then integrals expressible in terms of elliptic integrals of modulus
The explicit expressions are not by themselves very illuminating, since they involve combinations of elliptic integrals of different kinds; as a simple exponent, we have
(57)  
We use, as before, the notation of ref.[16].
An important point is that there is a minimum value for the reduced action and for the mass, corresponding to , as follows:
(59)  
(60) 
The classical action for this configuration vanishes.
The period has no upper bound. For large we find very long strings with
and the mass spectrum
The Regge trajectory is well behaved, and we portray it in Fig. 1.
iii.3.4 Schwarzschild black hole in de Sitter spacetime
We shall now find competing effects due to the presence of one cosmological and one black hole horizons. The function equals , and . We are presented with three cases:

(and, a fortiori, ); the positive roots of the potential are the cosmological horizon, the black hole horizon, and two others, which we examine later.

, when only strings inside the black hole horizon and outside the cosmological horizon are present within our ansatz.

, which entails that there is no horizon and no strings of the form of our ansatz.
We shall now study the first of these cases, when there are four positive roots of the potential , using a parametrization analogous to the one before. Let
(62)  
(64)  
(66)  
(68)  
(70)  
(72)  
(74)  
(76)  
(77) 
with . It follows that
(80)  
Take . The four positive roots are ordered as follows: . There are thus strings of the form of our ansatz extending from to , and outside the cosmological horizon and inside the black hole horizon. The strings outside the cosmological horizon are of infinite length, mass and action. The really relevant ones for our purposes are those extending from to , in complete analogy with the results for Schwarzschild’s black hole. We portray a numerical computation of the classical Regge trajectory in Fig. 2 for the case , that is, . Clearly to be seen are the two branches which had previously appeared for the rotating string in de Sitter spacetime. Surprisingly enough, there is no minimum value for and greater than zero in one of the branches, although it does appear in the second one. This is due to the numerical integration, which is very inexact in the limit , and the fact is that there is a minimum value for , independent of and given by , as can be found by computing the adequate limit ; the mass also has a minimum value, but this time dependent: . Notice that we recover the results previously obtained for Schwarzschild spacetime.
Iv Conclusions
We have seen that the study of the planetoid solutions to the classical equations of motion of a string provides us with a variety of effects due to the structure of the target spacetime. In particular there are two main effects that we have uncovered:

the existence of a maximum value for the angular momentum of (equatorially) moving strings in spacetimes with particle horizons (de Sitter and Schwarzschildde Sitter in particular), which reflects itself on the existence of two branches in the Regge plot. This means that the number of bound states is finite in the semiclassical quantization. (But this finiteness must be exact, beyond the semiclassical approximation).

the presence of a minimum value for the angular momentum in the case of a black hole event horizon, as in Schwarzschild and Schwarzschildde Sitter spacetimes.
It is not difficult to understand this phenomenon in the light of elementary quantum mechanics. In spacetimes with particle horizons it is necessary for the preservation of causality that if a string extends beyond the horizon that it be infinite. The length is quantized in the same manner that the angular momentum is, as can be read out from eq.(6); it is thus the case that there are a finite number of possible quantum planetoid strings.
As to the minimum value, given that if a string does penetrate into a (Schwarzschild) event horizon and is to maintain its linearity it must extend to infinity, we see that the “cutting out” of part of the spacetimes is what forces a minimum value even for classical values (quantum mechanically that was only to be expected).
String solutions that generalize noncircular point particle trajectories should also exist in the spacetimes considered here. However, the and dependence probably cannot be separated as we did in the planetoid strings presented in this paper.
We want to stress that the Regge trajectories are no longer linear (even for weak curvature) in the spacetimes considered here. We thus infer from this classical test string calculations that the fundamental string spectrum will get strongly modified in these nontrivial gravitational backgrounds.
Acknowledgements.
ILE has to thank the LPTHE for their hospitality on several occasions. Figure CaptionsFig.1: The reduced action (where is the angular momentum) in units of as a function of the string mass in Schwarzschild spacetime.
Fig. 2: The reduced action (where is the angular momentum) as a function of the string mass in Schwarzschildde Sitter spacetime.
References
 []
 [†] email:
 [3] H. J. de Vega and N. Sánchez Phys. Lett. B 197 320, (1987). H. J. de Vega and N. Sánchez, Nucl. Phys. B 309, 552 and 577 (1988).
 [4] H. J. de Vega and N. Sánchez in Proceedings of the Erice Schools: “String Quantum Gravity and Physics at the Planck Energy Scale”, 2128 June 1992, Edited by N. Sánchez, World Scientific, 1993, and Third D.Chalonge School, 416 September 1994.
 [5] H.J. de Vega and N. Sánchez in the Proceedings of the D. Chalonge School, 1995, N. Sánchez and A. Zichichi, editors, Kluwer 1996.
 [6] F. Combes, H. J. de Vega, A. V. Mikhailov and N. Sánchez, Phys. Rev. D50, 2754 (1994). I. Krichever, Funct. Anal. and Appl. 28, 21 (1994),
 [7] P.S. Letelier, P.R. Holvorcem and G. Grebot, Class. Quantum Grav. 7, 597610 (1990).
 [8] H.J. de Vega and I.L. Egusquiza, Phys. Rev. D49 763778 (1994).
 [9] A.L. Larsen and N. Sánchez, Phys. Rev. D50 74937518 (1994).
 [10] A.L. Larsen and N. Sánchez, Phys. Rev. D51 69296948 (1995).
 [11] H. J. de Vega, A.L. Larsen, and N. Sánchez, Phys. Rev. D 51 6917, (1995).

[12]
A.L. Larsen and N. Sánchez,
“SinhGordon, CoshGordon and Liouville Equations for Strings and MultiStrings in Constant Curvature Spacetimes”, hepth/96003049.  [13] W.G. Dixon, Proc. Roy. Soc. Lond. A 314 499527 (1970).
 [14] S. Chandrasekhar, The Mathematical Theory of Black Holes, Oxford University Press, New York (1992).
 [15] V.P. Frolov, V.D. Skarzhinsky, A.I. Zelnikov and O. Heinrich, Phys. Lett. B 224 225 (1989).
 [16] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, London (1990).