Comment on ‘‘Stationary rotating strings as relativistic particle mechanics’’

Conrad J. Burden*

Centre for Bioinformation Science, John Curtin School of Medical Research and Mathematical Sciences Institute,Australian National University, Canberra, ACT 0200, Australia

(Received 29 August 2008; published 8 December 2008)

The set of rigidly rotating solutions to the relativistic string equation with subluminal azimuthal

velocity has recently been reported by Ogawa et al. This set, together with a second set of rigidly rotating

strings with superluminal azimuthal velocity, was reported in earlier papers by Burden and Tassie, and

independently by Embacher. We believe the second set is also physically relevant to modelling cosmic

strings. We write both sets of rigidly rotating string solutions in the aligned standard gauge and give a

geometrical interpretation.

DOI: 10.1103/PhysRevD.78.128301 PACS numbers: 98.80.Cq

In Ref. [1] Ogawa et al. have set out to determine thecomplete set of stationary rotating Nambu-Goto strings inMinkowski spacetime. They define a stationary string asone whose world surface is tangent to a timelike Killingvector of the embedding Minkowski spacetime. For thecase of rotating stationary strings, this is equivalent todetermining the set of rigidly rotating string configurationswhose azimuthal component of velocity is less than thespeed of light at each point on the string. Their set ofsolutions is precisely that described in detail in an earlypaper by Burden and Tassie [2] in the context of the stringmodel of hadrons. The two parameters ðl; qÞ parametrizingthe family of solutions reported in Ref. [1] are related to theparameters ð�; AÞ in Ref. [2] by l ¼ A�=� and q ¼ A,where � is the string’s angular velocity. We will refer tothis set of solutions as type I solutions.

As the motivation of Ogawa et al.’s study is to determinecandidate stable trajectories for cosmic strings, we argueon physical grounds that it is not the azimuthal velocity ofthe string, but the velocity normal to the string’s motionthat should be less than the speed of light [3], and that thefocus on world surfaces whose tangents are timelike andnot spacelike Killing vectors is unnecessarily restrictive. Asecond set of rigidly rotating string configurations, whoseazimuthal velocity is greater than the speed of light wasreported by Burden and Tassie [4]. The transverse velocityof these trajectories is subluminal, and their energy andmomentum per unit string length is finite. We will refer tothese solutions as type II solutions. The complete set ofboth type I and type II solutions has also been foundindependently by Embacher [5,6].

A minor error occurs in Sec. 3 of Ref. [1], in which thetype I string configurations lying wholly within the x-yplane are considered. It is stated that these strings have endpoints on the light-cylinder r� ¼ 1. However, the stringtension has a nonzero transverse component at this point,which can only be balanced by extending the string pastthis cusp [2,7]. The full solution is a hypocycloid rotating

about its center, with the cusps moving at the speed of light.The corresponding type II x-y planar solution is an epicy-cloid rotating about its center, with the cusps moving at thespeed of light.Burden and Tassie [4] demonstrated that these configu-

rations are in fact hypocycloids and epicycloids by rewrit-ing the x-y planar trajectories in the aligned standard gauge[8,9]

x2� þ x2� ¼ 0; x� � x� ¼ 0; x0 ¼ �; (1)

in which the string trajectory takes the form x� ¼ð�;xð�; �ÞÞ with

x ð�; �Þ ¼ rð12ð�þ �ÞÞ þ qð12ð�� �ÞÞ; (2)

where r and q have the property that

r 02 ¼ q02 ¼ 1: (3)

In this gauge the x-y planar solutions are given by

r ð�Þ ¼ ð!�11 cos!1�;!

�11 sin!1�; 0Þ;

qð�Þ ¼ ð!�12 cos!2�;!

�12 sin!2�; 0Þ:

(4)

The string shape at � ¼ 0, namely xð0; �Þ ¼ rð�=2Þ þqð��=2Þ is a hypocycloid if !1 and !2 have the samesign, and an epicycloid if they have opposite signs.This form of the planar solutions readily generalizes to

the full set of rigidly rotating strings expressed in thealigned standard gauge (see also [5]). The functions

r ð�Þ ¼� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� �2!21

q!1

cos!1�;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� �2!2

1

q!1

� sin!1�; �!1�

�;

qð�Þ ¼� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1� �2!22

q!2

cos!2�;

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� �2!2

2

q!2

� sin!2�;��!2�

�;

(5)

*Conrad.Burden@anu.edu.au

PHYSICAL REVIEW D 78, 128301 (2008)

1550-7998=2008=78(12)=128301(2) 128301-1 � 2008 The American Physical Society

are easily seen to satisfy the gauge constraint Eq. (3).Furthermore, the string shape at time � is related to thatat time zero by

x ð�;�Þ ¼ Rz

��

!�11 þ!�1

2

�x

�0; �þ!1 �!2

!1 þ!2

�

�; (6)

where Rzð�Þ is a rotation through an angle � about the zaxis. Thus, up to a shift in the � parameter, the stringrotates rigidly about the z axis with angular velocity � ¼ð!�1

1 þ!�12 Þ�1.

The projection of the string onto the x-y plane at fixed� ¼ 0 is a hypotrochoid if !1 and !2 have the same sign,and an epitrochoid if they have opposite signs (see Fig. 1).A hypotrochoid (epitrochoid) is the plane curve traced out

by a point fixed in a disc rolling without slipping on theinterior (exterior) of a fixed circle [10]. Because of varioussymmetries present in the parametrization, the full set ofstring shapes is covered by the parameter range 0 � !1 �j!2j, 0 � � � j!2j�1. The fixed circle has a radius of

ð!�11 þ!�1

2 Þð1� �2!1Þ1=2, the rolling disc a radius of

j!�12 jð1� �2!1Þ1=2, and the point tracing out the string

is a distance j!�12 jð1� �2!2Þ1=2 from the center of the

disc.Hypotrochoids are evident in Fig. 6 of Ref. [1]. The sine

wave in Fig. 4 of Ref. [1] is the case!1 ¼ !2. Setting � ¼j!2j�1 gives a helix, the type I (i.e. !2 > 0) subcases ofwhich are illustrated in Fig. 3 of Ref. [1].

[1] K. Ogawa, H. Ishihara, H. Kozaki, H. Nakano, and S.Saito, Phys. Rev. D 78, 023525 (2008).

[2] C. J. Burden and L. J. Tassie, Aust. J. Phys. 35, 223 (1982).[3] V. Frolov, S. Hendy, and J. P. De Villiers, Classical

Quantum Gravity 14, 1099 (1997).[4] C. J. Burden and L. J. Tassie, Aust. J. Phys. 37, 1 (1984).[5] F. Embacher, Phys. Rev. D 46, 3659 (1992).[6] F. Embacher, Phys. Rev. D 47, 4803 (1993).

[7] C. J. Burden and L. J. Tassie, Phys. Lett. B 110, 64 (1982).[8] M. R. Anderson, The Mathematical Theory of Cosmic

Strings (Institute of Physics Publishing, Bristol, UK,2003), 1st ed.

[9] P. Goddard, J. Goldstone, C. Rebbi, and C. B. Thorn, Nucl.Phys. B56, 109 (1973).

[10] J. D. Lawrence, A Catalog of Special Plane Curves(Dover, New York, 1972), 1st ed.

FIG. 1. Hypotrochoid (a) and epitrochoid (b) traced out by the projection onto the x-y plane of the curve xð0; �Þ ¼ rð�=2Þ þqð��=2Þ, with r and q given by Eq. (5). The curves are the end points of the sum of two vectors executing uniform circular motionwith angular velocities with respect to � of !1=2 and �!2=2, while progressing uniformly along the z axis. Parameter values are(a) !1 ¼ 0:5, !2 ¼ 1, � ¼ 0:75 and (b) !1 ¼ 1=3, !2 ¼ �1, � ¼ 0:75. Because !1=!2 is rational in both examples the projectedcurves close on themselves, though this will not be the case in general.

COMMENTS PHYSICAL REVIEW D 78, 128301 (2008)

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