Planetoid strings : solutions and perturbations
ABSTRACT A novel ansatz for solving the string equations of motion and constraints in generic curved backgrounds, namely the planetoid ansatz, was proposed recently by some authors. We construct several specific examples of planetoid strings in curved backgrounds which include Lorentzian wormholes, spherical Rindler spacetime and the 2+1 dimensional black hole. A semiclassical quantisation is performed and the Regge relations for the planetoids are obtained. The general equations for the study of small perturbations about these solutions are written down using the standard, manifestly covariant formalism. Applications to special cases such as those of planetoid strings in Minkowski and spherical Rindler spacetimes are also presented. Comment: 24 pages (including two figures), RevTex, expanded and figures added
arXiv:hep-th/9701173v3 5 May 1997
Planetoid Strings : Solutions and Perturbations
Inter–University Centre for Astronomy and Astrophysics,
Post Bag 4, Ganeshkhind, Pune, 411 007, INDIA
Institut f¨ ur Physik, Humboldt Universit¨ at zu Berlin,
Invalidenstr. 110, D-10115 Berlin, GERMANY
A novel ansatz for solving the string equations of motion and constraints in
generic curved backgrounds, namely the planetoid ansatz, was proposed re-
cently by some authors. We construct several specific examples of planetoid
strings in curved backgrounds which include Lorentzian wormholes, spherical
Rindler spacetime and the 2+1 dimensional black hole. A semiclassical quan-
tisation is performed and the Regge relations for the planetoids are obtained.
The general equations for the study of small perturbations about these so-
lutions are written down using the standard, manifestly covariant formalism.
Applications to special cases such as those of planetoid strings in Minkowski
and spherical Rindler spacetimes are also presented.
PACS number(s) : 11.25.-w, 04.70.-s, 98.80.Cq
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∗Electronic Address : email@example.com
†Electronic Address : firstname.lastname@example.org
In the context of cosmic as well as fundamental strings, the analysis of the classical
string equations of motion and constraints in generic curved backgrounds  has become
an active area of research over the last decade or so (for a recent review and references
see ) Solutions representing string configurations, which essentially correspond to timelike
embedded minimal surfaces, are difficult to obtain largely due to the nonlinear and coupled
nature of the relevant equations. Therefore, the attitude has been to proceed by proposing
a generic ansatz based on symmetries or simplifying assumptions which reduce the com-
plicated set of equations to a tractable form. Among the various proposals till date, we
have the stationary string ansatz , dynamic circular strings  and more recently, plan-
etoid string configurations  as well as rigidly rotating strings . It is worth mentioning
that the planetoid and rigidly rotating strings are both special cases of an ansatz proposed
earlier by Larsen and Sanchez . Target spaces with metrics such as the Schwarzschild,
Kerr-Newman, Robertson–Walker, cosmic strings, wormholes etc. have been chosen and
explicit string configurations obtained in these backgrounds. Once specific configurations
are known, the obvious next question that emerges is about their stability. This turns out to
be related to the second variation of the action (Nambu–Goto or its generalisations) and the
corresponding Jacobi equations . Perturbative stability depends crucially on the analysis
of these equations. String propagation in an exact, stringy four-dimensional black hole back-
ground and the perturbations about extremal configurations have also been studied recently
. Furthermore, nonperturbative effects which include the formation of cusps and kinks on
the world surface of the string is governed by the character of solutions of the generalised
Raychaudhuri equations .
This paper deals with planetoid strings. First we obtain specific solutions in certain well
known backgrounds. Thereafter, we discuss small perturbations about these configurations.
The backgrounds chosen include the Ellis geometry (a Lorentzian wormhole), the spherical
Rindler spacetime, the Minkowski spacetime and the 2+1 dimensional BTZ black hole .
We also consider semi-classical quantization of strings in these backgrounds and compute
physical quantities such as the classical action, mass, reduced action and angular momentum.
The quantisation condition for each case is written explicitly.
Notations and sign conventions in the paper follow the norms of Misner, Thorne and
II. PLANETOID STRINGS: FORMALISM
We first briefly discuss general planetoid strings, quote the ansatz and the resulting
equations which we solve for specific backgrounds later.
The generic background metric (taking a θ =π
2section) is taken to be of the form :
ds2= gttdt2+ grrdr2+ 2gtφdtdφ + gφφdφ2
The planetoid ansatz is given as ,
t = t0+ ατ
φ = φ0+ βτ
r = r(σ) (2)
where, τ and σ are the time-like and space-like coordinates on the worldsheet respectively.
α and β are two arbitrary constants. Assuming β = 0 would give us the usual stationary
strings. Note that the planetoid ansatz is a special case of the one proposed by Larsen and
Sanchez  where the constants t0and φ0are replaced by general functions t0(σ) and φ0(σ)
respectively. A word about the name ‘planetoid’. The ansatz above is a sort of generalisation
of the ansatz one would take if one deals with the embedded curves along which planets move
in their orbit. Hence it is perhaps appropriate to call these kinds of worldsheets ‘planetoids’
– a name which drives home the message that these are related to planetary orbits while
being surfaces as opposed to curves.
From the bosonic string equations of motion and constraints one arrives at the first order
equation, which one needs to solve in order to get a planetoid string. This is given as :
α2gtt+ 2αβgtφ+ β2gφφ
where the right hand side can be identified with the negative of a potential V (r). However,
it is more convenient to work with˜V (r) which is defined as,
˜V (r) =V (r)
The induced metric on the world-sheet of the string is given as :
By choosing a conformal gauge in which the induced metric is diagonal and conformal to
Minkowski spacetime in two dimensions, we automatically satisfy the constraint equations
(gµν˙ xµ˙ xν+gµνx′µx′ν= 0;
gµν˙ xµx′ν= 0), where dot and prime denote differentiations with
respect to world-sheet coordinates τ and σ respectively and µ, ν are space-time indices. We
confine ourselves largely to spherically symmetric, static backgrounds for which the basic
equation to solve turns out to be :
[α2e2ψ(r)− r2β2] (6)
where our background metric is now assumed as diagonal and for a θ =π
2section, it is given
When is the induced metric on a planetoid string Minkowskian? By looking at the
expression for the induced metric one can easily say that this happens if :
For spherically symmetric, static metrics this turns out to be a very stringent constraint
on the red–shift function ψ(r), which should satisfy,
Additionally, we observe that the existence of a zero in the conformal factor in the
induced metric would indicate the existence of a singularity on the worldsheet. Specifically,
if r = r0is a zero of the expression for the conformal factor we must have :
If r0coincides with the horizon e2ψ(r0)= 0 then we can only have r0= 0. There maybe
other points in the geometry where this could be satisfied too regardless of whether the
geometry has a horizon or not. On the other hand if e2ψ= 1 (i.e. an ultrastatic metric) we
can clearly see that r =
βis the point where the worldsheet will become singular. These
facts will be generic features of all the solutions to be discussed below.
Let us also consider planetoids in generic time–dependent backgrounds of the form :
It can be shown that there will be no planetoid solutions in time-dependent backgrounds
of the above type (which includes the FRW models too). To see this let us look at the string
equation of motion for the coordinate φ. With the substitution of the planetoid ansatz, we
find that the equation reduces to the requirement :
Since α or β cannot be taken to be zero one needs Ω(t) to be a constant.
Also note that the planetoid ansatz is incompatible with null (tensionless) strings as has
been pointed out in .
We now move on towards solving the string equation of motion to obtain specific plane-
toid string configuration in some well–known backgrounds.
III. SOLUTIONS IN SPECIFIC BACKGROUNDS
(1) Spherically symmetric coordinate representation of Minkowski spacetime
In this case, the background metric (as given in the form in eqn. (7)) has b(r) = 0 and
ψ(r) = 0. The planetoid solution is :
β|sinβ (σ − σ0)| (13)