# Gravitational field and lensing of a circular chiral vorton

Leonardus B. Putra<EMAIL_ADDRESS>H. S.
Ramadhan111Corresponding author<EMAIL_ADDRESS>Departemen Fisika, FMIPA,
Universitas Indonesia, Depok, 16424, Indonesia.

###### Abstract

We derive the metric of a circular chiral vorton in the weak field limit. The
object is self-supporting by means of its chiral current. A conical
singularity with deficit angle, identical to that of straight string with the
same linear mass density, is present at the vorton’s core. We find that the
metric is akin to the electromagnetic $4$-potential of a circular current wire
loop, illustrating the concept of gravito-electromagnetism. Surprisingly we
find that the solution asymptotically mimics a Kerr-like naked singularity
with mass $M_{v}=4\pi R\mu$ and spin parameter $a=R/2$. Finally, we also
simulate the gravitational lensing images by solving the corresponding null
geodesic equations. This reveals interesting properties of the images, such as
the simultaneous creation of a minimally distorted source image and its
Einstein ring, as well as the formation of double images of the back side of
the ring.

##  I Introduction

Kibble mechanism suggests that cosmic strings could have formed in the early
universe, most likely during the Grand Unified phase transition, when the
vacuum manifold $\mathcal{M}$ of the corresponding symmetry is non simply-
connected, $\pi_{1}\left(\mathcal{M}\right)\cong\mathbb{Z}$ Kibble:1976sj ;
Vilenkin:1984ib ; CSBookVilenkin:2000jqa . Topology dictates that, upon
formation, strings must either extend infinitely or form closed loops. In both
cases, assuming the string’s curvature is significantly larger than its
thickness, its dynamics can then be effectively approximated by the Nambu-Goto
action Nambu ; Goto:1971ce .

Witten was the first to suggest that strings could be superconducting
Witten:1984eb , which he showed specifically within a $U(1)\times U(1)$ gauged
theory. The dynamics of superconducting strings in general can be very
complicated Copeland:1987th , but Carter and Peter showed that in the chiral
limit, $j^{a}j_{a}=0$, the corresponding equations of motion can be exactly
solved Carter:1999hx  (see also Blanco-Pillado_PhysRevD.63.103513 ;
Davis_PhysRevD.62.083516 ). In particular, the current flowing through the
string generates angular momentum that, when sufficiently large, can stabilize
the loop against collapse due to its tension. This spinning soliton-like
object is known as a vorton Davis:1988jq ; DAVISShellardVorton1989209 .
Extensive studies have been conducted on their formation Martins:1998gb ;
Battye:2008mm  and stability Carter:1990sm ; Lemperiere:2003yt ; Carter:1993wu
.

Once hypothesized to seed galaxy formation long ago Vilenkin:1981iu , cosmic
strings have recently regained interest due to the discovery of gravitational
waves LIGOScientific:2013tfe ; Belahcene:2019ric . This stems from predictions
that oscillating loops of cosmic strings could be efficient sources of
gravitational radiation Vachaspati:1984gt ; Vachaspati:1987sq . The
gravitational field for a circular loop was studied by Frolov, Israel, and
Unruh (FIU) Frolov:1989er . They show that the metrics suffer from deficit
angle $\delta\varphi=8\pi G\mu$, with $\mu$ is the string tension. The
enormous tension forces a loop to oscillate, while the FIU metric, on the
other hand, is stationary. Some authors postulate the need for an external
pressure Hughes_PhysRevD.47.468  or a revolving mechanism
McManus_PhysRevD.47.1491  to prevent the loop from collapsing.

In our view, the formation of a vorton, which does not require any external
mechanism, is the most natural stabilization scenario for a stationary string
loop. Surprisingly, there has been very little study on the gravitational
field of vortons. To the best of our knowledge, the only study on the
gravitational field of vortons to date is by Kunz, Radu, and Subagyo
KunzRaduBintoro_PhysRevD.87.104022 . They obtained a gravitating $U(1)\times
U(1)$ global vorton by numerically solving the full Einstein and scalar field
equations. For certain parameter values, the vorton has an ergoregion
surrounding it, indicating instability Cardoso:2007az . Asymptotically, the
metric approaches that of a Kerr black hole.

The purpose of this work is to further elaborate on the gravitational field of
vortons and its astrophysical impact. In this paper, we study the spacetime
around a circular chiral vorton in the weak-gravitational field limit and its
properties. These linearized solutions will then be used to calculate its
gravitational lensing effects. The paper is organized as follows. In Sec. II,
we derive the metric of a circular chiral vorton using the weak-field and
thin-string approximations. We begin by discussing the appropriate metric
ansatz for a rotating loop, the energy-momentum tensor for the chiral vorton
in the thin-string approximation, and finally the weak-field solution
including its asymptotic behavior. In Sec. III, we explore the physical
properties of the metric solution we obtain. Sec. IV addresses the lensing
signatures of the vorton. We numerically solve the corresponding null geodesic
equations to determine the light-bending patterns caused by a circular vorton
with various plane orientations relative to the optical axis. Finally, we
summarize our findings in Sec. V.

##  II Gravitational Field of Vorton

###  II.1 The metric ansatz

In general, a cosmic string loop cannot be stationary due to its tension.
However, its collapse and re-expansion can be prevented by the right amount of
angular momentum along the string. Therefore, the most general metric for a
rotating string loop is axially-symmetric with two commuting Killing vectors
$\partial_{t}$ and $\partial_{\phi}$. In cylindrical coordinate it reads
McManus_PhysRevD.47.1491 ; Hughes_PhysRevD.47.468

$ds^{2}=-e^{2\nu(r,z)}dt^{2}+e^{2\zeta(r,z)-2\nu(r,z)}r^{2}\left(d\phi-A(r,z)dt\right)^{2}+e^{2\eta(r,z)-2\nu(r,z)}\left(dr^{2}+dz^{2}\right).$
(1)

Assuming that all the metric functions ($\nu$, $\zeta$, $\eta$, and $A$) are
weak field of order $G\mu$, the Einstein tensor components are
Hughes_PhysRevD.47.468

$\displaystyle G_{t}^{t}$ $\displaystyle=$
$\displaystyle-2\nabla^{2}\nu+\nabla^{2}\zeta+\frac{1}{r}\zeta_{r}+\tilde{\nabla}^{2}\eta,$
(2a) $\displaystyle G_{\phi}^{t}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}r^{2}\left(\nabla^{2}A+\frac{2}{r}A_{r}\right),$
(2b) $\displaystyle G_{\phi}^{\phi}$ $\displaystyle=$
$\displaystyle\tilde{\nabla}^{2}\eta,$ (2c) $\displaystyle G_{r}^{r}$
$\displaystyle=$ $\displaystyle\zeta_{zz}+\frac{1}{r}\eta_{r},$ (2d)
$\displaystyle G_{r}^{z}$ $\displaystyle=$
$\displaystyle-\zeta_{zr}-\frac{1}{r}\left(\zeta_{z}-\eta_{z}\right),$ (2e)
$\displaystyle G_{z}^{z}$ $\displaystyle=$
$\displaystyle\zeta_{r}r-\frac{1}{r}\left(\eta_{r}-2\zeta_{r}\right).$ (2f)

Here,
$\nabla^{2}\equiv\partial_{r}^{2}+\frac{1}{r}\partial_{r}+\partial_{z}^{2}$ is
the Laplacian in cylindrical coordinate, while
$\tilde{\nabla}^{2}\equiv\partial_{r}^{2}+\partial_{z}^{2}$ is the Laplacian
in Cartesian coordinate with $(r,z)$ in place of $(y,z)$. In the toroidal
coordinates $(t,\phi,\sigma,\psi)$, using transformations

$\displaystyle z$ $\displaystyle=$
$\displaystyle\frac{R}{N(\sigma,\psi)^{2}}\sin{\psi},$ (3) $\displaystyle r$
$\displaystyle=$ $\displaystyle\frac{R}{N(\sigma,\psi)^{2}}\sinh{\sigma},$ (4)

with

$N(\sigma,\psi)^{2}\equiv\cosh{\sigma}-\cos{\psi},$ (5)

the metric becomes

$ds^{2}=-e^{2\nu}dt^{2}+R^{2}N^{-4}\sinh^{2}{\sigma}e^{2\zeta-2\nu}\left(d\phi-
Adt\right)^{2}+R^{2}N^{-4}e^{2\eta-2\nu}\left(d\sigma^{2}+d\psi^{2}\right).$
(6)

The weak field assumption of the ansatz means that the solutions of function
$\nu$, $A$, $\eta$, and $\zeta$, are of order of $O(G\mu)$.

###  II.2 The energy momentum tensor

In the thin-string approximation, the string’s dynamics is well-described by
the Nambu-Goto equations

$\ddot{x}^{\mu}-x^{\mu\prime\prime}=0,$ (7)

The corresponding energy momentum tensor is Blanco-Pillado_PhysRevD.63.103513
; Davis_PhysRevD.62.083516

$T^{\mu\nu}=\mu\int
d\eta\left(\dot{x}^{\mu}\dot{x}^{\nu}-x^{\prime\mu}x^{\prime\nu}\right)\delta^{(3)}\left(\vec{y}-\vec{x}(t,\eta)\right).$
(8)

The solutions for the chiral string loop are of the form

$x^{\mu}=\left(t,\vec{x}\right),$ (9)

where

$\vec{x}=\vec{a}(q)+\vec{b}(\eta).$ (10)

Here $\vec{a}$ and $\vec{b}$ are two 3-vectors satisfying the constraints

$|\vec{a}|^{2}=1,\ \ \ \ |\vec{b}|^{2}=k^{2},$ (11)

where $k$ is some constant, with $q=t+\sigma$ and $\eta=t-\sigma$. Vorton is
the class of stationary string loop solutions satisfying $k=0$.

A circular vorton solution in cylindrical coordinate $(t,\phi,r,z)$ can be
written as (similar to the axial coordinate in CCSLKerrPhysRevD.79.065029 )

$x^{\mu}=\bigg{(}t,\ \frac{q}{2R}+f(t),\ R,\ 0\bigg{)},$ (12)

which describes a vorton of constant radius $R$ at $z=0$, with its center at
the origin. The axial coordinate is the only one that depends on $\sigma$, and
the relation is linear to account for the axial symmetry of the system. Here
the function $f(t)$ is added to take into account the constraints

$\dot{x}^{\mu}x^{\prime}_{\mu}=\frac{1}{4}(1-k^{2}),$ (13)

where we take $k=0$ for the case of vorton

$\dot{x}^{\mu}x^{\prime}_{\mu}=\frac{1}{4}.$ (14)

Substituting the solution (12) to the constraint (14), and using the weak
field limit assumption of the ansatz (1), we have

$\dot{f}(t)=\mathcal{O}(G\mu).$ (15)

And therefore the (time dependent) shift of the spacelike string axial
coordinate is of order $G\mu$.

Ignoring the higher order term of $G\mu$, the nonzero mixed energy-momentum
tensor components in this coordinate would be

$GT_{t}^{t}=-2G\mu\delta(r-R)\delta(z),$ (16)

and

$GT^{t}_{\phi}=G\mu R\delta(r-R)\delta(z).$ (17)

Note how in our case here, the tension of the string approaches zero, so are
all the other $ij$ components of the stress-energy tensor, just like the case
of the infinite straight chiral string GravityStraightChiralStringSteer:2000jn
.

###  II.3 The weak-field solution

The Einstein’s equations read

$\displaystyle-2\nabla^{2}\nu+\nabla^{2}\zeta+\frac{1}{r}\zeta_{r}+\tilde{\nabla}^{2}\eta$
$\displaystyle=$ $\displaystyle-16\pi G\mu\delta(r-R)\delta(z),$ (18)
$\displaystyle-\frac{1}{2}r^{2}\left(\nabla^{2}A+\frac{2}{r}A_{r}\right)$
$\displaystyle=$ $\displaystyle 8\pi G\mu R\delta(r-R)\delta(z),$ (19)
$\displaystyle\tilde{\nabla}^{2}\eta$ $\displaystyle=$ $\displaystyle 0,$ (20)
$\displaystyle\zeta_{zz}+\frac{1}{r}\eta_{r}$ $\displaystyle=$ $\displaystyle
0,$ (21) $\displaystyle-\zeta_{zr}-\frac{1}{r}\left(\zeta_{z}-\eta_{z}\right)$
$\displaystyle=$ $\displaystyle 0,$ (22)
$\displaystyle\zeta_{rr}-\frac{1}{r}\left(\eta_{r}-2\zeta_{r}\right)$
$\displaystyle=$ $\displaystyle 0.$ (23)

We can convince ourselves that the Eqs. (18)-(23) are consistent. Eqs. (22)
and (23) gives

$\eta_{z}=\zeta_{z}+r\zeta_{zr},$ (24)

and

$\eta_{r}=r\zeta_{rr}+2\zeta_{r},$ (25)

respectively. Differentiating Eq. (21) and (24) wrt $r$ gives

$\displaystyle\eta_{rr}+\partial_{z}\left(\zeta_{z}+r\zeta_{zr}\right)$
$\displaystyle=$ $\displaystyle 0,$ (26)
$\displaystyle\eta_{zz}-\partial_{z}\left(\zeta_{z}+r\zeta_{zr}\right)$
$\displaystyle=$ $\displaystyle 0,$ (27)

respectively. This immediately leads to

$\eta_{rr}+\eta_{zz}=\tilde{\nabla}^{2}\eta=0,$ (28)

which is consistent with (20).

Integrating Eq. (25) yields

$\eta=r\zeta_{r}+\zeta+f(z),$ (29)

while integrating Eq. (24) gives

$\eta=r\zeta_{r}+\zeta+g(r).$ (30)

Thus,

$\eta=r\zeta_{r}+\zeta+\eta_{0},$ (31)

with $\eta_{0}$ a constant. From (20), the ansatz can be written in the form

$\eta=X(r)Z(z),$ (32)

whose solution can only either be a linear or a combination of harmonic and
hyperbolic functions. Either way, it violates the asymptotic flatness.
Therefore, the function must be constant, which can be absorbed into
$\eta_{0}$,

$\eta=\eta_{0}.$ (33)

Hence, with (31) we have

$\eta_{0}=\partial_{r}\left(r\zeta\right)\ \ \ \rightarrow\ \ \
\eta_{0}r=\zeta r-h(z)$ (34)

and thus

$\zeta=\eta_{0}+\frac{h(z)}{r}.$ (35)

Further, regularity at the core demands $h(z)=0$, thus

$\zeta=\eta_{0}.$ (36)

This is also evident from (21), where we find $\zeta_{zz}=0$. Consequently,
$h(z)$ must be a constant to satisfy the asymptotic flatness condition.

Using Eqs. (33) and (36), Eq. (18) becomes

$\nabla^{2}\nu=8\pi G\mu\delta(r-R)\delta(z).$ (37)

This is the Poisson’s equation for the electrostatic potential of a ring
charge. Following Hughes_PhysRevD.47.468 , the solution in the toroidal
coordinate is

$\nu(\sigma,\psi)=-4\sqrt{2}G\mu
N(\sigma,\psi)\frac{K\left(\tanh{\frac{\sigma}{2}}\right)}{\cosh{\frac{\sigma}{2}}},$
(38)

where

$\displaystyle
K\left(a\right)\equiv\int_{0}^{\frac{\pi}{2}}\frac{d\theta}{\sqrt{1-a\sin^{2}{\theta}}}$
(39)

is the complete elliptic integral of the first kind. In cylindrical coordinate
it takes the form

$\nu(\sigma,\psi)=-4\sqrt{2}G\mu\sqrt{\frac{R}{r}\sinh\left(\frac{1}{2}\ln\left(\frac{(r+R)^{2}+z^{2}}{(r-R)^{2}+z^{2}}\right)\right)}\frac{K\left(\tanh\left(\frac{1}{4}\ln\left(\frac{(r+R)^{2}+z^{2}}{(r-R)^{2}+z^{2}}\right)\right)\right)}{\cosh\left(\frac{1}{4}\ln\left(\frac{(r+R)^{2}+z^{2}}{(r-R)^{2}+z^{2}}\right)\right)}.$
(40)

Figure 1: 3D Plot of $\nu(r,z)$ for $G\mu=0.03$.

The function $A(r,z)$ is not necessarily needed for calculating the deficit
angle, however we could also solve it in order to have the full spacetime
profile of circular vorton. Eq. (19) can be re-expressed into a nicer form

$\nabla^{2}A+\frac{2}{r}A_{r}=-16\pi\frac{G\mu}{R}\delta(r-R)\delta(z).$ (41)

Defining $W\equiv rA$, it becomes

$\nabla^{2}W-\frac{1}{r^{2}}W=-16\pi G\mu\delta(r-R)\delta(z).$ (42)

One can easily recognize that the equation above just resembles the
$\varphi$-component of the Ampere equation for the magnetic vector potential
with a current source $\vec{J}=I\delta(r-R)\delta(z)\hat{\varphi}$. Following
Jackson Jackson:1998nia , we have

$\displaystyle A(r,z)$ $\displaystyle=$ $\displaystyle\frac{16G\mu
R}{r\sqrt{z^{2}+(r+R)^{2}}}\bigg{[}\left(\frac{z^{2}+(r+R)^{2}}{2rR}-1\right)K\left(\frac{4rR}{z^{2}+(r+R)^{2}}\right)$
(43)
$\displaystyle-\frac{z^{2}+(r+R)^{2}}{2rR}E\left(\frac{4rR}{z^{2}+(r+R)^{2}}\right)\bigg{]},$

or, in the toroidal coordinate

$A\left(\sigma,\psi\right)=\frac{8\sqrt{2}G\mu}{R}\frac{N^{3}\left(\sigma,\psi\right)}{\sinh^{2}{\sigma}}\bigg{[}e^{-\frac{\sigma}{2}}\cosh{\sigma}K\left(2e^{-\sigma}\sinh{\sigma}\right)-e^{\frac{\sigma}{2}}E\left(2e^{-\sigma}\sinh{\sigma}\right)\bigg{]},$
(44)

with

$\displaystyle
E\left(k^{2}\right)\equiv\int_{0}^{\frac{\pi}{2}}\sqrt{1-k^{2}\sin^{2}{\theta}}d\theta,$
(45)

the complete elliptic integral of the second kind. This solution can also be
obtained by solving Eq. (42) directly using the Green’s function method, as
shown in the Appendix A.

Figure 2: 3D Plot of $A(r,z)$ for $G\mu=0.03$.

(a) $g_{tt}(r,z)$

(b) $g_{t\phi}(r,z)$

(c) $g_{rr}(r,z)=g_{zz}(r,z)$

(d) $g_{\phi\phi}$

Figure 3: Complete metric of the circular vorton for $G\mu=0.03$ and $R=1$.

In Figs. 1-3 we show the profiles of the metric functions. It can be seen that
all functions diverge at $(r,z)=(1,0)$, precisely at the string cross section
of $R=1$. This is unexpected, as it is the natural behavior of a field with
delta function distributioncoming from the thin string approximation. It does
indicate the presence of ring singularity (the so-called ringularity) without
a horizon, and therefore we dub it the naked ringularity. The ringularity is
expected to be smoothed out with the order of the string thickness $\delta$
once we consider the full gravitating thick vorton (like in the case of
KunzRaduBintoro_PhysRevD.87.104022 ).

###  II.4 Asymptotic behaviour of the metric

The asymptotic expansion of $\nu$ near the string ($\sigma\rightarrow\infty$
in the toroidal coordinate) can be obtained using the asymptotic behavior of
the hyperbolic functions and elliptic integral,

$\cosh{\frac{\sigma}{2}}\rightarrow\frac{1}{2}e^{\sigma/2},\ \ \
N(\sigma,\psi)\rightarrow\sqrt{\frac{1}{2}}e^{\sigma/2},$ (46)

and

$K\left(\tanh{\frac{\sigma}{2}}\right)\rightarrow\frac{\sigma}{2}.$ (47)

Thus,

$\nu(\sigma\rightarrow\infty)=-4G\mu\sigma.$ (48)

Meanwhile, the asymptotic behavior far from the vorton would be

$\nu\left(\sqrt{r^{2}+z^{2}}\rightarrow\infty\right)\rightarrow-\frac{4\pi
G\mu R}{\sqrt{r^{2}+z^{2}}}.$ (49)

We can therefore deduce from the ansatz metric (1) that at infinity, we get

$\eta_{0}\to 0,$ (50)

which implies

$g_{rr}\rightarrow\left(1-\frac{8\pi G\mu R}{\sqrt{r^{2}+z^{2}}}\right)^{-1}.$
(51)

Defining $\rho=\sqrt{r^{2}+z^{2}}$ and expressing the vorton mass as

$M_{v}\equiv 4\pi R\mu,$ (52)

we get

$g_{rr}\rightarrow\left(1-\frac{2GM_{v}}{\rho}\right)^{-1},$ (53)

which is just the Schwarzschild metric of a point mass $M_{v}$, with $M_{v}$
the mass of a ring of linear density $\mu$ and radius $2R$. This is expected
from the formulation of chiral vorton, where the current energy is equal to
the mass term $\mu L$. Hence the total rest energy is twice the mass term
$E_{v}=2\mu L$, where $L$ is the invariant length of the vorton, which in the
circular case would be $2\pi R$.

The asymptotic expansion of $A$ near the string $(\sigma\rightarrow\infty)$
can be found by taking

$K\left(1-e^{-2\sigma}\right)\rightarrow\sigma,\ \ \
E\left(1-e^{-2\sigma}\right)\rightarrow 1,$ (54)

which gives

$A\left(\sigma\rightarrow\infty,\psi\right)\to\frac{8G\mu\sigma}{R}.$ (55)

This confirms the behavior that $A(r,z)$ diverges at the vorton core.

The asymptotic expansion at infinity of $A$ is

$A(\rho\rightarrow\infty)\to\frac{4\pi G\mu
R^{2}}{\rho^{3}}+O\left(\frac{1}{\rho^{4}}\right),$ (56)

where $\rho=\sqrt{r^{2}+z^{2}}$ is the radial position in spherical
coordinate. Thus, the expansion of $g_{t\phi}$ at infinity in spherical
coordinate is

$g_{t\phi}(\rho\rightarrow\infty)\to-\frac{4\pi G\mu
R^{2}}{\rho}\sin^{2}{\theta}+O\left(\frac{1}{\rho^{2}}\right),$ (57)

which coincides with the expansion of Kerr metric at infinity

$g_{t\phi}(\rho\rightarrow\infty)\to-\frac{2GJ}{\rho}\sin^{2}{\theta}+O\left(\frac{1}{\rho^{2}}\right),$
(58)

with

$J=aM,$ (59)

the angular momentum of the black hole of spin parameter $a$ and mass $M$. The
vorton’s angular momentum can then be identied as

$J_{v}=\frac{1}{2}RM_{v},$ (60)

which is similar to Kerr (59), only with

$a=\frac{R}{2}.$ (61)

We can conclude that the circular vorton spacetime is asymptotically identical
to the Kerr metric of a spinning black hole with mass $M_{v}=4\pi R\mu$ and
spin parameter $a=R/2$. This raises an intriguing possibility: some
observations of rotating black holes, particularly those involving nearby
orbits, could potentially be attributed to vortons with the same mass. It is
also worth noting that for the same amount of mass, the vorton would have a
radius of:

$\displaystyle R_{v}$ $\displaystyle=$ $\displaystyle\frac{GM_{v}}{4\pi
c^{2}\alpha}=\frac{1}{8\pi\alpha}R_{s},$ (62)

where $R_{s}$ is the Schwarzschild radius of a black hole of the same mass,
and $\alpha$ is the $G\mu$ value of the string. For a Grand Unified Theory
(GUT) scale string, $\alpha$ is on the order of $10^{-6}$. This implies that
the vorton would have a radius $\sim 10^{5}$ times that of a black hole of the
same mass, which still falls within the far-field (Schwarzschild)
approximation.

Interestingly, for a typical GUT-scale vorton, Eqs. (61)-(62) imply

$R_{s}^{2}-4a^{2}<0.$ (63)

Thus, the Kerr metric solution that the vorton asymptotically approaches
corresponds to a naked singularity. A distant observer would find it
challenging to distinguish between a vorton and a naked singularity. However,
the spin of a black hole is typically measured using X-Ray spectroscopy of its
accretion disk, as seen in SpinMeasurement10.1093/mnras/staa606 , rather than
by the orbits of nearby stars. This is because spin measurement is a strong-
field effect, while star orbits are a far-field effect
ObservingSpinReynolds:2019uxi ; SpinConstraintsReynolds:2020jwt . Furthermore,
the proposed method for detecting a supposed naked singularity typically
involves observing the object’s strong field effects
NakedSingularityObsMethodPhysRevD.95.084024 . Unless a vorton could have an
accretion disk—potentially formed from star matter orbiting too close to the
vorton string core—it is not feasible to detect this apparent naked
singularity, or more precisely, to distinguish it from a black hole of the
same mass. Therefore, the discrepancy between the length scales at which the
far-field approximation breaks down could help identify a vorton from nearby
orbits. Since, like a black hole, a vorton produces no light on its own, this
method could be crucial for distinguishing between the two objects. For
example, the star S2 orbits the Sgr A* supermassive black hole with a closest
approach of about $1400$ times the black hole’s Schwarzschild radius
S2SgrAGRAVITY:2018ofz . This distance would not adhere to the far-field
approximation if the object were a vorton. However, objects farther away from
the vorton would. To accurately identify a vorton, the near vorton orbits must
be studied more closely using the geodesic equations for massive objects.
There could be several unique orbital solutions around a vorton, potentially
including toroidal orbits, that could distinguish it from other dark sources.
Further research in this area is needed to identify a vorton without relying
solely on gravitational lensing, especially since resolving smaller scale
vortons could be challenging.

##  III The Spacetime Properties

###  III.1 Frame dragging and ergoregion

The metric solution represents a rotating spacetime, where an inertial frame
of reference will rotate with angular velocity $\Omega(r,z)$ together with the
spacetime. The angular frequency in the inertial frame is

$\Omega(r,z)=A(r,z),$ (64)

and thus Eq. (43) dictates the angular velocity of the inertial frame at point
$(r,z)$. A polar grid representation can be constructed by considering the
parametric curve for several (equally spaced) value of $\phi$ as illustrated
in Fig. 4.

(a) $t=0$

(b) $t=1$

(c) $t=2$

(d) $t=4$

Figure 4: Polar grid of the spacetime with angular velocity $\Omega(r,z)$ at
the vorton plane $z=0$ for $G\mu=0.2$ from $t=0$ to $t=4$.

The figure represents the dragging of inertial frame around a circular vorton
at $z=0$, which has the current running counter clockwise. As expected, the
frame-dragging effect is more extreme near the string, and that the current
drags the inertial frame in the same direction as the current. We also note
that the analogue of this part of the metric and the magnetic vector potential
of a circular current loop, together with the analogue of $\nu$ with the
electric scalar potential of the same object is a manifestation of gravito-
electromagnetism.

The ergoregion is the region between the event horizon and the ergosurface,
where the event horizon is the surface $(r,z)$ of $g_{rr}=0$, and the
ergosurface is the surface $(r,z)$ of $g_{tt}=0$. Since

$g_{rr}=e^{-2\nu(r,z)}>0,\forall\ r,z,$ (65)

the solution contains no event horizon. However, the metric component $g_{tt}$
does admit null solution

$-e^{2\nu(r,z)}+e^{-2\nu(r,z)}r^{2}A^{2}(r,z)=0,$ (66)

since $g_{tt}<0$ for both $r\rightarrow\infty$ (condition (49)) and $r=0$,
while $g_{tt}>0$ for both $r\to R$ and $z=0$. We can see this from the
behavior of $\nu$ and $A$, where

$\displaystyle\nu\left(\sqrt{r^{2}+z^{2}}\rightarrow 0\right)$
$\displaystyle=$ $\displaystyle-4\pi G\mu,$ (67) $\displaystyle
r^{2}A(r\rightarrow 0)$ $\displaystyle=$ $\displaystyle rW(r\rightarrow 0)=0,$
(68)

and $A(r,z)\rightarrow+\infty$ for $(r\approx R,z=0)$. Since $g_{tt}$ changes
sign from $(-+-)$ as $r$ increases, and the vorton cross-section lies within
the region where $\text{sign}(g_{tt})=+1$, it is clear that the ergosurface
must have a toroidal shape. Consequently, we refer to the ergoregion of this
system as the ergotorus. They are illustrated in Fig 5.

(a)

(b)

Figure 5: Curve of the surface of ergotorus cross section in $rz$-plane for
various value of $G\mu$ (a), and 3D curve of ergotorus surface for $G\mu=0.03$
(b).

The ergotorus is the region where the rotational energy of the vorton could
potentially be extracted. However, its presence suggests potential instability
against scalar and electromagnetic perturbations, as discussed in
ErgoregionInstabilityFriedman:1978ygc . It is expected that the ergotorus
would vanish when considering the full field equations with realistic
parameters, beyond the thin string approximation, as demonstrated in
KunzRaduBintoro_PhysRevD.87.104022 . Fig. 5(a) clearly illustrates that the
ergotorus surface diminishes as $G\mu$ decreases. For a GUT-scale string with
$G\mu\sim 10^{-6}$, the ergotorus region becomes nearly indistinguishable from
the vorton core. At this scale, specifically within a distance less than the
core thickness $\delta$ from the vorton core, our weak-field assumption fails.
Consequently, the ergotorus can be disregarded, much like how the
Schwarzschild radius of the Sun is meaningless.

###  III.2 Angular deficit

As with any cosmic string loop, the spacetime around a vorton exhibits a
conical singularity. The deficit angle around the string core in toroidal
coordinates is defined as

$\delta\psi=2\pi-\lim_{\sigma_{0}\rightarrow\infty}\frac{\int_{-\pi}^{\pi}N^{-2}e^{C}\big{|}_{\sigma=\sigma_{0}}d\psi}{\int_{\sigma_{0}}^{\infty}N^{-2}e^{C}d\sigma},$
(69)

where $C(\sigma,\psi)\equiv\eta-\nu$. Applying l’Hopital’s rule for the limit
gives

$\delta\psi=2\pi\lim_{\sigma_{0}\rightarrow\infty}\frac{\partial
C}{\partial\sigma}\bigg{|}_{\sigma=\sigma_{0}}.$ (70)

In our case, with $\eta=0$, we have $C\to-\nu$. Therefore, using (48) we
obtain

$\displaystyle\lim_{\sigma_{0}\rightarrow\infty}\frac{\partial
C}{\partial\sigma}\bigg{|}_{\sigma=\sigma_{0}}$ $\displaystyle=$
$\displaystyle-\frac{\nu(\sigma\rightarrow\infty)}{\partial\sigma}$ (71)
$\displaystyle=$ $\displaystyle 4G\mu,$

which yields the deficit angle around the circular vorton as

$\delta\psi=8\pi G\mu.$ (72)

This result is exactly the same as for the case of a straight string, a
straight chiral string GravityStraightChiralStringSteer:2000jn , and circular
string loops (both static and rotating) Hughes_PhysRevD.47.468 ;
McManus_PhysRevD.47.1491 .

##  IV Gravitational Lensing by Cosmic Vorton

The gravitational lensing by string loops in the thin-lens approximation was
first studied by de Laix and Vachaspati deLaix:1996vc . They calculated the
lensing images produced by a circular loop as well as some non-circular Turok
loops with higher frequency modes. With the (weak-field) metric solution in
hand, we can improve upon this for the vorton. Specifically, we calculate the
lensing around a circular vorton using the geodesic equations. Assuming that
the full Einstein field solutions behave similarly, the solution we obtained
should be sufficient to visualize the gravitational lensing around the vorton.
This approach allows for a more accurate and detailed understanding of the
lensing effects produced by vortons.

The acceleration of light from the null geodesic equation is given by

$\frac{d^{2}x^{\mu}}{d\lambda^{2}}=-\Gamma^{\mu}_{\alpha\beta}\frac{dx^{\alpha}}{d\lambda}\frac{dx^{\beta}}{d\lambda}.$
(73)

Given the constraints

$g_{tt}\dot{t}^{2}+2g_{t\phi}\dot{t}\dot{\phi}+\vec{v}^{2}=0,$ (74)

with

$\vec{v}^{2}\equiv g_{ij}\frac{dx^{i}}{d\lambda}\frac{dx^{j}}{d\lambda},$ (75)

we can express $\dot{t}$ as

$\dot{t}=\frac{-g_{t\phi}\dot{\phi}+\sqrt{g_{t\phi}^{2}\dot{\phi}^{2}-g_{tt}\vec{v}^{2}}}{g_{tt}}.$
(76)

This expression for $\dot{t}$ can be substituted into Eq. (73), yielding
equations that can be solved using spatial initial conditions alone.
Consequently, the required acceleration is reduced to the three-acceleration
$\vec{a}=\frac{d^{2}\vec{x}}{d\lambda^{2}}$ of the null particle. This
simplification streamlines the analysis, allowing us to focus solely on the
spatial dynamics of the particle’s motion.

The lensing calculation was conducted in Wolfram Mathematica and HLSL
programming language, while the simulation and visualization were crafted
using Unreal Engine 5. However, Unreal Engine 5 demands an analytic
approximation of the functions. To facilitate this, a celestial sphere was
employed in the scene setup. This celestial sphere, an inverted sphere with a
celestial texture cubemap assigned to it, utilized the deep starmap
NASAStarmap  as its cubemap texture. Positioned at the center of the sphere,
the vorton was represented by an image derived from the circular (chiral)
vorton metric obtained previously. In this setup, the celestial sphere boasted
a radius of $5000$ units. The rendering in Unreal Engine 5 yielded Figs. 6-11.
We show lensing images for various tension and angle $\phi$ (the angle between
the vorton’s plane and the detector’s axis). We also provide the unlensed
image for comparison.

(a) $4\pi\mu=0.06$

(b) $4\pi\mu=0.2$

Figure 6: Lensing images by a chiral vorton at $\phi=0.035\pi$.

(a) Unlensed

(b) $4\pi\mu=0.06$

(c) $4\pi\mu=0.1$

Figure 7: Lensing images by a chiral vorton at $\phi=0.35\pi$.

(a) Unlensed

(b) $4\pi\mu=0.05$

(c) $4\pi\mu=0.06$

Figure 8: Lensing images by a chiral vorton at $\phi=\pi/4$.

(a) $4\pi\mu=0.06$

(b) $4\pi\mu=0.025$

(c) $4\pi\mu=0.05$

(d) $4\pi\mu=0.075$

Figure 9: Visualization of vorton gravitational lensing from perpendicular
point of view ($\phi=\pi/2$).

(a) Unlensed

(b) $4\pi\mu=0.2$

(c) $4\pi\mu=0.1$

Figure 10: Lensing images by a chiral vorton from the center of the loop.

(a) Unlensed

(b) $4\pi\mu=0.02$

(c) $M=3$, $R=50$

Figure 11: Visualization of Einstein ring together with the weakly distorted
source inside the loop generated by vorton.

The lensing images produced by a circular vorton vary significantly with
different angles $\phi$, representing potential detections from various
observational viewpoints. Figs. 6, 7, 8, and 9 illustrate typical image
distortions as $\phi$ increases up to $\pi/2$. Fig. 10 illustrates the image
detected when the detector is positioned at the center of the vorton loop.
Fig. 11 depicts an Einstein ring, occurring when the source, vorton loop, and
detector are aligned. As $\mu$ increases, image distortion becomes more
pronounced.

For an observer viewing a perpendicular circular vorton (Fig. 9), there would
be a weakly distorted image inside the string, and a highly distorted image,
or even an Einstein ring (Fig. 11), outside the string, similar to that in
deLaix:1996vc , though it was obtained using different approach and different
scale. At every angle, there is an apparent ring-shaped discontinuity in the
image, a generic feature of cosmic strings due to their conical geometry. The
combination of a weakly distorted source at the center of an Einstein ring and
the image discontinuity can be a signature of cosmic string loop lensing,
suggesting the possibility of observation. This is true since typically the
straight path to the source of an Einstein ring would be blocked by the lens
(usually a galaxy) if not distorted to form the ring. Additionally, if the
observer views the string from near the loop plane, a double image of the
loop’s back appears, one on top and one on the bottom, similar to the double
images of an accretion disk around a black hole. From the loop’s center (Fig.
10), the outside image appears as a double image of a weakly-lensed source
with a central discontinuity, resembling the image of a straight string. This
indicates that a circular vorton seen from the center of the loop looks like a
straight string.

##  V Conclusions

In this paper, we study the gravitational field and lensing of a circular
chiral vorton. Using a weak-field approximation, we solve the corresponding
Einstein field equations. The metric solutions $\nu(r,z)$ and $A(r,z)$ are
analogous to the $A_{t}$ and $A_{\varphi}$ components, respectively, of the
vector potential of a circular current wire loop. Around the string core, the
spacetime exhibits conical singularity with deficit angle $\delta\psi=8\pi
G\mu$, as in the straight string. There exists ergosurface with $S^{1}\times
S^{1}$ topology, dubbed the ergotorus, between the ergoregion and the external
space. The appearance of the ergotorus indicates instability, which we expect
to be smoothed out in the thick string approach.

Asymptotically, the spacetime around a vorton loop of radius $R$ is identical
to a Kerr black hole with mass $M_{v}=4\pi R\mu$, spin parameter $a=R/2$, and
angular momentum $J_{v}=2\pi R^{2}\mu$. This is intriguing because what we
previously detected as a distant rotating black hole might actually be a
vorton. More surprisingly, for a typical GUT-scale string ($R_{v}\sim
10^{5}-10^{6}R_{S}$), Eq. (63) is always satisfied, saturating the extremal
bound for Kerr. Thus, a GUT-scale vorton is indistinguishable from a Kerr
naked singularity to a distant observer. It is worth noting that recently
there has been a growing interest in the study of Kerr naked singularities,
spurred by the possible existence of Kerr superspinars within the string
theory framework Gimon:2007ur . Researchers are particularly focused on photon
orbits Charbulak:2018wzb  and the shadows Nguyen:2023clb  of Kerr naked
singularities. Studying the null and timelike geodesics around a vorton could
provide insights into distinguishing it from a Kerr naked singularity.

Recently, a suspected gravitational lensing detection of a cosmic string has
been reported arxivsafonova2023deep . Using photometric and spectroscopic
observations, researchers propose that the galaxy pair SDSSJ110429 are
actually duplicate images of a single galaxy lensed by a cosmic string. Given
that a vorton is postulated to be a stable string loop configuration, our
study of lensing images by a vorton becomes particularly relevant in this
context. Our study reveals unique properties of vorton-generated lensing
images. They have an apparent discontinuity that separates minimally distorted
regions from highly distorted ones from the same source, as can be seen, for
example, in Figs. 7-8. Moreover, there are similarities between the lensing
images due to vorton and a black hole with an accretion disk. This can be
observed, for example in Fig. 6, where the ring image could form twice
simultaneously, with light from both the upper and lower sides of the ring
being deflected back towards a camera positioned near the plane of the vorton.
An Einstein ring could form around a vorton, while the original image of the
source remains visible with minimal distortion at the center of the ring, as
shown in Fig. 11. Typically, an Einstein ring forms when another object, which
creates the ring, blocks the direct view of the source image, distorting it
into the ring shape. The simultaneous appearance of both the ring and the
almost undistorted source can be a distinctive signature of a string loop in
the sky.

This preliminary work opens up possibilities for studying the general
gravitational lensing and dynamics of vortons. Here, we focus on the geometry
and geodesics of a circular chiral vorton, as this loop shape possesses the
highest symmetry, making the metric solution analytically tractable. Other
loop shape configurations, reminiscent of the Kibble-Turok two-parameter
family of solutions Kibble:1982cb ; Turok:1984cn  in the case of Nambu-Goto
cosmic string loops, certainly exist. We will discuss the gravitational
lensing in the thin-lens approximation by arbitrary vorton loop shapes in our
next publication. Another issue that can be addressed is the dynamics of
vorton loop around a massive object, for example a supermassive black hole.
The dynamics of a charged string loop around Reinssner-Nordstrom black hole
was discussed in CCSLRNOteev:2018gth . In Jacobson:2008cw , the authors study
the dynamics of a current-carrying string loop around a Kerr black hole,
suggesting that such a system might help solve the problem of collimated jets
in astrophysics. It is interesting to consider the effects of using a vorton
instead. Further research is also needed on the dynamics of light and massive
objects around a vorton loop. We shall explore these possibilities in
subsequent publications.

###### Acknowledgements.

We thank Steven Holme and Akif Pervez for the enlightening discussions. This
work is funded by the Hibah Riset FMIPA UI No.
PKS-026/UN2.F3.D/PPM.00.02/2023.

##  Appendix A Derivation of $A(r,z)$

The solution of $W(r,z)$ is just the Green’s function of operator

$\hat{L}\equiv\nabla^{2}-\frac{1}{r^{2}},$ (77)

in cylindrical coordinate, which can be solved by the elementary method using
Fourier transform. Multiplying (42) with an arbitrary function $f(\theta)$
gives

$f(\phi)\nabla^{2}W(r,z)-\frac{1}{r^{2}}W(r,z)f(\phi)=-16\pi
G\mu\delta(r-R)\delta(z)f(\phi).$ (78)

We would like to find a function $f(\phi)$, such that

$f(\phi)\nabla^{2}W(r,z)-\frac{1}{r^{2}}W(r,z)f(\phi)=\nabla^{2}\left(W(r,z)f(\phi)\right).$
(79)

Therefore,

$\frac{d^{2}f(\phi)}{d\phi^{2}}+f(\phi)=0,\ \ \rightarrow\ \ \
f(\phi)=C_{1}\cos{\phi}+C_{2}\sin{\phi},$ (80)

$C_{1}$ and $C_{2}$ being two integration constants which can be set $C_{1}=1$
and $C_{2}=0$. Thus,

$\nabla^{2}\left(W(r,z)\cos{\phi}\right)=\left(\nabla^{2}W-\frac{W}{r^{2}}\right)\cos{\phi}.$
(81)

Substituting 42, and redefining

$V(r,z,\phi)=W(r,z)\cos{\phi},$ (82)

yields

$\nabla^{2}V(r,z,\phi)=-16\pi G\mu\delta(r-R)\delta(z)\cos{\phi}.$ (83)

This is the Poisson’s equation for the electrostatic potential of a ring
charge distribution with radius $R$ centered at $z=0$, with azimuthal charge
distribution of $\cos{\phi}$. Using the Green’s function for the Laplacian
$G(\vec{r},\vec{r}^{\prime})=1/4\pi|\vec{r}-\vec{r}^{\prime}|$, the solution
is

$V(r,z,\phi)=-\int_{R}d^{3}\vec{r}^{\prime}\frac{\rho(\vec{r}^{\prime})}{4\pi|\vec{r}-\vec{r}^{\prime}|},$
(84)

where

$\rho(\vec{r}^{\prime})=-16\pi
G\mu\delta(r^{\prime}-R)\delta(z^{\prime})\cos{\phi^{\prime}}.$ (85)

The point $P$ at which the potential is measured has the position vector

$\vec{r}\equiv\vec{z}+\vec{\rho},$ (86)

where $\vec{\rho}=r\hat{\rho}$. Suppose $\hat{\rho}$ make an angle $\phi$ with
the $x$ axis, and $\hat{r}^{\prime}$ make an angle $\phi^{\prime}$ with the
$x$ axis. Then,

$\displaystyle|\vec{r}-\vec{r}^{\prime}|^{2}$ $\displaystyle=$
$\displaystyle|\vec{z}+\vec{\rho}-\vec{r}^{\prime}|^{2}$ (87) $\displaystyle=$
$\displaystyle z^{2}+\left(\vec{\rho}-\vec{r}^{\prime}\right)^{2}$
$\displaystyle=$ $\displaystyle z^{2}+r^{2}+r^{\prime
2}-2rr^{\prime}\cos{(\phi^{\prime}-\phi)},$

and the potential becomes

$\displaystyle V(r,z,\phi)$ $\displaystyle=$ $\displaystyle 16\pi
G\mu\int\int\int
r^{\prime}dr^{\prime}dz^{\prime}d\phi^{\prime}\frac{\delta(r^{\prime}-R)\delta(z^{\prime})\cos{\phi^{\prime}}}{4\pi\sqrt{z^{2}+r^{2}+r^{\prime
2}-2rr^{\prime}\cos{(\phi^{\prime}-\phi)}}}$ (88) $\displaystyle=$
$\displaystyle 4G\mu
R\int_{0}^{2\pi}d\phi^{\prime}\frac{\cos{\phi^{\prime}}}{\sqrt{z^{2}+r^{2}+R^{2}-2rR\cos{(\phi^{\prime}-\phi)}}}.$

Defining

$\phi^{\prime}=\pi+\phi+2\theta\ \ \ \rightarrow\ \ \
d\phi^{\prime}=2d\theta,$ (89)

we have

$\displaystyle V(r,z,\phi)$ $\displaystyle=$ $\displaystyle 8G\mu
R\int_{0}^{\pi}d\theta\frac{\cos{(\pi+\phi+2\theta)}}{\sqrt{z^{2}+(r+R)^{2}-4rR\sin^{2}{\theta}}}$
(90) $\displaystyle=$ $\displaystyle-8G\mu
R\bigg{(}\cos{\phi}\int_{0}^{\pi}d\theta\frac{\cos{(2\theta)}}{\sqrt{z^{2}+(r+R)^{2}-4rR\sin^{2}{\theta}}}$
$\displaystyle-\sin{\phi}\int_{0}^{\pi}d\theta\frac{\sin{(2\theta)}}{\sqrt{z^{2}+(r+R)^{2}-4rR\sin^{2}{\theta}}}\bigg{)}.$

The second term vanishes due to being an odd function. Thus,

$\displaystyle V(r,z,\phi)$ $\displaystyle=$ $\displaystyle-8G\mu
R\cos{\phi}\int_{0}^{\pi}d\theta\frac{1-2\sin^{2}{\theta}}{\sqrt{z^{2}+(r+R)^{2}-4rR\sin^{2}{\theta}}}$
(91) $\displaystyle=$ $\displaystyle-8G\mu
R\cos{\phi}\bigg{(}\int_{0}^{\pi}d\theta\frac{1}{\sqrt{z^{2}+(r+R)^{2}-4rR\sin^{2}{\theta}}}$
$\displaystyle-2\int_{0}^{\pi}d\theta\frac{\sin^{2}{\theta}}{\sqrt{z^{2}+(r+R)^{2}-4rR\sin^{2}{\theta}}}\bigg{)}.$

From (82), we obtain

$\displaystyle W(r,z)$ $\displaystyle=$ $\displaystyle-8G\mu
R\bigg{(}\int_{0}^{\pi}d\theta\frac{1}{\sqrt{z^{2}+(r+R)^{2}-4rR\sin^{2}{\theta}}}$
(92)
$\displaystyle-2\int_{0}^{\pi}d\theta\frac{\sin^{2}{\theta}}{\sqrt{z^{2}+(r+R)^{2}-4rR\sin^{2}{\theta}}}\bigg{)}.$

Taking the fact that both terms are even, and using the definitions of
complete elliptic integrals, we have

$\displaystyle W(r,z)$ $\displaystyle=$ $\displaystyle\frac{16G\mu
R}{\sqrt{z^{2}+(r+R)^{2}}}\bigg{(}\left(\frac{z^{2}+(r+R)^{2}}{2rR}-1\right)K\left(\frac{4rR}{z^{2}+(r+R)^{2}}\right)$
(93)
$\displaystyle-\frac{z^{2}+(r+R)^{2}}{2rR}E\left(\frac{4rR}{z^{2}+(r+R)^{2}}\right)\bigg{)}.$

Thus, the metric function $A(r,z)$ in terms of complete elliptic integral as
well as the hypergeometric function reads

$\displaystyle A(r,z)$ $\displaystyle=$ $\displaystyle\frac{16G\mu
R}{r\sqrt{z^{2}+(r+R)^{2}}}\bigg{(}\left(\frac{z^{2}+(r+R)^{2}}{2rR}-1\right)K\left(\frac{4rR}{z^{2}+(r+R)^{2}}\right)$
(94)
$\displaystyle-\frac{z^{2}+(r+R)^{2}}{2rR}E\left(\frac{4rR}{z^{2}+(r+R)^{2}}\right)\bigg{)}$
$\displaystyle=$ $\displaystyle\frac{8\pi G\mu
R}{r\sqrt{z^{2}+(r+R)^{2}}}\bigg{(}\left(\frac{z^{2}+(r+R)^{2}}{2rR}-1\right){}_{2}F_{1}\left(\frac{1}{2},\frac{1}{2};1;\frac{4rR}{z^{2}+(r+R)^{2}}\right)$
$\displaystyle-\frac{z^{2}+(r+R)^{2}}{2rR}{}_{2}F_{1}\left(\frac{1}{2},-\frac{1}{2};1;\frac{4rR}{z^{2}+(r+R)^{2}}\right)\bigg{)}.$

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