The Shadow of Charged Traversable Wormholes M ario Raia Neto12 Daniela P erez3and Joaqu n Pelle4

2025-05-06 0 0 1015.04KB 19 页 10玖币
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The Shadow of Charged Traversable
Wormholes
M´ario Raia Neto1,2*, Daniela P´erez3* and Joaqu´ın Pelle4*
1Departamento de F´ısica, Universidade Federal de S˜ao Carlos,
S˜ao Paulo, Brazil.
2Instituto Nacional de Pesquisas Espaciais, S˜ao Paulo, Brazil.
3Instituto Argentino de Radioastronom´ıa,(IAR,
CONICET/CIC/UNLP), Buenos Aires, Argentina.
4Facultad de Matem´atica, Astronom´ıa, F´ısica y Computaci´on,,
Universidad Nacional de C´ordoba, Argentina.
*Corresponding author(s). E-mail(s):
mraianeto@estudante.ufscar.br;daniela.perez2812@gmail.com;
jpelle@mi.unc.edu.ar ;
Abstract
We compute the shadow cast by a charged Morris-Thorne wormhole
when the light source is a star located beyond the mouth which is
opposite to the observer. First, we provide an extensive analysis of the
geodesic properties of the spacetime, both for null and massive particles.
The geometrical properties of this solution are such that independently
of the viewing angle, some light rays always reach the observer. Addi-
tionally, the structure of the images is preserved among the different
values of the charge and scales proportionally to the charge value.
Keywords: Wormhole; shadow; black hole; general relativity.
1 Introduction
Wormholes are exact solutions of Einstein Field Equations (EFE), which con-
nects two spacetime regions by a throat. The mouths are not hidden by event
horizons, as in the case of black holes, and there are no singularities.
1
arXiv:2210.14106v1 [gr-qc] 25 Oct 2022
2The Shadow of Charged Traversable Wormholes
The first work on what we now call wormhole was published by A. Einstein
and N. Rosen [5] in 1935. They tried to construct a model for an elemen-
tary particle that could be everywhere finite and free of singularities. Though
their particle model was a complete failure, they arrived to the first wormhole
spacetime model that is known as Einstein-Rosen bridge.
Almost twenty years passed until the subject was revived by J.A.Wheeler
[6] in 1955 when he published a work about “geons”. This is the first time that
a wormhole space diagram appears in the scientific literature. Two years later,
C.W. Misner and J.A.Wheeler [7] analyzed the geometry of manifolds with
non-trivial topology with the aim to explain completely all classical physics.
The word “wormhole” is employed for the first time in the context of General
Relativity.
In the seventies, H.G. Ellis introduced[10,11] a kind of wormhole solution,
called Ellis drainhole; Bronnikov[12] noticed that the solution was free from
event horizons and singularities, geodesically complete and able to be “crossed”
independent of direction1. During the sixties and seventies, there was great
progress in different aspects of General Relativity. However, not much work
was focused on the analysis of Lorentzian wormholes.
The pioneer work of Michael Morris and Kip Thorne [13] on traversable
wormholes, also called Morris-Thorne Wormholes (MTWH), caused a major
revival on the subject. In their famous paper published in 1988, they focused
on the necessary conditions in order to have a wormhole geometry that could
connect two flat asymptotically spacetime regions; these solutions must not
contain horizons or real singularities, and are connected by a “throat” that is
kept open due to presence of exotic matter (matter that violates the energy
conditions) that exerts gravitational repulsion2.
Much has been discovered about these astonishing solutions. Roman,
proposed a mechanism to “inflate” the microscopic wormholes discussed
by Wheeler[15]; Kim has found cosmological traversable wormholes[16], Teo
discovered the rotational version of MTWH[17] and A. Alias proposed a
slowly rotational version of Teo’s rotating metric[18]. Furthermore, Kim also
constructed a form of charged MTWH[19]. A scalar field supporting the
traversable wormhole was introduced by L.Butcher[20] and a few years ago
Lobo, Quinet and Oliveira discovered the deSitter MTWH[21]. Traversable
wormholes have also been studied in alternative theories of gravitation
such as f(R) and f(R, T ) gravity[22–34] and in analogue gravity[38]. There
are works on traversable wormholes in cosmology[36,37], astrophysics [43],
thermodynamics[39,40,42] and their shadows were also computed [35].
Though several wormhole solutions have been extensively analyzed during
the years, the first charged transversable wormhole geometry found more than
twenty years ago[19] has barely been investigated. One interesting feature of
1There are two different solutions of the Ellis drainhole, with matter [10] and no matter flow
[11], being the later the simplest. Notice that both Bronnikov and Ellis arrived independently to
these solutions, so these are referred as “Ellis-Bronnikov Wormholes”.
2In 1989, G. Cl´ement [14] demonstrated that the Ellis drainhole is type of a traversable
wormhole.
The Shadow of Charged Traversable Wormholes 3
this solution is that the presence of charge allows to minimize the content of
exotic matter at the throat
In this article we analyze various properties of the charged traversable
wormhole solution derived by Kim and Lee[19]. In particular, our main goal is
to compute the shadow of the wormhole, which is relevant from an astrophys-
ical point of view. Though shadow of wormholes were calculated for various
wormhole geometries [50–52], Kim and Lee’s solution has not been explored
in the literature yet.
We first provide a short review of the main properties of the charged
transversable wormhole. Then, by solving the geodesic equations for null par-
ticles, we determine the location of the photon ring (Section 2.1.1). We also
study the trajectories of massive particles (Section 2.3) and in Section 2.3.1
we extend these calculations to a more general charged wormhole geometry.
Finally, we present in Section 3the images of the shadow of the wormhole when
the light source is a star located beyond the mouth opposite to the observer’s.
We produce images for several values of the charge and viewing angles. The
final section of the article is devoted to the conclusions. Throughout this work,
we employ geometrized units G=c= 1.
2 Charged Traversable Wormholes
The solution for a charged traversable wormhole was found by Kim and Lee
[19] and was generalized in general relativity and modified gravity [53–55]. The
line element is given by:
ds2=− 1 + Q2
r2!dt2+1
1−b(r)
r+Q2
r2
dr2+r2dθ2+sin2(θ)dφ2.(1)
The corresponding components of the energy-momentum tensor3are [19], .
ρ(r) = 1
8π(b0(r)
r2−Q2
r4).(2)
σ(r) = −1
8π(b(r)
r3−2 1−b(r)
r!Φ0(r)
r+Q2
r4).(3)
p(r) = 1
8π( 1−b(r)
r!"Φ00(r)+(Φ0(r))2−b0(r)r−b(r)
2r(r−b(r)) Φ0(r)−b0(r)r−b(r)
2r2(r−b(r))+
+Φ0(r)
r−Q2
r4#).(4)
3A didactic derivation was presented by Kimet [56]
4The Shadow of Charged Traversable Wormholes
Where ρ(r) is the energy density, σ(r) is the tension and p(r) is the pressure.
In the present work, we will focus on the shape function:
b(r) = b2
0
r.(5)
When Q= 0, the solution describes a neutral traversable wormhole, and
when b(r) = 0 the spacetime geometry corresponds to the massless Reissner-
Norstr¨om solution.
The throat of the wormhole is located at
1−b2
0
r2+Q2
r2= 0,(6)
that is
r=±qb2
0−Q2.(7)
Notice that the coordinate ris defined in the range r∈(−∞,−pb2
0−Q2]∪
[+pb2
0−Q2,+∞).
2.1 Geodesics of Charged Traversable Wormholes
2.1.1 Trajectories of photons
We begin by computing the trajectories of null particles. From the line element
(31), the Lagrangian of the charged wormhole is
2L=− 1 + Q2
r2!˙
t2+1
1−b2
0
r2+Q2
r2
˙r2+r2˙
θ2+r2sin2(θ)˙
φ2,(8)
where the dot is a derivative with respect to the affine parameter λ:˙
xα=:
dxα/dλ. Since the metric coefficients are independent of tand φ, the Killing
vectors of the spacetime are k0= (1,0,0,0) and k3= (0,0,0,1). Therefore,
the constants of motion for tdirection and φdirection are:
k0u0=−E , (9)
k3u3=` . (10)
where uµ=gµν uν. The geodesic equations for tcoordinate and φcoordinate
now take the simple form
dt
dλ=u0=g0νuν=g00u0=E
1 + Q2
r2
,(11)
dφ
dλ=u3=g3νuν=g33u3=`
r2sin2(θ),(12)
摘要：

TheShadowofChargedTraversableWormholesMarioRaiaNeto1,2*,DanielaPerez3*andJoaqunPelle4*1DepartamentodeFsica,UniversidadeFederaldeS~aoCarlos,S~aoPaulo,Brazil.2InstitutoNacionaldePesquisasEspaciais,S~aoPaulo,Brazil.3InstitutoArgentinodeRadioastronoma,(IAR,CONICET/CIC/UNLP),BuenosAires,Argentina...

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