Shadows of rotating Einstein-Maxwell-dilaton black holes surrounded by a plasma Javier Bad ıa12and Ernesto F. Eiroa1

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Shadows of rotating Einstein-Maxwell-dilaton black holes

surrounded by a plasma

Javier Bad´ıa1,2∗and Ernesto F. Eiroa1†

1Instituto de Astronom´ıa y F´ısica del Espacio (IAFE, CONICET-UBA),

Ciudad Universitaria, 1428, Buenos Aires, Argentina

2Departamento de F´ısica, Facultad de Ciencias Exactas y Naturales,

Universidad de Buenos Aires, Ciudad Universitaria Pabell´on I, 1428, Buenos Aires, Argentina

Abstract

We consider rotating charged black holes with a scalar dilaton ﬁeld and surrounded by

plasma, with the purpose of studying their shadows. The corresponding metric has been previ-

ously obtained in the literature from the static solution by using the Newman-Janis algorithm.

Assuming a well known form for the pressureless and nonmagnetized plasma distribution, which

is suitable for the separation of the Hamilton-Jacobi equation for light, we derive an expression

that determines the shape of the shadow. We present some examples of contours and we analyze

their observable properties as functions of the charge and the dilaton coupling. We ﬁnd that

the presence of plasma introduces a dependency on the frequency, with the shadow becoming

smaller as the frequency decreases.

1 Introduction

In the last few years, it has been observed for the ﬁrst time that black holes cast shadows on

their surroundings, as predicted by the theory of general relativity [1–3]. The Event Horizon

Telescope (EHT) Collaboration has produced reconstructed images of both the supermassive

black hole M87* at the center of the elliptical galaxy M87 [4] as well as Sgr A*, the black hole

at the center of our galaxy [5]; they show a dark region surrounded by a bright ring of light,

which for Sgr A* has a diameter of ∼50 µas. These observations are consistent with the well-

known theoretical scenario in which the trajectories of light rays emitted by the accretion disk

of the black hole are deﬂected by its strong gravitational ﬁeld, forming a region in a distant

observer’s sky from which no light arrives. The size and shape of the shadow depend on the

various parameters characterizing the black hole and the observer, which for the Kerr solution

in general relativity are the mass and the angular momentum of the black hole as well as the

inclination angle of the observer. Modiﬁed theories of gravity or theories in which general

relativity is coupled to additional ﬁelds can produce a shadow that is modiﬁed with respect to

the Kerr shadow, possibly depending on additional parameters. This has motivated the study of

black hole shadows as a way of distinguishing Einstein gravity from its alternatives; see Ref. [6]

for a review of analytical studies of black hole shadows and Ref. [7] for a thorough testing of

alternative geometries against the EHT image of Sgr A*. There have been many publications

exploring the present and future possibilities for observing black hole shadows [8, 9], as well

as using them to constrain values of physical parameters [10] and to test alternative theories

of gravity [11]. Among the many other interesting works in the literature we can mention

∗e-mail: jbadia@iafe.uba.ar

†e-mail: eiroa@iafe.uba.ar

arXiv:2210.03081v2 [gr-qc] 16 Jun 2023

Refs. [12–14] concerning shadows in Einstein gravity and Refs. [15–19] in theories of modiﬁed

gravity.

One of these alternatives is the Einstein-Maxwell-dilaton (EMD) gravity, in which a scalar

ﬁeld ϕ(the dilaton) is coupled to the electromagnetic ﬁeld Fµν through a term exp(−2λϕ)F2in

the action, with λa coupling constant. When λ= 1 this theory arises as a low energy limit of

string theory, though here we consider a generalization of this limit, where the dilaton is allowed

an arbitrary coupling parameter. Due to the presence of the dilaton, charged black holes in string

theory do not approach the Reissner-Nordstr¨om solution of general relativity at low energies [20],

which in turn can lead to an observable diﬀerence between the shadows of charged black holes

for both theories [21]. The static black hole solution in this theory is well known [20, 22], and

its shadow is studied in Ref. [23]. However, ﬁnding a rotating solution has proven signiﬁcantly

more diﬃcult; closed form solutions are only known for λ= 0 (which is simply Einstein-Maxwell

theory) and λ=√3, corresponding to the Kaluza-Klein action [24,25]. Therefore, it is necessary

to turn to the Newman-Janis algorithm (NJA) [26] in order to generate rotating metrics from

static solutions—or rather, the so-called modiﬁed Newman-Janis algorithm [27], which removes

some of the ambiguity present in the original method. We follow Ref. [28], in which the modiﬁed

algorithm is used to obtain a rotating solution for arbitrary λ. It has been shown [29] that any

metric obtained through the modiﬁed NJA admits a separable Hamilton-Jacobi equation for

light rays, and thus allows for the analytic calculation of the black hole shadow.

It is expected that astrophysical black holes are surrounded by a plasma medium, and there

has been much interest in studying how the properties of the shadow change in the presence

of the plasma; see for example Refs. [30, 31]. It has also been shown that if the density of a

pressureless and nonmagnetized plasma obeys a certain condition [29], then the Hamilton-Jacobi

equation for light rays is still separable. The properties of the shadow in this case are chromatic,

since the eﬀect of the plasma on the propagation of light depends on the frequency. For low

enough frequencies, the black hole develops a “forbidden region” which light cannot penetrate,

leading to a dramatic decrease of the shadow size [32]. In this work, we arrive at an expression

for the contour of the shadows of the rotating black holes obtained from the static solutions of

EMD gravity through the modiﬁed NJA and surrounded by a plasma obeying the separability

condition. We then adopt a simple plasma model and we present the shadow and its geometric

properties for various values of the coupling parameter of the theory and the photon frequency,

as well as the angular momentum and charge of the black hole. The paper is organized as

follows: in Sec. 2, we brieﬂy present Einstein-Maxwell-dilaton theory and discuss its static and

rotating black hole solutions. In Sec. 3, we introduce the Hamilton-Jacobi equation for light

rays in a plasma and ﬁnd the contour of the black hole shadow, of which we show some examples

in Sec. 4. Finally, in Sec. 5 we conclude and discuss our results. Throughout this work we

adopt units such that G=c=ℏ= 1.

2 Einstein-Maxwell-dilaton gravity

We consider the theory deﬁned by the action [20,22]

S=Zd4x√−gR−2(∇ϕ)2−e−2λϕFµν Fµν ,(1)

where Ris the scalar curvature associated to the metric tensor gµν and λis an arbitrary

coupling parameter between the electromagnetic ﬁeld tensor Fµν and the dilaton ﬁeld ϕ. Note

that changing the sign of λis equivalent to changing the sign of ϕ, so we can take λ≥0 without

loss of generality. When λ= 0 the action reduces, up to an unimportant overall factor of

1/16π, to the usual Einstein-Maxwell action together with a minimally coupled scalar ﬁeld. As

mentioned before, this action is part of the low-energy limit of string theory when λ= 1. The

ﬁeld equations resulting from Eq. (1) read

∇µe−2λϕFµν = 0,(2)

∇2ϕ+λ

2e−2λϕFµν Fµν = 0,(3)

Rµν = 2∇µϕ∇νϕ+ 2e−2λϕ FµαFα

ν−1

4gµν FαβFαβ .(4)

2.1 Static solution

The static and spherically symmetric solution to EMD gravity with an arbitrary coupling pa-

rameter λhas the form [20, 22]

ds2=−f(r)dt2+dr2

f(r)+h(r)dΩ2,(5)

with

f(r) = 1−r1

r1−r2

r(1−λ2)/(1+λ2)(6)

and

h(r) = r21−r2

r2λ2/(1+λ2),(7)

where r1and r2are two parameters related to the mass Mand charge Qof the black hole by

M=r1

2+1−λ2

1 + λ2r2

2(8)

and

Q2=r1r2

1 + λ2.(9)

The dilaton and the Maxwell ﬁelds are given by

e2ϕ=1−r2

r2λ/(1+λ2)(10)

and

Ftr =Q

r2.(11)

Equations (8) and (9) can be inverted to give the radii r1and r2in terms of the mass and

charge:

r1=M+pM2−(1 −λ2)Q2(12)

r2=1 + λ2

1−λ2M−pM2−(1 −λ2)Q2; (13)

these equations are quadratic, and the signs have been chosen to give positive solutions. Note

that the radii are real, and thus the metric (5) is well deﬁned, only if (1 −λ2)Q2≤M2. This

condition is automatically satisﬁed if λ≥1, but if λ < 1 it places an upper limit

Q2≤1

1−λ2M2(14)

on the charge. If the above condition is met, the spacetime may still contain a naked singularity.

For λ= 0, the solution reduces to the Reissner-Nordstr¨om metric of general relativity, which

has a pair of horizons at r±=r1,2and a point singularity at r= 0. For any λ > 0 the horizons

are still located at r±=r1,2but the geometry at r=r2becomes singular, so we demand

that r1> r2in order to avoid a naked singularity [20]. In terms of the charge and mass, this

translates into the condition

Q2≤(1 + λ2)M2(15)

for an event horizon to exist.

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ShadowsofrotatingEinstein-Maxwell-dilatonblackholessurroundedbyaplasmaJavierBad´ıa1,2∗andErnestoF.Eiroa1†1InstitutodeAstronom´ıayF´ısicadelEspacio(IAFE,CONICET-UBA),CiudadUniversitaria,1428,BuenosAires,Argentina2DepartamentodeF´ısica,FacultaddeCienciasExactasyNaturales,UniversidaddeBuenosAires,Ciuda...

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Shadows of rotating Einstein-Maxwell-dilaton black holes surrounded by a plasma Javier Bad ıa12and Ernesto F. Eiroa1

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