Shadows of Einstein-dilaton-Gauss-Bonnet black holes

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Physics

Letters

B

www.elsevier.com/locate/physletb

Shadows

of

Einstein–dilaton–Gauss–Bonnet

black

holes

Pedro V.P. Cunha

a

,

b

,

∗

,

Carlos

A.R. Herdeiro

a

,

Burkhard Kleihaus

c

,

Jutta Kunz

c

,

Eugen Radu

a

a_{DepartamentodeFísicadaUniversidadedeAveiroandCentreforResearchandDevelopmentinMathematicsandApplications(CIDMA),CampusdeSantiago,}

3810-183Aveiro,Portugal

b_{CENTRA,DepartamentodeFísica,InstitutoSuperiorTécnico,UniversidadedeLisboa,AvenidaRoviscoPais 1,1049,Lisboa,Portugal} c_{InstituteofPhysics,UniversityofOldenburg,Oldenburg,26111,Germany}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received12January2017

Receivedinrevisedform11March2017 Accepted11March2017

Availableonline16March2017 Editor:M.Trodden

Westudytheshadowsofthefullynon-linear,asymptoticallyflatEinstein–dilaton–Gauss–Bonnet(EdGB) blackholes (BHs),forbothstaticand rotatingsolutions.We findthat,inall cases,theseshadowsare

smaller thanforcomparable KerrBHs,i.e. withthesametotalmassandangularmomentumundersimilar observation conditions.In order to compare both cases we provide quantitative shadow parameters, observinginparticularthatthedifferencesintheshadowsmeanradiiareneverlargerthanthepercent level.Therefore,generically,EdGBBHscannot beexcludedby(nearfuture)shadowobservationsalone.On thetheoreticalside,wefindnoclearsignatureofsomeexoticfeaturesofEdGBBHsonthecorresponding shadows,suchastheregionsofnegative(Komar,say)energydensityoutsidethehorizon.Wespeculate that thisis dueto thefact that the Komarenergy interior to thelight rings(or more precisely, the surfaces ofconstant radialcoordinate thatintersect the lightrings inthe equatorial plane)is always smaller thanthe ADMmass,and consequently the corresponding shadows are smallerthanthose of comparable Kerr BHs. The analysis herein provides a clear example that it is the light ring impact parameter,ratherthanits“size”,thatdeterminesaBHshadow.

©2017TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

UltraviolettheoreticalinconsistenciesofEinstein’sGeneral Rel-ativity,suchasitsnon-renormalizability[1–3]andtheexistenceof singularities,havesincelongmotivatedthesuggestionthathigher curvaturecorrectionsshouldbetakenintoaccount,inanimproved theory of gravity (see e.g. [4]). Inclusion of a finite set of such higher curvature corrections, however, generically leads to run-awaymodes(Ostrogradskyinstabilities[5]) intheclassicaltheory anda breakdownofunitarityduetoghosts, inthequantum the-ory.Theseundesirable propertiescan besimplydiagnosed,atthe level of the classic field equations, by the presence of third or-dertime(andconsequently alsospace,bycovariance)derivatives. A naturalwayaroundthisproblemis torequirea self-consistent model,obtainedasatruncationofthehighercurvatureexpansion, toyielda setoffieldequationswithoutsuchhigherorder deriva-tives.

*

Correspondingauthor.

E-mailaddresses:pintodacunha@tecnico.ulisboa.pt(P.V.P. Cunha),herdeiro@ua.pt (C.A.R. Herdeiro),b.kleihaus@uni-oldenburg.de(B. Kleihaus),

jutta.kunz@uni-oldenburg.de(J. Kunz),eugen.radu@ua.pt(E. Radu).

Lovelock[6]firstestablished,forvacuumgravity,whatare the allowed curvature combinations so that the field equations have nohigherthansecondordertimederivatives.Itturnsout that,in a Lagrangian, these combinations are simply the Euler densities, particularscalarpolynomialcombinationsofthecurvaturetensors oforder n.Sincethenth Eulerdensityisatopologicalinvariantin spacetime dimension D

=

2n and yields a non-dynamical contri-butiontotheactionindimensionsD



2n,animmediatecorollary is that,in D

=

4 vacuum gravity, the mostgeneralLovelock the-oryisa combinationofthe0th and1st Euler density,orinother words, General Relativity with a cosmological constant. The 2nd Eulerdensity,knownastheGauss–Bonnet (GB) combination,isa topologicalinvariant in D

=

4 anddoesnot contribute tothe dy-namicalequationsofmotionifincludedintheaction.

There is, however,a simpleand naturalway to make the GB combinationdynamicalina D

=

4 theory:coupleittoa dynami-cal scalarfield.Thisisactually amodelthatemerges naturally in stringtheory[7](seealso[8]foradiscussiononthispoint),where the scalarfield isthe dilaton,andcanbe considered asa simple effective model to investigate the consequences of higher curva-turecorrectionsin D

=

4 gravity.The corresponding modeltakes

http://dx.doi.org/10.1016/j.physletb.2017.03.020

0370-2693/©2017TheAuthors.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

manifestationofthiseffectiveexoticmatter isthattheBHsolutions haveregions of negative energydensityoutside thehorizon. An-othermanifestationisthatthereisaminimalmassforBHs, deter-minedby theGBcoupling.Weremarkthatthescalarhairofthis BHs hasno-independentconserved charge, thusbeingcalled sec-ondary.See,e.g.[13–17]forfurtherdiscussionsofthesespherically symmetricsolutionsandsomechargedgeneralizations.1

Rotating BHs in EdGB theory were found, fully non-linearly in [20,21] (see also [22–25] forperturbative studies). A minimal massdepending on theGBcouplingstill exists fortheserotating solutionsand, asa novelphysicalfeature,some (small)violations oftheKerrboundintermsofADMquantitiesareobserved.Again, regionswithnegativeenergydensityexistoutsidethehorizon.

Inthis paper,we shall investigatehow thedGB termimpacts ononeparticularobservable featureofa BH:itsshadow[26].BH shadowscan be roughly described asthe silhouetteproduced by the BH when placed in front of a bright background. They are determined by the BH absorption cross section forlight at high frequencies.Overthelastfewyearstherehasbeenarenewed the-oreticalinterestinthisoldconcept,firstdiscussedfortheKerrBH byBardeen[27],duetoobservationalattemptstomeasuretheBH shadow of the supermassive BHs in our galactic centre as well as that in the centre of M87 [28]. In particular, in [29–31], the shadowsofatypeofhairyBHsthatconnect continuouslytoKerr, withinGeneralRelativityandwithmatterobeyingallenergy con-ditions,calledKerrBHs withscalarhair [32–34],havebeen stud-ied. It has beenpointed out that, generically, theseshadows are smallerthanthoseofacomparableKerrBH,i.e. avacuumrotating BHwiththesametotalmassandangularmomentum.A possible interpretationofthisqualitativebehaviouristhefollowing:the to-talmass(andangularmomentum)ofthehairyBHsisnowpartly storedinthescalarfieldoutsidethehorizon;inparticularthe ex-istence of some energy outside the region of unstable spherical photon orbits,also referred to asphoton region (see section 3.1)

[35], impliesthat lessenergyexists inside thisregion andhence thelightringsshouldbe smaller(withinanappropriate measure) as compared to their vacuum counterparts and consequently so shouldbetheshadows.

Theaboveinterpretation raisesan interestingquestion in rela-tiontotheBHs inEdGBtheory.Sincethesehavenegative energy densitiesoutsidethehorizon,howdotheseregionsofeffective ex-oticmatter impacton theirshadows? Inparticularcould therebe a negative energy contribution outsidethe photon region that is sufficientlylargetoincreasetheshadowsizewithrespecttoa vac-uumcounterpart?Weremarkthatforothernon-vacuumsolutions withphysicalmatter, i.e. obeyingallenergyconditions,thesizeof

1 _{BHsolutionsofacloselyrelatedHorndeskimodelcanbefoundin[18,19].}

also provided there, together with the corresponding domain of existence and limiting cases. Then, in Section 3 we present the shadowsforarepresentativesampleofsolutionsandinterpretthe patternsobtained.WeclosewithadiscussioninSection4.

2. Themodelandsolutions

2.1. Thefieldequationsandgeneralresults

WeconsidertheEinstein–dilaton–Gauss–Bonnet(EdGB)model, describedbythefollowingaction2

S

=

1 16

π

ˆ

d4x

√

−

g



R

−

1 2

(∂

μ

φ)

2

₊

_α

_e−γφ_R2 GB



,

(1)

where

φ

isthedilatonfield,

α

isaparameterwithunits(length)2 and R2_GB

=

Rμνρσ Rμνρσ

−

4Rμν Rμν

+

R2 isthe GB combination. Also,

γ

is an input parameter of the theory,3 with most of the studiesassuming

γ

=

1.Both

γ

and

φ

aredimensionless.

Varyingtheaction(1)withrespectto gμν ,weobtain4_the

Ein-steinequations:

Gμν

=

Tμν(eff)

,

(2)

where Gμν is the standard Einstein tensor and the effective

energy-momentumtensorreads

T(μνeff)

≡

1 2



∇

μ

φ

∇

ν

φ

−

1 2gμν

(

∇φ)

2



−

α

e−γφTμν(GBd)

,

(3) wherethefullexpressionforTμν(GBd)canbefoundin[21].Varying theaction(1)withrespecttothedilatonfield,ontheotherhand, yieldsthescalarequationofmotion,whichreads:

2φ =

αγ

e−γφR2_GB

.

(4)

The EdGB model possesses BH and wormhole [49] solutions,

but no particle-like solitonicconfigurations are known (for a re-view, seethe recentwork [50]), although the couplingto matter leads,e.g.,toneutronstars[51,52].NotethatincontrasttotheGR case, allEdGBsolutions(with

α

=

0)havebeenobtained numeri-cally.

In terms of the spherical-type coordinates r, θ and

ϕ

, all knownEdGBsolutionspossessatleasttwoKillingvectors

ξ

= ∂/∂

t and

ζ

= ∂/∂

ϕ

(where t is the time coordinate). Then a generic metricansatzcanbewrittenas

2 _{Inthisworkweshallusegeometricunitsc=G=1.}

3 _{Sincethesystempossessesthesymmetryγ→ −γ,φ→ −φ,itisenoughto} considerstrictlypositivevaluesofγ.Furthermore,inordertohaveanon-trivial couplingtothedilatonfield,γ=0.

Fig. 1. (Left)DomainofexistenceofstaticEdGBBHsinaD/M vs.α/M2_{diagramwithseveralvaluesofγ.Thepointsa andb depictthelimitingandcriticalsolutions} respectivelyforγ=10.(Right)Domainofexistenceofspinningsolutionswithγ=1.Thesetofconsidered(spinning)solutionsinFig. 3andFig. 4areshownhereas highlightedpoints.

ds2

=

grrdr2

+

gθ θd

θ

2

+

gϕϕd

ϕ

2

+

2gϕtd

ϕ

dt

+

gttdt2

,

(5)

where gμν andthescalar

φ

are functionsof

(r,

θ ).

Moreover, we canset

φ (

∞) =

0 withoutanylossofgenerality(anyotherchoice would correspond to a rescaling of the radial coordinate in (5) [21]).TheADM(Arnowitt–Deser–Misner)massM andangular mo-mentum J arereadoff,asusual,fromtheasymptoticexpansion

gtt

= −

1

+

2M r

+ . . . ,

gϕt

= −

2 J r sin 2

_θ

_{+ . . . .}

₍₆₎

One can also define a global dilaton measure D from the

asymptotic expansion of the scalar field,

φ

= −

D/r

+ . . .

which however is not an independent quantity, since the dilaton field doesnotqualifyasprimaryhair[8,21].

2.2.ThestaticEdGBblackholes

Considerforthe momentthe static,spherically symmetric so-lutions( J

=

0).Closetotheeventhorizon,thesesolutionspossess anapproximateexpressionasapowerseriesinr

−

rH,whererH is

theradialcoordinateofthehorizon.Inparticular,inSchwarzschild coordinatesonefinds

φ (r)

= φ

H

+ φ

1

(r

−

rH

)

+ . . .

,where

φ

1 sat-isfiesa quadraticequation (seee.g. [8,13,14,21]). Since thescalar fieldisreal,thediscriminantofthequadraticequation isrequired tobepositive,yieldingthecondition:

1

−

96

α

2

γ

2 e

−2γφH

A2_H

/(16

π

2

₎

≥

0, (7)

whereAH istheeventhorizonarea.Eventually,thisconditionwill

be violated aftersome limitingsolution is reached, beyondwhich solutionsceasetoexistintheparameterspace.Foragiven

γ

,all solutions can be obtained continuously in the parameter space. When appropriately scaled they form a line, starting from the smooth GR limit (φ

→

0 as

α

→

0), and ending at the limiting solution.Theexistence ofthe latterplaces alower bound onthe BHhorizonradius.Itactuallyalsoimpliestheexistenceofalower bound on the BH mass. In particular, as discussed in [8,22], the static EdGB solutions with

γ

=

1 are limited to the parameter range 0

≤

α

/M

2



0.1728. A rather similar behaviour holds5 for

γ

=

1.

Solutions no longer exist if the ratio

α

/M

2 _{is larger than a} criticalvalue, whichdecreases withincreasing

γ

.The configura-tion at this maximal value is dubbed the criticalsolution, which

5 _{Notethatsolutionsseemtoexistforanynonzerovalueofγ.}

needs nottocoincidewiththelimitingsolution.Inparticular,for large enough

γ

, the solutionline can be extended backwards in

α

/M

2_{, into a “secondary branch”, after the critical configuration} is reached [48]; this secondary branch eventually terminates at the limiting solution. Some of these features can be seen in an

(

α

,

D)-diagram of solutions with different

γ

, as shownin Fig. 1

(left).Inparticular,noticehow forsufficientlylarge

γ

valuesitis possibletohavetwodifferentvaluesof D/M forthesame

α

/M

2, whichindicates thepresenceoftwobranches. Accordingto argu-mentsfromcatastrophe theory,thestability shouldchangeatthe criticalsolution,sothatthesolutions alongthesecondary branch willbeunstable[13].

2.3. ThespinningEdGBblackholes

Spherically symmetric BHs typically possessspinning general-izations.However,sofaronlythe

γ

=

1 casehasbeenexploredin theliterature.TheseBHswerefirstobtainedatthefullynon-linear levelin[20] (seealso[22–25] forperturbative results).Similar to theGRcase,theseBHspossessa

Z

2symmetryalongtheequatorial plane(θ

=

π

/2)

andareobtainedbysolvingthefieldequations(2)

and (4) subject to appropriate boundary conditions that are de-tailedin[21].

ThedomainofexistenceofEdGBBHsisboundedbyfoursetsof solutions:i)thesetofstatic(i.e. sphericallysymmetric)EdGBBHs with J

=

0; ii) the set ofextremal (i.e., zero temperature) EdGB BHs;iii)thesetofcriticalsolutions;andiv)thesetofGRsolutions –theKerr/Schwarzschild BHswith

α

=

0.InFig. 3andFig. 4the boundarylinedisplayedincludesthesetsii)andiii).

Thegeneralcriticalsolutions aretherotatinggeneralization of thestaticcase,whiletheextremalsetdoesnotappeartobe regu-laronthehorizon.6 Moreover,themassoftheEdGBrotatingBHs

is always bounded frombelow, whereas the angularmomentum

can (slightly) exceed the Kerrbound, which is givenby J



M2. Furtherdetailsontheseaspectstogetherwithvariousplotsofthe domainofexistencearefoundin[21].Herewegivethedomainof existencein

(

α

,

D

)-variables

[Fig. 1(right)]andin

(

α

,

J)-variables (Fig. 3).

6 _{Perturbingtheextremalvacuumsolutioninα,thescalarfield/metricdevelops} singularitiesatthepolesinfirst/secondorderinα.

shadow edge in observations restricted to the equatorial plane’s line of sight (providedboth exist). Following [31], the light ring positionscan beobtainedbyanalysingthe followingconditionin theequatorialplane:

∂

rh±

=

0, with h_±

=

−

gtϕ

±



g_t2_ϕ

−

gttgϕϕ gtt

.

(8)

Recalling the Kerr case, each sign

±

leads to one of the two light rings. Curiously, although the EdGB BHs discussed in this paperare fullynon-linear solutions (rather thanperturbations of Kerr), the light ring qualitative structure still appears to be the same as in Kerr. However, notice that for other families of so-lutions this is not always the case. For instance, multiple light ringscan appearforBHs withscalarhair,someofwhichare sta-ble[31].

3.2. Characterizingtheshadow

Assuming that a suitable light source is present to provide contrast, a BH casts a black region in an observer’s sky, com-monly called the BH shadow. Although some characteristics are observer dependent [30], the size and shape of the shadow are essentially a manifestation of the spacetime properties close to the BH, depending for instance on the light ring characteristics. Consequently,instructivephysicscanbeinferredfromsuch obser-vations.

ConsiderthedummyshadowinFig. 2,representedintheimage plane ofthe observer. ACartesian parametrization

(x,

y) is used, wherethex-axisisdefinedtobeparalleltotheazimuthal Killing vector

ζ

= ∂/∂

ϕ

attheobserver’sposition.Theorigin

(0,

0)ofthis coordinatesystem,definedaspoint O inFig. 2,correspondstothe directionpointingtowardsthe centre oftheBH

−∂/∂

r (fromthe readerintothepaper).

ThepointC inthefigure,takentobethecentre oftheshadow, is such that its abscissa is givenby xC

= (

xmax

+

xmin

)/2,

where xmaxandxminarerespectivelythemaximumandminimum abscis-saeoftheshadow’sedge.Iftheobserverisintheequatorialplane (θ

=

π

/2),

whichwillbeassumedthroughoutthepaper,thenthe shadowinheritsalongthex-axisthespacetimereflection symme-try,givingyC

=

0.SincethepointsC andO neednottocoincide,

aspecificfeatureofashadowisthedisplacementxC betweenthe

shadowandthecentre oftheimageplane O .

A generic point P on the shadow’s edge is at a distance r from C ,whichisdefinedasr

≡



yP2

+ (

xP

−

xC

)

2.Giventheline

elementds2

₌

_dx2

₊

_dy2_{, theperimeter}

_P

_{ofthe shadow,its} av-erage radius

¯

r andthe deviation from sphericity

σ

r are defined

by:

˛

ds

≡

P

,

¯

r

≡

1

P

˛

rds

,

σ

r

=



1

P

˛ 

1

−

r

¯

r



2 ds

.

(9)

Fig. 2. Representation of a BH shadow in the(x,y)image plane of the observer. AlltheseparametersareexpressedinunitsoftheADMmass M. Insomecases,itispossibletocomparetheshadowparameters ofa givenEdGBsolutionwiththe onesfromaKerrBH withthe sameADMmass M andangularmomentum J .Hence,let usalso definetherelativedeviationstotheKerrcase7_:

δ

r

=

¯

r

− ¯

rkerr

¯

rkerr

,

δ

σ

=

σ

r

−

σ

kerr

σ

kerr

,

δ

xC

=

xC

−

xC kerr xC kerr

.

(10) 3.3. RotatingEdGBBHs

Due to the existence of a hidden constant of motion – the Carter constant – the edge of the Kerrshadow can be obtained inaclosedanalyticalform[27,35,53].However,EdGBBHsarenot expected to havesuch a property, since they all appearto be of Petrov type I[21].Thisisconsistent withtheperturbative results in[24].Asaconsequence,ingeneraltheshadowofthelatterhas to be obtained numerically through the standard backwards ray-tracing framework [45,54]. In order to generate a virtual image of the shadow, this method requires propagating null geodesics “backwardsintime”,whereahighfrequencyapproximationis as-sumed, startingfromtheobserver’s positionanddeterminingthe source ofeach light ray. Different points inthe image plane cor-respondtodifferentdirectionsintheobserver’ssky,andhenceto differentinitial conditionsofthe geodesic equations.Theshadow is precisely the set of all those initial conditions which induce geodesicswithendpointson theeventhorizon,whenpropagated backwardsintime.Sincetheeventhorizonisnot asourceofany light (classically), the shadow actually embodies a lack of radia-tion.8

Thegeodesicpropagationmethoddescribedaboveisnecessary tocomputemostoftheshadowedge.However,thepointsx1 and x2 in Fig. 2, where the edge intersects the x-axis, can be com-puted using a highly precise local method. In particular, for an observerintheequatorialplane,lightringsaretheorbits respon-siblefortheseintersectionpoints.Theimpact parameter

η

=

L/E will play herea crucial role, where E and L are respectively the photon’s energy andaxial angularmomentum withrespect to a staticobserveratinfinity.Moreover,thesequantitiesareconstants ofgeodesic motion,connectedtotheKillingvectorsofthe space-time

ξ

= ∂/∂

t and

ζ

= ∂/∂

ϕ

.Thefunctionh_±willnowbehelpful again,asthevalueof

η

inagivenlightringorbitisprovided sim-plyby

η

=

h_±,computedatthatposition[31].

7 _{AnanalyticalexpressionfortheKerrshadow,asseenbyanobserverwithzero} angularmomentum(ZAMO),canbefoundin[53].

Fig. 3. Representationof(¯r−4.68M)(left)andδr(right)forEdGBsolutionswithγ=1,inaα/M2vs. J/M2 diagram.Eachcircleradiusisproportionaltothequantity

represented,withsomevaluesalsoincludedforreference.Allthevaluesofδrarenegative,withthemaximumdeviationtoKerrbeingaround −1.5%.

Fig. 4. (Left)Representationof|δσ|forEdGBsolutionswithγ=1,inaα/M2vs.J/M2diagram.Eachcircleradiusisproportionaltothequantityrepresented,withsome valuesalsoincludedforreference.Allthevaluesofδσ arenegative.(Right)DepictionoftheshadowedgeofaEdGBBHwithγ=1 and(α/M2,J/M2) (0.172,0.41), yielding¯r4.85,σ=0.3,xC=0.84;theradialdeviationδrwithrespecttothecomparableKerrcaseis −1.35%.Theobserverisataperimetralradius15M.

The precise relation betweenthe image coordinate x and the impact parameter

η

depends on the choice for the observer’s frame, but also on how x is constructed in terms of observa-tion angles. Following [31,53], the x coordinate is definedto be directly proportional to an observation angle

β

along that axis: x

= − ˜

R

β

, where the perimetral radius R

˜

≡ √

gϕϕ is computed at the observer’s position. By computing the projection of the photon’s 4-momentum onto a ZAMO frame [31,53], the relation sin

β

=

η

/(A

0

+

η

B0

)

can be derived (if y

=

0), where the fol-lowing quantities are computed at the position of the observer: A0

=

gϕϕ/

√

D, B0

=

gtϕ

/

√

D,with D

≡

g2 tϕ

−

gttgϕϕ .Thisleads

totherelation(with y

=

0):

x

= − ˜

R arcsin



η

A0

+

η

B0



.

(11)

Forthe sake of the argument,consider alsoa very faraway ob-server (r

→ ∞

). In these conditions we obtain the very simple relationx

= −

η

.Bycomputing

η

1 and

η

2 foreachofthetwo light

rings, we can obtain the shadow radius r

¯

x on the x-axis simply

with r

¯

x

= |

x1

−

x2

|/

2, where each x is evaluated from the re-spective

η

. Notice that this is a local method,in the sense that it does not require the evolution of a geodesic throughout the spacetime.Hence, obtaining avery preciser

¯

x value onlydepends

on knowing

η

at the light rings with sufficiently high accuracy. Furthermore, by comparing this

¯

rx value with the one obtained

withray-tracing,wecanestimatethattheprecisionofthelatteris around

∼

0.08%.

The data ofthe EdGBshadows, computedwithray-tracing, is representedinFig. 3andFig. 4,whereadilatoncoupling

γ

=

1 is assumed.Theobserverisalwaysplacedintheequatorialplane,at aradialcoordinatesuchthatR

˜

= √

gϕϕ

=

15M.

Intheleft ofFig. 3,thesizeofeachcirclerepresentsthevalue oftheshadowradius

¯

r forseveralEdGBsolutions.Inordertomake thedifferencesacrossthesolutionspacemoreapparent,thecircle radius is proportional to

¯

r

−

4.68M. In other words, a vanishing circle (in this plot only) represents

¯

r

=

4.68M. With this depic-tion, it is clear that – as a rule of thumb – increasing either J

arerepresentedintheleftofFig. 4.Curiously,allvaluesof

δ

σ are negative,whichmeansthatEdGBshadowsaremore“circular”than thecorrespondingKerrcase.Hence,theGBtermappearstosoften the spin deformationsthat exist on theKerr shadows.Moreover, noticehowthelargest

|δ

σ

|

valuescanbefoundclosetothecritical boundaryinsolutionspace.Additionally,thedeviations

δ

xC canbe

bothpositiveandnegative,althoughaplotforthisquantityisnot shown.

In order to display an illustrative shadow case, in the right of Fig. 4 we have the representation of a EdGB shadowedge in theimageplane,togetherwiththecomparableKerrone.Although thedifference betweenthecurves isbarelyvisible, amountingto a variation of only

 −

1.35% in the shadow size, the case here depictedhas one ofthe largestvalues of

|δ

r

|

for

γ

=

1. Such an

examplereinforcestheideathatshadowobservationsarevery un-likelytoconstrainEdGBBHmodelsinthenearfuture.

3.4. StaticEdGBBHs

Untilthispoint we discussedonly theshadowsofEdGB solu-tionsfordilatoniccoupling

γ

=

1.Repeatingtheaboveanalysisfor other valuesof

γ

wouldbe rather cumbersome. Nevertheless,as discussedintheprevioussubsection,some ofthelargestr devia-

¯

tionsoccurwithinthestaticcase.Thereforethiscanbeconsidered asanincentivetoexploreothervaluesof

γ

,whilerestricting our-selvesto J

=

0.Thiswillprovidesomeinsightontheeffectofthe

γ

parameterwithoutmuchmoreeffort.

For the staticcase ( J

=

0) the shadowis a circledue to the sphericalsymmetryofthespacetime.Usingthisproperty,wehave

¯

r

= ¯

rx,whichallowsustousethehighprecisionmethoddescribed

before,thus obtaining the shadowedge withouthaving to resort toanyray-tracing.Noticethatinthiscase

σ

r andxC arebothzero

duetothesphericalsymmetry.

Theradialdeviations

δ

rofstaticEdGBshadowswithrespectto

thoseofacomparableSchwarzschildBHarerepresentedinFig. 5, fordifferent

γ

values. The data suggestsa scenario where fora fixed value of

α

/M

2 _{the deviations on the stable branches are} largerifweincrease

γ

;however,afterenteringthedomainofthe secondary(unstable)branches,

γ

hastodecreaseinordertoyield largerdeviations.Furthermore,foragiven

γ

,themaximum devi-ation always appears to occur at the limitingsolution, with this maximaldeviationbeinglargerforsmaller

γ

values.Forinstance,

γ

=

0.5 canleadtoshadows



2% smallerthanforSchwarzschild, whereasfor

γ

=

1 alldeviationsarebelow 1.5%.

9 _{ForagivenBHunderobservation,thequantitiesM,J andR are˜ allassumedto} beknown.

10 _{TheshadowsarecomparableifM, J andtheobservationdistance R}˜_{= √g} ϕϕ arethesame.

11 _{AdditionalmeasuresofEdGBshadowshapesarepossible,buttheyresemble} closelyKerrones.

Fig. 5. Representation ofδr for staticEdGBBHs, computed with respectto the

Schwarzschildcase.Datafordifferentγvaluesisdisplayedasafunctionofα/M2_. All deviations are negative. The displayed linesonly interpolate the numerical data, with colours red, green, blue, pink and light blue respectively for γ = {0.5,1,2,5,10}.Theobserver’sperimetralradiuswassetat 15M.(For interpreta-tionofthereferencestocolourinthisfigurelegend,thereaderisreferredtothe webversionofthisarticle.)

4. Discussion

The shadow of a EdGB BH is always smaller than the

com-parable Kerr one. However, the deviations observed are always smaller(inmodulus)thanafewpercent(

∼

1%).Sincesuch differ-ences are belowthe expectedresolution ofplanned observations (

∼

6% asanticipatedin[55]),itisunlikely thatinthenearfuture

any shadow measurement can exclude or restrict EdGB models.

Nevertheless, thepresentstudywas not exhaustive; itleaves, for instance,studies fordifferentinclinationsanddistances asfuture work.

Since EdGBtheory possessesunusualfeatures such aseffective exoticmatter, it might come as a surprise that there are no sig-nificant effectsatthe levelof theshadow.However, thiseffective exoticmatter isconcentratedclosetothehorizon, suchthat there isnonegative energycontributionoutsidethephotonregion that could significantlyaffecttheshadow’ssize.Atthe sametimeany near-horizonoddeffectsareconcealedfromaremoteobserverby theshadow.

It may comeas anothersurprise, that the light ring size12 of EdGBBHscan,forinstance,changebyasmuchas



4%,when con-sideringthestaticcasewith

γ

=

0.5,andthiseffectwillincrease withfurther decreasing

γ

.The naturalquestionis then: whyare the deviationsintheshadowsize notlarger?Forthesake ofthe argumentconsiderthestaticcase,whereitbecomesclearthatthe criticalingredientfortheshadowradiusistheimpactparameter

η

, andnotthelight ringsize.Naturally,thereisa strongcorrelation betweenbothconcepts,butattheendofthedaywhatmattersis thevalueoftheimpactparameter.Wewouldliketopointoutthat thisobservationisoftennotclearenoughintheliterature:alarge variation of the light ring size does not have to lead to equally largevariationsoftheshadowradius.

12 _{Theperimetralradius}√_g

ϕϕinM unitscanbeusedasaninvariantmeasurefor thelightringsize.

Acknowledgements

P.C. is supported by Grant No. PD/BD/114071/2015 under the

FCT–IDPASC Portugal Ph.D. program and by the Calouste

Gul-benkian Foundation under the Stimulus for Research Program

2015. C.H. and E.R. acknowledge funding from the FCT–IF

pro-gramme. This project has received funding from the European

Union’s Horizon 2020 research and innovation programme

un-derthe MarieSklodowska-Curie grantagreement No 690904and by the CIDMA project UID/MAT/04106/2013. B.K. and J.K. grate-fullyacknowledge discussions withArne Grenzebach, Claus Läm-merzahlandVolkerPerlick,aswellassupportbytheDFGResearch Training Group 1620 “Models of Gravity” and by the grant FP7, MarieCurieActions,People,InternationalResearchStaffExchange Scheme(IRSES-606096).

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