Light paths of normal and phantom Einstein-Maxwell-dilaton black holes

0.3 MB·English

by  Mustapha Azreg-Aïnou

Light paths of normal and phantom Einstein-Maxwell-dilaton black
holes
MustaphaAzreg-A¨ınou
2
1 Bas¸kentUniversity,Department ofMathematics,Bag˘lıcaCampus, Ankara, Turkey
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2

p
e Abstract
S
NullgeodesicsofnormalandphantomEinstein-Maxwell-dilatonblackholesaredeterminedanalytically
4
2 bytheWeierstrassellipticfunctions.Theblackholeparametersotherthanthemassenter,withtheappropri-

atesigns,theformulafortheangleofdeflectiontothesecondorderintheinverseoftheimpactparameter
]
c allowing for the identification of the nature of matter (phantom or normal). Observation of the so called
q
- relativisticimagesonphotonspheresalsoallowsforsuchadetermination.Scatteringexperiencesmayfavor
r
g blackholesofEinstein-anti-Maxwell-dilatonictheoryfortheirhighrelativediscrepancywithrespecttothe
[
Schwarzschild value. For the cases we restrict ourselvesto, phantomblack holesare characterizedby the

1 absenceofmany-worldandtwo-worldnullgeodesics.
v
2
PACSnumbers:04.20.Gz,04.50.Gh,04.70.–s
3
2
5
.
9 1 Introduction
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2
1 Most experimental settings for testing gravitational theories are designed to evaluate trajectories of light rays.
:
v Accuracy in this field is a growing interest. From this point of view, the leading experimental settings are
i
X aiming to achieve high accuracies beyond the known first order level and to reach a sensitivity of 1 part in 109
r inmeasuring theEddington parameterg [1],whichisamimportant parameterinpost-Newtonian formalism.
a
On the theoretical front, workers have been striving to evaluate exactly light paths using (hyper)-elliptic
functions mainly the Weierstrass elliptic functions denoted byˆ [2,3]. From the one hand, this has provided
answerstosomeopenquestions,forinstance,whetherthecosmologicalconstantcouldbeacauseofthePioneer
anomaly [4], has raised the question to whether lensing could be used as a test of the Cosmic Censorship [5]
and has lead to discover new light paths, the Pascal Limac¸on trajectories for black holes with cosmological
constant [6]. From the other hand, the analytical solutions derived so far, [7–23] to mention but a few, could
be useful for any of the experimental settings aiming to test gravitational theories. Moreover, they provide
new academic techniques for tackling motion of massive and massless particles in the geometries of various
gravitational fieldsandmayserveasreferences fortestingtheaccuracy ofnumerical methods[24]andprovide
uniquebenchmarksfortestingandimprovingperturbationanddecomposition methods[25]. Forthatpurposeit
isveryhelpfultohaverelativelysimplesolutions.
1
Incaseofspherical symmetry,oneoftheequations governinggeodesic motionreduces to
dr 2
=P(r) (1.1)
df
(cid:16) (cid:17)
where P(r) is a polynomial function of the radial variable r, the parameters of the solution and the constants
of motion. Depending on the dimension of the space-time, P(r) may be reduced, as described in [16], to a
polynomial ofdegree3or5. Weareinterested intheformercaseandweassumethat(1.1)isbrought to
dr 2
=4r 3 g r g (1.2)
dQ − 2 − 3
(cid:16) (cid:17)
bycoordinatetransformations. Hereg ,g dependontheparametersofthesolutionandtheconstantsofmotion.
2 3
So far no special terminology has been introduced to simply notations and expressions. We introduce the fol-
lowingterminologytodescribe(1.2)andtherelatedpolynomialandcoordinates. Weshallcall(1.2)Weierstrass
differential equation1,w(r )=4r 3 g r g Weierstrasspolynomial and(r ,Q )Weierstrass coordinates.
2 3
− −
Webring(1.1)to(1.2)byaseriesofcoordinatetransformationsrelatingrtoWeierstrassradialcoordinater
wherer (r)isanon-trivial andnonlinear transformation; however,Q (f )isalineartransformation andinmany
casesQ =f ,wheref istheazimuthalangle.
Most workers prefer to use the effective potential approach by which they determine all planar trajectories
(absorbed paths (captured photons), scattering paths, trapped or confined paths, (un)stable circular paths, spi-
ral paths approaching the circular paths from above and/or below and some other special closed curves). The
method we shall apply is entirely based on the properties of Weierstrass differential equation and of its poly-
nomial. We shall develop and use this method, which has been used in [9,19] (and partially used in [6,8]),
leading toasystematic approach forall problems governed by (1.2). Thiswillallowsus todetermine alltypes
oftrajectories.
None of the papers mentioned above has ever dealt with light paths of normal black holes of Einstein-
Maxwell-dilaton(EMD).Oneofthepurposesofthispaperistoaddressthisquestion;theotheroneistoextend
theanalysistothatoflightpathsofphantomblackholesofEMDandtodrawacomparisonbetweenthesystems
oftrajectories foragivenratioofchargetomasssquared(a2 =q2/M2).
In a phantom gravitating field theory one or more of the matter fields appear in the action with an unusual
signofthekinetic term,sothattheyarecoupled repulsively togravity. Inthecaseof“phantom EMD”theory,
which is also a short term for the theory, we may have a number of ways the matter fields are coupled to
gravity: Einstein-anti-Maxwell-anti-dilaton, Einstein-Maxwell-anti-dilaton, Einstein-Maxwell-dilaton and so
on. Thepresenceofphantomfieldscontinuestoreceivesupportfrombothcollectedobservationaldata[26]and
theoretical models[27].
Thestatic,sphericallysymmetricblackholesolutionstoEMDtheorywithphantomMaxwelland/ordilaton
field were derived and their causal structure was analyzed, among which one finds nine classes of asymptoti-
callyflatandtwoclassesofnon-asymptotically flatphantom blackholes[28]. Inasubsequent work[29],these
solutionshavebeengeneralizedtomulticentersolutionsofphantomEMD.Recently,theirthermodynamicprop-
1ThisisjustifiedsincethedifferentialequationsatisfiedbyWeierstrassellipticfunctionsˆ (Q )isjust(1.2)uponreplacingr byˆ .
2
erties and stability were investigated too [30]. One of the remaining tasks is to investigate their null geodesics
to see how phantom fields may affect the light paths, particularly the angles of deflection, the photon spheres
andrelatedlensingeffects. DeviationsoftheangleofdeflectionfromtheSchwarzschildvaluearegenerallydue
to extensions in the theory (inclusion of Maxwell fields or and scalar ones, cosmological constant and so on),
departure from spherical symmetry ormotion ofthesolution itself (mostly rotation). Inthis paper weexamine
thecaseduetotheinclusion of(anti)-dilatonic and/or(anti)-Maxwell fields.
In Section 2 we consider the cosh and sinh black hole solutions of the generalized phantom EMD, which
dependonthreeparameters(M,q,g ). Weevaluate,anddiscuss,theangleofdeflectiondf tothesecondorderof
approximationintheinverseoftheimpactparameterasfunctionoftheblackholethreeparameters. Figures,re-
lyingonexactformulas,depictingdf forphantomandnormalblackholesareplottedagainsttheSchwarzschild
angleofdeflectionfordifferent valuesoftheparameters. Therelativediscrepancy isalsodiscussed andplotted
showinghighvaluesfromsomesetoftheparameters.
InSection3weintroducetheWeierstrassellipticfunctionsanduseanddevelopthemethodbasedonWeier-
strass polynomial to determine exactly all kind of null geodesics to any spherically symmetric geometry pro-
vided the equation of (planar) motion of light rays may be brought to (1.2). Applications are considered in
Sections 4 and 5. In the former section we consider the case g =1 and show that the problem of determining
the null geodesics ofnormal Reissner-Nordstro¨m black holes by the method based onWeierstrass polynomial,
which was initiated in [19], is tractable analytically and extend the analysis to phantom Reissner-Nordstro¨m
black holes upon applying the results of Section 3, and in the latter one we consider the case g =0 and deter-
mine all the null geodesics of the phantom cosh and normal sinh EMDblack holes by mere comparison to the
work done in Section 3. In Sections 3 to 5, we do not aim to go into the details of each null geodesic motion;
rather,wepresentageneralprocedure(Section3)bywhichwediscusssometypeofnullgeodesicmotions,the
natureofexistingdivergencies andpresentexactreferenceandstandardformulasforspecificgeodesicsandfor
theangleofdeflection. ThepaperendswithaConclusion SectionandanAppendixone.
2 The deflection angle of light paths in the cosh-sinh solutions of EMD
TheactionforEMDtheorywithphantomMaxwelland/ordilaton fieldreads
S= d4x√ g [R 2h 1gmn ¶ m j¶ n j +h 2e2lj Fmn Fmn ], (2.1)
− − −
Z
where l is the real dilaton-Maxwell coupling constant, and h = 1, h = 1. Normal EMD corresponds to
1 2
± ±
h =h =+1, while phantom couplings of the dilaton field j or/and Maxwell field F =dA are obtained for
2 1
h = 1or/andh = 1.
1 2
− −
3
Themetricsofthesocalledcoshandsinhsolutions, derivedin[28],taketheform
ds2= f+f−g dt2− f+−1f−−g dr2−r2f−1−g dW 2 (2.2)
F =−rq2dr∧dt, e−2lj = f−1−g , f±=1−rr±, g = 11−+hh 11ll 22
h (1+h l 2)<0forcosh, h (1+h l 2)>0forsinh (2.3)
2 1 2 1
g ( ¥ , 1) [1,+¥ )ifh = 1 (2.4)
1
∈ − − ∪ −
g ( 1,+1]ifh =+1 (2.5)
1
∈ −
wherewehaveintroduced theparameterg followingthenotation of[30]2.
These are asymptotically flat spherically symmetric black holes of mass M, electric charge q and event
horizonr >0relatedby[28]
+
r +g r
+
M= − (2.6)
2
1+g
q= h r r (2.7)
2 +
±r 2 −
wherewehavesubstituted,intotheoriginalformulaofq,1+h l 2=2/(1+g ). Sinceqisreal,r andh (1+g )
1 2
−
musthave thesamesign. Usingthis fact in(2.3),wehave r <0for thecosh solution and r >0forthe sinh
− −
one.
Asshowninsubsection4.1case2. (d)of[28],r=0correspondstoasingularityforthecoshsolutionwhere
geodesics terminate(aPenrose diagram isgiveninfigure1of[28]). Similarly, insubsection 4.3case1. (a)(ii)
of [28] it is established that, for generic values of 1+h l 2 as this is the case for g =0 (to which we restrict
1
ourselves inSection 5),r=r isanullsingularity forthesinhsolution (aPenrose diagram isgiveninfigure3
−
of[28]). Thecurvature scalarof(2.2)divergesatthesetwopointsforg =0
r (r r )
R = − − + .
−2r3(r r )2
− −
Expressing (r ,r )intermsofM anda2=q2/M2,oneobtains
+
−
r =2M, r =h Ma2, ifg =0 (2.8)
+ 2
−
M M 2h M2a2 2h g a2
r =M+M, r = − = 2 , M =M 1 2 , g > 1. (2.9)
+ − g (1+g )r+ s − (1+g ) ∀ −
[Thelimitg 0in(2.9)leadsto(2.8)].
→
2Forthesinhsolutionthecaseh 2(1+h 1l 2)<0,whichwouldleadtor <0,isnotpossible[28].
−
4
RelatedtothetwoKillingvectors(¶ t,¶ f )arethetwoconstants ofmotion(E,L)givenby
g dt
f f =E (2.10)
+ dt
−
r2sin2q f1−g ddft =L. (2.11)
−
Since(2.2)isendowed withspherical symmetry, the motion happens inaplane through the origin. Letting the
planebeq =p /2andinserting (2.10,2.11)intothelineelement(2.2):
f+f−g ddtt 2− f+−1f−−g ddtr 2−r2f−1−g ddft 2=e
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
(withe =1,0formassive,masslessparticles, respectively) webringitto
dr 2=E2 f fg e + L2 . (2.12)
(cid:18)dt (cid:19) − + −(cid:20) r2f1−g (cid:21)
−
Forscattering statesE2 e >0. Eliminatingt in(2.11,2.12)wearriveat
−
(cid:18)ddfr(cid:19)2= r4f−L2(21−g)"E2− f+f−g (cid:20)e +r2Lf12−g (cid:21)#
−
whichwere-writeintermsofu=1/r as
(cid:18)ddfu(cid:19)2= f−2(L12−g)"E2− f+f−g (cid:20)e +Lf12−u2g (cid:21)#. (2.13)
−
Fromnowonwetakee =0sothattheconditionforlightscatteringisE2>0. Now,letg(u)bethefunction
g(u):=L2u2f+f2g−1
−
andu =1/r bethepointonthelightscatteringgeodesicnearesttheoriginwhere du(u )=0. SinceE2=g(u ),
n n df n n
theangleofdeflection, whichistwicethevariation off minusp ,takestheform
un du
df =2L p . (2.14)
Z0 f1−g (u) g(un) g(u)−
− −
p
Settingu=u x,webring(2.14)totheform
n
1 Lu √1 x2 dx
df =2 n − p . (2.15)
Z0 f−1−g (u) g(un)−g(u) √1−x2 −
p
5
Withu 1forscattering withlargevaluesoftheimpactparameter(b=L/E),weobtain
n
≪
Lu √1 x2 [(g 1)r +2M] 1 x3
f−1−g (u)n g(u−n)−g(u) =1+(cid:20) − 2− 1−−x2 −(g −1)r−x(cid:21)un
1
p + 3r2(1+x+x2)2+2r r [2g 1+2g x+(6g 1)x2+2x3+x4]
8(1+x)2 + − + − −
(cid:26)
+r2[4g 2 1+6(2g 1)x+(8g 2+1)x2+6x3+3x4] u2+O[u ]3
n n
− − −
(cid:27)
wherewehaveused(2.6): r +g r =2M. Performingtheintegrations overxweobtain
+
−
4M p 1
df = + 2M[r +r (2g 1)]+ [15r2 +6r r (4g 1)+r2(16g 2 1)] +O[1/r ]3. (2.16)
rn − + − − 16 + − + − − − rn2 n
(cid:8) (cid:9)
Since the values of g depend on h according to (2.4, 2.5) and the sign of r is that of h (1+g ) by (2.7),
1 2
−
the deflection angle depends on the type of EMD under investigation. From (2.8, 2.9) one sees that, for both
cases g =0 and g =0, the limit case q=0 corresponds to r =0 and r =2M. Thus in the limit q 0, df
+
6 − →
approaches thevaluedf (r =0,r =2M),whichistheSchwarzschild angleofdeflection df :
+ S
−
4M (15p 16)M2 1
limdf =df (r =0,r =2M)=df = + − +O[1/r ]3. (2.17)
q→0 − + S rn 4 rn2 n
Usingthisalongwith(2.8,2.9)in(2.16)weexpressdf intermsofthecharges (M,q)anddf
S
p M2a2 4(3p 8) 1
df =df h − +a2 + , ifg =0 (2.18)
S− 16 2 p r2 ···
n
(g (cid:2)1)[16g p (g +1)(cid:3)]M M M +pg (7g 1)q2 1
df =df h − − | − | − + , g > 1 (2.19)
S− 2 8g 2 r2 ··· ∀ −
n
4M (cid:2) 4(1 2g )a2M3 (cid:3)
= + 2M(M+M)+h − (2.20)
r − 2(1+g )(M+M)
n
(cid:26)
p 4(16g 2 1)a4M4 12(4g 1)a2M2 1
+ 15(M+M)2+ − +h − + , g > 1
16 (1+g )2(M+M)2 2 1+g r2 ··· ∀ −
(cid:27) n
h i
where we have made use of M M = h M M and (2.9). [The limit g 0 in (2.19) or in (2.20) leads
2
− | − | →
to(2.18)]. Thustothefirstorderofapproximationin1/r allnormalandphantomblackholesdeflectlightpaths
n
inthesamewaywithdf =4M/r + . Tothesecond orderofapproximation in1/r ,theaddedcontribution
n n
···
to the Schwarzschild one (second terms in (2.18, 2.19)) does not depend on the sign of q but depends on the
signsof(h ,h ).
1 2
First consider the special case g =0, which corresponds to h =+1. In normal EMD (h =+1) we have
1 2
df <df . In phantom EMD (h = 1), which is E-anti-MD theory, we have df >df provided we restrict
S 2 S
−
ourselvestothephysicalcasea2<1(4(3p 8)/p 1.8). Thus,inthepresenceofphantomfields,lightraysare
− ≃
moredeflectedthaninthenormalcase. Phantomfieldscauseslightraystobentwithanangle(3p 8)q2/(2r2)
n
−
larger than the angle of deflection caused by normal fields. This difference is independent on the sign of q but
6
R∆Φ R∆Φ
Γ 0.0035
-0.9 -0.7 -0.4 -0.2
-0.01
0.0025
0.0015 HΗ =-1,M=1L
HΗ =-1,M=1L -0.03 2
2
Ha2=1(cid:144)16,u =0.05L
n
Ha2=1(cid:144)16,un=0.05L 0.0005
.
-0.05 Γ
2 4 6 8 10
Figure1: TherelativediscrepancyRdf =(df df S)/df Svs. g forfixed(M=1,a2=1/16,un=0.05)forE-anti-M-(anti)-D.Rdf
isnotalwaysnegativeforg <0:itvanishesthen−becomespositiveforsomeg0between 0.1and 0.05.
− −
R
∆Φ R
∆Φ
0.025
Γ
HΗ2=1,M=1L . 2 4 6 8 10
0.02 -0.0005
Ha2=1(cid:144)16,un=0.05L HΗ2=+1,M=1L
0.015
-0.0015 Ha2=1(cid:144)16,u =0.05L
n
0.01
-0.0025
0.005
Γ -0.0035
-0.9 -0.7 -0.4 -0.2
Figure2: TherelativediscrepancyRdf =(df df S)/df Svs.g forfixed(M=1,a2=1/16,un=0.05)forEM-(anti)-D.Rdf isnot
alwayspositiveforg <0:itvanishesthenbecom−esnegativeforsomeg0between 0.1and 0.05.
− −
dependsonthemassoftheblackholethroughr . Using(2.17)weobtain
n
3p 8 3p 8
− M2a2 − a2(df )2 (2.21)
2r2 ≃ 32 S
n
whichis0.22% oftheSchwarzschild valuedf ifM=1,a2 =1/4,u =0.05and0.89%ofit(theexactvalue
S n
is1.03)ifM=1,a2=1,u =0.05.
n
Thecaseg =1isphantom ornormalReissner-Nordstro¨m blackhole. With
3p M2a2 1
df =df h + , ifg =1 (2.22)
S− 2 4 r2 ···
n
weconfirmtheprevious conclusions: df <df fornormalReissner-Nordstro¨m blackholesanddf >df for
S S
phantom ones. A phantom Reissner-Nordstro¨m black hole deflects light with an angle 3p q2/(2r2) larger than
n
thedeflectionanglecaused byanormalReissner-Nordstro¨m blackhole
3p 3p
M2a2 a2(df )2 (2.23)
2r2 ≃ 32 S
n
whichis1.47%oftheSchwarzschildvaluedf ifM=1,a2=1/4,u =0.05and5.89%ofit(theexactvalue
S n
is6.56)ifM=1,a2=1,u =0.05.
n
7
R∆Φ R∆Φ
Γ 0.008
-0.8 -0.6 -0.4 -0.2
-0.02
0.006
HΗ =-1,M=1L -0.04
2
0.004
Ha2=1(cid:144)16,u =0.1L -0.06
n HΗ =-1,M=1L
2
-0.08 0.002 Ha2=1(cid:144)16,un=0.1L
.
-0.1
Γ
2 4 6 8 10
Figure3: TherelativediscrepancyRdf =(df df S)/df Svs.g forfixed(M=1,a2=1/16,un=0.1)forE-anti-M-(anti)-D.Rdf is
notalwaysnegativeforg <0:itvanishesthenbe−comespositiveforsomeg0between 0.1and 0.05.
− −
R∆Φ R∆Φ
0.7 Γ
2 4 6 8 10
0.6 .
0.5 -0.02 HΗ2=+1,M=1L
Ha2=1(cid:144)16,un=0.3L 0.4 Ha2=1(cid:144)16,un=0.3L
-0.04
HΗ =1,M=1L 0.3
2
0.2
-0.06
0.1
Γ
-0.8 -0.6 -0.4 -0.2 -0.08
Figure 4: TherelativediscrepancyRdf =(df df S)/df Svs. g forfixed(M=1,a2=1/16,un=0.3)forEM-(anti)-D.Rdf isnot
alwayspositiveforg <0:itvanishesthenbecom−esnegativeforsomeg0between 0.15and 0.1.
− −
Now, consider the case g =0 and g =1. Here again we confirm the previous conclusions: df <df for
S
6 6
normal black holes and df >df for phantom ones provided g is large enough. This is no longer true if g is
S
closerto 1asthecoefficientof1/r2 becomestoolargeinvalidating (2.20). Letuslookatthederivativeofthe
n
−
relativediscrepancy function Rdf
df df
S
Rdf = df− , (2.24)
S
dRdf 2LE2 un ln(f )du
dg = df S Z0 f1−g (u)[E2− g(u)]3/2 , (2.25)
− −
which is positive for phantom black holes (f >1) and negative for normal ones (f <1). Thus, Rdf (g ) (all
− −
other parameters being fixed) increases for phantom black holes and decreases for normal ones. Figures (1
to 4)), which have been plotted using the exact formula (2.15), show the existence of a zero g beyond which
0
Rdf >0(df >df S)forphantom blackholesandRdf <0(df <df S)fornormalones.
Figures (1 to 4) have been plotted for fixed (M = 1,a2 = 1/16) and relatively large values of r (u =
n n
0.05,0.1)orsmallvaluesofr uptothephotonsphere(u =0.3). Basedontheseandonsomeotherfigures(not
n n
showninthispaper)forlargervaluesofa2 upto1anddifferentvaluesofu ,wecandrawageneralconclusion:
n
All the other parameters being fixed, there is always a root g0 in the interval ( 0.2,0) to Rdf (g )= 0. Said
−
otherwise, forsome values ofg intheinterval ( 0.2,0), itseemsthere isalways acritical value r =r , larger
n c
−
thanthephotonsphere, whereRdf =0.
8
∆Φ ∆Φ
4
HΓ=0,M=1,a2=1(cid:144)4L
4
upper plot: Η =-1 HΓ=0,M=1,a2=1L
3 2 upper plot: Η =-1
intermediate: ∆ΦS 3 2
intermediate: ∆Φ
lower plot: Η =+1 S
2 2 2 lower plot: Η2=+1
1 1
HaL HbL
u u
n n
0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.1 0.15 0.2 0.25 0.3
Figure 5: Theangleofdeflection(Eq.(2.15))inradiansvs. un=1/rn (rn isthepointonthenullgeodesicnearesttheorigin). In
bothplotstheintermediategraphistheSchwarzschildvaluedf S.(a):PhantomEMDcosh(h 2= 1)andnormalEMDsinh(h 2=+1)
blackholesforg =0>g0,M=1anda2=1/4. (b):PhantomEMDcosh(h 2= 1)andnormal−EMDsinh(h 2=+1)blackholesfor
g =0,M=1anda2=1. −
∆Φ ∆Φ
4 5
HΓ=1,M=1,a2=1(cid:144)16L
3 upper plot: Η2=-1 4 HΓ=1,M=1,a2=1L
intermediate: ∆ΦS upper plot: Η2=-1
2 lower plot: Η2=+1 3 intermediate: ∆ΦS
lower plot: Η =+1
2 2
1
1
HaL HbL
un un
0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.1 0.15 0.2 0.25
Figure 6: Theangleofdeflection(Eq.(2.15))inradiansvs. un=1/rn (rn isthepointonthenullgeodesicnearesttheorigin). In
bothplotstheintermediategraphistheSchwarzschildvaluedf S. (a): PhantomReissner-Nordstro¨m(h 2= 1)andnormalReissner-
Nordstro¨m(h 2=+1) blackholesfor g =1>g0, M=1anda2=1/16. (b): PhantomReissner-Nordstro¨−m(h 2= 1) andnormal
Reissner-Nordstro¨m(h 2=+1)blackholesforg =1,M=1anda2=1. −
Asg +¥ ,Rdf approaches thelimit
→
2 1 eK xdx p +df
− S
df S Z0 eK−(1−K+)−x2eK−x(1−K+x)− df S
K =Mupn( 1 2h 2a2 1).
± − ±
p
As Figures (5 to 8) reveal, the vertical spacing df df depends slightly on g , which itself depends on
S
| − |
h , and increases with a2. In the extreme case (a2 =1), the winding number for phantom black holes with
1
h = 1andg >g (regardless thesignofh )divergesnearu 0.3,avalueforwhichtheangleofdeflection
2 0 1 n
− ≃
forphantom(h = 1)ornormal(h =+1)blackholeswithh =+1islessthanfewradians. Asweshallsee
1 1 2
−
later, this is a consequence of the fact that the photon sphere for black holes with h = 1(black holes where
2
−
theMaxwellfieldF iscoupledrepulsively togravity) andg >g islarger than3M,whichistheSchwarzschild
0
limit, allowing photons to orbit the hole at larger, ever-decreasing, radii. TheSchwarzschild limit 3M is larger
than the photon sphere for black holes with h =+1 and g >g . If g <g , all that said for black holes with
2 0 0
h = 1(respectively h =+1)appliestoblackholeswithh =+1(respectively h = 1).
2 2 2 2
− −
It is useful to express df in terms of the charges (M,q) and the impact parameter b=L/E. Solving the
9
∆Φ ∆Φ
5
4
HΓ=10,M=1,a2=1(cid:144)16L HΓ=10,M=1,a2=1L
upper plot: Η =-1,Η =-1 4
3 1 2 upper plot: Η1=-1,Η2=-1
intermediate: ∆Φ
S 3 intermediate: ∆Φ
lower plot: Η =-1,Η =+1 S
2 1 2 lower plot: Η =-1,Η =+1
2 1 2
1
1
HaL HbL
un un
0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.1 0.15 0.2 0.25
Figure7: Theangleofdeflection(Eq.(2.15))inradiansvs.un=1/rn(rnisthepointonthenullgeodesicnearesttheorigin).Inboth
plotstheintermediategraphistheSchwarzschildvaluedf S. (a): E-anti-M-anti-DandEM-anti-Dblackholesforg =10>g0,M=1
anda2=1/16.(b):E-anti-M-anti-DandEM-anti-Dblackholesforg =10,M=1anda2=1.
∆Φ ∆Φ
4 3.5 HΓ=-1(cid:144)2,M=1,a2=1L
HΓ=-1(cid:144)2,M=1,a2=1(cid:144)16L
3 upper plot: Η1=+1,Η2=+1 3 upper plot: Η1=+1,Η2=+1
intermediate: ∆Φ 2.5
S intermediate: ∆Φ
lower plot: Η =+1,Η =-1 2 S
2 1 2
1.5 lower plot: Η1=+1,Η2=-1
1 1
HaL 0.5 HbL
un un
0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.1 0.15 0.2 0.25
Figure 8: Theangleofdeflection(Eq.(2.15))inradiansvs. un=1/rn (rn isthepointonthenullgeodesicnearesttheorigin). In
bothplotstheintermediategraphistheSchwarzschildvaluedf S.(a):E-anti-MDandEMDblackholesforg = 1/2<g0,M=1and
a2=1/16.(b):E-anti-MDandEMDblackholesforg = 1/2,M=1anda2=1. −
−
equation E2=g(u )foru =1/r tothesecondorder, weobtain
n n n
1 r +r (2g 1) 1
un= b+ + −2 − b2 +O[1/b]3.
Usingthisin(2.16)wederivethedesired equation
4M p 1
df = + [15r2 +6r r (4g 1)+r2(16g 2 1)] +O[1/b]3. (2.26)
b 16 + − + − − − b2
TheSchwarzschild value,corresponding tothelimitq 0,isagaingivenby
→
4M 15p M2 1
df =df (r =0,r =2M)= + +O[1/b]3. (2.27)
S − + b 4 b2
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