Gravitational lensing by a black hole in effective loop quantum gravity Qi-Ming Fu 1, 2, * and Xin Zhang †1, 3, ‡ 1 Department of Physics, College of Sciences, Northeastern University, Shenyang 110819, China 2 Institute of Physics, Shaanxi University of Technology, Hanzhong 723000, China 3 Key Laboratory of Data Analytics and Optimization for Smart Industry (Northeastern University), Ministry of Education, China It is well known that general relativity is an effective theory of gravity at low energy scale, and actually quantum effects cannot be ignored in the strong-field regime. As a strong gravitational object, black hole plays a key role in testing the quantum effects of gravity in the strong-field regime. In this paper, we focus on black hole in effective loop quantum gravity and investigate what the influences are of the quantum effects on the weak and strong bending angles of light rays. We find that this black hole could be a Schwarzschild black hole, a regular black hole, a one-way traversable wormhole, or a two-way traversable wormhole for the different values of the quantum parameter, and the strong bending angle for this compact object exhibits two different divergent behaviors, i.e., the logarithmic divergence and non-logarithmic divergence. There are a series of relativistic images on both sides of the optical axis. Only the outermost one can be resolved as a single image, and all others are packed together at the limiting angular position. It is interesting to note that the angular separation between the outermost relativistic image and the others initially increases and then decreases as the quantum parameter increases, indicating that there is a maximum in the angular separation. The maximum is reached after the black hole becomes a wormhole, which can be taken as a signal for the formation of the wormhole. Moreover, the limiting angular position decreases as the quantum parameter increases but a little bounce occurs after the formation of the wormhole, and the relative magnification in magnitudes first decreases and then increases as the quantum parameter increases. PACS numbers: 95.30.Sf, 98.62.Sb, 97.60.Lf I. INTRODUCTION It is widely known that general relativity (GR) has been very successful in modern physics. It has passed an array of tests on solar systems [1] as well as on binary pulsars [1, 2] and cosmology [3, 4]. Besides, the direct detections of gravitational waves [5–7] and the image of the black hole shadow [8] further confirm the validity of GR at the scale of high curvature, such as near neutron stars or black holes. However, GR may not be the com- plete theory of gravity, since its nonrenormalization, the disability of explaining dark matter and dark energy, and so on. One of the most important issues is the prediction of the spacetime singularity inside a black hole or at the beginning of the universe, which suggests that the theory breaks down there. Usually, it is believed that the sin- gularity can be circumvented by some quantum effects. To address the issues like this, one need to resort to the quantum theory of gravity. Although a well-developed quantum theory of gravity is still out of reach, many efforts have been made to un- derstand the spacetime singularity. One of the most suc- cessful attempts is the phase space quantization, i.e., the so-called polymerization schemes developed in loop quan- tum gravity (LQG), which has been used to resolve the † Corresponding author * Electronic address: [email protected]‡ Electronic address: [email protected]big-bang singularity [9–12]. In this quantization scheme, a small parameter called the polymer scale is introduced. When approaching this scale, the quantum effects can no longer be neglected. On the other hand, as an important candidate of the strong gravitational regime, black hole plays a key role in understanding the nature of space- time at high-energy scales. It is natural to describe the black-hole spacetimes by considering the quantum cor- rections, such as the polymerization schemes. Over the past few years, a lot of effective polymerized black holes have been constructed [13–27], most of which focus on the spherically symmetric cases. The common features of these black holes are that the classical singularities inside them are replaced by a transition surface, which connects a black hole to a white hole, and that the space- time is regular everywhere. Besides, since the resulting quantum corrected spacetimes can still be described by an explicit line element, one can study such nonsingular effective geometries by using the well-developed mecha- nism for semiclassical black holes. There have already been some investigations on various aspects of the polymerized black holes in the literature. For example, the authors of Ref. [28] studied the particle production by a regular LQG black hole, and showed that the unitarity is recovered for the black hole evaporation. The relation between the polymerized black holes and the non-singular black holes in mimetic gravity was in- vestigated in Ref. [29]. The exact solutions for the static spherically symmetric black hole interior were obtained in Ref. [30]. In a status report [31], the authors gave a review about the possible experimental or observational arXiv:2111.07223v2 [gr-qc] 17 Feb 2022
Transcript
Gravitational lensing by a black hole in effective loop quantum gravity
Qi-Ming Fu1, 2, ∗ and Xin Zhang†1, 3, ‡
1Department of Physics, College of Sciences, Northeastern University, Shenyang 110819, China2Institute of Physics, Shaanxi University of Technology, Hanzhong 723000, China
3Key Laboratory of Data Analytics and Optimization for SmartIndustry (Northeastern University), Ministry of Education, China
It is well known that general relativity is an effective theory of gravity at low energy scale, andactually quantum effects cannot be ignored in the strong-field regime. As a strong gravitationalobject, black hole plays a key role in testing the quantum effects of gravity in the strong-fieldregime. In this paper, we focus on black hole in effective loop quantum gravity and investigate whatthe influences are of the quantum effects on the weak and strong bending angles of light rays. We findthat this black hole could be a Schwarzschild black hole, a regular black hole, a one-way traversablewormhole, or a two-way traversable wormhole for the different values of the quantum parameter,and the strong bending angle for this compact object exhibits two different divergent behaviors,i.e., the logarithmic divergence and non-logarithmic divergence. There are a series of relativisticimages on both sides of the optical axis. Only the outermost one can be resolved as a single image,and all others are packed together at the limiting angular position. It is interesting to note thatthe angular separation between the outermost relativistic image and the others initially increasesand then decreases as the quantum parameter increases, indicating that there is a maximum in theangular separation. The maximum is reached after the black hole becomes a wormhole, which canbe taken as a signal for the formation of the wormhole. Moreover, the limiting angular positiondecreases as the quantum parameter increases but a little bounce occurs after the formation of thewormhole, and the relative magnification in magnitudes first decreases and then increases as thequantum parameter increases.
PACS numbers: 95.30.Sf, 98.62.Sb, 97.60.Lf
I. INTRODUCTION
It is widely known that general relativity (GR) hasbeen very successful in modern physics. It has passed anarray of tests on solar systems [1] as well as on binarypulsars [1, 2] and cosmology [3, 4]. Besides, the directdetections of gravitational waves [5–7] and the image ofthe black hole shadow [8] further confirm the validity ofGR at the scale of high curvature, such as near neutronstars or black holes. However, GR may not be the com-plete theory of gravity, since its nonrenormalization, thedisability of explaining dark matter and dark energy, andso on. One of the most important issues is the predictionof the spacetime singularity inside a black hole or at thebeginning of the universe, which suggests that the theorybreaks down there. Usually, it is believed that the sin-gularity can be circumvented by some quantum effects.To address the issues like this, one need to resort to thequantum theory of gravity.
Although a well-developed quantum theory of gravityis still out of reach, many efforts have been made to un-derstand the spacetime singularity. One of the most suc-cessful attempts is the phase space quantization, i.e., theso-called polymerization schemes developed in loop quan-tum gravity (LQG), which has been used to resolve the
big-bang singularity [9–12]. In this quantization scheme,a small parameter called the polymer scale is introduced.When approaching this scale, the quantum effects can nolonger be neglected. On the other hand, as an importantcandidate of the strong gravitational regime, black holeplays a key role in understanding the nature of space-time at high-energy scales. It is natural to describe theblack-hole spacetimes by considering the quantum cor-rections, such as the polymerization schemes. Over thepast few years, a lot of effective polymerized black holeshave been constructed [13–27], most of which focus onthe spherically symmetric cases. The common featuresof these black holes are that the classical singularitiesinside them are replaced by a transition surface, whichconnects a black hole to a white hole, and that the space-time is regular everywhere. Besides, since the resultingquantum corrected spacetimes can still be described byan explicit line element, one can study such nonsingulareffective geometries by using the well-developed mecha-nism for semiclassical black holes.
There have already been some investigations on variousaspects of the polymerized black holes in the literature.For example, the authors of Ref. [28] studied the particleproduction by a regular LQG black hole, and showed thatthe unitarity is recovered for the black hole evaporation.The relation between the polymerized black holes andthe non-singular black holes in mimetic gravity was in-vestigated in Ref. [29]. The exact solutions for the staticspherically symmetric black hole interior were obtainedin Ref. [30]. In a status report [31], the authors gave areview about the possible experimental or observational
consequences of black holes in LQG. The greybody fac-tors for the scalar and fermion fields in the backgroundspacetime of a polymerized black hole were explicitlycalculated in Ref. [32]. The quasinormal modes of thepolymerized spherically symmetric black holes were stud-ied in Refs. [33, 34]. In Ref. [35], the authors analyzedthe thermodynamic properties of the quantum correctedpolymerized black holes and investigated the effects ofthe corresponding quantum corrections as functions ofthe masses of the black and white holes. The Hawkingradiation spectra and evaporation of spherically symmet-ric black holes in LQG were investigated in Refs. [36–41].In Ref. [42], the authors constructed the rotating poly-merized black hole using the revised Newman-Janis algo-rithm and investigated the shadow cast by this black hole.One can see Refs. [43–49] for more investigations on thepolymerized black holes. In this paper, we mainly focuson the polymerized black hole proposed in Refs. [25, 26]and investigate the strong and weak deflection angles oflight rays by this black hole.
On the other hand, light rays will be bent as they passthrough a massive object, which is known as gravita-tional lensing, and is a prediction of GR. Since the firstobservation of the deflection of starlight near the Sun in1919, gravitational lensing by other astronomical objectshas been observed many times. In modern astronomy,gravitational lensing plays a key role in many aspects inastrophysical and cosmological observations. Most stud-ies of gravitational lensing are in the weak deflectionlimit, and the bending angle is at most a few arcsec-onds. However, when passing close to a black hole, lightrays may revolve many times around the black hole be-fore arriving at the observer. This phenomenon is knownas the so-called strong gravitational lensing, which wasintroduced a few decades ago for Schwarzschild space-time [50]. An analytical expression for the strong de-flection angle of the Schwarzschild black hole was givenin Ref. [51]. Subsequently, the observables, such as thepositions, magnifications, and time delays of the imagesfor the Schwarzschild black hole in the strong deflectionlimit were investigated systematically in Refs. [52–55]. InRef. [56], the author introduced an analytical frameworkto investigate the strong gravitational lensing effect fora generic static spherically symmetric metric, and thenapplied it to Reissner-Nordstrom black hole and otherspherically symmetric objects with a photon sphere. Nu-merical investigations on the gravitational lensing wereperformed in Refs. [53, 55, 57]. Other interesting studieson gravitational lensing in spherically symmetric space-times can be found in Refs. [58–82]. Besides, gravita-tional lensing by a rotating black hole has been analyzedin Refs. [83–86].
Although there are many studies on polymerized blackholes, the deflection of light rays by this type of blackholes has not yet been studied. The deflection angle oflight rays can be used as an effective way to study blackholes and can provide us with valuable information aboutblack holes. For example, the mass of the black hole and
the quantum parameters resulting from quantum effectscan be determined by analyzing the angular position ofthe relativistic images. In addition, the angular sepa-ration between the outermost relativistic image and theother images can present interesting features. Thus, itis necessary to study the gravitational lensing effect of apolymerized black hole in detail, and it is interesting toknow what the effects are of the quantum corrections onthe deflection angle.
This paper is organized as follows. In Sec. II, we firstgive a brief review of a LQG black hole and then analyzethe deflection angle of light rays by such a black hole. InSecs. III and IV, we calculate the bending angles of lightrays in the weak and strong deflection limits, respectively.The observables of the relativistic images are investigatedin Sec. V. Section VI comes with the conclusion.
II. BLACK HOLES IN EFFECTIVE LOOPQUANTUM GRAVITY
Let us start with the quantum extension of the staticand spherically symmetric metric by solving the loopquantum gravity (LQG) effective equations [25, 26, 42]:
ds2 = −A(y)dτ2 +dy2
A(y)+ C2(y)dΩ2
2, (1)
where the metric functions in terms of the radial coordi-nate y ∈ (−∞,∞) are given by
A(y) =
(1− 1√
2Aλ(1 + y2)
)1 + y2
C2(y), (2)
C2(y) =Aλ√1 + y2
M2B(y +
√1 + y2)6 +M2
W
(y +√
1 + y2)3, (3)
with MB and MW two Dirac observables in the model.The dimensionless and non-negative parameter Aλ is de-fined as Aλ ≡ (λk/(MBMW ))2/3/2, where λk is a quan-tum parameter originated from holonomy modifications[25, 26]. An important feature of this black hole is thatinside the black hole, the areal radius C reaches a mini-mum, which stands for a spacelike transition surface thatsmoothly connects an asymptotically Schwarzschild blackhole to a white hole with mass MB and MW , respectively[42]. In this paper, we focus on the physically interest-ing case with MB = MW = M corresponding to thesymmetric bounce, i.e., the spacetime is symmetric un-der y → −y. Without loss of generality, we concentrateon the branch of y > 0. After redefining two new coordi-nates r ≡
1− 2Aλ.Near the horizon r = rh + ε, A(r) can be expanded as
A(rh + ε) =
√1− 2Aλε
(2− 3Aλ)M+O(ε2). (7)
Obviously, if Aλ 12 , A(r) ' ε
2M near the horizon,which is the same as the Schwarzschild case. Besides,when r → ∞, we have A(r) → 1 − 2M
r . Namely, thespacetime reduces to the Schwarzschild black hole whenit approaches infinity. Thus, the metric (4) can not bedistinguishable from the Schwarzschild black hole at orbeyond the horizon for Aλ 1
2 .It can be easily shown that the metric (4) represents
(i) a Schwarzschild black hole if Aλ = 0 and M 6= 0,(ii) a regular black hole if 0 < Aλ <
12 , (iii) a one-way
traversable wormhole with a null throat if Aλ = 12 , (vi)
a traversable wormhole with a two-way throat at r = 0if Aλ >
12 , and (v) a flat spacetime if M = 0. From the
line element (4), the Lagrangian for the geodesics can beexpressed as
2L = −A(r)t2 +B(r)r2 + C(r)(θ2 + sin2 θφ2), (8)
where the dot denotes the derivative with respect to an
affine parameter along the trajectory. Since the indepen-dence of the Lagrangian on t and φ, one can obtain twoconserved quantities:
pt =∂L∂t
= −A(r)t = −E, (9)
pφ =∂L∂φ
= C(r)φ sin2 θ. (10)
The lightlike geodesics in the equatorial plane θ = π2
are given by
−A(r)t2 +B(r)r2 + C(r)φ2 = 0. (11)
Inserting the two conserved quantities into Eq. (11), thenull geodesics can be cast as
r2 + V (r) = 0, (12)
where the effective potential is V (r) ≡ L2
C(r)B(r) − E2
and L ≡ C(r)φ. Obviously, light rays can only exist ina region for V (r) ≤ 0. The effective potential and itsderivatives with respect to the radial coordinate r can becalculated as
V (r) = E2
(b2(8M2Aλ − 2M
√8M2Aλ + r2 + r2
)(2M2Aλ + r2) 2
− 1
), (13)
V ′(r) = − 2b2E2r
(2AλM2 + r2)3√
8AλM2 + r2
(2AλM
2(
7√
8AλM2 + r2 − 15M)
+ r2(√
8AλM2 + r2 − 3M))
, (14)
V ′′(r) =2b2E2
(2AλM2 + r2)4
(8AλM2 + r2)3/2
(32A3
λM6(
15M − 7√
8AλM2 + r2)
+ 44A2λM
4r2(
11√
8AλM2 + r2
− 24M
)+ 3r6
(√8AλM2 + r2 − 4M
)+ 8AλM
2r4(
11√
8AλM2 + r2 − 30M))
, (15)
V ′′′(r) = − 24b2E2r
(2AλM2 + r2)5
(8AλM2 + r2)5/2
(16A4
λM8(
351M − 160√
8AλM2 + r2)
+ 40A3λM
6r2(32√
8AλM2 + r2
− 67M)
+ 72A2λM
4r4(
7√
8AλM2 + r2 − 18M)
+ r8(√
8AλM2 + r2 − 5M)
+ 2AλM2r6(23√
8AλM2 + r2
− 75M)), (16)
V ′′′′(r) =24b2E2
(2M2Aλ + r2) 6 (8M2Aλ + r2) 7/2
(5r12
(√8M2Aλ + r2 − 6M
)+ 80M2r10Aλ
(4√
8M2Aλ + r2 − 15M)
+ 256M12A6λ
(160√
8M2Aλ + r2 − 351M)
+ 1920M10r2A5λ
(301M − 136
√8M2Aλ + r2
)+ 128M8r4A4
λ
(5√
8M2Aλ + r2 + 168M)
+ 240M6r6A3λ
(117√
8M2Aλ + r2 − 280M)
+ 60M4r8A2λ
(87√
8M2Aλ + r2 − 254M))
, (17)
4
where the impact parameter is defined as b(r0) ≡ LE =√
C(r0)A(r0)
, and r0 is the closest distance between the light
ray and the black hole. Light rays with the critical im-pact parameter bm ≡ b(rm) form unstable circular orbitscalled photon sphere, where rm is the radius of the pho-ton sphere, which is determined by V (rm) = V ′(rm) = 0and V ′′(rm) 6 0. Light rays with the impact parameterb < bm are captured by the black hole or the wormholewhile light rays with b > bm are deflected. The dimen-sionless effective potential V (r)/E2 for the light ray withthe critical impact parameter bm is plotted in Fig. 1.Obviously, for 0 < Aλ < 225
392 , there are two unstablephoton spheres located at r = ±rm, respectively. ForAλ > 225
392 , these two unstable photon spheres merge intoa marginally unstable photon sphere located at rm = 0.Besides, it should be noted that the light orbits at r = 0satisfy V ′(0) = 0 and V ′′(0) > 0 for 1
2 6 Aλ <225392 , how-
ever V (0) 6= 0, which indicates r 6= 0. Thus there is noantiphoton sphere at r = 0 according to its definition inRefs. [78, 79].
-10 -5 0 5 10-1.0-0.8-0.6-0.4-0.20.0
r
V/E2
FIG. 1: Dimensionless effective potential V/E2 in terms ofthe radial coordinate r. The parameters are set to M = 1,Aλ = 0 for red line (Schwarzschild metric), Aλ = 0.2 forgreen dashed line (regular black hole), Aλ = 1
line (traversable wormhole with a marginally unstable photonsphere), and Aλ = 2 for blue dotted dashed line (traversablewormhole).
From Eq. (11), the deflection angle α(r0) can be com-puted to be
α(r0) = I(r0)− π, (18)
where
I(r0) ≡ 2
∫ ∞r0
dr√CB
(CAb2 − 1
) . (19)
III. WEAK DEFLECTION ANGLE
In this section, we investigate the weak deflection an-gle of light rays in this LQG black hole spacetime. Bydefining x = r/2M and T = t/2M , the line element (4)can be transformed into
dS2 = (2M)−2ds2 = −A(x)dT 2 +B(x)dx2
+ C(x)(dθ2 + sin2 θdφ2
), (20)
where
A(x) =1
B(x)=
2√
2Aλ + x2(√
2Aλ + x2 − 1)
Aλ + 2x2, (21)
C(x) =Aλ2
+ x2. (22)
Then, the integral (19) reduces to
I(x0) = 2
∫ ∞x0
(Aλ + 2x2)−1(−2Aλ +
√2Aλ + x2 − x2
(Aλ + 2x2) 2
+2Aλ −
√2Aλ + x20 + x20
(Aλ + 2x20) 2
)−1/2dx, (23)
and the impact parameter is given by
b
2M=
(Aλ2
+ x20
)(2Aλ + x20 −
√2Aλ + x20
)−1/2.(24)
After introducing a new variable z = x0/x, the aboveintegral can be rewritten as
I(x0) =
∫ 1
0
2x0z2
(Aλ +
2x20z2
)−1(2Aλ−
√2Aλ + x20+x20
(Aλ + 2x20) 2
+−2Aλz
2 + z√
2Aλz2 + x20 − x20z (Aλz2 + 2x20) 2
)−1/2dz, (25)
which can not be integrated out analytically. However,in the weak field limit, i.e., x0 1, the integrand can beexpanded in terms of 1/x0, which can be integrated outterm by term as follows:
α(x0) = I(x0)− π =2
x0+
(π
(15
16−Aλ
)− 1
)1
x20
+
((π − 25
3
)Aλ +
61
12− 15π
16
)1
x30
+
(1
64Aλ(152πAλ − 465π + 928
)+
3465π
1024
− 65
8
)1
x40+
(− 1
480Aλ(24(135π − 746)Aλ
− 7875π + 30940)
+7783
320− 3465π
512
)1
x50
+
(1
512Aλ(− 3064πA2
λ + 3(8601π − 21376)Aλ
− 30765π + 86560)
+310695π
16384− 21397
384
)1
x60
+ O(
1
x70
). (26)
Besides, the inverse solution of Eq. (24) also can be ob-tained in the weak field limit as
5
1
x0=
2M
b+
1
2
(2M
b
)2
+1
8(5− 4Aλ)
(2M
b
)3
+
(1− 3Aλ
2
)(2M
b
)4
+3
128
(64A2
λ − 168Aλ
+ 77)(2M
b
)5
+
(57A2
λ
8− 10Aλ +
7
2
)(2M
b
)6
+ O
((2M
b
)7). (27)
Inserting the above relation into Eq. (25), the weakdeflection angle as a function of b is found to be
α =4M
b+
15πM2
4b2− 4πAλM
2
b2+
(128− 224Aλ)M3
3b3
+ O(b−4). (28)
It is obvious that the first two terms come from the weakdeflection angle of light rays by a spherically symmet-ric black hole [87], while the third term originates fromthe quantum effects, which is of the same order as thecharge or spin of a black hole [88, 89]. Besides, the mi-nus sign in front of Aλ indicates that the quantum effectsmake a negative contribution to the weak deflection an-gle. What’ s more, the primary quantum correction tothe weak deflection angle from the LQG is larger than thenon-loop quantum one investigated in Ref. [90], where theprimary correction is only at the order of b−3.
IV. STRONG DEFLECTION LIMIT
In this section, we mainly focus on the bending anglein the strong deflection limit, i.e., x0 → xm or b → bm,for different values of Aλ.
A. Case of 0 6 Aλ <225392
For 0 6 Aλ < 225392 , there are two unstable photon
spheres at x = ±xm, respectively. Without loss of gener-ality, we focus on the unstable photon sphere at x = xmand investigate the deflection angle of light rays in thestrong deflection limit. After redefining a new variable zas
z = 1− x0x, (29)
the integral (19) can be rewritten as
I(x0) =
∫ 1
0
R(z, x0)f(z, x0)dz, (30)
where
R(z, x0) =2x0√ABC0
C(1− z)2, (31)
f(z, x0) =1√
A0 − C0
C A. (32)
It can be easily shown that the function R(z, x0) is regu-lar for all values of z and x0, while f(z, x0) diverges whenz → 0. Then, following Ref. [56], the integral I(x0) canbe divided into two parts:
I(x0) = ID(x0) + IR(x0), (33)
i.e., the divergent part ID(x0) and the regular partIR(x0). The divergent part including the divergence isgiven by
ID(x0) =
∫ 1
0
R(0, xm)fD(z, x0)dz, (34)
where fD(z, x0) is the series expansion of the argumentof the square root in f(z, x0) up to the second order inz:
fD(z, x0) =1√
c1(x0)z + c2(x0)z2, (35)
with the expansion coefficients c1 and c2 given by
c1(x0) = x0
(A0C
′0
C0−A′0
), (36)
c2(x0) =x0
2C20
(− 2x0A0C
′20 − C2
0 (2A′0 + x0A′′0)
+ C0(2x0A′0C′0 +A0(2C ′0 + x0C
′′0 ))), (37)
where the functions with subscript 0 denote that they arecomputed at x0.
After substracting the divergent part, one can obtainthe regular part:
IR(x0) =
∫ 1
0
gR(z, x0)dz, (38)
where gR(z, x0) ≡ R(z, x0)f(z, x0) − R(0, xm)fD(z, x0).Then, the deflection angle of light rays in the strong fieldlimit can be expressed as
α(b)=−a ln
(b
bm− 1
)+u+O[(b− bm) ln(b− bm)], (39)
where
a ≡ R(0, xm)
2√c2(xm)
, (40)
u ≡ a ln δ + IR(x0)− π, (41)
δ ≡ x2m
(C ′′(xm)
C(xm)− A′′(xm)
A(xm)
). (42)
After inserting Eqs. (21) and (22) into Eqs. (40) and (42),the coefficients in the deflection angle (39) can be calcu-lated as
6
a(xm) =
(2Aλ + x2m
)3/4(Aλ + 2x2m
)√4x6m
√2Aλ + x2m + 12Aλx4m + 3A2
λx2m
(4− 3
√2Aλ + x2m
)+A3
λ
(14√
2Aλ + x2m − 15) , (43)
δ(xm) = −2x2m
(4x6m
√2Aλ + x2m + 12Aλx
4m + 3A2
λx2m
(4− 3
√2Aλ + x2m
)+A3
λ
(14√
2Aλ + x2m − 15))
(2Aλ + x2m) 3/2 (Aλ + 2x2m) 2(−2Aλ +
√2Aλ + x2m − x2m
) . (44)
Expanding the integrand of the regular part (38) inpowers of x0 − xm:
IR(x0) =
∞∑j=0
1
j!(x0 − xm)j
∫ 1
0
∂jg
∂xj0
∣∣∣∣∣x0=xm
dz, (45)
and focusing on the primary term j = 0, the regular partcan be approximated as
IR(x0) ∼ IR(xm) =
∫ 1
0
g(z, xm)dz, (46)
which can be calculated numerically. Figure 2 shows thedeflection angle of light rays (39) in terms of b for sev-eral values of Aλ. It can be seen that the strong deflec-tion angle is an excellent approximation for photons pass-ing close to the photon sphere, as shown in Fig. 2. Theweak deflection angle (28) is also a good approximationto the exact deflection angle (18). Besides, for Aλ = 0,the strong deflection angle of light rays by Schwarzschildblack hole is recovered [56]:
α(b) = − ln
(b
3√
3M− 1
)+ ln 6 + 0.9496− π
+ O[(b− bm) ln(b− bm)]. (47)
B. Case of Aα >225392
In the case of Aα >225392 , two unstable photon spheres
merge into one unstable photon sphere located at thewormhole throat x = 0. Defining a new radial coordinate
ρ =√
Aλ2 + x2, the line element (20) can be transformed
into
dS2 =−A(ρ)dT 2+B(ρ)dρ2+C(ρ)(dθ2+sin2 θdφ2), (48)
where
A(ρ) = 1 +3Aλ2ρ2−√
6Aλ + 4ρ2
2ρ2, (49)
B(ρ) =2ρ2
(2ρ2 −Aλ)A(ρ), (50)
C(ρ) = ρ2. (51)
Since the variable z defined in Eq. (29) would not workwell in this case, we introduce a new definition z = 1− ρ0
ρ .
Then, the integral (19) can be expressed as
I(ρ0) =
∫ 1
0
R(z, ρ0)f(z, ρ0)dz, (52)
where
R(z, ρ0) =ρ0
√C(ρ0)
C(ρ)(1− z)2, (53)
f(z, ρ0) =
√√√√ A(ρ)B(ρ)
A(ρ0)− C(ρ0)
C(ρ)A(ρ)
, (54)
and ρ0 =√
Aλ2 + x20. After inserting A(ρ), B(ρ) and
C(ρ), the above expressions can be simplified to
R(z, ρ0) = 2, (55)
f(z, ρ0) =2ρ20√G(z, ρ0)
, (56)
where
G(z, ρ0) =(2ρ20 − (z − 1)2Aλ
) [2ρ20 −
√6Aλ + 4ρ20
+ 3Aλ −(
(z − 1)2((z − 1)
√6(z − 1)2Aλ + 4ρ20
+ 3(z − 1)2Aλ + 2ρ20))]
. (57)
Obviously, R(z, ρ0) is regular and f(z, ρ0) is divergentwhen z → 0. To solve the integral (52), we also split itinto two parts I(ρ0) = Id(ρ0) + Ir(ρ0). The divergentpart Id(ρ0) is given by
Id(ρ0) =
∫ 1
0
2fd(z, ρ0)dz, (58)
fd(z, ρ0) =2ρ20√
c3(ρ0)z + c4(ρ0)z2 + c5(ρ0)z3, (59)
where ρm =√
Aλ2 + x2m, and the expansion coefficients
of the function G(z, ρ0) up to the third order in z are
7
Aλ = 0.46
1.6 1.8 2.0 2.2 2.4 2.6024681012
b
αAλ = 0.5
1.4 1.6 1.8 2.0 2.2 2.4012345
b
α
Aλ = 0.52
1.4 1.6 1.8 2.0 2.2 2.4012345
b
α
FIG. 2: Plots of the exact deflection angle (18) (the black full lines), the corresponding strong deflection limit (39) (the reddashed lines) and weak deflection limit (28) (the blue dashed lines) as a function of the impact parameter b for some values ofAλ.
given by
c3(ρ0) = 2ρ30(Aλ − 2ρ20
)(A′0 −
A0C′0
C0
), (60)
c4(ρ0) = − ρ30
C20
(C2
0
(2A′0
(Aλ + 2ρ20
)+ρ0A
′′0
(2ρ20 −Aλ
))+ C0
(A0
(ρ0C
′′0
(Aλ − 2ρ20
)− 2C ′0
(Aλ + 2ρ20
))+ 2ρ0A
′0C′0
(Aλ − 2ρ20
) )+ 2ρ0A0C
′20
(2ρ20 −Aλ
)), (61)
c5(ρ0) = − ρ503C3
0
(C3
0
(− A0
(3)(Aλ − 2ρ20
)+ 12ρ0A
′′0
+ 12A′0)− C2
0
(− 3A′′0 C
′0
(Aλ − 2ρ20
)+ 3A′0
(8ρ0C
′0 − C ′′0
(Aλ − 2ρ20
))+ A0
(−C0
(3)(Aλ − 2ρ20
)+ 12ρ0C
′′0 + 12C ′0
))+ 6C0C
′0
(A0
(4ρ0C
′0 − C ′′0
(Aλ − 2ρ20
))− A′0C ′0
(Aλ − 2ρ20
) )+ 6A0C
′30
(Aλ − 2ρ20
)),(62)
where the functions with subscript 0 denote that they arecomputed at ρ0.
Expanding c3(ρ0), c4(ρ0), and c5(ρ0) in powers of ρ0−ρm:
c3(ρ0) =(
28√
2A3/2λ − 30Aλ
)(ρ0 − ρm) +
(44Aλ
− 39√Aλ√2
)(ρ0 − ρm)2 +O
((ρ0 − ρm)3
), (63)
c4(ρ0) =(
28A2λ − 15
√2A
3/2λ
)+
(153Aλ
4
− 30√
2A3/2λ
)(ρ0 − ρm) +O
((ρ0 − ρm)2
),(64)
c5(ρ0) =
(231A
3/2λ
4√
2− 52A2
λ
)+O
((ρ0 − ρm)
), (65)
one can easily see that fd(z, ρ0) diverges as z−1 in thestrong deflection limit ρ0 → ρm.
Subtracting the divergent part from I(ρ0), the regularpart can be obtained as
Ir(ρ0) =
∫ 1
0
g(z, ρ0)dz, (66)
where g(z, ρ0) ≡ 2(f(z, ρ0)− fd(z, ρ0)
). Then, the de-
flection angle of light rays in the strong deflection limitρ0 → ρm is
α(b)=−a ln
(b
bm− 1
)+u+O[(b− bm) ln(b− bm)], (67)
where
a ≡ 4ρ2m√c4(ρm)
, (68)
u ≡ a ln δ + Ir(ρ0)− π, (69)
δ ≡ 2ρm
(C ′(ρm)
C(ρm)− A′(ρm)
A(ρm)
). (70)
By inserting A(ρ), B(ρ) and C(ρ), the parameters a and
δ reduce to
a(ρm) = ρm√
6Aλ + 4ρ2m
(18A2
λ − 3ρ2m√
6Aλ + 4ρ2m
− 6Aλ
(√6Aλ + 4ρ2m − 3ρ2m
)+ 4ρ4m
)−1/2,(71)
δ(ρm) = 4
(6Aλ −
3√
2(2Aλ + ρ2m
)√3Aλ + 2ρ2m
+ 2ρ2m
)(3Aλ
−√
6Aλ + 4ρ2m + 2ρ2m)−1
. (72)
Expanding Ir(ρ0) in powers of ρ0 − ρm:
Ir(ρ0) =
∞∑j=0
1
j!(ρ0 − ρm)j
∫ 1
0
∂j g
∂ρj0
∣∣∣∣∣ρ0=ρm
dz, (73)
8
and focusing on the term of j = 0, the regular part canbe approximated in the strong deflection limit as
Ir(ρ0) ∼ Ir(ρm) =
∫ 1
0
g(z, ρm)dz, (74)
which can be calculated numerically. Figure 3 shows thedeflection angle of light rays (67) in terms of b for the caseof Aλ >
225392 . Obviously, the strong and weak deflection
angles also approximate well to the exact deflection angle,as can be seen in the left panel of Fig. 3. The middleand right panels of Fig. 3 show that the critical impactparameter has a minimum bm = 0.6495 at Aλ = 1.125,which indicates that the radius of the photon sphere alsoreaches a minimum there.
C. Case of Aα = 225392
In the case of Aλ = 225392 , the unstable photon sphere
is located at x = xm = 0 or ρ = ρm =√Aλ/2, which is
marginally unstable since the light rays satisfy V (xm) =V ′(xm) = V ′′(xm) = V ′′′(xm) = 0 and V ′′′′(xm) < 0.
When Aλ = 225392 , f(z, ρm) reduces to
f(z, ρm) =
√15
z(2− z)
(4− (z − 1)2
(45(z − 1)2
+ 28√
3(z − 2)z + 4(z − 1) + 15))−1/2
,(75)
and Eqs. (63), (64) and (65) become
c3(ρ) =855
196(ρ0 − ρm)2 +O
((ρ0 − ρm)3
), (76)
c4(ρ) =38475
10976(ρ0 − ρm) +O
((ρ0 − ρm)2
), (77)
c5(ρ) =192375
307328+O
((ρ0 − ρm)
). (78)
Then, from Eq. (59), one can conclude that f(z, ρm) di-verges as z−2/3 in this case, and the divergent part off(z, ρm) becomes
fd(z, ρm) ≡√
10
19z3, (79)
which can be integrated out as
Id =
∫ 1
0
2fd(z, ρ0)dz =
4√
1019√z
∣∣∣∣∣z=0
− 4
√10
19,
=
4√
1019√
ρ0
ρm− 1
∣∣∣∣∣ρ0=ρm
− 4
√10
19. (80)
From the expression of the impact parameter in terms
of the new radial coordinate, i.e., b =
√A(ρ0)
C(ρ0), one can
obtain
b− bm =133√
35
16(ρ0 − ρm)2 +O
((ρ0 − ρm)3)
). (81)
Then, the divergent term Id as a function of the impactparameter b can be expressed as
Id =2 4
√619
√5
4
√bbm− 1− 4
√10
19. (82)
Subtracting the divergent part, the regular part is
Ir(ρm) =
∫ 1
0
2(f(z, ρm)− fd(z, ρm)
)dz, (83)
which can be calculated numerically. Now, the deflectionangle of light rays in the strong deflection limit ρ0 → ρmcan be given by
α =c
4
√bbm− 1
+ d+O[(b− bm) ln(b− bm)], (84)
where
c = 24
√6
19
√5 ∼ 3.35247, (85)
d = −4
√10
19+ Ir − π ∼ −3.24981. (86)
Figure 4 shows that the deflection angles in the strongand weak deflection limits also approximate well to theexact deflection angle.
V. OBSERVABLES IN THE STRONGDEFLECTION LIMIT
In the previous section, we have obtained the deflec-tion angles as a function of the impact parameter. In thissection, we will investigate the observables of this lensingsystem. We consider a beam of light emanated from asource (S), deflected by a lens (L) of mass M , and thenarriving at an observer (O). Both the source and the ob-server are in asymptotically flat spacetime. DLS standsfor the distance between the lens L and the source S,and DOS = DOL + DLS is the distance between the ob-server O and the source S. Defining the optical axis asthe line connecting the lens and the observer, and focus-ing on a small-angle approximation, i.e., the source, lens,and observer are highly aligned, the lens equation can bewritten as
β = θ − DLS
DOSα∗, (87)
where β and θ denote the angular position of the sourceand image with respect to the optical axis, respec-tively. α∗ is the effective deflection angle defined by
FIG. 3: Plots of the exact deflection angle (18) (the black full lines), the corresponding strong deflection limit (67) (the reddashed lines) and weak deflection limit (28) (the blue dashed lines) as a function of the impact parameter b for some values ofAλ.
Aλ = 225392
1.0 1.2 1.4 1.6 1.8 2.00
5
10
15
b
α
FIG. 4: Plots of the exact deflection angle (18) (the black full lines), the corresponding strong deflection limit (84) (the reddashed lines) and weak deflection limit (28) (the blue dashed lines) as a function of the impact parameter b for some values ofAλ.
α∗ ≡ α − 2nπ with n ∈ N. The deflection angle α(θ)can be expanded around θ = θ0n as
α(θ) = α(θ0n) +dα
dθ
∣∣∣∣θ=θ0
n
(θ − θ0n) +O((θ − θ0n)2
), (88)
where θ0n is determined by
α(θ0n) = 2nπ. (89)
A. Aλ 6= 225392
In the case of Aλ 6= 225392 , both of the deflection angles
(39) and (67) are logarithmic divergence in the strongdeflection limit, and they can be rewritten as a functionof θ:
α(θ) = −˜a ln
(θ
θ∞− 1
)+ ˜u
+ O((
θ
θ∞− 1
)ln
(θ
θ∞− 1
)), (90)
where ˜X denotes X or X, and we have used the first-order approximations θ ≈ b/DOL and θ∞ ≈ bm/DOL.From the above expression, one can easily obtain
dα
dθ
∣∣∣∣θ=θ0
n
= −˜a
θ0n − θ∞. (91)
Solving Eq. (90) with α(θ0n) = 2nπ, we find
θ0n =(
1 + e˜u−2nπ
˜a
). (92)
Then, from Eqs. (88), (91) and (92), the effective deflec-tion angle α∗ can be calculated as
α∗(θn) =˜a
θ∞e˜u−2nπ
˜a
(θ0n − θn). (93)
After inserting the effective deflection angle (93) intothe lens equation (87), the angular position of the n-thimage can be calculated as
θn(β) = θ0n +θ∞e
˜u−2nπ˜a DOS(β − θ0n)
˜aDLS, (94)
where the second term is a small correction to θ0n. In thecase of the observer, the lens, and the source perfectlyaligned in a line, i.e., β = 0, the point images become aninfinite series of concentric rings with angular position
θEn ≡ θn(0) =
(1− θ∞e
˜u−2nπ˜a DOS
˜aDLS
)θ0n, (95)
which are the so-called Einstein rings.
10
With the approximation of θn ∼ θ0n, the magnificationof the n-th relativistic image can be given by
µn ≡1
βθdβdθ
∣∣∣∣∣θ=θ0
n
' θ2∞DOS
˜aβDLSe
˜u−2nπ˜a
(1 + e
˜u−2nπ˜a
). (96)
Obviously, the magnification gets exponentially sup-pressed with increasing n, which means that the firstimage is the brightest one and the brightness of otherrelativistic images are highly demagnified unless the lensis highly aligned with the source, i.e., β ∼ 0.
In the simplest situation, we consider that only theoutermost relativistic image θ1 can be resolved separatelyand all other relativistic images are packed together atthe limiting angular position θ∞ = bm/DOL. Then, theangular separation s between the outermost one and theothers can be defined by
s ≡ θ1 − θ∞ ∼ θ01 − θ0∞ = θ∞e˜u−2π
˜a , (97)
and the quotient of the flux of the outermost relativisticimage to that of all other relativistic images is
r ≡ µ1∑∞n=2 µn
∼(e
4π˜a − 1
)(e
2π˜a + e
˜u˜a
)e
4π˜a + e
2π˜a + e
˜u˜a
, (98)
where the cumulative flux from all other relativistic im-ages except the outermost one is
∞∑n=2
µn ∼θ2∞DOS
(e
4π˜a + e
2π˜a + e
˜u˜a
)e
˜u−4π˜a
β˜aDLS
(e
4π˜a − 1
) . (99)
B. Aλ = 225392
In the case of Aλ = 225392 , the deflection angle as a func-
tion of θ can be derived as
α(θ) =c(
θθ∞− 1)1/4 + d, (100)
which yields
dα
dθ
∣∣∣∣θ=θ0
n
= − c
4θ∞
(θ0nθ∞− 1
)−5/4. (101)
Then, from Eqs. (89) and (100), one can obtain
θ0n =
[1 +
(c
2nπ − d
)4]θ∞, (102)
and the effective deflection angle can be calculated as
α∗(θn) =(2nπ − d)5
4θ∞c4(θ0n − θ). (103)
Inserting the effective deflection angle into the lensequation (87), the angular positions of the relativisticimages are solved as
θn(β) = θ0n +4c4DOSθ∞(β − θ0n)
DLS(2nπ − d)5. (104)
Then, the relativistic Einstein rings are given by
θEn(0) =
[1− 4c4DOSθ∞
DLS(2nπ − d)5
]θ0n. (105)
With the approximation of θn ∼ θ0n, the magnificationof the n-th image is
µn '4θ2∞c
4DOSFnβDLS
, (106)
where
Fn =1 +
(c
2nπ−d
)4(2nπ − d)5
. (107)
The separation of the angular positions between theoutermost relativistic image and the others is
s = θ1 − θ∞ =
(c
2π − d
)4
θ∞, (108)
and the luminosity ratio of the flux of the outermost oneto that of all others is
r =µ1∑∞n=2 µn
∼ F1∑∞n=2 Fn
= 9.95368, (109)
where
F1 ∼ 1.28957× 10−5, (110)
and
∞∑n=2
Fn ∼ 1.29557× 10−6. (111)
From the observation of the relativistic images, onecan obtain the limiting angular position θ∞, the angularseparation s, and the flux ratio r. By inverting Eqs. (97)and (98) or Eqs. (108) and (109), one then obtains thecoefficients of the strong deflection angle, which containthe information about the parameters of the lensing blackhole.
C. Applications: Black holes in our Galaxy andM87
To evaluate the observables defined above, we respec-tively take the black holes with mass M = 4.3× 106Mand 6.5 × 109M at the centers of our Galaxy [91] and
11
M87 [92] as examples and apply the strong deflectionlimit. The distances between the observer and the lensare taken as DOL = 8.35 kpc and 16.8 Mpc, respectively.The numerical estimates for the observables with differ-ent Aλ are given in Tables I and II. It can be seen thatthe limiting angular position θ∞ decreases rapidly withincreasing Aλ until Aλ ∼ 0.7 for both black holes, indi-cating that the limiting relativistic images are closer tothe optical axis than the Schwarzschild case, and thenthere is a little bounce at Aλ = 1.125, as shown in theleft panels of Figs. 5 and 6. Since the independences ofthe black hole mass M and the distance DOL, the rela-tive magnifications are the same for both black holes asshown in Tables I and II or the right panels of Figs. 5and 6. Obviously, the relative magnifications in magni-tudes rm first decrease and then increase as Aλ increases,which suggests that the outermost relativistic image firstbecomes fainter and then brighter compared to the oth-ers. Besides, we also find that there is a maximum of theobservable s at Aλ = 0.58 for both black holes, as shownin the middle panels of Figs. 5 and 6, which means thatthe angular separation between the outermost relativis-tic image and the others first becomes wider and thendecreases with increasing Aλ. This feature can be eas-ily recognized in the astronomical observation. What’smore, since the lensing parameters are associated with awormhole when Aλ > 1
2 , the maximum of the angularseparation s signals that a wormhole has formed. Re-cently, from the image of the black hole shadow [42],the perihelion advance, and Shapiro time Delay [93], thetightest constraint on Aλ is 0 < Aλ < 4 × 10−6. Withthe maximum Aλ = 4 × 10−6, the observables for theblack hole SgrA∗ are θ∞ = 26.4232322, s = 0.0331672and rm = 6.8218757, which correspond to a 4.46× 10−6
deviation from the case of the Schwarzschild black holeat best.
VI. CONCLUSION
In this paper, we investigated the deflection angles oflight rays by a black hole in effective loop quantum grav-ity (LQG) in the strong and weak deflection limits. Itwas found that the weak deflection angle is consistentwith the Schwarzschild black hole at the lowest order, andthe corrections originating from quantum effects make anegative contribution to the weak deflection angle. Theprimary quantum correction is of the same order as thespin or charge of a black hole.
Then, we investigated the bending angle in the strongdeflection limit. It was found that the LQG blackhole reduces to a Schwarzschild black hole for vanish-
ing Aλ, a regular black hole for 0 < Aλ <12 , a one-way
traversable wormhole with a null throat for Aλ = 12 , and
a traversable wormhole with two-way throat for Aλ >12 .
Besides, if M = 0, the LQG black hole disappears andthe spacetime is flat as expected, while the spacetime ofthe black-bounce black hole reduces to an Ellis worm-hole with vanishing M , which seems unnatural [78, 94].There are two unstable photon spheres for 0 < Aλ <
225392
located at r = ±rm, respectively. When Aλ = 225392 , these
two unstable photon spheres merge into one marginallyunstable photon sphere. It was found that the behaviorof the strong deflection angle exhibits logarithmic diver-gence in terms of the impact parameter for Aλ 6= 225
392 and
nonlogarithmic divergence for Aλ = 225392 .
Next, we calculated the observables of this lensing sys-tem, such as the angular positions of the relativistic im-ages, the angular separation between the outermost rel-ativistic image and the others, and also their flux ratio.These observables were estimated numerically, using theblack holes at the centers of our Galaxy and M87 as ex-amples. We showed that the observables for both blackholes have the similar behaviors. To be specific, the lim-iting angular position θ∞ and the relative magnificationin magnitudes rm decrease with Aλ for 0 6 Aλ 6 225
392 .
However, when Aλ > 225392 , there is a little bounce at
Aλ = 1.125 in the limiting angular position, and the rel-ative magnification in magnitudes rm increases with Aλ,suggesting that the outermost relativistic image gradu-ally becomes brighter relative to the others. Besides, it isinteresting to find that there is a maximum in the angularseparation at Aλ = 0.58, which means that the angularseparation first becomes wider and then decreases withincreasing Aλ. The maximum is reached after the worm-hole is formed, which can be considered as a signal forthe formation of a wormhole. It is expected that thesefeatures will be used to distinguish LQG from generalrelativity.
Acknowledgments
This work was supported by scientific research pro-grams funded by the National Natural Science Foun-dation of China (Grants No. 11975072, 11835009,11875102, and 11690021), the Liaoning RevitalizationTalents Program (Grant No. XLYC1905011), the Fun-damental Research Funds for the Central Universities(Grant No. N2005030), the National Program forSupport of Top-Notch Young Professionals (Grant No.W02070050), the National 111 Project of China (GrantNo. B16009), and Research Foundation of Education Bu-reau of Shaanxi Province, China (Grant No. 19JS009).
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TABLE I: Numerical estimations for the main observables for the black hole at the center of our Galaxy.
FIG. 5: Plots of the observables θ∞ (left), s (middle), and r in magnitudes (rm = 2.5 log r, right) as a function of Aλ for thesupermassive black hole at the center of our Galaxy.
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TABLE II: Numerical estimations for the main observables for the black hole at the center of M87.Observable Schwarzschild Loop Quantum Black Hole
FIG. 6: Plots of the observables θ∞ (left), s (middle), and r in magnitudes (rm = 2.5 log r, right) as a function of Aλ for thesupermassive black hole at the center of M87.
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