Strong Deflection Gravitational Lensing by a Modified Hayward Black Hole

Eur. Phys. J. C (2017) 77:272 DOI 10.1140/epjc/s10052-017-4850-5

Regular Article - Theoretical Physics

Strong deﬂection gravitational lensing by a modiﬁed Hayward black hole

Shan-Shan Zhao1,2,YiXie1,2,a 1 School of Astronomy and Space Science, Nanjing University, Nanjing 210023, China 2 Key Laboratory of Modern Astronomy and Astrophysics, Nanjing University, Ministry of Education, Nanjing 210093, China

Received: 8 January 2017 / Accepted: 20 April 2017 © The Author(s) 2017. This article is an open access publication

Abstract A modified Hayward black hole is a nonsingu- Various models of static and spherically symmetric nonsin- lar black hole. It is proposed that it would form when the gular black holes are reviewed in [4,5] and rotating nonsin- pressure generated by quantum gravity can stop matter’s col- gular black holes were also studied [6]. Lots of work [7Ð lapse as the matter reaches the Planck density. Strong deflec- 21] in string theory and supergravity suggest that the sin- tion gravitational lensing occurring nearby its event horizon gularity in the black hole might be replaced with a horizon might provide some clues of these quantum effects in its sized “fuzzball”, whose nonsingular geometries are related central core. We investigate observables of the strong deflec- to microstates of the black hole, so that the information loss tion lensing, including angular separations, brightness dif- paradox might be evaded. ferences and time delays between its relativistic images, and A nonsingular black hole with a “Planck star” replacing we estimate their values for the supermassive black hole in the singularity was recently proposed [22]. This scenario the Galactic center. We find that it is possible to distinguish suggests that when matter collapses toward the center and the modified Hayward black hole from a Schwarzschild one, reaches the Planck density, the pressure generated by quan- but it demands a very high resolution, beyond current stage. tum gravity becomes so large that it could resist the collapse and finally form a bouncing instead of a singularity in the center of the black hole. In the process of Planck star form- 1 Introduction ing, the explosion may produce detectable short gamma ray bursts [23,24] and fast radio bursts [25]. The Hayward metric Einstein’s general relativity (GR) breaks down at singulari- was originally chosen to describe the Planck star [22]. It is ties. Quantum gravity plays an important role in the space- an effective metric for a nonsingular black hole proposed by time of a singularity and its surroundings, since energy den- Hayward [26] and has been well studied [27Ð29]. In order sity and curvature become very large in a tiny region [1]. to incorporate the 1-loop quantum corrections and a finite Therefore, sufficient knowledge of quantum gravity is nec- time delay between the center and infinity, a modified Hay- essary for studying the physics in the vicinity of a gravi- ward metric was then presented as the effective metric of the tational singularity. But the required energy is too high to Planck star [30]. Accretion and evaporation of the modified test quantum gravity effects by any Earth-based experiments. Hayward black hole was studied [31] and its thermodynamics Although we can lower the required energy by testing the was also discussed [32]. As another important and (possibly) effects predicted by some specific models of quantum gravity, observable aspect, investigation of gravitational lensing by there have not been any significant experimental results yet the modified Hayward black hole is still absent in the lit- [2]. Astronomical observations on singularities might also erature, although the gravitational lensing versions by other test quantum gravity and relate to two objects: the starting kinds of nonsingular black holes have aroused a lot of con- point of the Big Bang and the center of a black hole. cerns [33Ð37]. By studying its gravitational lensing effects, It is believed that, in a quantum theory of gravity, the sin- especially those in a strong gravitational field, we can have gularities in black holes under GR can be removed. A black a better understanding of the modified Hayward black hole. hole without singularity is called a nonsingular black hole Testing quantum gravity effects on a black hole would not or a regular black hole; it was first proposed by Bardeen [3]. be easy because an observer usually has to access the region extremely close to its event horizon. It was suggested that a e-mail: [email protected] the quantum effects might occur at 7/6 Schwarzschild radius 123 272 Page 2 of 10 Eur. Phys. J. C (2017) 77:272

[38]. Gravitational lensing caused by a strong field in the 2 Modified Hayward black hole vicinity of a black hole can provide an opportunity for this purpose. Gravitational lensing in a strong gravitational field The modified Hayward metric discussed in this work was is dramatically different from the one in a weak gravitational proposed in [30]. In order to fix its original two shortcomings, field. The unique phenomenon of the lensing effects in a the original Hayward metric is modified by including the 1- strong field was firstly discussed by Darwin in 1959 [39]. loop quantum correction on the Newton potential [116,117] A set of infinite discrete images, called relativistic images, and by allowing for a non-trivial time dilation between a will be generated in a very close area on the two sides of clock in the center of the black hole and a clock at infinity. the lens, due to photons winding several loops around the Thus, the modified Hayward metric reads (in units of G = lens before reaching the detector, which belongs to strong c = 1) [30] deflection gravitational lensing. The relativistic images are a new observational phenomenon which cannot be predicted ds2 = A(x)dt2 − B(x)dx2 − C(x)(dθ 2 + sin2 θdφ2), (1) by classical gravitational lensing in a weak field. Strong deflection lensing by a Schwarzschild black hole has been wherewetake2M as the measure of distances and set it to well studied [40Ð43] and such effects caused by other static unity and and symmetric black holes were also investigated [44Ð73]. A x2 κλ more complicated scenario is the gravitational lensing in the A(x) = 1 − 1 − , (2) strong field around a rotating black hole, which was widely l2 + x3 λ + 2κx3 − discussed [74Ð87]. These strong deflection lensing can be x2 1 B(x) = 1 − , (3) used to determine different black holes [88Ð92], naked sin- l2 + x3 gularities [93Ð96] and wormholes [97Ð101]aswellastest C(x) = x2. (4) gravity [102,103]. If the source of light has time signals, time delays between the relativistic images can also reveal Here, l is a parameter with a length dimension as the same some information as regards the lens [104Ð107]. Reviews of one in the Hayward metric and it can introduce a repulsive strong deflection lensing can be found in [108,109]. force for the avoidance of the singularity [26]; κ and λ are Direct observation of gravitational lensing in a strong field parameters used to modify the Hayward metric [30]. The is still challenging since it requires a very high angular resolu- parameter κ ∈[0, 1) relates to the time delay between x = 0 tion. The most possible candidate to realize this observation and x →∞, and a larger time delay corresponds to a larger is the supermassive black hole in the center of our Galaxy, κ. When κ = 0, there is no time delay between the center called Sagittarius A* (Sgr A*). The apparent angular diam- and infinity, then the modified Hayward metric (1) degener- eter (shadow) of Sgr A* is ∼50 microarcseond (µas), which ates to the Hayward metric. The parameter λ indicates the is largest among all the known black holes [110,111]. The strength of the 1-loop correction on the Newtonian gravita- first real image of Sgr A* will probably be detected by the tional potential [116,117]. 1 Event Horizon Telescope which is an international sub-mm The values of l should be limited in a certain range to guar- very long baseline interferometry (VLBI) network, and when antee the existence of the event horizon(s), whose definition it comes true, a new fundamental physics laboratory can be of B−1(x) = 0 gives a cubic equation of x as provided for testing black hole physics as well as gravity in the strong field regime [112Ð115]. x3 − x2 + l2 = 0. (5) In this work, we will study the strong deflection gravitational lensing by the modified Hayward black hole. Its space- According to Descartes’ rule of signs, this equation has either time and the domains of its model parameters are discussed two positive roots or none. In order to ensure that the discrimi- in Sect. 2. By using the strong deflection limit (SDL) method nant of this cubic equation, =−27l4 +4l2, be nonnegative [46], we analytically describe the gravitational lensing in we must have SDL in Sect. 3. In Sect. 4, the observables, including angular separations, brightness differences and time delays between 2 l ≤ √ 0.385. (6) the resulting relativistic images will be directly obtained. 3 3 Then we estimate these observables for a modified Hayward black hole which has the same mass and distance of Sgr A* In the following work on estimations of the strong deflection√ in Sect. 5. Finally, conclusions and discussion are presented gravitational lensing observables, we set l ∈[0, 2/(3 3)] in Sect. 6. based on the above inequality and take κ ∈[0, 1) according to [30]. These domains of the parameters render A(x)>0 and B(x)>0 for any x outside the (outer) event horizon of 1 http://www.eventhorizontelescope.org/. the modified Hayward black hole. 123 Eur. Phys. J. C (2017) 77:272 Page 3 of 10 272

It is worth emphasizing that, for the modified Hayward lens equation for more general cases [49,52,53,78,129Ð131]. black hole, the geodesic rule is violated inside the event hori- However, the lens equation (7) is predominant, because its zon and such a violation is essential and critical to avoid all of brief form make it possible to analyze the observational lens- the matter falling into the singularity. In fact, a quantum the- ing effects of the modified Hayward black hole in a clear ory of gravity might induce interaction between electromag- physical picture. netic and gravitational fields beyond the standard EinsteinÐ The deflection angle of a photon moving on the equato- Maxwell theory, such as 1-loop vacuum polarization on the rial plane (θ = 2π) in a static and spherically symmetric photon for quantum electrodynamics in which a photon might spacetime is [44,132] travel “faster than light” [118Ð128]. It can make the world- √ line of a photon deviate from its geodesic. Strong deflection ∞ 2 B(x) α(x ) =−π + dx, (8) lensing of photons which do not follow geodesics was inves- 0 √ ( ) x C x A0 0 C(x) ( ) − 1 tigated [71,107] and this deviation is characterized by a con- C0 A x stant. In this work, we assume that the photons in the strong deflection gravitational lensing are following geodesics in the where x0 represents the closest approach distance of the ( ) ( ) spacetime outside the photon sphere of the modified Hayward winding photon; A0 and C0 are the values of A x and C x = black hole so that the SDL method can be applied. Although at x x0. The exact deflection angle of modified Hayward this assumption is still open for testing, it would give a base- black holes could be found by substituting (2)Ð(4)into(8). line for future work which takes the violation of the geodesic The integral in (8) has an approximated form in the rule into account. weak deflection limit (WDL) by assuming that the deflection angle is small. However, this classic WDL method fails in describing the deflection in the strong gravitational field. 3 Gravitational lensing under SDL The divergence occurs when x0 approaches the photon sphere [45,133]. An effective way to handle this problem is to To analyze strong deflection gravitational lensing by the expand the deflection angle near the photon sphere in the SDL modified Hayward black hole, we need two ingredients. One [46]. This method can provide an explicit physical picture and is a lens equation to define the geometrical relationships of a straightforward connection to the observables, which will the observer, the lens and the light source; the other is a be discussed in Sect. 4. deflection angle determined by the spacetime of the lens. The radius of the photon sphere xm is defined as the largest The lens equation given in [43] is adopted in our work for positive root of the following equation [45,133]: its physical feasibility and widespread usage. There are three C (x) A (x) assumptions in applying this lens equation: = . (9) C(x) A(x) 1. The spacetime is asymptotically flat at infinity. 2. The observer and the source are far from the lens. By assuming the closest distance x0 is not too larger than xm, 3. The observer, lens and the source are nearly in alignment, the deflection angle can be expanded in the SDL as [46] and the source locates behind the lens. θ DOL ¯ α(θ) =−¯a log − 1 + b + O(u − um), (10) Under such assumptions, the lens equation could be written um as [43] where u is the impact parameter given by [45,132] DLS β = θ − αn, (7) DOS C u = 0 , (11) A0 where αn = α(θ) − 2nπ is the extra deflection angle of a photon winding n loops and having a deflection angle α; β and u is the impact parameter evaluated at x . The impact and θ are the angular separation between the source and the m m parameter u and the angular separation θ could be related by lens and the angular separation between the image and the u ≈ θ D when the lens equation (7) is adopted. Meanwhile, lens; D is the distance of the lens to the source and D OL LS OS a¯ and b¯ are the SDL coefficients and their expressions are [46] is the distance of the observer to the lens; both of them are projected along the optical axis. R a¯ = √m , (12) This asymptotically approximated lens equation might 2 βm also be defined in other ways, which were summarized and β ¯ =−π + +¯ 2 m , discussed in [61]. In some work one has tried to define the b bR a ln (13) Am 123 272 Page 4 of 10 Eur. Phys. J. C (2017) 77:272 wherewehave The angular separation between the lens and a n-loop rel- ( − )2 − ativistic image can be written as a combination of two parts Cm 1 Am AmCm Cm Am βm = , (14) [45,46], 2A2 C 2 √ m m 2(1 − Am) Am Bm θ = θ 0 + θ , Rm = √ , (15) n n n (21) A C ⎡ m m ⎤ √ 0 1 ( − ) ( ) ( ) where θ is the angle corresponding to the relativistic image ⎣ 2 1 Am A z B z Rm ⎦ n bR = − √ dz, (16) π θ ( ) β with the photon winds completely 2n and n is the extra 0 A (z)C(z) Am − A z z m π Cm C(z) part exceeding 2n . They have expressions as and z is a new variable deduced from x by θ 0 = um + ( ¯ − π)/¯ , n 1 exp b 2n a (22) DOL A(x) − Am (β − θ 0) z = . (17) θ = um n DOS ( ¯ − π)/¯ , − A n exp b 2n a (23) 1 m aD¯ LS DOL θ 0 θ All the quantities with subscript m refer to their correspond- in which n n. The brightness of the relativistic images ing values at x = xm; and and mean taking the derivative will be magnified by the lensing. For the n-loop relativistic of x once and twice. Therefore, a¯ and b¯ can be directly deter- image, its magnification is [134,135] mined by the metric (2)Ð(4) of the modified Hayward black hole after fixing the model parameters l, κ and λ. 1 μn = . (24) Apart from the deflection angle, other important observ- (β/θ)∂β/∂θ θ0 n ables are the time delays between relativistic images if the light source is variable with time. The total travel time of a In practice, if the 1-loop relativistic image can be dis- photon moving from the source to the observer is [104] tinguished from other inner packed ones, we can find three ∞ ∞ characteristic observables [46]: ˜ dt dt T = T (x0) − dx − dx, (18) um θ∞ = , DOL dx DLS dx (25) DOL ¯ where D is the projected distance between the source and b 2π LS s = θ − θ∞ = θ∞ − , 1 exp ¯ ¯ (26) the lens. The last two terms can easily be calculated since the a a ˜ photon is far from the lens. The first term T (x0) is [95,104, μ r = 2.5log 1 = 2.5log exp (2π/a¯) . (27) 132] 10 ∞ μ 10 n=2 n √ ∞ ( ) ( ) Here, θ∞ is the asymptotic position of the images with ˜ 2 B x C x A0 T (x0) = dx, (19) n →∞, i.e., angular radius of the photon sphere; s is the C(x) A x0 A(x) 0 − 1 C0 A(x) angular separation between the 1-loop relativistic image and the packed others (n = 2,...,∞); and r is the magnitude which is divergent by x0 → xm, and it can also be manipu- difference of their brightness. lated with the method of the SDL as The time delay between different relativistic images can also be calculated. If we can distinguish the time signals of u T˜ (u) =−ã ln − 1 + b˜ + O(u − u ), (20) the 1-loop relativistic image and the 2-loop one, the delay u m m between them, T2,1, is given by [104] ˜ where a˜ and b are coefficients in the SDL. For a spherically = 0 + 1 , T2,1 T2,1 T2,1 (28) symmetric spacetime, it is found that a˜ =¯aum [104]. 0 1 where T2,1 and T2,1 are, respectively, the leading and correction term and they are 4 Observables T 0 = 2πu , (29) 2,1 m Combining the lens equation (7) with the deflection angle ¯ (10) and the time delay (20) in the SDL, we can find 1 = Bm um b T2,1 2 exp the observables of the strong deflection lensing, including Am cm a¯ angular separations, brightness differences and time delays π 2π × exp − − exp − (30) between the relativistic images. a¯ a¯ 123 Eur. Phys. J. C (2017) 77:272 Page 5 of 10 272 √ with 25.11 µas when κ = 0, l → 2/(3 3) and λ → 0. The value θ∞ = 26.54 µas, which corresponds to the angular

A C 2 radius of the photon sphere of a Schwarzschild black hole c = β m m . (31) m m 3 ( − )2 Cm 2 1 Am with the same mass and distance, is also permitted. It means that the measurement of θ∞ itself cannot distinguish the mod- After taking the metric of the modified Hayward black hole ified Hayward black hole from the Schwarzschild one. Other and fixing the parameters l, κ and λ, we can obtain the observ- observables are needed for this purpose. We find that, based ables θ∞, s, r and T2,1 for its strong deflection gravitational on Fig. 1, the angular separation s ranges from about 30 to lensing. about 200 nanoarcsecond (nas). For the smallest s,ifthe angular resolution can reach about 10nas or better, which is far beyond current capabilities, the 1-loop relativistic image 5 Estimations for Sgr A* and the packed others can be separated so that it is possible to measure their brightness difference and the time delay. In this section, all of the observables obtained by the According to Fig. 1, it is found that r ranges from 4.7 to SDL method will be estimated by taking the supermas- 6.8 mag; T2,1 can reach from 11.3 to 16.2min and its cor- sive black hole in the Galactic center, Sgr A*, as an rection term can have values of tens seconds, which means 1 example of the modified Hayward black hole. With the s and T2,1 and its correction T2,1 might be accessible coefficients of the modified Hayward metric (2)Ð(4) and under such an extremely high resolution. These additional specifying values for the parameters l, κ and λ, we can constraints imposed by s, r and T2,1 can be helpful for estimate the observables according to Eqs. (25)Ð(30)in pinning down the modified Hayward black. which the SDL coefficients are calculated numerically. As The modified Hayward black hole we have discussed shown in Eqs. (25) and (29), both the radius of the pho- above is a non-rotating one. An astrophysical black hole ton sphere θ∞ and the leading term of the time delay is very likely spinning. In order to describe the spacetime 0 T2,1 are directly proportional to um so that new infor- of a rotating modified Hayward black hole and the gravi- mation provided by the time delay would be contributed tational lensing occurring in its vicinity in a self-consistent 1 1 0 from the correction term T2,1 although T2,1 T2,1. way, its metric is indispensably needed. Although some rotat- All of these observables are represented in color-indexed ing regular black holes are known [6], it was found [137] that Fig. 1. the metric of a rotating modified Hayward black hole is not Figure 1 shows that, when λ and κ are fixed, the increase unique and has no closed causal curves for any positive radial of l can make θ∞, r and T2,1 decrease but causes s and coordinates. As suggested in [137], studying its geodesics 1 T2,1 to grow, which physically means shrinking of the pho- equation might be able to provide helpful insights on the ton sphere, stretching the gap between the 1-loop relativistic properties of this nonsingular rotating metric. Investigations image and the packed others, weakening their brightness dif- on gravitational lensing by the rotating modified Hayward ference, lessening the time delay between them and enlarging black hole should be proceeded with caution. Nevertheless, the correction term in the time delay. When a bigger value based on the work on strong deflection lensing by a Kerr black of λ is taken, all of the observables will become less sensi- hole [74], we can intuitively expect that the angular momen- tive to the variation of l for a given κ.Ifwefixl and λ,Fig. tum of a modified Hayward black hole would also drift its 1 demonstrates that the role of κ will be played in a more caustics away from the optical axis, make the caustic have a complex way. In the case of λ = 0.1, the augmentation of κ finite extension and cause only one image visible instead of can barely affect the values of all the observables. However, two sets of relativistic images. Direct imaging might not be for a bigger λ, while the growth of κ causes θ∞ and T2,1 to able to independently and simultaneously determine the spin 1 increase monotonically, it can make s and T2,1 increase first and its inclination relative to the observer, whose degeneracy and then decrease and their locally maxima can be found at a would be broken by observing its high order effects [76,77]. larger l. A similar behavior occurs for r but the consequence A detailed study on the strong deflection lensing by a rotating of a growing κ make it decrease and then increase, which modified Hayward black hole will be left for our future work 1 is opposite to the tendency of s and T2,1. It is also clear given the fact that knowledge of such a rotating metric is still that the patterns of these color-indexed figures are strongly limited for now. affected by the values of λ, especially in the cases of s, r Accretion flow and its emission around Sgr A* will sig- 1 λ and T2,1. In principle, can enhance the maximum values nificantly affect the observations in the wavelength range of the observables given by the domain of l and κ, except of millimeters on the angular radius of the photon sphere for r. (“shadow”) and the relativistic images. However, since cur- More specifically, θ∞ can vary widely, ranging from 25.2 rent understanding of accretion physics is still incomplete to 36.8µas in Fig. 1. It has a theoretically lower limit of and the emission from Sgr A* is expected to have time- 123 272 Page 6 of 10 Eur. Phys. J. C (2017) 77:272

λ=0.1 λ=0.5 λ=1.0 λ=2.0 λ=10 1 1 1 1 1

0.8 0.8 0.8 0.8 0.8