Eur. Phys. J. C (2019) 79:930 https://doi.org/10.1140/epjc/s10052-019-7464-2 Regular Article - Theoretical Physics Black hole shadow with a cosmological constant for cosmological observers Javad T. Firouzjaee1,2,a, Alireza Allahyari3,b 1 Department of Physics, K.N. Toosi University of Technology, P.O. Box 15418-49611, Tehran, Iran 2 School of Physics, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran 3 School of Astronomy, Institute for Research in Fundamental Sciences (IPM), P. O. Box 19395-5531, Tehran, Iran Received: 24 June 2019 / Accepted: 6 November 2019 / Published online: 15 November 2019 © The Author(s) 2019 Abstract Weinvestigate the effect of the cosmological con- cosmological expansion [12]. Motivated by the accelerated stant on the angular size of a black hole shadow. It is known expansion of the universe, we seek to investigate the effect that the accelerated expansion which is created by the cos- of the cosmological constant on the shadow shape. mological constant changes the angular size of the black hole Dark energy, one of the most important problems in cos- shadow for static observers. However, the shadow size must mology, is assumed to drive the current cosmic acceler- be calculated for the appropriate cosmological observes. We ated expansion. The cosmological constant, , is among calculate the angular size of the shadow measured by cosmo- the favored candidates responsible for this acceleration. The logical comoving observers by projecting the shadow angle cosmological constant has imprints on various phenomena to this observer rest frame. We show that the shadow size in structure formation and astrophysics. These effects can tends to zero as the observer approaches the cosmological appear as the integrated Sachs-Wolfe effect in the Cosmic horizon. We estimate the angular size of the shadow for a Microwave Background and large scale structure observ- typical supermassive black hole, e.g M87. It is found that the ables. Moreover, the cosmological constant can change the angular size of the shadow for cosmological observers and luminosity distance in the gravitational wave observations. static observers is approximately the same at these scales Recently, the role of the cosmological constant in gravita- of mass and distance. We present a catalog of supermassive tional lensing has been the subject of focused studies [13–22]. black holes and calculate the effect of the cosmological con- It seems there is no general agreement on the final results. The stant on their shadow size and find that the effect could be black hole shadow arises as a result of gravitational lensing in 3 precent for distant sources. strong gravity regime. Thus, one can investigate the influence of the cosmological constant on the shadow of black holes. The first attempt to derive the shadow size as measured by 1 Introduction the comoving observers was performed by Perlick et al. [23] using the Kottler metric. Authors in [12] presented an approx- Recent observations of black holes shadow at a wavelength imate method based on the angular size-redshift relation for of 1.3 mm spectrum succeeded in observing the first image of the general case of any multicomponent universe. Although the black hole in the center of the galaxy M87 [1,2]. Although the cosmological constant correction on the shadow shape of the first image of the black hole does not distinguish the black black holes has been studied in [24–27], choosing the appro- hole geometry precisely, the future improvement of measure- priate (cosmological) observers is another issue that one has ments will lead to a much higher resolution. This will allow to consider. us to study different aspects of cosmological black holes [3– Tostudy the effect of the cosmological constant in the lens- 8]. To this end, one needs to calculate the influence of dif- ing of a cosmological structure or shadow of a black hole, ferent parameters on the shape of cosmological (astrophys- one has to model a cosmological structure like a black hole in ical) black holes shadow, e.g accretion plasma effects [9], the expanding background. There have been many attempts semi-classical quantum effects [10], dark matter [11] and the to model a cosmological black hole (CBH). The authors in [3–8,30,31] have studied the effect of the expanding back- a e-mail: fi[email protected] ground. Since a cosmological structure like a galaxy, a cluster b e-mail: [email protected] of galaxies or a supermassive black hole in the center of a 123 930 Page 2 of 7 Eur. Phys. J. C (2019) 79 :930 quasar have dynamical evolution due to their dynamics or metric (1) will be written as cosmological expansion, one has to use a dynamical cosmo- 2M logical model to study its lensing [32,33]. It’s been shown ds2 =−dτ 2 + + R2 dr2 + R2d2. (4) [30,31] that the background expansion for CBHs leads to R 3 the formation of voids. These voids are formed between the We need to find R as a function of (r,τ). Using the coordinate black hole and the expanding background and prevent the transformation (3) yields black hole’s matter flux from increasing. After this phase of dR the black hole evolution, the black hole can be approximated τ + r = as a point mass [29]. As a result, the final stage of evolution R2 + 2M can be approximated by a point mass located in a dark energy 3 R 3 √ dominated background. 2 R 2 + 6M + 2 R3 = √ ln . (5) There is a well-known solution of the Einstein equation 3 κ for a point mass black hole in the presence of the cosmo- logical constant known as Schwarzschild-de Sitter/Kottler Solving Eq. (5) for R, we get spacetime. This solution is written in static coordinates. To √ 2 − (r+τ) ( +τ) 3 find the geodesic observers we have to apply an appropri- e 3 e 3 r κ2 − 6M = , ate coordinate transformation to present the Schwarzschild- R 2 (6) de Sitter/Kottler spacetime in the cosmological synchronous (2κ)3 gauge [28,29]. where κ is a constant. When M = 0, one can define a new This paper is organized as follows. In Sect. 2 we introduce radial coordinate as a point mass cosmological black hole. In Sect. 3, we calcu- late the angular size of the shadow in the Schwarzschild-de 2 κ 3 r R˜ = 3 . Sitter spacetime as measured by a cosmological comoving e (7) observer and calculate the different limits. The conclusion and discussion are given in Sect. 4. In this coordinate we obtain the de Sitter metric in the syn- chronous coordinates as 2 Point mass black holes metrics ds2 =−dτ 2 + a(τ)2 d R˜2 + R˜2 d2 , (8) It is known that the Schwarzschild-de Sitter is a standard met- τ ric for the CBH, but Schwarzschild-de Sitter has been written where a(τ) = e 3 . The four-velocity of the comoving in the static coordinates. To study the point mass metric in an observer, dr = 0, who measures the proper time τ can be expanding background, one needs to find the Schwarzschild- written using transformation (3) in the static coordinates as de Sitter metric in the cosmological coordinates. Since the √ standard cosmological metrics are written in the synchronous μ 1 U = , 1 − , 0, 0 , (9) coordinate, one has to transform the Schwarzschild-de Sitter to the synchronous coordinate [28,29]. The Schwarzschild- Four-acceleration of this observer is zero. Hence, it is called de Sitter metric in the static coordinates is given by the comoving cosmological observer. In contrast to the met- − ds2 =−dt2 + 1dR2 + R2 d2, (1) ric introduced in [23], the metric (4) is the appropriate met- ricinwhichr=constant world lines are attached to the where geodesic (comoving) observers. The position of each comov- 2M ing observer r = constant in static coordinates is given by = 1 − R2 − . (2) 3 R Eq. (6). In the next section, we calculate the angular shadow size seen by the cosmological comoving observers. 2 In the weak field limit, one can interpret 3 R as the cosmo- logical constant potential and compare it with the gravita- tional potential of the black hole, M/R for the gravitational 3 Shadow for cosmological comoving observer interactions. Using coordinate transformations √ 1 − Before calculating the angular size of the shadow for the dτ = dt − dR, cosmological observers, let us present the result for the static (3) dR observers from [23]. The shadow is locally seen by a static dr =−dt + √ , ( , ,θ = π/ ,φ = ) 1 − observer at a spacetime point tO RO O 2 O 0 123 Eur. Phys. J. C (2019) 79 :930 Page 3 of 7 930 outside the black hole horizon and inside the cosmologi- the photon trajectory denoted by d. In these coordinates, 1 2 cal horizon. The Lagrangian for geodesics in the equatorial δμ = ( , R , , ) dμ = kμ = L ER , A, , 0 o 0 0 and R2 L 0 1 where plane gives = dR A dφ on the null geodesics. As Ishak and Rindler have 1 R˙2 shown [14,15], one can calculate the bending angle using the L(x, x˙) = − (R)t˙2 + + R2φ˙2 . (10) 2 (R) invariant formula for the cosine of the angles between two directions d and δ. This gives The Euler–Lagrange equations for t and φ coordinates give two constants of motion, A cos ψ = √ , (17) E = (R) t˙, L = R2 φ˙ (11) A2 + R2 where E and L represent the energy and angular momentum where dot denotes the inner product. For the static observers of the geodesic. For null geodesics we have of the Schwartzchild-de Sitter spacetime, one can show ˙2 ˙2 R 2 ˙2 1 − 2M − R2 − (R) t + + R φ = 0.