The Shadow of a Black Hole from Heterotic String Theory at the Center of the Milky Way

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1 EJTP 1, No. 3 (015) 31 4 Electronic Journal of Theoretical Physics The Shadow of a Black Hole from Heterotic String Theory at the Center of the Milky Way Alexis Larrañaga National Astronomical Observatory. National University of Colombia Received 4 October 014, Accepted 0 December 014, Published 10 January 015 Abstract: We study the shadow produced by a charged rotating black hole arising from the heterotic string theory. The size and the shape of the shadow, depending on the mass, electric charge and angular momentum, are calculated. We find that very-long baseline interferometry (VLBI) in the near future will be enough to extract the information about the electric charge of this kind of black hole. c Electronic Journal of Theoretical Physics. All rights reserved. Keywords: Black Holes; Strings; Branes PACS (010): s; w; Jk 1. Introduction Black holes are one of the most interesting predictions of general relativity, as well as of other gravitational theories, and it is widely believed that there exist supermassive black holes in the center of many galaxies. Due to the rotation of the host galaxies, these black holes are generally thought to possess a spin parameter. In fact, the center of our galaxy host the best constrained supermassive black hole known up to date, Sagittarius A*. Its mass and the distance from us have been accurately determined but, unfortunately, other characteristics such as its spin or electric charge are not known. Recently, a possible method to constrains such parameters has emerged: the observation of black hole shadows. The shadow of a black hole corresponds to the lensed image of the event horizon and it is a two-dimensional dark zone seen from the observer, so it can be, in principle, easily measured. For a non-rotating black hole, the shadow is a perfect circle while for the rotating case, it has an elongated shape in the direction of the rotation axis due to the dragging effect [1, ]. ealarranaga@unal.edu.co

2 3 Electronic Journal of Theoretical Physics 1, No. 3 (015) 31 4 This subject has gained attention recently [3, 4, 5, 6, 7, 8] because it is expected that very long baseline radio interferometry can resolve it and observation results may be obtained in the near future [9]. In the literature, the optical properties and apparent shape of different rotating black holes have been investigated. For example, the shadow of a Kerr black hole and a Kerr naked singularity was examined by Hioki and Maeda [10] where it was suggested that the spin parameter of the black hole can be determined from the observation. Schee and Stuchlik[11] studied the phenomena in a braneworld Kerr black hole, and applied their results to Sagittarius A (see also [1]). The same treatment was extended to the rotating black hole in extended Chern-Simons modified gravity [13], the Kaluza-Klein rotating dilaton black hole [14], rotating traversable wormholes [15], the Einstein-Maxwell-Dilaton-Axion black hole [16], some cosmic strings solutions [17] and regular black holes [18]. These results show that, beside the spin, other parameters (as for example coupling constants or tidal charges) also affect the shadow of a black hole, and therefore, observation of its characteristics provides a possible way to determine such parameters. On the other hand, it has been realized that the low-energy effective field theory describing heterotic string theory contains black hole solutions which can have properties which are qualitatively different from those that appear in Einstein gravity. In this work, we propose that the observation of the shadow will be very useful to determine the true nature of a particular black hole and to decide whether it comes from general relativity or from string theory. Using the observables proposed in [10], we will study the apparent shape of the shadow of the Sen black hole, which comes from heterotic string theory. Assuming that the supermassive black hole Sagittarius A can be described by the Sen black hole metric, the astronomical observables and angular radius of the shadow are obtained. The result shows that the resolution of 1 μas expected with the very-long baseline interferometry (VLBI) in the near future [19, 0] will be enough to extract the information about the electric charge of the black hole and thus discriminating the string theory solution from the corresponding general relativity counterpart.. The Sen Black Hole Sen [1, ] found a charged, stationary, axially symmetric solution of the field equations of heterotic string theory by using target space duality, applied to the classical Kerr solution. The line element of this solution can be written, in generalized Boyer-Lindquist coordinates, as where ( ds = 1 Mr ) ( ) dr dt + ρ ρ Δ + dθ ( + r (r + r α )+a + Mra sin θ ρ 4Mrasin θ dtdϕ ρ ) sin θdϕ, (1)

3 Electronic Journal of Theoretical Physics 1, No. 3 (015) Δ=r (r + r α ) Mr + a () ρ = r (r + r α )+a cos θ. (3) Here M is the mass of the black hole, a = J is the specific angular momentum of the M black hole and the electric charge is represented by the quantity r α = Q M. (4) Note that taking a = 0, the metric (1) coincides with the (Gibbons-Maeda-Garfinkle- Horowitz-Strominger) GMGHS solution [3, 4, 5, 6] while setting r α = 0 in (1) gives the Kerr solution. The Sen spacetime has a spherical event horizon given as the biggest root of the equation Δ = 0, r H = M r α + (M r α ) 4a or in terms of the black hole parameters M,Q and J, r H = M Q M + ( ) M Q J M M. (5) Here one distinguishes three interesting cases. First, for a< 1 M r α the event horizon exists and the solution represents a black hole. Second, when a> 1 M r α the metric represents a naked singularity. Finally, the critical condition a = 1 M r α results in an extremal black hole. In this work we will restrict our considerations to the existence of the event horizon. For simplicity, in what follows we will adopt M =1inall equations, which is equivalent to adimensionalize all physical quantities with the mass of the black hole. 3. Photon Orbits When a photon coming from infinity passes near a black hole, there are two cases that can be considered. The first one corresponds to a photon with large orbital angular momentum. At some turning points, it will turn back to be received by an observer located at infinity. The second case corresponds to a photon with small orbital angular momentum, which falls into the black hole, without reaching the observer. This simple analysis shows that there is a dark zone in the sky called the shadow of the black hole and the boundary of such region is determined by the two cases described above. In order to determine the motion of the photons in the black hole background, we will use the Hamilton-Jacobi equation, S λ = 1 S S gμν x μ x, (6) ν

4 34 Electronic Journal of Theoretical Physics 1, No. 3 (015) 31 4 to determine the geodesic equation. In this equation λ is an affine parameter along the geodesics, and S is the Jacobi action. In general, for a particle with rest mass m 0 in the background of the Sen black hole described by the metric in equation (1), the Hamilton-Jacobi equation is separable and it possesses a solution of the form S = 1 m 0λ Et + Lϕ + S r (r)+s θ (θ), (7) where S r and S θ are the radial and angular contributions, respectively. The constants of motion E and L are the energy at infinity and the angular momentum with respect to the axis of symmetry of the spacetime, respectively. Obviously, the rest mass for the photon is zero, m 0 = 0, and therefore replacing the action (7) into equation (6) gives the null geodesic equation ρ dt dλ = [a + r (r + r α )] [a E al + r(r + r α )E] + al a E sin θ Δ (8) ρ dϕ dλ = a [a E al + r (r + r α ) E] + L csc θ ae Δ (9) ρ dr dλ = ± R (r) (10) where we have defined ρ dθ dλ = ± Θ(θ) (11) R (r) = Δ [ K +(L ae) ] + [ a E al + Er (r + r α ) ] (1) and ] Θ(θ) =K + [a E L sin cos θ, (13) θ and K is a separation constant. Note that taking r α = 0, we recover the null geodesic equations for the Kerr black hole [10]. In order to obtain the boundary of the shadow of the black hole, we need to study the radial motion. Equation (10) can be rewritten as where the effective potential V eff is ρ 4 ṙ + V eff =0, (14) V eff E = [ a + r (r + r α ) aξ ] [ r + a +r ( rα 1 )] [η +(ξ a) ], (15) with the impact parameters defined as ξ = L and η = K. The motion of the photon E E is parameterized only by ξ and η which are conserved quantities according to the null

5 Electronic Journal of Theoretical Physics 1, No. 3 (015) geodesics. Since the boundary of the shadow of the black hole is determined by the circular photon orbits, we impose the conditions defining these, { Veff =0 dv eff dr =0. (16) 3.1 The non-rotating case For a non-rotating black hole (a = 0) and using equation (15), the parameters ξ and η satisfy the condition ξ + η = r3 0 +r α r0 + rαr 0, (17) r 0 + r α where r 0 is the solution of equations (16), [ r 0 = 1 3 3r ] (rα ) α + 5rα +9. (18) Note that when r α = 0, the radius of the circular orbit becomes the well known radius of the Schwarzschild s photon sphere, r 0 =3. 3. The rotating case For the rotating black hole (a 0), we obtain the parameters ( r + r α +1 ) a + r [ Δ a + ( r α 1 ) (r + r α ) ] ξ = a ( (19) r + rα ( 1) ) [ η = r 4a r + r α Δ a + ( r α 1 ) (r + r α ) ] a ( r + r α 1 ). (0) Taking r α = 0 in this equation reproduces the result for the Kerr black hole obtained by Hioki and Maeda [10]. 4. The Shadow of the Black Hole Now we will analyze the above equations in order to determine the shadow of the Sen black hole. As discussed in the first section, the existence of the event horizon is determined by the condition a 1 r α. Since our interest goes into the astrophysical observation, we introduce the celestial coordinates using the functions (8) and (11), ( ) α = lim r dϕ sin θ 0 r dr = ξ csc θ 0, (1) ( θ θ0 β = lim r dθ ) r dr = ± η + a cos θ 0 ξ cot θ 0, () θ θ0

6 36 Electronic Journal of Theoretical Physics 1, No. 3 (015) 31 4 where θ 0 is the angle between the rotation axis of the black hole and the line of sight of the observer, i. e. the observer is located at infinity with angular coordinate θ 0. The coordinate α corresponds to the apparent perpendicular distance of the image as seen from the axis of symmetry and β is the apparent perpendicular distance of the image from its projection on the equatorial plane (a detailed derivation of such coordinates for the Kerr metric can be found in [7]). These coordinates give the apparent position of the image in the plane that passes through the center of the black hole and that is orthogonal to the line joining the observer and the black hole. The shadow of the black hole corresponds to the region in the parameter space (α, β) not illuminated by the photon sources located at infinity and distributed uniformly in all directions. From the dependence of equations (1) and (), it can be seen that the gravitational effects on the shadow are larger when the observer is situated in the equatorial plane. Also, it is clear that for the supermassive black hole Sagittarius A* the inclination is expected to lie close to θ 0 = π, as seen from the Earth. Hence, the celestial coordinates take the simpler form α = ξ (3) β = ± η (4) for equatorial obervers. Note that from this equation it is clear that α and β are independent of the angular coordinate ϕ (because the black hole is axisymmetric). The shadow of the non-rotating black hole, a = 0, is determined by the radius of the photon sphere. Hence, the condition in equation (17) for the parameters ξ and η describing the photon sphere gives α + β = r3 0 +r α r0 + rαr 0 = R r 0 + r α C. (5) This expression implies that the shadow of the black hole in the (α, β) parameter r0 space is a circle with the radius R C = 3+rαr 0 +r αr 0 r 0 +r α depending on the electric charge of the Sen black hole r α. In Figure 1(a) we plot this shadow for the electric charge r α = 0 (continuous line) which corresponds to the circle with radius R C =3 3. For nonvanishing electric charge, the radius of the shadow decreases until it shrinks to a point, R C = 0, when the extremal black hole is reached, at the value r α =. This behavior distinguishes the GMGHS black hole from the Reissner-Nordström solution because in the latter case it reaches a minimum radius of R s = 4 in the extremal case. However this behavior of the GMGHS black hole is similar to the obtained for the Kaluza-Klein rotating dilaton black hole by Amarilla and Eiroa [14]. For the rotating black hole, a 0, we replace the parameters ξ and η from equations (19) and (0) into the parameters α and β. The result is a shadow that will be a deformed circle, as plotted in Figure 1 (b) and (c). In all cases, the size of the shadow reduces when there is a non-vanishing electric charge r α for a fixed spin a. The behavior with the spin is such that when a increases, the shadow will shift to the right. It is important to note that when the black hole approaches the extremal case, the shadow will be distorted more away from a circle and it will reach different sizes for a

7 Electronic Journal of Theoretical Physics 1, No. 3 (015) Figure 1 Shadows of Sen black hole situated at the origin of coordinates with inclination angle θ 0 = π/. (a) Spin a = 0 and electric charge r α = 0 (full line), r α =0.5 (dashed line) and r α =0.8 (dashed-dotted line). (b) Spin a =0.5 and electric charge r α = 0 (full line), r α =0. (dashed line) and r α =0.5 (dashed-dotted line). (c) Spin a =0.8 and electric charge r α =0 (full line), r α =0.1 (dashed line) and r α =0.. fixed value of a when compared with those obtained with the Kerr-Newman solution. Even more, the maximum allowed charge for a fixed mass is larger for the Sen black hole than for Kerr-Newman ones. This means that the Sen black hole can harbor larger amounts of electric charge before becoming naked singularities than their general relativity counterparts. 5. Astronomical Observables In this section we will introduce two observables in order to extract the information of an astronomical object from this shadow. The size and the form of the shadow are characterized by using the two astronomical observables introduced in [10]: The first observable R s is defined as the radius of a reference circle passing by three points: the top position and the bottom position of the shadow, (α t,β t )and(α b,β b ) respectively; and the point (α r, 0) corresponding to the unstable retrograde circular orbit seen from an observer located in the equatorial plane. The second observable is the distortion parameter, defined as δ s = D R s,whered is the difference between the endpoints of the circle and of the shadow, both of them correspond to the prograde circular orbit and locate at the opposite side of the point (α r, 0). As the name of the two observables indicate, R s measures the approximate size of the shadow while δ s measures its deformation with respect to the reference circle. By knowing the inclination angle θ 0, and with a precise enough measurements of R s and δ s, one can obtain the value of the physical rotation parameter a and the electric charge r α of the black hole. This information can be obtained by plotting the contour curves with constant R s and δ s in the plane (a, r α ). Hence, the intersection point in this plane gives the corresponding rotation and charge. The geometry of the shadow let us write the observable R s as (see [10] for a detailed analysis) R s = (α t α r ) + βt, (6) (α r α t ) where we have used the relations α b = α t and β b = β t. Similarly, the distortion

8 38 Electronic Journal of Theoretical Physics 1, No. 3 (015) 31 4 Figure Astronomical observables as function of the electric charge of the black hole. (a) R s for the black hole with zero spin, a = 0, as function of r α. See text for details about the non-zero spin case. (b) δ s as a function of r α for spin a = 0 (full line), a =0.5 (dashed line), and a =0.8 (dashed-dotted line). parameter δ s is δ s = ( α p α p ), (7) R s with ( α p, 0) and (α p, 0) the points where the reference circle and the contour of the shadow cut the horizontal axis at the opposite side of (α r, 0), respectively. Writing α p = α r R s, the parameter δ s can be expressed as δ s = D s R s, (8) where we defined the diameter of the shadow along β =0asD s = α r α p. Note that this equation says that for smaller D s and larger R s we obtain larger values of δ s.also, if D s =R s (i.e. in the non-rotating case) we get δ s = 0, as expected. For a non-rotating black hole, the behavior of the coordinate R s as function of the electric charge r α is shown in Figure (a). As it can be seen, R s decreases with r α, which means that the approximate size of the shadow decreases with the existence of electric charge. When the black hole is rotating, the behavior of R s is almost the same for different values of the rotation parameter a. In fact, for a fixed value of the electric charge, the difference of R s between a =0anda = 1 is of order 10 3, which causes such a small variation in the size of the shadow that the curves are indistinguishable (and therefore in the plot we see just one curve). The same behavior is obtained for the Kerr black hole [10] as well as for the Einstein-Maxwell-Dilaton-Axion black hole and some braneworld [1] and Kaluza-Klein solutions [14]. On the other hand, the distortion parameter δ s is plotted as a function of the electric charge for different values of the spin of the black hole in Figure (b). For the nonrotating black hole (continuous curve), a = 0, the distortion is always zero, which means that the contour of the shadow is a perfect circle with no deformation as expected. In

9 Electronic Journal of Theoretical Physics 1, No. 3 (015) Figure 3 Contour plot of the astronomical observables in the plane (a, r α ). R s is described by the full lines (taking the values 4.5, 4.75 and 5 from top to bottom) while δ s is described by the dashed lines (taking the values 0.00, 0.003, 0.006, 0.010, 0.05, 0.050, 0.1, and 0. from left to right). The gray zone at the right of the figure represents naked singularities which are not considered here. the rotating case, δ s as function of the electric charge is a monotonically increasing curve which comes to a maximal value when the black hole approaches the extremal case. For a fixed electric charge, the distortion of the shadow increases with the spin a. Finally, we plot the contour curves of constant R s and δ s in the parameter (a, r α ) space in Figure 3.The grey zone represents naked singularities, which are outside the scope of this work. In the figure, each point is characterized by four values, (a, r α, R s, and δ s ) and therefore, from the observational point of view, if we measure and fix the values of R s and δ s, we are allowed to directly read out the spin and electric charge of the black hole through the corresponding point of intersection in Figure Angular Size of the Shadow for Sagittarius A* The observable R s can be used to estimate the angular radius of the shadow as θ s = RsM D 0 where D 0 corresponds to the distance between the observer and the black hole. As observers, we are located in the equatorial plane of the Milky way, at a distance of D 0 =8.3 kpc from the supermassive black hole Sagittarius A which have an estimated mass of M = M [8]. This data gives the function θ s = R s μas which, together with the values of the observable R s, let us calculate the angular size of the shadow for different values of spin and electric charge. Note that using the values shown in Figure (a), the resolution of 1 μas will be enough to extract the information of the electric charge from further observations, while for the spin a, the resolution of less than 1 μas is needed. These are out of the capacity of the current astronomical observations but can be likely to be observed with the Event Horizon Telescope at wavelengths around 1mm based on VLBI, and with the space-based radio telescopes RadioAstron [19, 0].

10 40 Electronic Journal of Theoretical Physics 1, No. 3 (015) Conclusions In this paper, we studied the size and shape of the shadow cast by a charged rotating black hole from the low-energy effective field theory describing heterotic string theory, which have properties which are qualitatively different from those that appear in Einstein gravity. We have found that, for a fixed mass and rotation parameter, the presence of of electric charge leads to a shadow that is slightly smaller and more deformed. Supposing that the black hole is situated at the origin of the coordinate system and the observer is located in the equatorial plane, we use the null geodesics to obtain the celestial coordinates α and β in order to visualize the black hole s shadow. To analyze the shadow in detail, we define two astronomical observables: R s that describes the approximate size of the shadow and δ s that measures its deformation. These quantities are functions of the electric charge and the spin of the black hole. Our study shows that, for a fixed value of the spin, the size of the shadow described by R s decreases when increasing the electric charge but its value is always smaller than that of the Kerr black hole. On the other hand, the observable δ s is a monotonically increasing function of the electric charge and takes its maximal value for the extremal black hole. Also, for a fixed value of r α, an increase in the value of the spin results in a greater distortion of the shadow. Finally, we plot the contour curves of constant R s and δ s in the (a, r α ) plane because careful measurements of the astronomical observables allow to read out the spin and the electric charge of the black hole from this figure. It is expected that direct observation of the shadow of black holes will be possible in the near future by using the very long baseline interferometer technique in (sub) millimeter wavelengths [9, 30]. The Event Horizon Telescope will reach a resolution of 15 μas at 345 GHz, while the RadioAstron mission will be capable to resolve1 10 μas and the Millimetron mission may provide a resolution of 0.3 μas or less at 0.4 mm. Following our estimates for the angular size of the shadow, these instruments will be capable of observing the shadow of the supermassive galactic black hole Sagittarius A* to distinguish the presence of electric charge as described by the Sen black hole and maybe to discriminate the string theory solution from the corresponding general relativity counterpart. Acknowledgements This work was supported by the Universidad Nacional de Colombia. Hermes Project Code References [1] J. Bardeen. École d ete de Physique Théorique, Les Houches, 197, edited by C. De Witt and B. S. De Witt, (Gordon and Breach Science Publishers, New York, 1973). [] S. Chandrasekhar, The Mathematical Theory of Black Holes, (Oxford University Press, New York, 199). [3] H. Falcke, F. Melia, and E. Agol. Astrophys. J. 58, L13 (000)

11 Electronic Journal of Theoretical Physics 1, No. 3 (015) [4] A. de Vries. Class. Quant. Grav. 17, 13 (000). [5] R. Takahashi. Astrophys. J. 611, 996 (004) [6] C. Bambi and K. Freese. Phys. Rev. D 79, (009) [7] C. Bambi and N. Yoshida. Class. Quant. Grav. 7, (010) [8] G. V. Kraniotis. Class. Quant. Grav. 8, (011) [9] A. F. Zakharov, A. A. Nucita, F. DePaolis, and G. Ingrosso. New Astron. Rev. 10, 479 (005) [10] K. Hioki and K. I. Maeda. Phys. Rev. D 80, 0404 (009) [11] J. Schee and Z. Stuchlik. Int. J. Mod. Phys. D 18, 983 (009) [1] L. Amarilla and E. F. Eiroa. Phys. Rev. D 85, (01) [13] L. Amarilla, E. F. Eiroa, and G. Giribet. Phys. Rev. D 81, (010) [14] L. Amarilla and E. F. Eiroa. Phys. Rev. D 87, (013) [15] P. G. Nedkova, V. Tinchev, and S. S. Yazadjiev. [arxiv: [gr-qc]] [16] S.W. Wei and Y.X. Liu. JCAP 11, 063 (013) [17] V. K. Tinchev and S. S. Yazadjiev. [arxiv: [gr-qc]] [18] Z. Li and C. Bambi. JCAP 01, 041 (014) [19] [0] [1] A. Sen, Phys. Rev. Lett. 69, 1006 (199) [] S. Yazadjiev, Gen. Relativ. Gravit. 3, 345 (000) [3] G.W. Gibbons, Nucl. Phys. B 07, 337 (198) [4] G.W. Gibbons and K. Maeda. Nucl. Phys. B 98, 741 (1988); [5] D. Garfinkle, G.T. Horowitz, and A. Strominger, Phys. Rev. D 43, 3140 (1991) [6] D. Garfinkle, G.T. Horowitz, and A. Strominger, Phys. Rev. D 45, 3888(E) (199) [7] S. E. Vázquez and E. P. Esteban. Nuovo Cim. B 119, (004) [8] S. Gillessen, F. Eisenhauer, S. Trippe, T. Alexander, R. Genzel, F. Martins, and T. Ott. Astrophys. J. 69, 1075 (009) [9] V. L. Fish and S. S. Doelman, in Proceedings of the International Astronomical Union, Symposium S Cambridge University Press, Cambridge, England. Vol (010) [30] T. Johannsen, D. Psaltis, S. Gillesen, D. P. Marrone, F. Özel,S.S.DoelmanandV. L. Fish. Astrophys. J. 758, 30 (01)

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