A Virtual Trip to the Black Hole Computer Simulation of Strong Gravitional Lensing in Schwarzschild-de Sitter Spacetimes Pavel Bakala Petr Čermák, Stanislav Hledík, Zdeněk Stuchlík, Kamila Truparová and Milan Uhlár Institute of Physics Faculty of Philosophy and Science Silesian University in Opava Czech Republic INTERNATIONAL CONFERENCE ON "Black Holes: Power behind the Scene" October 23-27, 2006, Kathmandu, Nepal

Motivation This work is devoted to the following “virtual astronomy” problem: What is the view of distant universe for an observer (static or radially falling ) in the vicinity of the black hole (neutron star) like? Nowadays, this problem can be hardly tested by real astronomy, however, it gives an impressive illustration of differences between optics in a strong gravity field and between flat spacetime optics as we experience it in our everyday life. We developed a computer code for fully realistic modelling and simulation of optical projection in a strong, spherically symmetric gravitational field. Theoretical analysis of optical projection for an observer in the vicinity of a Schwarzschild black hole was done by Cunningham (1975) an Nemiroff (1993). Recent observation indicated cosmological expansion accelerated by dark energy as an effective value of cosmological constant. Therefore analysis of optical projection was extended to spacetimes with repulsive cosmological constant (Schwarzschild – de Sitter spacetimes). In order to obtain whole optical projection we considered all direct and undirect rays - null geodesics - connecting sources and the observer. The simulation takes care of frequency shift effects (blueshift, redshift), as well as the amplification of intensity.

Formulation of the problem Schwarzschild – de Sitter metric Black hole horizon Cosmological horizon Static radius Critical value of cosmological constant

Formulation of the problem The spacetime has a spherical symmetry, so we can consider photon motion in equatorial plane ( θ=π/2 ) only. Constants of motion are time and angle covariant componets of 4- momentum of photons. Impact parameter Contravariant components of photons 4-momentum Direction of 4-momentum depends on an impact parameter b only, so the photon path (a null geodesic) is described by this impact parameter and boundary conditions.

Formulation of the problem There arises an infinite number of images generated by geodesics orbiting around the black hole in both directions. In order to calculate angle coordinates of images, we need impact parameter b as a function of Δφ along the geodesic line „Binet“ formula for Schwarzschild – de Sitter spacetime Condition of photon motion

Consequeces of photons motion condition Existence of maximal impact parameter for observers above the circular photon orbit. Geodesics with b>b max never achieve r obs. Existence of limit impact parameter and location of the circular photon orbit Turning points for geodesics with b>b lim. Nemiroff (1993) for Schwarzschild spacetime Geodesics have b<b lim for observers under the circular photon orbit. (b≤b lim for observers just on the circular photon orbit).

Three kinds of null geodesics Geodesics with b<b lim, photons end in the singularity. Geodesics with b>b lim and |Δφ(u obs )| b lim and |Δφ(u obs )| < |Δφ(u turn )|, the observer is ahead of the turning point. Geodesics with b>b lim a |Δφ(u obs )|> |Δφ(u turn )|, the observer is beyond the turning point. These integral equations express Δφ along the photon path as a function:

Starting point of the numerical solution We can write the final governing equation for observers on polar axis in a following way : Final equation expresses b as an implicit function of the boundary conditions and cosmological constant. However, the integrals have no simple analytic solution and there is no explicit form of the function. Numerical methods can be used to solve the final equation. We used Romberg integration and trivial bisection method. Faster root finding methods (e.g. Newton-Raphson method) may unfortunately fail here. Parameter k takes values of 0,1,2…∞ for geodesics orbiting clokwise, -1,-2, …∞ for geodesics orbiting counter-clokwise. Infinite values of k correspond to a photon capture on the circular photon orbit.

Solution for static observers In order to calculate direct measured quantities, one has to transform the 4-momentum into local coordinate system of the static observer. Local components of 4-momentum for the static observer in equatorial plane can be obtained using appropriate tetrad of 1-form ω (α) Transformation to a local coordinate system

Solution for static observers As 4-momentum of photons is a null 4-vector, using local components the angle coordinate of the image can be expressed as: π must be added to α stat for counter-clockwise orbiting geodesics (Δφ>0). Frequency shift is given by the ratio of local time 4-momentum components of the source and the observer. In case of static sources and static observers, the frequency shift can be expressed as :

Solution for static observers above the photon orbit Impact parameter b increases according to Δ φ up to b max,, after which it decreases and asymptotically aproaches to b lim from above. The angle α stat monotonically increases according to Δ φ up to its maximum value, which defining the black region on the observer sky. The size of black region expands with decreasing radial coordinate of observer but decreases with increasing value of cosmologival constant The size of black region expands with decreasing radial coordinate of observer but decreases with increasing value of cosmologival constant. Δφ at Impact parameter as function of Δφ at r obs =6M Δφ at Directional angle as function of Δφ at r obs =6M

Simulation : Saturn behind the black hole, r obs =20M Nondistorted view

Simulation : Saturn behind the black hole, r obs =5M Outward direction view Small part of image is moving into an opposite hemisphere of observers sky Blueshift

Solution for static observers under the photon orbit Solution for static observers under the photon orbit Impact parameter b monotonically increases with Δ φ and, asymptotically nears to b lim from below. The angle α stat monotonically increases with Δ φ up to its maximum value, which defines a black region on the observer sky. The black region occupies a significant part of the observer sky now. The size of black region now expands with increasing value of cosmologival constant The angle α stat monotonically increases with Δ φ up to its maximum value, which defines a black region on the observer sky. The black region occupies a significant part of the observer sky now. The size of black region now expands with increasing value of cosmologival constant. In case of an observer near the event horizon, the whole universe is displayed as a small spot around the intersection point of the observer sky and the polar axis. Δφ at Impact parameter as function of Δφ at r obs =2.7M Δφ at Directional angle as function of Δφ at r obs =2.7M

Simulation : Saturn behind the black hole, r obs =3M Observer on the photon orbit would be blinded and burned by captured photons. Outward direction view, whole image is moving into opposite hemisphere of observers sky Strong blueshift Black region occupies more than one half of the observers sky.

Simulation : Saturn behind the black hole, r obs =2.1M The observer is very close to the event horizon. Outward direction view Most of the visible radiation is blueshifted into UV range. Black region occupies a major part of observer sky, all images of an object in the whole universe are displayed on a small and bright spot.

Simulation : Influence of the cosmological constant M31, r obs =27M, Λ=0 M31, r obs =27M, Λ=10 -5 Sombrero, r obs =25M, Λ=0 Sombrero, r obs =25M, Λ=10 -5 Sombrero, r obs =5M, Λ=0 Sombrero, r obs =5M, Λ=10 -5

Behavior Behavior of angular size depend of the position of the static observer. From the static observers above the photon orbit angular size is anticorrelated with cosmological constant, the largest angular size in given radius matches pure Schwarzschild case. Under the photon orbit dependency on cosmological constant has opossite behavior. For static observers just on the photon orbit the angular size of the black hole is independent on the cosmological constant and it is allways π, one half ( all inward hemisphere ) of the observer sky. For static observers under the circular photon orbit A size is given as Apparent angular size of the black hole as a function of the cosmological constant Apparent angular size A size of the black hole can be considered as the angular size of the black region of the observer´s sky, thus is given by maximum value of the directional angle α. For static observers above the circular photon orbit A size is given as

Apparent angular size of the black hole for static observers as a function of the cosmological constant Zoom near event horizons Zoom near the photon orbit

Apparent angular size of the black hole for radially free falling observers as a function of the cosmological constant Observers are free falling from apropriate static radius angular size matches The angular size increases with cosmological constant, the smallest angular size matches pure Schwarzschild case. For observers just in the singularity the angular size is independent on the value of cosmological constant and it is allways π, one half ( all inward hemisphere ) of the observer sky. Situation is similar to static observers on the photon orbit. Angular size for free falling observers is always smaller then for the static one in the same radial coordinate.

Simulation : Free-falling observer from infinity to the event horizon in pure Schwarzschild case. The virtual black hole is between observer and Galaxy M104 „Sombrero“. Nondistorted image of M104 r obs =100M r obs =40M r obs =50M r obs =15M

Simulation : Observer falling from 10M to the rest on the event horizon Galaxy „Sombrero“ is in the observer sky.

Computer implementation The code BHC_IMPACT is developed in C language, compilated by GCC and MPICC compilers, OS LINUX. Libraries NUMERICAL RECIPES, MPI and LIGHTSPEED! were used. We used IBM BladeCenter with 8 AMD Dual Core Opteron 64 CPUs for simulation run. One bitmap image of nondistorted objects is the input for the simulation. We assume that it is projection of part of the observer sky in direction of the black hole in flat spacetime. Two bitmap images are generated as an output. The first image is the view in direction of the black hole, the second one is the view in the opposite direction. Only the first three images are generated by the simulation. The intensity of higher order images rapidly decreases and their positions merge with the second Einstein ring. However, the intensity ratio between images with different orders is unrealistic. Computer displays have not required bright Only the first three images are generated by the simulation. The intensity of higher order images rapidly decreases and their positions merge with the second Einstein ring. However, the intensity ratio between images with different orders is unrealistic. Computer displays have not required bright resolution.

Future plans Simulation for rotating spacetimes with cosmological constant, e.g. Kerr–de Sitter spacetimes. Using core of this code for modeling of light curves – in preparation.

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