A Virtual Trip to the Black Hole

1. Eleventh Marcel Grossmann Meeting on General Relativity A Virtual Trip to the Black Hole Pavel Bakala Petr Čermák , Kamila Truparová , Stanislav Hledíkand Zdeněk Stuchlík Institute of Physics Faculty of Philosophy and ScienceSilesian University in Opava Czech Republic Computer Simulation of Strong Gravitional Lensing in Schwarzschild-de Sitter Spacetimes This presentation can be downloaded from www.physics.cz/research in section News

2. 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 forfully 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). This analysiswas extended to spacetimes with repulsive cosmological constant (Schwarzschild – de Sitter spacetimes). In order to obtain whole optical projection we consideredall 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.

3. Formulation of the problem • Schwarzschild – de Sitter metric • Black hole horizon • Cosmological horizon • Static radius • Critical value of cosm. constant

5. 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. • Contravariant components of photons 4-momentum • Impact parameter • Direction of 4-momentum depends on an impact parameter b only, sothe photon path (a null geodesic) is described by this impact parameter and boundary conditions.

6. 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

7. Consequeces of photons motion condition • Existence of limit impact parameter and location of the circular photon orbit • Existence of maximal impact parameter for observers above the circular photon orbit. Geodesics with b>bmaxnever achieve robs. • Geodesics have b<blim for observers under the circular photon orbit. (b≤blim for observers on the circular photon orbit). • Turn points for geodesics with b>blim. • Nemiroff (1993) for Schwarzschild spacetime

8. Three kinds of null geodesics • Geodesics with b<blim , photons end in the singularity. • Geodesics with b>blim and|Δφ(uobs)|< |Δφ(uturn)|, the observer is ahead of the turn point. • Geodesics with b>blim a |Δφ(uobs)|> |Δφ(uturn)|, the observer is beyond the turn point. • These integral equations expressΔφ along the photon path as a function:

9. Starting point of the numerical solution • We can rewrite the final equation for observers on polar axis in a following way : • Parameter k takes values of 0,1,2…∞ for geodesics orbiting clokwise , -1,-2, …∞ for geodesics orbiting counter-clokwise. Infinite value of k corresponds to a photon capture on the circular photon orbit. • 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.

10. 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-momentumfor the static observer in equatorial plane can be obtained using appropriate tetrad of 1-form ω(α) • Transformation to a local coordinate system

11. 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 αstatfor 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 :

12. Solution for static observers above the photon orbit Impact parameter as function of Δφat robs=6M Directional angle as function of Δφat robs=6M • Impact parameter b increases according to Δφ up to bmax,, after which it decreases and asymptotically aproaches to blim from above. • The angle αstat monotonically increases according toΔφup 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.

13. Simulation : Saturn behind the black hole, robs=20M Nondistorted view

14. Simulation : Saturn behind the black hole, robs=20M

15. Simulation : Saturn behind the black hole, robs=5M View of outward direction • Some parts of image are moving into an opposite hemisphere of observers sky • Blueshift

16. Solution for static observers under the photon orbit Impact parameter as function of Δφat robs=2.7M Directional angle as function of Δφat robs=2.7M • Impact parameter bmonotonically increases with Δφ and, asymptotically nears to blim from below. • The angle αstat monotonically increases withΔφup 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 withincreasing 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.

17. Simulation : Saturn behind the black hole, robs=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.

18. Simulation : Saturn behind the black hole, robs=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.

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

20. Apparent angular size of the black holeas a function of the cosmological constant • Apparent angular Asize can be considered as border of the black region of the static observer´s sky, thus is given by maximum value of the angle αstat . • From observers above the photon orbit angular size is given as • From observers under and on the photon orbit angular size is given as • Behavior of angular size depend of the position of the observer. From the 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 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.

21. Apparent angular size of the black holeas a function of the cosmological constant Zoom near event horizons Zoom near the photon orbit

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

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

24. Computer implementation • The code BHC_IMPACTis developed in C language, compilated by GCC and MPICC compilers, OS LINUX. Libraries NUMERICAL RECIPES, MPI and LIGHTSPEED! were used. We used IBM BladeCenter and SGI ALTIX 350 with 8 Itanium II 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 resolution.

25. This presentation can be downloaded from www.physics.cz/research in section News End