A Virtual Trip to the Black Hole Computer Simulation of Strong Lensing near Compact Objects RAGTIME 2005 Pavel Bakala Petr Čermák, Kamila Truparová, Stanislav Hledík and Zdeněk Stuchlík Institute of Physics Faculty of Philosophy and Science Silesian University at Opava

Motivation This work is devoted to the following “virtual astronomy” problem: What is the view of distant universe for an observer 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. Our pursuit was to develop a maximally realistic model and computer 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). Based on work of Institut of Physics, Silasian University in Opava ( Stuchlík, Hledík, Plšková), this analysis was extended to spacetimes with repulsive cosmological constant (Schwarzschild – de Sitter spacetimes). In this simulation we consider also geodesics beyond the turn point which was neglected by Cunningham cutting of a part of optical projection. Simulation takes care of frequency shift effects (blueshift, redshift), as well as the amplification and deamplification of intensity. RAGTIME 2005

Formulation of the problem RAGTIME 2005 Schwarzschild – de Sitter metric Event horizon Cosmological horizon Static radius Critical value of cosm. constant

Formulation of the problem The spacetime has a spherical symmetry, so we can study 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 only, so the photon path (a null geodesic) is described by this impact parameter and boundary conditions. RAGTIME 2005

Formulation of the problem There arises an infinite number of images generated by geodesics orbiting around the gravitational center 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, which is not a wholly trivial problem. „Binet“ formula for Schwarzschild – de Sitter spacetime Condition of photon motion RAGTIME 2005

Condition of photon motion : C(u,b,Λ)≥0 Graph for Λ=0 a M=1 Cosmological constant Λ compensates the influence of impact parameter only

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 Turn 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 on the circular photon orbit). RAGTIME 2005

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 turn point. Geodesics with b>b lim a |Δφ(u obs )|> |Δφ(u turn )|, the observer is beyond the turn point. These integral equations express Δφ along the photon path as a function: RAGTIME 2005

Starting point of the numerical solution We can rewrite the final 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 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 value of k corresponds to a photon capture on the circular photon orbit. RAGTIME 2005

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 RAGTIME 2005

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 : RAGTIME 2005

Solution for static observers above the photon orbit, Λ=0 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 black region increases together with a decrescent radial coordinate of observer The black region increases together with a decrescent radial coordinate of observer r obs. r obs =6M r obs =50M RAGTIME 2005

Simulation : Saturn near the black hole RAGTIME 2005 r obs =20M

Simulation : Saturn near the black hole, r obs =5M View of outward direction Some parts of image are moving into an opposite hemisphere of observers sky Blueshift RAGTIME 2005

Solution for static observers under the photon orbit, Λ=0 Impact parameter b increases according to Δ φ up to b max, asymptotically nears to b lim from below. The angle α stat monotonically increases according to 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. 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. r obs =3M – on the photon orbit r obs =2.1M RAGTIME 2005

Simulation : Saturn near 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 one half of the observers sky. RAGTIME 2005

Simulation : Saturn near 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. RAGTIME 2005

Others optical effects Einstein rings A source on polar axis has no defined plane of photons motion and it is displayed as infinitesimaly thin rings. As it is in case of standard images, there is an infinite number of Einstein rings generated, but higher order rings merge in the bright ring on border of the black region in the observer sky. A source on polar axis has no defined plane of photons motion and it is displayed as infinitesimaly thin rings. As it is in case of standard images, there is an infinite number of Einstein rings generated, but higher order rings merge in the bright ring on border of the black region in the observer sky. The physical reason for this is an infinite number of equivalent null geodesics. The physical reason for this is an infinite number of equivalent null geodesics. Intensity changes The strong gravity fields make time, frequency and space redistribution of radiation flux from the whole observer sky. Intensity of higher order images decrease very rapidly, except for Einstein rings where intensity teoretically ( in geometrical approach) goes to infinity. Einstein rings are very well detectable and observable. The strong gravity fields make time, frequency and space redistribution of radiation flux from the whole observer sky. Intensity of higher order images decrease very rapidly, except for Einstein rings where intensity teoretically ( in geometrical approach) goes to infinity. Einstein rings are very well detectable and observable. Geometry of optical projection The optical projection conserves spherical symmetry. The rings in imaginary sky in flat spacetime is transformed into rings with diferrent radius. Images with Δφ>0 (counter-clockwise orbiting geodesics) is angulary inverted in coordinates φ and θ. The optical projection conserves spherical symmetry. The rings in imaginary sky in flat spacetime is transformed into rings with diferrent radius. Images with Δφ>0 (counter-clockwise orbiting geodesics) is angulary inverted in coordinates φ and θ. RAGTIME 2005

Simulation : M31 in the observer sky, r obs =27M RAGTIME 2005

Simulation : M31 in the observer sky, r obs =10M RAGTIME 2005

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

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“. Nondistorted image of M104 r obs =100M r obs =40M r obs =50M r obs =15M

Simulation : Free-falling observer from 10M to the event horizon Galaxy „Sombrero“ is in the observer sky. RAGTIME 2005

Computer implementation The code BHC_IMPACT is developed in C language, compilated by GCC and MPICC compileres, OS LINUX. Libraries NUMERICAL RECIPES and MPI were used. We used SGI ALTIX 350 with 8 Itanium II CPUs for simulation run. Bimpat image of nondistorted objects is the input for the simulation. We assume it is projection of the observer sky in direction of the black hole – in infinity for Schwarzschild field or at the static radius in case of Schwarzschild –de Sitter field. Two bitmap images are 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. Intesity of higher order images rapidly decrease and its positions merge with the second Einstein ring. However, the intensity ratio between images with different orders is nonrealistic. Computer displays have no required bright Only the first three images are generated by the simulation. Intesity of higher order images rapidly decrease and its positions merge with the second Einstein ring. However, the intensity ratio between images with different orders is nonrealistic. Computer displays have no required bright resolution. We used software frequency shift routine from LIGHTSPEED! (R.G. Daniel) special relativity simulator. RAGTIME 2005

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