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A Virtual Trip to the Black Hole

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Simulation : Saturn behind the black hole, robs=20M ... The observer is very close to the event horizon. Outward direction view ... – PowerPoint PPT presentation

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Title: A Virtual Trip to the Black Hole


1
A Virtual Trip to the Black Hole
Eleventh Marcel Grossmann Meeting on General
Relativity
  • Computer Simulation of Strong Gravitional Lensing
    in Schwarzschild-de Sitter Spacetimes

Pavel Bakala Petr Cermák , Kamila Truparová
, Stanislav Hledík and Zdenek Stuchlík Institute
of Physics Faculty of Philosophy and
ScienceSilesian University in Opava Czech
Republic
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 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). This analysis 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.

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

4
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5
Formulation of the problem
  • The spacetime has a spherical symmetry, so we
    can consider photon motion in equatorial plane (
    ?p/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, so the 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 ?f 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 bgtbmax never achieve robs.
  • Geodesics have bltblim for observers under the
    circular photon orbit. (bblim for observers on
    the circular photon orbit).
  • Turn points for geodesics with bgtblim.
  • Nemiroff (1993) for Schwarzschild spacetime

8
Three kinds of null geodesics
  • Geodesics with bltblim , photons end in the
    singularity.
  • Geodesics with bgtblim and ?f(uobs) lt
    ?f(uturn), the observer is ahead of the turn
    point.
  • Geodesics with bgtblim a ?f(uobs)gt ?f(uturn) ,
    the observer is beyond the turn point.
  • These integral equations express ?f 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 0,1,28 for
    geodesics orbiting clokwise , -1,-2, 8 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-momentum for the static observer
    in equatorial plane can be obtained using
    appropriate tetrad of 1-form ?(a)
  • 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
  • p must be added to astat for counter-clockwise
    orbiting geodesics (?fgt0).
  • 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 ?f at robs6M
Directional angle as function of ?f at robs6M
  • Impact parameter b increases according to ?f up
    to bmax,, after which it decreases and
    asymptotically aproaches to blim from above.
  • The angle astat monotonically increases according
    to ?f 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.

13
Simulation Saturn behind the black hole,
robs20M
Nondistorted view
14
Simulation Saturn behind the black hole,
robs20M
15
Simulation Saturn behind the black hole, robs5M
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 ?f at robs2.7M
Directional angle as function of ?f at robs2.7M
  • Impact parameter b monotonically increases with
    ?f and, asymptotically nears to blim from below.
  • The angle astat monotonically increases with ?f
    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.

17
Simulation Saturn behind the black hole,
robs3M
  • 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,
robs2.1M
  • 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, ?0
M31, robs 27M, ?0
Sombrero, robs 5M, ?0
M31, r obs27M, ?10-5
Sombrero, robs 25M, ?10-5
Sombrero, robs 5M, ?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
    observers sky, thus is given by maximum value
    of the angle astat .
  • 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 p , 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 horizon Galaxy Sombrero is
in the observer sky.
24
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 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
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