A Virtual Trip to the Black Hole

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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
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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.

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Formulation of the problem

• Schwarzschild de Sitter metric

• Black hole horizon

• Cosmological horizon

• Static radius

• Critical value of cosm. constant

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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.

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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

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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

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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

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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.

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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

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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

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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.

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Simulation Saturn behind the black hole,
robs20M
Nondistorted view
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Simulation Saturn behind the black hole,
robs20M
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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

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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.

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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.

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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.

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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
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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.

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Apparent angular size of the black holeas a
function of the cosmological constant
Zoom near event horizons
Zoom near the photon orbit
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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
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Simulation Observer falling from 10M to the
rest on the event horizon Galaxy Sombrero is
in the observer sky.
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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.

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