Title: A Virtual Trip to the Black Hole
1A 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 HledÃk and Zdenek StuchlÃk 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
2Motivation
- 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.
3Formulation of the problem
- Schwarzschild de Sitter metric
- Critical value of cosm. constant
4(No Transcript)
5Formulation 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
- 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. -
6Formulation 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
7Consequeces 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
8Three 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
9Starting 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.
10Solution 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
11Solution 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
12Solution 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.
13Simulation Saturn behind the black hole,
robs20M
Nondistorted view
14Simulation Saturn behind the black hole,
robs20M
15Simulation 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.
17Simulation 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.
18Simulation 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.
19Simulation 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
20Apparent 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.
21Apparent angular size of the black holeas a
function of the cosmological constant
Zoom near event horizons
Zoom near the photon orbit
22Simulation 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
23Simulation Observer falling from 10M to the
rest on the event horizon Galaxy Sombrero is
in the observer sky.
24Computer 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.
25This presentation can be downloaded from
www.physics.cz/research in section News
End