A Virtual Trip to the Black HoleA Virtual Trip to the Black HoleComputer Simulation of Strong Gravitional Lensing in Computer Simulation of Strong Gravitional Lensing in

Schwarzschild-de Sitter SpacetimesSchwarzschild-de Sitter Spacetimes

Pavel BakalaPetr Čermák , Stanislav Hledík, Zdeněk Stuchlík,

Kamila Truparová and Milan Uhlár

Institute of PhysicsFaculty of Philosophy and Science

Silesian University in Opava Czech Republic

INTERNATIONAL CONFERENCE ON "Black Holes: Power behind the Scene"October 23-27, 2006, Kathmandu, Nepal

MotivationMotivationThis work is devoted to This work is devoted to the followingthe following “virtual astronomy” problem: What is the “virtual astronomy” problem: What is the view of distant universe for an observerview of distant universe for an observer (static or radially falling ) (static or radially falling ) in the in the vicinity of the black hole (neutron star) like? vicinity of the black hole (neutron star) like? Nowadays, this problem can be hardly tested by real astronomy, however, it Nowadays, this problem can be hardly tested by real astronomy, however, it gives an impressive illustration of differences between optics in a strong gives an impressive illustration of differences between optics in a strong gravity field and between flat spacetime optics as we experience it in our gravity field and between flat spacetime optics as we experience it in our everyday life. everyday life.

We developed a computer code forWe developed a computer code for fullyfully realistic model realistic modellingling and simulation of and simulation of optical projection in a strong, spherically symmetric gravitational field. optical projection in a strong, spherically symmetric gravitational field. Theoretical analysis of optical projection for an observer in the vicinity of a Theoretical analysis of optical projection for an observer in the vicinity of a Schwarzschild black hole was done by Cunningham (1975)Schwarzschild black hole was done by Cunningham (1975) an Nemiroff an Nemiroff (1993)(1993). . Recent observation indicated cosmological expansion accelerated by Recent observation indicated cosmological expansion accelerated by dark energy as an effective value of cosmological constant. Therefore dark energy as an effective value of cosmological constant. Therefore analysisanalysis of optical projection of optical projection was extended to spacetimes with repulsive was extended to spacetimes with repulsive cosmological constant (cosmological constant (Schwarzschild – de Sitter Schwarzschild – de Sitter spacetimes). In spacetimes). In order to order to obtain whole optical projection obtain whole optical projection we considerwe considereded all direct and undirect rays - all direct and undirect rays - nullnull geodesics geodesics - - connecting sources and the observer.connecting sources and the observer. The sThe simulation takes imulation takes care of frequency shift effects (care of frequency shift effects (blueshift, redshift)blueshift, redshift), as well as the amplification , as well as the amplification of intensity.of intensity.

Formulation of the problemFormulation of the problem

222221

2222 sin3

213

21 ddrdrrrMdtr

rMds

Schwarzschild – de Sitter metricSchwarzschild – de Sitter metric

Black hole horizonBlack hole horizon

Cosmological horizon Cosmological horizon

Static radiusStatic radius

Critical value of cosmCritical value of cosmologicalological constant constant

3

43arccoscos2 Mrevent

33

Mrstat

291Mcrit

33arccoscos2 Mrcosm

Formulation of the problemFormulation of the problem

Eb

The spThe spaacetime has a sphericcetime has a sphericalal symmetry, so we can consider symmetry, so we can consider photon photon motion motion in equatorial planein equatorial plane ( ( θθ==ππ/2/2 ) ) o onlynly..Constants of motion are time aConstants of motion are time andnd angle covariant componets of 4- angle covariant componets of 4-momentummomentum of photons of photons. .

Impact parameter Impact parameter Contravariant components of photons 4-momentumContravariant components of photons 4-momentum

ppEpt ,0,

0

3211

321

2

22

12

p

Erbp

ErrM

rbp

ErrMp

r

t

Direction of Direction of 4-momentum4-momentum depends odepends on ann an impact parameter impact parameter bb only, only, soso the photon the photon path (path (aa null null geodesic) is described bygeodesic) is described by this this impact parameter and boundary impact parameter and boundary conditions.conditions.

Formulation of the problemFormulation of the problemThere arises an iThere arises an infinite number of images generated by geodesics nfinite number of images generated by geodesics orbiting around the orbiting around the black holeblack hole in both directions. in both directions.In order to In order to calcula calculatete angle coordinates of images angle coordinates of images, w, we need impact parameter e need impact parameter bb as as a a function of function of ΔφΔφ along the geodesic along the geodesic line line

„„Binet“ formula for Schwarzschild – de Sitter spacetime Binet“ formula for Schwarzschild – de Sitter spacetime bb

32

1,1,322

Muub

dud

ru

pp

drd

r

Condition of photon motionCondition of photon motion

3

2,, 322 MuubbuC

0,, buC

Consequeces of photons motion conditionConsequeces of photons motion condition

Existence of maximal impact parameter for observers above the circular photon orbit. Existence of maximal impact parameter for observers above the circular photon orbit. Geodesics with Geodesics with bb>>bbmaxmax never achieve never achieve rrobsobs. .

Existence of limit impact parameter and location of the circular photon orbitExistence of limit impact parameter and location of the circular photon orbit

32

1,32

max

obsobs

obs

Muuub

MrM

b pho 3,9127

2lim

Turning points for geodesics with Turning points for geodesics with bb>>bblimlim..

327arccos

31cos

33

2, 2

2

bMb

br turn

bMbbr turn

27arccos31cos

32

Nemiroff (1993) for Schwarzschild spacetimeNemiroff (1993) for Schwarzschild spacetime

Geodesics have Geodesics have bb<<bblimlim for observers under the circular photon orbit. ( for observers under the circular photon orbit. (bb≤≤bblimlim for observers just on for observers just on the the circular photon orbit).circular photon orbit).

Three kinds of null geodesicsThree kinds of null geodesicsGeodesics with Geodesics with bb<<bblimlim , photons end in the singularity. , photons end in the singularity.Geodesics with Geodesics with bb>b>blimlim a andnd ||ΔφΔφ((uuobsobs)|)| < |< |ΔφΔφ((uuturnturn)|)|, , the observer is ahead the observer is ahead ofof the turning point. the turning point.

obs

source

u

uobs

Muub

duu

32 322

Geodesics with Geodesics with bb>b>blimlim a a ||ΔφΔφ((uuobsobs)|> |)|> |ΔφΔφ((uuturnturn)|)| , , the observer is beyond the turning the observer is beyond the turning point.point.

turn

u

uobs u

Muub

duuobs

turn

32 322

ThTheseese integral equations expres integral equations expresss ΔφΔφ along the photon path as along the photon path as a a function:function:

,,, sourceobs uubF

turn

source

u

uturn

Muub

duu

32 322

Starting pointStarting point of of thethe numerical solution numerical solutionWe can write the final governing equation for observers We can write the final governing equation for observers onon polar axis polar axis in a following wayin a following way : :

Final equation expresses Final equation expresses bb as an implicit function of the boundary as an implicit function of the boundary conditions and cosmological constant. However, the integrals have no conditions and cosmological constant. However, the integrals have no simple analytic solution and there is no explicit form of the function.simple analytic solution and there is no explicit form of the function. Numerical methods can be used to solve the final equation.Numerical methods can be used to solve the final equation. We used We used Romberg integration and Romberg integration and trivial bisectiontrivial bisection method. method. Faster root finding Faster root finding methods (e.g. methods (e.g. Newton-RaphsonNewton-Raphson method) may method) may unfortunately failunfortunately fail here here..

02,,, krrbF sourcesourceobs

ParameterParameter k k takes values of takes values of 0,1,2…0,1,2…∞∞ for geodesics orbiting clokwise , for geodesics orbiting clokwise , --11,-2, …∞,-2, …∞ for geodesics orbiting counter-clokwise. Infinite values of k for geodesics orbiting counter-clokwise. Infinite values of k correspond to a photon capture on the circular photon orbit.correspond to a photon capture on the circular photon orbit.

Solution for static observersSolution for static observersIn order to calculate direct measured quantitiesIn order to calculate direct measured quantities,, one has to transform the 4-momentum into local coordinate system of the one has to transform the 4-momentum into local coordinate system of the static observer. static observer. Local components of Local components of 4-4-momentummomentum for the static observer in equatorial plane can be obtained using for the static observer in equatorial plane can be obtained using appropriate tetrad of 1-form appropriate tetrad of 1-form ωω((αα))

dr

rd

drrrM

dtrrM

r

t

sin

321

321

21

2

2

rp

p

rrM

rrM

rb

p

rrMEp

r

t

03

21

3211

321

2

22

21

2

pp

TransformaTransformation to tion to a a lolocal coordinate systemcal coordinate system

Solution for static observersSolution for static observersAs 4-momentum of photons is a null 4-vector, using local components the angle As 4-momentum of photons is a null 4-vector, using local components the angle coordinate of the image can be expressed as:coordinate of the image can be expressed as:

2

2

3211arccosarccos rrM

rb

ppt

r

stat

ππ must be added to must be added to ααstatstat for counter-clockwise orbiting geodesics (for counter-clockwise orbiting geodesics (ΔφΔφ>0>0).).

Frequency shift is given by the ratio of local time 4-momentum components of the source and the Frequency shift is given by the ratio of local time 4-momentum components of the source and the observer.observer. In case of static sources and static observers, the frequency shift can be expressed as :In case of static sources and static observers, the frequency shift can be expressed as :

2

2

321

321

obsobs

sourcee

tsource

tobs

source

obs

rrM

rrM

pp

Solution for static observers above the photon orbitSolution for static observers above the photon orbit

Impact parameter Impact parameter bb increases according to increases according to ΔΔφφ up to up to bbmaxmax,,, after, after which which it decreases and it decreases and asymptotically asymptotically aproachesaproaches to to bblimlim from above. from above.

The angle The angle ααstat stat monotonically monotonically increases according toincreases according to ΔΔφφ up to its maximum value, which up to its maximum value, which defining the black region on the observer sky.defining the black region on the observer sky.

The size of black region expands with decreasing radial coordinate of observer but decreasesThe size of black region expands with decreasing radial coordinate of observer but decreases with with increincreaasing value of cosmologival constantsing value of cosmologival constant.

Impact parameter as function of ΔΔφφ atat robs=6M Directional angle as function of ΔΔφφ atat robs=6M

Simulation : Saturn behind the black hole, Simulation : Saturn behind the black hole, rrobsobs=20M=20M

Nondistorted viewNondistorted view

Simulation : Saturn behind the black hole, Simulation : Saturn behind the black hole, rrobsobs=5M=5MOutward direction view

Small part of image Small part of image is moving into is moving into an an opposite opposite hemisphere of hemisphere of observerobserverss sky sky

BlueshiftBlueshift

Solution for static observers under the photon orbitSolution for static observers under the photon orbit

Impact parameter Impact parameter bb monotonically monotonically increases with increases with ΔΔφφ and and, , asymptotically nears to asymptotically nears to bblimlim from below.from below.The angle The angle ααstat stat monotonically monotonically increases withincreases with ΔΔφφ up to its maximum value, which definup to its maximum value, which defineses a black region on the observer sky. The black region occupies a black region on the observer sky. The black region occupies a significanta significant part of the part of the observer sky now. The size of black region now expands withobserver sky now. The size of black region now expands with increincreaasing value of sing value of cosmologival constantcosmologival constant.In case of In case of anan observer near observer near the the event horizonevent horizon,, the the whole universe is displayed as whole universe is displayed as a a small spot around the intersection point of the observer sky and the polar axis.small spot around the intersection point of the observer sky and the polar axis.

Impact parameter as function of ΔΔφφ at at robs=2.7M Directional angle as function of ΔΔφφ atat robs=2.7M

Simulation : Saturn behind the black hole, Simulation : Saturn behind the black hole, rrobsobs=3M=3MObserver on the Observer on the photon orbit would be photon orbit would be blinded and burned by blinded and burned by captured photons.captured photons.

Outward direction Outward direction view, whole image is view, whole image is moving into opposite moving into opposite hemisphere of hemisphere of observerobserverss sky sky

Strong blueshiftStrong blueshift

Black region occupies Black region occupies more than more than one one half of half of the observerthe observerss sky. sky.

Simulation : Saturn behind the black hole, Simulation : Saturn behind the black hole, rrobsobs=2.1M=2.1MThe observer is very The observer is very close toclose to the event the event horizon.horizon.

Outward direction Outward direction viewview

MMost ofost of the the visible visible radiation is radiation is blueshifted into UV blueshifted into UV range.range.

Black region Black region ococccupiesupies a a major major part of observer sky, part of observer sky, all images of all images of an an object inobject in the the whole whole universe are universe are displayed on displayed on a a small small and bright spot. and bright spot.

Simulation : Influence of the cosmological constantSimulation : Influence of the cosmological constant

M31, robs =27M, Λ=0

M31, r obs=27M, Λ=10-5

Sombrero, robs =25M, Λ=0

Sombrero, robs =25M, Λ=10-5

Sombrero, robs =5M, Λ=0

Sombrero, robs =5M, Λ=10-5

BehaviorBehavior of angular size depend of the position of the static observer. From the static 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 static 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.

For static observers under the circular photon orbit For static observers under the circular photon orbit AAsizesize is given asis given as

Apparent angular size of the black holeApparent angular size of the black holeas a function of the cosmological constantas a function of the cosmological constantApparent angular size AApparent angular size Asizesize of the black hole can be considered as the of the black hole can be considered as the angular size of the black region of the observer´s sky, thus is given by angular size of the black region of the observer´s sky, thus is given by maximum value of the directional angle maximum value of the directional angle αα . .

For For static observers static observers above the circular photon orbit above the circular photon orbit AAsize size is given asis given as

22

lim

3211arccos2)),,(lim(2

lim

rrM

rbrbA obsstat

bbsize

22

lim

3211arccos2)),,(lim(2

lim

rrM

rbrbA obsstat

bbsize

Apparent angular size of the black hole for static Apparent angular size of the black hole for static observers as a function of the cosmological constantobservers as a function of the cosmological constant

Zoom near event horizons

Zoom near the photon orbit

Apparent angular size of the black hole for radially free falling Apparent angular size of the black hole for radially free falling observers as a function of the cosmological constantobservers as a function of the cosmological constant

Observers are free falling from apropriate static radius

The angular size increases with cosmological constant, the smallest angular size angular size matches matches pure Schwarzschild case.

For observers just in the singularity the angular size is independent on the value of cosmological constant and it is allways π , one half ( all inward hemisphere ) of the observer sky. Situation is similar to static observers on the photon orbit.Angular size for free falling observers is always smaller then for the static one in the same radial coordinate.

Simulation : Free-falling observer from infinity to Simulation : Free-falling observer from infinity to the the event horizon event horizon in pure Schwarzschiin pure Schwarzschilld cased case.. The virtual black hole is between observer and Galaxy M104 „Sombrero“. The virtual black hole is between observer and Galaxy M104 „Sombrero“.

Nondistorted image of M104 robs =100M

robs =40M

robs =50M

robs =15M

Simulation : Observer falling from 10M to the rest Simulation : Observer falling from 10M to the rest on on thethe event horizon event horizon

Galaxy „Sombrero“ is Galaxy „Sombrero“ is inin the observer sky. the observer sky.

Computer implementationComputer implementation The cThe code ode BHC_IMPACTBHC_IMPACT is developed iis developed in C language, compilated by n C language, compilated by GCC and MPICC compilers, OS LINUX. Libraries NUMERICAL GCC and MPICC compilers, OS LINUX. Libraries NUMERICAL RECIPES, MPI and RECIPES, MPI and LLIGHTSPEED! were used. We used IGHTSPEED! were used. We used IBM IBM BladeCenter BladeCenter with 8 with 8 AMD Dual Core Opteron 64 AMD Dual Core Opteron 64 CPUs for simulation CPUs for simulation run.run.

One bitmap image of nondistorted One bitmap image of nondistorted objects objects is is the ithe input for nput for the the simulation.simulation. We assumeWe assume that that it is projection it is projection of part of part of the observer sky in direction of of the observer sky in direction of the black holethe black hole in flat spacetime. in flat spacetime.

Two bitmap images are Two bitmap images are generated as generated as an output. The first image is the an output. The first image is the view in direction of the black hole, the second one is the view in the view in direction of the black hole, the second one is the view in the opposite direction. opposite direction.

Only the first three images are generated by the simulation. Only the first three images are generated by the simulation. The iThe intentensnsity ity of higher order images rapidly decreaseof higher order images rapidly decreasess and and theirtheir positions merge with positions merge with the second Einstein ring. the second Einstein ring. However, tHowever, the intensity ratio between images he intensity ratio between images with different orders is with different orders is ununrealistic. Computer displays have norealistic. Computer displays have not t required required bright bright resolution.

Future plansFuture plans

Simulation for rotating spacetimes with Simulation for rotating spacetimes with cosmological constant, e.g. Kerr–de Sitter cosmological constant, e.g. Kerr–de Sitter spacetimes.spacetimes.

Using core of this code for modeling of Using core of this code for modeling of light curves – in preparation.light curves – in preparation.

End

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