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 , Kamila Truparová ,

Stanislav Hledík and Zdeněk Stuchlík

Institute of PhysicsFaculty of Philosophy and Science

Silesian University in Opava Czech Republic

Eleventh Marcel Grossmann Meeting on General Relativity

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

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

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

Formulation of the problemFormulation of the problem

222221

2222 sin3

21

3

21 ddrdrr

r

Mdtr

r

Mds

Schwarzschild – de Sitter metricSchwarzschild – de Sitter metric

Black hole horizonBlack hole horizon

Cosmological horizon Cosmological horizon

Static radiusStatic radius

Critical value of cosm. constantCritical value of cosm. constant

3

43arccoscos

2 Mrevent

33

Mrstat

29

1

Mcrit

3

3arccoscos

2 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 in equatorial planemotion in 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

3

211

3

21

2

22

12

p

Er

bp

Err

M

r

bp

Err

Mp

r

t

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

Muubdu

d

ru

p

p

dr

dr

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 Existence of maximal impact parameter for observers above the circular photon orbit. Geodesics with orbit. 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,

32max

obsobs

obs

Muu

ub

MrM

b pho 3,91

272lim

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

327arccos

3

1cos

33

2, 2

2

bM

b

br turn

bM

bbr turn

27arccos

3

1cos

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 on for observers 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 turn point. the turn point.

obs

source

u

u

obs

Muub

duu

32 322

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

turn

u

u

obs u

Muub

duu

obs

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

u

turn

Muub

duu

32 322

Starting pointStarting point of of thethe numerical solution numerical solutionWe can rewrite the final equation for observers We can rewrite the final 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 obssourceobs

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 value of k for geodesics orbiting counter-clokwise. Infinite value of k corresponds to a photon capture on the circular photon orbit.corresponds 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 one has to transform the 4-momentum into local coordinate system of the static observer. of the static observer. Local components of Local components of 4-4-momentummomentum for the static observer in equatorial plane can be obtained for the static observer in equatorial plane can be obtained using appropriate tetrad of 1-form using appropriate tetrad of 1-form ωω((αα))

dr

rd

drrr

M

dtrr

M

r

t

sin

3

21

3

21

2

1

2

2

rp

p

rrM

rrM

rb

p

rr

MEp

r

t

0

32

1

32

11

3

21

2

22

2

1

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

2

2

3

211arccosarccos r

r

M

r

b

p

pt

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

32

1

32

1

obsobs

sourcee

tsource

tobs

source

obs

rrM

rrM

p

p

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 The size of black region expands with decreasing radial coordinate of observer but decreasesdecreases 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=20M=20M

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

View of outward direction

Some parts of Some parts of image are moving image are moving into 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 up to its maximum value, which defindefineses 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 observer sky now. The size of black region now expands withpart of the observer sky now. The size of black region now expands with increincreaasing sing value of cosmologival constantvalue of cosmologival 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=3M

Observer on the photon Observer on the photon orbit would be blinded orbit would be blinded and burned by captured and burned by captured photons.photons.

Outward direction view, Outward direction view, whole image is moving whole image is moving into opposite 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 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 π , one half ( all inward hemisphere ) of the observer sky.

From observers under and on the photon orbit angular size is given asFrom observers under and on the photon orbit angular size is 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 AApparent angular Asizesize can be considered as border of the black region can be considered as border of the black region

of the static observer´s sky, thus is given by of the static observer´s sky, thus is given by maximum value of the maximum value of the angle angle ααstatstat . .

From observers above the photon orbit angular size is given asFrom observers above the photon orbit angular size is given as

22

lim

3

211arccos)),,(lim(2

lim

rr

M

r

brbA obsstat

bbsize

22

lim

3

211arccos)),,(lim(2

lim

rr

M

r

brbA obsstat

bbsize

Apparent angular size of the black holeApparent angular size of the black holeas a function of the cosmological constantas a function of the cosmological constant

Zoom near event horizons

Zoom near the photon orbit

Simulation : Free-falling observer from infinity to Simulation : Free-falling observer from infinity to the the event horizon event horizon in pure Schwarzschid casein pure Schwarzschid 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 andand SGI ALTIX 350 SGI ALTIX 350 with 8 Itanium II CPUs for simulation with 8 Itanium II 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.

End

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