Black holes and tests of gravitation
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1 Black holes and tests of gravitation Éric Gourgoulhon Laboratoire Univers et Théories (LUTH) CNRS / Observatoire de Paris / Université Paris Diderot Meudon, France eric.gourgoulhon@obspm.fr École Polytechnique Palaiseau, 27 January 2015 Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
2 Outline 1 Black holes in general relativity 2 Astrophysical black holes 3 The near-future observations of black holes 4 Tests of gravitation Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
3 Outline Black holes in general relativity 1 Black holes in general relativity 2 Astrophysical black holes 3 The near-future observations of black holes 4 Tests of gravitation Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
4 Black holes in general relativity What is a black hole?... for the layman: A black hole is a region of spacetime from which nothing, not even light, can escape. The (immaterial) boundary between the black hole interior and the rest of the Universe is called the event horizon. [Alain Riazuelo, 2007] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
5 Black holes in general relativity What is a black hole?... for the mathematical physicist: black hole: B := M J (I + ) i.e. the region of spacetime where light rays cannot escape to infinity (M, g) = asymptotically flat manifold I + = future null infinity J (I + ) = causal past of I + event horizon: H := J (I + ) (boundary of J (I + )) H smooth = H null hypersurface Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
6 Black holes in general relativity What is a black hole?... for the mathematical physicist: black hole: B := M J (I + ) i.e. the region of spacetime where light rays cannot escape to infinity (M, g) = asymptotically flat manifold I + = future null infinity J (I + ) = causal past of I + event horizon: H := J (I + ) (boundary of J (I + )) H smooth = H null hypersurface Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
7 Black holes in general relativity What is a black hole? singularité temps horizon des événements tube d'univers de la surface de l'étoile spacetime diagram depicting the formation of a black hole from the gravitational collapse of the core of a massive star (supernova phenomenon) espace t = 0 singularity: curvature Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
8 Black holes in general relativity What is a black hole?... for the astrophysicist: a very deep gravitational potential well Release of potential gravitational energy by accretion on a black hole: up to 42% of the mass-energy mc 2 of accreted matter! NB: thermonuclear reactions release less than 1% mc 2 Matter falling in a black hole forms an accretion disk [Lynden-Bell (1969), Shakura & Sunayev (1973)] [J.-A. Marck (1996)] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
9 Black holes in general relativity Main properties of black holes (1/3) In general relativity, a black hole contains a region where the spacetime curvature diverges: the singularity (NB: this is not the primary definition of a black hole). The singularity is inaccessible to observations, being hidden by the event horizon. Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
10 Black holes in general relativity Main properties of black holes (1/3) In general relativity, a black hole contains a region where the spacetime curvature diverges: the singularity (NB: this is not the primary definition of a black hole). The singularity is inaccessible to observations, being hidden by the event horizon. The singularity marks the limit of validity of general relativity: to describe it, a quantum theory of gravitation would be required. Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
11 Black holes in general relativity Main properties of black holes (1/3) In general relativity, a black hole contains a region where the spacetime curvature diverges: the singularity (NB: this is not the primary definition of a black hole). The singularity is inaccessible to observations, being hidden by the event horizon. The singularity marks the limit of validity of general relativity: to describe it, a quantum theory of gravitation would be required. The event horizon H is a global structure of spacetime: no physical experiment whatsoever can detect the crossing of H. Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
12 Black holes in general relativity Main properties of black holes (2/3) The event horizon as a null cone t = T t = T temps cône de lumière horizon des événements espace t = 0 t = 0 espace-temps plat espace-temps avec trou noir Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
13 Black holes in general relativity Main properties of black holes (3/3) Viewed by a distant observer, the horizon approach is perceived with an infinite redshift, or equivalently, by an infinite time dilation A black hole is not an infinitely dense object: on the contrary it is made of vacuum (except maybe at the singularity); if one defines its mean density by ρ = M/(4/3πR 3 ), then for the Galatic center BH (Sgr A*): ρ 10 6 kg m ρ white dwarf for the BH at the center of M87: ρ 2 kg m ρ water! = a black hole is a compact object: M R large, not M R 3! Due to the non-linearity of general relativity, black holes can form in spacetimes empty of any matter, by collapse of gravitational wave packets. Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
14 Black holes in general relativity The no-hair theorem Dorochkevitch, Novikov & Zeldovitch (1965), Israel (1967), Carter (1971), Hawking (1972) Within 4-dimensional general relativity, a black hole in equilibrium in an otherwise empty universe is necessarily a Kerr-Newmann black hole, which is a vacuum solution of Einstein described by only three parameters: the total mass M the total angular momentum J the total electric charge Q = a black hole has no hair (John A. Wheeler) Astrophysical black holes have to be electrically neutral: Q = 0 : Kerr solution (1963) Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
15 Black holes in general relativity The Kerr solution Roy Kerr (1963) g αβ dx α dx β = where ρ 2 := r 2 + a 2 cos 2 θ, ( 1 2GMr ) c 2 ρ 2 c 2 dt 2 4GMar sin2 θ c 2 ρ ( 2 +ρ 2 dθ 2 + r 2 + a 2 + 2GMa2 r sin 2 θ c 2 ρ 2 c dt dϕ + ρ2 dr2 ) sin 2 θ dϕ 2 := r 2 2GM c 2 r + a 2, a := J cm event horizon (black hole) a GM c 2 Schwarzschild subcase (a = 0): ( g αβ dx α dx β = 1 2GM ) ( c 2 c 2 dt GM ) 1 r c 2 dr 2 +r 2 ( dθ 2 + sin 2 θ dϕ 2) r Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
16 Black holes in general relativity The black hole parameters The mass M is not some measure of the amount of matter inside the black hole, but rather a parameter characterizing the external gravitational field; it is measurable from the orbital period of a test particle in circular orbit around the black hole and far from it (Kepler s third law). Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
17 Black holes in general relativity The black hole parameters The mass M is not some measure of the amount of matter inside the black hole, but rather a parameter characterizing the external gravitational field; it is measurable from the orbital period of a test particle in circular orbit around the black hole and far from it (Kepler s third law). Similarly, the angular momentum J = cam is a parameter characterizing the gravito-magnetic part of the external gravitational field. It is measurable from the precession of a gyroscope orbiting the black hole (Lense-Thirring effect). Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
18 Black holes in general relativity The black hole parameters The mass M is not some measure of the amount of matter inside the black hole, but rather a parameter characterizing the external gravitational field; it is measurable from the orbital period of a test particle in circular orbit around the black hole and far from it (Kepler s third law). Similarly, the angular momentum J = cam is a parameter characterizing the gravito-magnetic part of the external gravitational field. It is measurable from the precession of a gyroscope orbiting the black hole (Lense-Thirring effect). Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
19 Black holes in general relativity The black hole parameters The mass M is not some measure of the amount of matter inside the black hole, but rather a parameter characterizing the external gravitational field; it is measurable from the orbital period of a test particle in circular orbit around the black hole and far from it (Kepler s third law). Similarly, the angular momentum J = cam is a parameter characterizing the gravito-magnetic part of the external gravitational field. It is measurable from the precession of a gyroscope orbiting the black hole (Lense-Thirring effect). Remark: the radius of a black hole is not a well defined concept: it does not correspond to some distance between the black hole centre (the singularity) and the event horizon. A well defined quantity is the area of the event horizon, A. The radius can be then defined from it: for a Schwarzschild black hole: A R := 4π = 2GM ( ) M c 2 3 km M Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
20 Black holes in general relativity Why is the Kerr metric special? Spherically symmetric (non-rotating) case: Birkhoff theorem Within 4-dimensional general relativity, the spacetime outside any spherically symmetric body is described by Schwarzschild metric = No possibility to distinguish a non-rotating black hole from a non-rotating dark star by monitoring orbital motion or fitting accretion disk spectra Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
21 Black holes in general relativity Why is the Kerr metric special? Spherically symmetric (non-rotating) case: Birkhoff theorem Within 4-dimensional general relativity, the spacetime outside any spherically symmetric body is described by Schwarzschild metric = No possibility to distinguish a non-rotating black hole from a non-rotating dark star by monitoring orbital motion or fitting accretion disk spectra Rotating axisymmetric case: No Birkhoff theorem Moreover, no reasonable matter source has ever been found for the Kerr metric (the only known source consists of two counter-rotating thin disks of collisionless particles [Bicak & Ledvinka, PRL 71, 1669 (1993)]) = The Kerr metric is specific to rotating black holes (in 4-dimensional general relativity) Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
22 Black holes in general relativity Lowest order no-hair theorem: quadrupole moment Asymptotic expansion (large r) of the metric in terms of multipole moments (M k, J k ) k N [Geroch (1970), Hansen (1974)]: M k : mass 2 k -pole moment J k : angular momentum 2 k -pole moment = For the Kerr metric, all the multipole moments are determined by (M, a): M 0 = M J 1 = am = J/c M 2 = a 2 M = J 2 J 3 = a 3 M M 4 = a 4 M c 2 M ( ) mass quadrupole moment Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
23 Black holes in general relativity Lowest order no-hair theorem: quadrupole moment Asymptotic expansion (large r) of the metric in terms of multipole moments (M k, J k ) k N [Geroch (1970), Hansen (1974)]: M k : mass 2 k -pole moment J k : angular momentum 2 k -pole moment = For the Kerr metric, all the multipole moments are determined by (M, a): M 0 = M J 1 = am = J/c M 2 = a 2 M = J 2 J 3 = a 3 M M 4 = a 4 M c 2 M ( ) mass quadrupole moment Measuring the three quantities M, J, M 2 provides a compatibility test w.r.t. the Kerr metric, by checking ( ) Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
24 Black holes in general relativity Other theoretical aspects The four laws of black hole dynamics Quantum properties (Bekenstein entropy, Hawking radiation) Black holes in higher dimensions Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
25 Outline Astrophysical black holes 1 Black holes in general relativity 2 Astrophysical black holes 3 The near-future observations of black holes 4 Tests of gravitation Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
26 Astrophysical black holes Known black holes Three kinds of black holes are known in the Universe: Stellar black holes: supernova remnants: M M and R km example: Cyg X-1 : M = 15 M and R = 45 km Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
27 Astrophysical black holes Known black holes Three kinds of black holes are known in the Universe: Stellar black holes: supernova remnants: M M and R km example: Cyg X-1 : M = 15 M and R = 45 km Supermassive black holes, in galactic nuclei: M M and R km 200 UA example: Sgr A* : M = M and R = km = 18 R = 0.09 UA = 1 4 radius of Mercury s orbit Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
28 Astrophysical black holes Known black holes Three kinds of black holes are known in the Universe: Stellar black holes: supernova remnants: M M and R km example: Cyg X-1 : M = 15 M and R = 45 km Supermassive black holes, in galactic nuclei: M M and R km 200 UA example: Sgr A* : M = M and R = km = 18 R = 0.09 UA = 1 4 radius of Mercury s orbit Intermediate mass black holes, as ultra-luminous X-ray sources (?): M M and R 300 km km example: ESO HLX-1 : M > 500 M and R > 1500 km Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
29 Astrophysical black holes Stellar black holes in X-ray binaries 20 identified stellar black holes in our galaxy Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
30 Astrophysical black holes Stellar black holes in X-ray binaries [McClintock et al. (2011)] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
31 Astrophysical black holes Supermassive black holes in active galactic nuclei (AGN) Jet emitted by the nucleus of the giant elliptic galaxy M87, at the centre of Virgo cluster [HST] M BH = M V jet 0.99 c Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
![Astrophysical black holes The black hole at the centre of our galaxy: Sgr A* [ESO (2009)] Measure of the mass of Sgr A* black hole by stellar dynamics: M BH = 4.](/docs-images/73/68537290/images/32-1.jpg "3 10 6 M Orbit of the star S2 around Sgr A* P = 16 yr, r per = 120 UA = 1400 R S, V per = 0.")
32 Astrophysical black holes The black hole at the centre of our galaxy: Sgr A* [ESO (2009)] Measure of the mass of Sgr A* black hole by stellar dynamics: M BH = M Orbit of the star S2 around Sgr A* P = 16 yr, r per = 120 UA = 1400 R S, V per = 0.02 c [Genzel, Eisenhauer & Gillessen, RMP 82, 3121 (2010)] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
33 Outline The near-future observations of black holes 1 Black holes in general relativity 2 Astrophysical black holes 3 The near-future observations of black holes 4 Tests of gravitation Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
34 The near-future observations of black holes Can we see a black hole from the Earth? Angular diameter of the event horizon of a Schwarzschild BH of mass M seen from a distance d: Θ = 6 3 GM c 2 d 2.602R S d Image of a thin accretion disk around a Schwarzschild BH [Vincent, Paumard, Gourgoulhon & Perrin, CQG 28, (2011)] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
35 The near-future observations of black holes Can we see a black hole from the Earth? Angular diameter of the event horizon of a Schwarzschild BH of mass M seen from a distance d: Θ = 6 3 GM c 2 d 2.602R S d Largest black holes in the Earth s sky: Sgr A* : Θ = 53 µas M87 : Θ = 21 µas M31 : Θ = 20 µas Image of a thin accretion disk around a Schwarzschild BH [Vincent, Paumard, Gourgoulhon & Perrin, CQG 28, (2011)] Remark: black holes in X-ray binaries are 10 5 times smaller, for Θ M/d Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
36 The near-future observations of black holes The solution to reach the µas regime: interferometry! Very Large Baseline Interferometry (VLBI) in (sub)millimeter waves Existing American VLBI network E ric Gourgoulhon (LUTH) [Doeleman et al. 2011] Black holes and tests of gravitation E cole Polytechnique, 27 January / 52
37 The near-future observations of black holes The solution to reach the µas regime: interferometry! Very Large Baseline Interferometry (VLBI) in (sub)millimeter waves The best result so far: VLBI observations at 1.3 mm have shown that the size of the emitting region in Sgr A* is only 37 µas [Doeleman et al., Nature 455, 78 (2008)] Existing American VLBI network E ric Gourgoulhon (LUTH) [Doeleman et al. 2011] Black holes and tests of gravitation E cole Polytechnique, 27 January / 52
38 The near-future observations of black holes The near future: the Event Horizon Telescope To go further: shorten the wavelength: 1.3 mm 0.8 mm increase the number of stations; in particular add ALMA Atacama Large Millimeter Array (ALMA) part of the Event Horizon Telescope (EHT) to be completed by 2020 Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
39 The near-future observations of black holes Near-infrared optical interferometry: GRAVITY GRAVITY instrument at VLTI (late 2015) Beam combiner (the four 8 m telescopes + four auxiliary telescopes) = astrometric precision of 10 µas [Gillessen et al. 2010] E ric Gourgoulhon (LUTH) Black holes and tests of gravitation E cole Polytechnique, 27 January / 52
40 The near-future observations of black holes X-ray observations (Athena) The accretion disk as a spacetime probe intensité aile rouge aile bleue Kα line: X fluorescence line of Fe atoms in the accretion disk (the Fe atoms are excited by the X-ray emitted from the plasma corona surrounding the disk) Redshift time dilatation laboratoire énergie (kev) Kα line in the nucleus of the galaxy MCG observed by XMM-Newton (red) and Suzaku (black) (adapted from [Miller (2007)]) Athena X-ray observatory selected in 2014 for ESA L2 mission = launch 2028 Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
41 The near-future observations of black holes Another way to see BHs: gravitational waves Link between black holes and gravitational waves: Black holes and gravitational waves are both spacetime distortions: extreme distortions (black holes) small distortions (gravitational waves) In particular, black holes and gravitational waves are both vacuum solutions of general relativity equations (Einstein equations) Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
42 The near-future observations of black holes VIRGO: a giant Michelson interferometer... Gravitational wave detector VIRGO in Cascina, near Pisa (Italy) [CNRS/INFN] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
43 The near-future observations of black holes Optical scheme of the VIRGO interferometer Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
44 The near-future observations of black holes VIRGO sensitivity curve Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
45 The near-future observations of black holes Advanced VIRGO Advanced VIRGO: dual recycled (power + signal) interferometer with laser power 125 W VIRGO+ decommissioned in Nov Construction of Advanced VIRGO underway First lock in 2015 Sensitivity 10 VIRGO = explored Universe volume 10 3 times larger! [CNRS/INFN/NIKHEF] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
46 The near-future observations of black holes International Pulsar Timing Array (IPTA) [Hobbs et al., CQG 27, (2010)] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
![org/] elisa scientific theme selected in Nov.](/docs-images/73/68537290/images/47-1.jpg "2013 for ESA L3 mission = launch in 2028 LISA Pathfinder: launch in Sept.-Nov.")
47 elisa The near-future observations of black holes Gravitational wave detector in space = low frequency range: [10 3, 10 1 ] Hz [ elisa scientific theme selected in Nov for ESA L3 mission = launch in 2028 LISA Pathfinder: launch in Sept.-Nov Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
48 Outline Tests of gravitation 1 Black holes in general relativity 2 Astrophysical black holes 3 The near-future observations of black holes 4 Tests of gravitation Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
49 Tests of gravitation Theoretical alternatives to the Kerr black hole Within general relativity The compact object is not a black hole but a boson star a gravastar a dark star... Beyond general relativity The compact object is a black hole but in a theory that differs from GR: Einstein-Gauss-Bonnet with dilaton Chern-Simons gravity Hořava-Lifshitz gravity Einstein-Yang-Mills... Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
50 Tests of gravitation How to test the alternatives to the Kerr black hole? Search for stellar orbits deviating from Kerr timelike geodesics (GRAVITY) accretion disk spectra different from those arising in Kerr metric (X-ray observatories) images of the black hole shadow different from that of a Kerr black hole (EHT) Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
51 Tests of gravitation How to test the alternatives to the Kerr black hole? Search for stellar orbits deviating from Kerr timelike geodesics (GRAVITY) accretion disk spectra different from those arising in Kerr metric (X-ray observatories) images of the black hole shadow different from that of a Kerr black hole (EHT) Need for a good and versatile geodesic integrator to compute timelike geodesics (orbits) and null geodesics (ray-tracing) in any kind of metric Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
52 Tests of gravitation Black hole images from null geodesics Simulated image of a thin accretion disk around a Schwarzschild black hole [Vincent, Paumard, Gourgoulhon & Perrin, CQG 28, (2011)] Light ray trajectories fr/~luminet/ Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
53 Tests of gravitation Flight to a black hole Images computed by J.-A. Marck [Marck, CQG 13, 393 (1996)] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
54 Tests of gravitation Flight to a black hole Images computed by J.-A. Marck [Marck, CQG 13, 393 (1996)] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
55 Tests of gravitation Flight to a black hole Images computed by J.-A. Marck [Marck, CQG 13, 393 (1996)] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
56 Tests of gravitation Flight to a black hole Images computed by J.-A. Marck [Marck, CQG 13, 393 (1996)] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
57 Tests of gravitation Flight to a black hole Images computed by J.-A. Marck [Marck, CQG 13, 393 (1996)] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
58 Tests of gravitation Flight to a black hole Images computed by J.-A. Marck [Marck, CQG 13, 393 (1996)] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
59 Tests of gravitation Flight to a black hole Images computed by J.-A. Marck [Marck, CQG 13, 393 (1996)] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
60 Tests of gravitation Flight to a black hole Images computed by J.-A. Marck [Marck, CQG 13, 393 (1996)] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
61 Tests of gravitation Flight to a black hole Images computed by J.-A. Marck [Marck, CQG 13, 393 (1996)] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
62 Tests of gravitation Flight to a black hole Images computed by J.-A. Marck [Marck, CQG 13, 393 (1996)] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
63 Tests of gravitation Flight to a black hole Images computed by J.-A. Marck [Marck, CQG 13, 393 (1996)] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
64 Tests of gravitation Isolated black hole in front of a stellar background Image computed by A. Riazuello (IAP) [Riazuelo, 2007] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
65 Tests of gravitation Isolated black hole in front of a stellar background Image computed by A. Riazuello (IAP) [Riazuelo, 2007] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
66 Tests of gravitation Isolated black hole in front of a stellar background Image computed by A. Riazuello (IAP) [Riazuelo, 2007] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
67 Gyoto code Tests of gravitation Main developers: T. Paumard & F. Vincent Integration of geodesics in Kerr metric Integration of geodesics in any numerically computed 3+1 metric Radiative transfer included in optically thin media Very modular code (C++) Yorick interface Free software (GPL) : [Vincent, Paumard, Gourgoulhon & Perrin, CQG 28, (2011)] [Vincent, Gourgoulhon & Novak, CQG 29, (2012)] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
68 Gyoto code Tests of gravitation Computed images of a thin accretion disk around a Schwarzschild black hole Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
69 Tests of gravitation Measuring the spin from the black hole silhouette Ray-tracing in the Kerr metric (spin parameter a) Accretion structure around Sgr A* modelled as a ion torus, derived from the polish doughnut class [Abramowicz, Jaroszynski & Sikora (1978)] Radiative processes included: thermal synchrotron, bremsstrahlung, inverse Compton Image of an ion torus computed with Gyoto for the inclination angle i = 80 : black: a = 0.5M red: a = 0.9M [Straub, Vincent, Abramowicz, Gourgoulhon & Paumard, A&A 543, A83 (2012)] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
70 Tests of gravitation Orbits around a rotating boson star Boson star = localized configurations of a self-gravitating complex scalar field Φ Klein-Gordon geons [Bonazzola & Pacini (1966), Kaup (1968)] Boson stars may behave as black-hole mimickers y [h _ /m] Solutions of the Einstein-Klein-Gordon system computed by means of Kadath [Grandclément, JCP 229, 3334 (2010)] Timelike geodesics computed by means of Gyoto x [h _ /m] Zero-angular-momentum orbit around a rotating boson star based on a free scalar field Φ = φ(r, θ)e i(ωt+2ϕ) with ω = 0.75 m/. [Granclément, Somé & Gourgoulhon, PRD 90, (2014)] Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
71 Tests of gravitation Another tool to explore black hole spacetimes: SageManifolds Symbolic differential geometry based on the modern free open-source mathematics software system Sage Sage is based on the Python programming language it makes use of many pre-existing open-sources packages, among which Maxima (symbolic calculations, since 1968!) GAP (group theory) PARI/GP (number theory) Singular (polynomial computations) matplotlib (high quality 2D figures) and provides a uniform interface to them William Stein (Univ. of Washington) created Sage in 2005; since then, 100 developers (mostly mathematicians) have joined the Sage team The mission Create a viable free open source alternative to Magma, Maple, Mathematica and Matlab. Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
72 Tests of gravitation The SageManifolds project Aim Implement the concept of real smooth manifolds of arbitrary dimension in Sage and tensor calculus on them, in a coordinate/frame-independent manner Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
73 Tests of gravitation The SageManifolds project Aim Implement the concept of real smooth manifolds of arbitrary dimension in Sage and tensor calculus on them, in a coordinate/frame-independent manner In practice, this amounts to introducing new Python classes in Sage, basically one class per mathematical concept, for instance: Manifold: differentiable manifolds over R, of arbitrary dimension Chart: coordinate charts Point: points on a manifold DiffMapping: differential mappings between manifolds ScalarField, VectorField, TensorField: tensor fields on a manifold DiffForm: p-forms AffConnection, LeviCivitaConnection: affine connections Metric: pseudo-riemannian metrics Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
74 Tests of gravitation Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
75 Tests of gravitation Éric Gourgoulhon (LUTH) Black holes and tests of gravitation École Polytechnique, 27 January / 52
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