A black hole has no-hair (no-hair theorem) Made only of spacetime warpage Mass M Spin: angular momentum J Charge Q J = a GM2/c 0 |a| 1 RS = 2GM c2 R. Nemmen
dependent on spin and inclination Bardeen 1973 R = 9/2M(a Nonspinning Spinning (a = 1) Black hole casts apparent shadow on light from surrounding accretion flow
solar masses ~107 per galaxy Stellar black holes ~1 Mpc ~100 kpc Active galactic nuclei Quasars Radio galaxies black holes Gamma- ray bursts Microquasars 1 pc = 3×1013 km
Fourier transform of T(l,m), the sky brightness distribution (for incoherent source, small field of view, far field, etc.) [for derivation from van Cittert-Zernike theorem, see TMS Ch. 14] ! • mathematically u,v are E-W, N-S spatial frequencies [wavelengths]" l,m are E-W, N-S angles in the tangent plane [radians]" (recall ) T(l,m) van Cittert-Zernike theorem, see TMS Ch. 14] al frequencies [wavelengths]" s in the tangent plane [radians]" ) 19 Interferometry basics visibility sky brightness Visibility and Sky Brightness • V(u,v), the complex visibility function, is the 2D Fourier trans sky brightness distribution (for incoherent source, small field [for derivation from van Cittert-Zernike theorem, see T ! • mathematically u,v are E-W, N-S spatial frequencies [wavelengths]" l,m are E-W, N-S angles in the tangent plane [radians]" (recall ) Visibility and Sky • V(u,v), the complex visibility fun sky brightness distribution (for i [for derivation from van Citt ! • mathematically u,v are E-W, N-S spatial freque l,m are E-W, N-S angles in the (recall ) Interferometer observes 2D components u,v of Fourier transform of sky brightness
MHD condition Kerr metric Conservation of Particle number Energy-momentum r⌫(⇢u⌫) = 0 r⌫Tµ⌫ = 0 r⌫ ⇤ Fµ⌫ = 0 r⌫Fµ⌫ = Jµ Maxwell equations r⌫ ⇤ Fµ⌫ = 0 r⌫Fµ⌫ = Jµ Fµ⌫u⌫ = 0 ds2 = ↵2dt2 + ij(dxi + p = ( 1)⇢✏ ;l s the stress energy tensor. In a coordinate basis, ffiffiffiffiffiffiffi À g p Tt Á ¼ À @i ffiffiffiffiffiffiffi À g p Ti À Á þ ffiffiffiffiffiffiffi À g p T À ; ð4Þ notes a spatial index and À is the connection. rgy momentum equations have been written with dex down for a reason. Symmetries of the metric conserved currents. In the Kerr metric, for exam- xisymmetry and stationary nature of the metric o conserved angular momentum and energy cur- eneral, for metrics with an ignorable coordinate rce terms on the right-hand side of the evolution or Tt l vanish. These source terms do not vanish quation is written with both indices up. ss energy tensor for a system containing only a id and an electromagnetic field is the sum of a Tl fluid ¼ ð þ u þ pÞulu þ pgl ð5Þ The rest of M and are not n MHD. Maxwell’s by taking the Here FÃ l ¼ 1 2 tensor (MTW which can be The comp blul ¼ 0. Fol where i denotes a spatial index and À is the The energy momentum equations have bee the free index down for a reason. Symmetrie give rise to conserved currents. In the Kerr me ple, the axisymmetry and stationary nature give rise to conserved angular momentum a rents. In general, for metrics with an ignora xl the source terms on the right-hand side o equation for Tt l vanish. These source terms when the equation is written with both indices The stress energy tensor for a system con perfect fluid and an electromagnetic field is fluid part, Tl fluid ¼ ð þ u þ pÞulu þ pgl (here u internal energy and p press electromagnetic part, Tl EM ¼ Fl F À 1 4 glF F :
tracing in curved spacetimes Geodesic equation Radiative transfer 3. GEODESIC INTEGRATION General relativistic radiative transfer differs from conven- ional radiative transfer in Minkowski space in that photon tra- ectories are no longer trivial; photons move along geodesics. Tracking geodesics is a significant computational expense in grmonty. The governing equations for a photon trajectory are dxα dλ = kα (11) which defines λ, the affine parameter, the geodesic equation dkα dλ = −Γα µν kµkν, (12) and the definition of the connection coefficients With ε = 0.04, grmonty integrates ∼16,700 geodesics s−1 on a single core of an Intel Xeon model E5430. If we use fourth- order Runge–Kutta exclusively so that the error in E, l, and Q is ∼1000 times smaller, then the speed is ∼ 6200 geodesics s−1. If we use the Runge–Kutta Prince–Dorman method in GSL with ε = 0.04 the fraction error is ∼ 10−10 and the speed is ∼1700 geodesics s−1. These results can be compared to the publicly available integral-based geokerr code of Dexter & Agol (2009), whose geodesics are shown as the (more accurate) solid lines in Figure 1. If we use geokerr to sample each geodesic the same number of times as grmonty (∼180), then on the same machine geokerr runs at ∼1000 geodesics s−1. It is possible that other implementations of an integral-of-motion- based geodesic tracker could be faster. If only the initial and final states of the photon are required, we find that geokerr computes ∼77,000 geodesics s−1. The adaptive Runge–Kutta Cash–Karp integrator in GSL computes ∼34,500 geodesics s−1 with fractional error ∼10−3. 4. ABSORPTION grmonty treats absorption deterministically. We begin with the radiative transfer equation written in the covariant form 1 C d dλ Iν ν3 = jν ν2 − (ναν,a ) Iν ν3 . (15) (see Mihalas & Mihalas 1984). Here Iν is specific intensity and for example, i Since Iν/ν3 along each ray emission) where is the different parentheses is with second-o τ and then set Since the com rest-mass ene opacity at the be reused as th Our treatm determines w
3. GEODESIC INTEGRATION General relativistic radiative transfer differs from conven- ional radiative transfer in Minkowski space in that photon tra- ectories are no longer trivial; photons move along geodesics. Tracking geodesics is a significant computational expense in grmonty. The governing equations for a photon trajectory are dxα dλ = kα (11) which defines λ, the affine parameter, the geodesic equation dkα dλ = −Γα µν kµkν, (12) and the definition of the connection coefficients Black hole appearance: Ray tracing in curved spacetimes publicly available integral-based geokerr code of Dexter Agol (2009), whose geodesics are shown as the (more accura solid lines in Figure 1. If we use geokerr to sample ea geodesic the same number of times as grmonty (∼180), th on the same machine geokerr runs at ∼1000 geodesics s−1 is possible that other implementations of an integral-of-moti based geodesic tracker could be faster. If only the initial and final states of the photon are requir we find that geokerr computes ∼77,000 geodesics s−1. T adaptive Runge–Kutta Cash–Karp integrator in GSL compu ∼34,500 geodesics s−1 with fractional error ∼10−3. 4. ABSORPTION grmonty treats absorption deterministically. We begin w the radiative transfer equation written in the covariant form 1 C d dλ Iν ν3 = jν ν2 − (ναν,a ) Iν ν3 . ( absorption absorption
Cafardo PhD Raniere Menezes PhD Ivan Almeida Msc Rodrigo Nemmen Apply to join my group Roberta Pereira undergrad (IC) blackholegroup.org +Caio Salgado, IC +Edson Ponciano, IC
mm- wavelengths Goals: Direct image of event horizons Test general relativity (strong field) Probe black hole “gastrophysics” First observation with full array: April 2017 Work in progress… R. Nemmen Rodrigo Nemmen IAG USP
The first picture of a black hole E v e n t H o r i z o n Telescope attains the impossible BURACO NEGRO NO CENTRO DE M87 FAZ DENÚNCIAS GRAVES CONTRA O PT 2017 EXCLUSIVO
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