PoS(P2GC)004 http://pos.sissa.it/ ce. t objects at galactic centers. hysics. The emphasis is made on the ive Commons Attribution-NonCommercial-ShareAlike Licen ∗ ´ c [email protected] phenomenology of X-ray binaries and of supermassive compac This set of lectures is an introduction to black-hole astrop Speaker. ∗ Copyright owned by the author(s) under the terms of the Creat c School on Particle Physics, Gravity andAugust Cosmology 21 -September 2, 2006 Dubrovnik

Rudjer Boškovi´cIntitute, Zagreb, Craotia E-mail: Neven Bili Black-Hole Phenomenology PoS(P2GC)004 ´ c 4 4 4 5 6 6 6 7 8 8 8 24 26 26 27 27 27 28 10 12 13 15 16 17 18 18 18 19 20 21 21 21 22 23 Neven Bili 2 ∗ 6.1.1 Gas Spectroscopy 6.1.2 Maser Interferometry 6.1.3 Virial Mass 3.3.1 Spherical Collapse 3.3.2 Eddington-Finkelstein Coordinates 4.1.1 Photon Circular4.1.2 Orbit Angular Velocity 4.1.3 Innermost Stable Circular Orbit 4.2.1 Penrose Process 5.1.1 Binary X-ray5.1.2 Pulsars Mass Measurements 5.2.1 Cygnus X-1: A Black-Hole Candidate 23 2.2.1 Birkhoff’s Theorem 2.4.1 Polytropes 3.1.1 Equation of State 6.1 Masses and Radii 6.2 Sagittarius A 5.3 Spin Estimates 2.1 Spherical Configurations 2.2 Schwarzschild Solution 2.3 Spherical Stars 2.4 Newtonian Limit 3.1 White Dwarfs 3.2 Neutron Stars 3.3 Black Holes 4.1 Geodesic Motion 4.2 Ergosphere 5.1 X-ray Binaries 5.2 Black-Hole Binaries 6. Supermassive Black Holes Contents 1. Introduction Black-Hole Phenomenology 2. Preliminaries 3. Compact Astrophysical Objects 4. Rotating Black Holes 5. Stellar Mass BHs versus NSs PoS(P2GC)004 ´ c 29 31 32 32 32 32 33 33 34 34 35 38 38 39 41 41 42 42 43 43 44 44 45 45 46 46 47 48 48 49 Neven Bili 3 6.3.1 Eddington Limit 6.3.2 Radius and6.3.3 Mass Estimate Estimate of the AGN Efficiency 9.3.1 De Sitter9.3.2 Gravastars Chaplygin Gravastars 6.3 Supermassive BHs in Active Galactic Nuclei 29 8.1 ADAF 8.2 X-ray Bursts 8.3 Direct Imaging 9.1 Neutrino Stars 9.2 Boson Stars 9.3 Dark-Energy Stars A.1 Geometry A.2 Covariant Derivative A.3 Geodesics A.4 Isometries and KillingA.5 Vectors Constants of Motion A.6 Einstein’s Equations B.1 Fluid Velocity B.2 Hydrostatic Equilibrium C.1 Tolman Equations C.2 Fermi Distribution C.3 Isentropic Fluid C.4 Degenerate Fermi Gas C.5 Polytropic Gases Black-Hole Phenomenology 7. Intermediate Mass BHs 8. Observational Evidence for the Horizon 9. Alternatives to Supermassive BHs A. Basic General Relativity B. Basic Fluid Dynamics C. Basic Thermodynamics PoS(P2GC)004 ) ´ c t (2.3) (2.1) (2.2) tween →− t and about Neven Bili , or Planck

3 c M / . At present we 1 ¯ hG p .

= M Pl 5 l . 9 , ) 2 etric generated by a spherical xt books Misner, Thorne, and - 10 φ d 6 by general relativity. There exist out by comparing BH candidates ion, i.e., (+,- - -) and we mostly equire the time reversal ( must vanish. θ trophysical BHs ction 9 which is based on original l and Ostlie [6], and on the review 10 e mass range 5 - 20 i 2 g talks may be found at the site of a 0 g sin ompact, which narrows the list of pos- e described by the metric coefficients s. Therefore, considerable attention will papers wherever appropriate but the list + , For a stellar-mass BH, the only standard ν , Planck length 2 , called the “lapse function", may be repre- θ G 1. In these units, the physical quantities are dx ξ , d / ) µ ( r = 2 ( ¯ hc r dx ϕ G e 4 p − µν 2 = g = = ¯ h dr ξ = Pl 2 (NS). It is therefore quite important to understand the = 2 m λ c ds − 2 dt 2 only. The function ξ is time independent. In general relativity we distingush be r = µν 2 neutron stars g ds are functions of . c λ / Pl l and = ξ Pl t Static fluid configurations are spherical. The most general m We use the positive-time negative-space signature convent These lecture notes are in large part based on the standard te Unfortunately, there exists as yet no direct evidence for as Black holes (BH) are the most fascinating objects predicted See A. Müller’s lecture at this School Consider the space time metric 1 mass distribution is of the form expressed in powers of the Planck mass astrophysical alternative is only hope that the black-holewith paradigm credible may alternatives. be Fortunately, proved BHs orsible are ruled alternatives dark among and standard c astrophysical objects. use the so-called natural units in which Wheeler [3], Shapiro and Teukolskyarticles [4], by Wald Townsend [5], [7] and and Carrol articles. Narayan [8]. Efforts are The made exception toof is references provide se is citations by to no original meansrecent complete. conference A on number supermassive black of holes interestin [9]. properties of NSs and their observational distinction to BH be devoted here to NSs. invariance of the metric. In this case, the mixed components three dozen supermassive BH candidates [2] in the mass range in which the metric tensor static and stationary metric.that do Stationary not depend configurations on ar time. For static configurations we also r sented in terms of the gravitational potential Black-Hole Phenomenology 1. Introduction about 20 confirmed candidates [1] for astrophysical BHs in th where time 2. Preliminaries 2.1 Spherical Configurations PoS(P2GC)004 ´ c . (2.8) (2.5) (2.7) (2.4) (2.6) (2.9) (2.11) (2.14) (2.16) (2.12) (2.15) (2.10) (2.13) Neven Bili , including a M event horizon . ) 2 φ d θ 2 sin + , 2 N θ traint (C.1), we have proper volume integral as , d . ( 1 to two nontrivial independent equa- , 2 1 )= . . r . ) r is the mass of the source. In the weak 0 − . 2 λ r ( − ρ r , r / , spherical object of mass 2 2 2 − n ( 2 1 r / M p 2 . ) λ 2 1 = − ρ λ 3 π . / 2 ) 0 1 r 0  r 4  dr ρ + M r 0 M r − . In this region equations (2.8) and (2.9) may − π ) 2 1 ( r 2 const. ) dr M 0 r r r T π r 4 R − r ( 0 2 4 R π T / − r = 0  M 0 + T > ≈− 4 2 r Z 2 − 5 M r ( λ dr M r ξ M M 2 = 1 r λ , r r M 2 = 2 r 0 p π ln  ξ − Z π µ 8 − − )= − 8 M 1 Σ dr = , at which the metric diverges, is the BH . The constant r 1 1 = − d d ( ( )=  = = M ϕ  )= 2 r µ ξ = r r 2 ( r u dr = r d ( M λ − ξ π Σ ξ dr = d T 2 λ 4 Z dr d M r 2 dt λ may indeed be interpreted as an enclosed mass. 2 ξ dr  R 0 M M r Z 2 − 1  Schwarzschild metric = is related to the enclosed mass 2 , we obtain the Newtonian potential λ ds M  r Assume the absence of matter in the region Finally, if we impose the particle number conservation cons where we have employed the spherical symmetry to replace the The metric takes the form black hole. The sphere with radius This metric describes the gravitational field outside of any 2.2 Schwarzschild Solution Using (2.4), Einstein’s field equations take the form Black-Hole Phenomenology and the function It may be easily showntions [10] that Einstein’s equations reduce For a perfect fluid The latter may be written in the form This is known as the which shows that the function be easily solved. One finds field limit PoS(P2GC)004 ´ c R . In ) to ) ρ (2.18) (2.22) (2.20) (2.21) (2.19) ρ ( through  p p ξ Neven Bili = p 1 , and 0 to some radius  = r r / 0. In the case when the = M o a chosen cutoff radius. , p ϕ 0 (2.17) = + given in the form ρ 1 )= which in turn depend on of weak gravity and low velocities. 0 , ' ( . µ p ) ϕ off (TOV) form 3 ined to give one 2nd-order differential the equation of hydrostatic equilibrium e M . . ) and (2.9) from r where M erical distribution of matter is obtained. π = and R 2 4 . dr πρ dp ; is just the rest mass density ξ T 4 ρ − 2 p + 2 / r ρ − . r 1 ( 1 + r π M  = ρ 4 mn ) ) 6 ρ R − = = TOV equations dr ( dp R + 2 ρ = ρ r p M spherical gravitating bodies (not necessarily static) is M dr ( ξ 2 satisfy an equation of state, e.g., in the form given by d d dr − ln dr all p − 2 1 d r 1 =  and dr dp ρ )= R ( ξ The exterior spacetime of 0 point does not exist, one must integrate up to infinity or up t = p General relativity reduces to Newtonian theory in the limit Generally, we assume that It is often more convenient (e.g., if the equation of state is = with the boundary conditions 2.4 Newtonian Limit The Newtonian limit is achieved by the approximation express the field equations in the Tolman-Oppenheimer-Volk The set of equations (2.18)-(2.20) is called the this limit, the two equations, (2.18)equation and (2.19), can be comb In this approximation, the relativistic energy density the Tolman equations (C.10). Numerical integration of (2.8 ρ Here, equation (2.18) is obtained from (2.8)(B.14) with which the may help of be written as Black-Hole Phenomenology 2.2.1 Birkhoff’s Theorem Theorem 1. (C.25) and (C.26), and are generally functions of local described by the Schwarzschild metric. The proof is simple. See, e.g., Misner et al. [3]. 2.3 Spherical Stars is rather straightforward and, as a result, a nontrivial sph The boundary is usually naturally provided at the radius PoS(P2GC)004 ; ´ c c 1 ρ z = (2.30) (2.27) (2.23) (2.28) (2.24) (2.26) (2.32) (2.31) (2.25) (2.29) z Neven Bili finite value . | 0. The radius is 0 . This equation can be 1 n θ = | 2 1 p is negative. Eliminating z ) ) n = 1 − , z , equilibrium configurations are 1 | ( ρ 1 ( 0 0 1 z / ) θ θ n | n 2 2 1 / , − ≡ ) z 3 n n 2 0 1 ( 1 2 / . z − θ / 1 r polytropes [4]: , ) 1 0 ) . ( itial conditions 1 c z # n dz n ρ − 1 − α θ n θ 3 2 − , ( )= ( dz c / d n / − n = 0 , . 1 / n ρ 2 ( / Γ 1  r c 0 z  Γ dz 2 = 1 / ρ ρ n θ K m 3 K K d π dz ) θ + θ K K 7 )  4 2 ) 1 dz 1 ; 1 d 1 π = z z π 1 K = 2 0 4 1; θ 1 4 + ) z + z c = Z K p + 1 0 n dr n c π ρ d ( Z Γ n ( dz 4 ρ ρ + )= ( c  3 =  2 2 0 1 ρ n " z r ) ( ( 3 n ρ = . Then the Newtonian equation (2.21) can be reduced to a π θ πα  − = 1 4 1 4 z ( π πα α / for the structure of a polytropic index Z 4 4 − α ) n = = = = − = 3 ( has been introduced because R M R | 0 1 π θ 4 | = polytropic index M 5), the solutions decrease monotonically and have a zero at a / . Owing to (2.22) we can rewrite the equation of state as 6 > Lane-Emden equation Γ is the central density. Then is called the 0. This point corresponds to the surface of the star, where 5 ( c n ρ polytropes < If the equation of state is in the polytropic form (C.32), the )= n 1 z ( while the mass is where where This is the θ between (2.30) and (2.31) gives the mass-radius relation fo Black-Hole Phenomenology 2.4.1 Polytropes called It is convenient to write where the absolute value numerically integrated starting from the center with the in where For simple form by writing PoS(P2GC)004 ´ c and

(2.33) (2.34) egimes, M Neven Bili of a WD is basically , 1 4 , 6 − − ρ 1 − 10 10 ∼ ity M/R kg and radii smaller than a ed plasma at a temperature 71406 ∼ ∼ 01824 . . 12 2 2 and their gravitational collapse = = They differ from normal stars in stic regime the pressure roughly | lapsed (BHs). With the exception | rt themselves against gravitational 0 1 from normal stars is their exceed- 0 1 their average kinetic energies in the θ ing to the momentum conservation, he final stages of stellar evolution: θ han 10 | ressure is dominated by the pressure | mass and radius are about 1 2 1 s, compact objects have much smaller 2 1 z z 3 3 7 15 R hey are prevented from collapsing by the − , / , 10 10 etime of the universe. 1 10 M < < ∼ ∼ g cm ∼ 65375 89685 . . 8 km 3 6 5 = =

1 10 1 z z R R × 5 2

− − M , R 2 , 10 10 / 3 9599 3 . correspond to the nonrelativistic and ultrarelativistic r ∼ ∼ 6 = = Γ n = n

,

M , R 3 3 3

M / / MR 4 − M 5 kg; < Mass Radius Mean Density Surface Potential ∼ 1 = = Arbitrary 2 30 ∼ Γ Γ 10 × 989 . 1 = . Hence, the electrons are assumed to be degenerate. The dens e

Black hole Object Sun White dwarf Neutron star m The second characteristic distinguishing compact objects Traditionally, compact astrophysical objects represent t First, since they do not burn nuclear fuel, they cannot suppo We assume that the interior of a WD is almost completely ioniz White dwarfs are stars that no longer burn their nuclear fuel M  fermi, all these objects are essentially static over the lif ingly small size. Relative to normalradii stars and of hence, comparable much mas stronger surface gravity. collapse by generating thermal pressure.degeneracy Instead, pressure either (WDs t and NSs)of or the they spontaneously are radiating completely “mini" col BHs with masses less t respectively (see appendices C.4 and C.5). 3. Compact Astrophysical Objects white dwarfs (WD), neutron starstwo basic (NS), ways. and black holes (BH). 5000 km, respectively. 3.1.1 Equation of State T is supported by the pressure of degenerate electrons. Their 3.1 White Dwarfs the density of barionic matterof (neutrons electrons. end protons). This Theequals p may the be average seen kinetic energy asthe (see electron follows. and equation proton (C.14)). average In momenta are the Ow equal nonrelativi and hence, Black-Hole Phenomenology The solutions we are particularly interested in are where the values 5/3 and 4/3 of PoS(P2GC)004 ´ c (3.2) (3.8) (3.3) (3.6) (3.1) (3.7) (3.4) (3.5) Neven Bili ree electron, i.e., 3 for the extreme / 4 , , = , km km Γ 3

6 / / 2 5 M − − 2 / trons are nonrelativistic. The .   5 , e e

− , 2 3 2 η η /  3 M 5 e /  . 2. The number of electrons is given 3 3 e  5 4 / / 2 e η 3 6 η

and extreme relativistic regimes sep- 4 2 − / 3 / = η 1 / 1 ,   π π 3 a polytropic form. The pressure in the M e n 5 3 3 3 − / e − 2 lytropes (2.30)-(2.34) we find the radius 2 ) / / η n 4 2 / ith (3.2) and numerically solve the TOV m η asses. Hence, the pressure of electrons is 2 1   1  e X m 3 3 3 3 e  e , C,  4 m 2 − 3 − η e 3 5 m 12 − n ( −  = n c c B  = ρ ρ m K gcm c g cm m e 9 6 e 6 K 457 ρ gcm . km η η R 6 4 1 10 10 2 = ; 3 for the nonrelativistic and 1 π 10 10  =  ; / 3 4 ρ 3 4 4 5 5 3   M = 10 = 10 = = Γ ρ × × Γ Γ 7011 4964 . . 0 0 122 347 . . 2000 larger than the pressure due to nucleons. is the Fermi momentum of the electrons. We could now use the = = 1 3 2 e ' M = = m e R R m − / 2 n . Hence, µ e m m p , to a good approximation, equals the number of nucleons per f = e = η F m q = . For example, for completely ionized pure X e Z m / Low-density (nonrelativistic) regime. High-density (ultrarelativistic) regime. A However, it is instructive to consider the nonrelativistic To determine the equation of state, first assume that the elec • • = e relativistic regime. Using this and the solutions for the po and the mass of the WD: relativistic expression (C.26) for theequations pressure to together find the w entire range of white-dwarf solutions. arately because, in these cases,two the regimes equation is of given state by takes (2.23) with where η Black-Hole Phenomenology nonrelativistic regime are inverse proportional to their m where the factor roughly by a factor of density is related to the number density of electrons as by (C.24), with PoS(P2GC)004 , ´ c = ∞ p (3.9) m → (3.12) (3.11) (3.10) (3.13) c − ρ n um with a m Neven Bili pport the star decay β ated by the degen- of the order of . The maximal mass n R m → e . At a density in the range p m the inverse , and represents the maximum m h , ekhar limit, the degeneracy pres- − 3 n − a density not very much higher than f the TOV equations using the exact . asymptotically approaches the value m h electrons, nuclei, and free neutrons miting value of l attraction and the collapse does not action ; one can describe the medium as one decay cannot occur because all electron ulting fluid consists mostly of neutrons > of the electrons increases according to km gcm , more pressure is provided by neutrons is of the order of 3 , 3 F β 14 µ e p R − E 10 ν reaches a critical level. Any further increase 10 = + 5km × + . p F × − 5 2 n / g cm e E 2 n ∼ ∼ 12 → 10 = → e 2 n Chandrasekhar limit p Pl e 3 10 Pl m ) m ν m n + m × m + − ∼ GeV , the Fermi energy of the electrons has risen to the ratio n e 2fm 1 3 ∼ ( R − 11 in the extreme relativistic limit. We conclude that as Ch = c 10 are occupied when R ρ g cm p × nucl 7 4 m ρ 10 ∼ − the medium is a composition of separated nuclei in equilibri n ρ × 3 m − = ) gas of neutrons. Can the degeneracy pressure of neutrons su E n g cm m 5%) fraction of electrons and protons. The pressure is domin 11 is independent of > < 10 n ∼ M 0. This mass limit is called the µ × , 0 4 → = ≤ R ρ T As the density reaches the normal nuclear density of about If the mass of the collapsing star is larger than the Chandras Assuming that the Newtonian calculations are still valid at At a density of about 2 Note that Of course, the limiting radius is not zero. The integration o , we can use the high-density results obtained for WDs but wit ≤ 7 29 MeV where electrons can now be absorbed by protons through nucl . remains the same as in (3.9) but the value of the critical 1 than by electrons. The neutron gas so controls the situation vast nucleus with lower-than-normal nuclear density. 3.2 Neutron Stars sure of the electrons canstop. no As longer the support density the increases, gravitationa the Fermi energy 10 relativistic electron gas. At there is a phase transition inwith which a nuclei small dissolve. ( The res erate ( because the neutrinos escape from the star and the normal against collapse? ρ (3.2). energy levels below Black-Hole Phenomenology This reaction cannot come to equilibrium with the reverse re the electrons become more and more(3.8) relativistic as and the mass leads to a “neutron drip" – thatcoexist. is When , a the two-phase density system exceeds in about whic 4 possible mass of a WD [11]. degenerate Fermi gas equation of state would also give the li PoS(P2GC)004 - ´ c (3.15) (3.14) Neven Bili iting values are ,

, 2 M / 2 1 / 1  of nuclear matter. The first  the central density becomes km g 2 realistic because at such large he general-relativistic framework g 2  es in part on being able to state  2 koff [12] who used the relativistic 2  er. Since the density inside the star eglect of GR effects is not justified.  re realistic equations of state predict e plot the mass of the NS as a function the equation of state is actually rather 2 t corresponding to the infinite central- ear matter. 2 m n the maximum allowed mass of a stable m 1 GeV roximation (dotted line). The dashed line is the 1 GeV   7 . 6 . Hence, the maximal mass is rather sensitive . 0

9 = 11 M = 2 / 2 1 / 1   2 g g 2   2 3 Pl 2 Pl m m m m 2 for neutrons. The part of the curve left from the maximum rep 2. 357 = . / 38426 3 . R g 0 = = = M with OV OV R 3 − M . The maximum of the curve corresponds to the OV limit. The lim gcm R 15 Mass versus radius for fermion stars at zero temperature in t 10 is the fermion degeneracy factor. This limit is reached when × g 5 The possibility to identify some compact objects as BHs reli However, the degenerate neutron gas equation of state is not We have to solve the TOV equations with the equation of states = c to the not very well-known equation of state for nuclear matt varies from very large centralcomplicated values as to the zero star at may the contain surface, different phases of nucl categorically that the observed object has a mass larger tha resents unstable configurations that curldensity up limit. around the poin densities the effects of nuclearthe forces maximum must NS be mass included. in Mo the range 1.5 - 2.7 where ρ Figure 1: Black-Hole Phenomenology (solid line) compared with the correspondingSchwarzschild Newtonian BH app limit which is close to the Schwarzschild radius of the sun, so thenumerical n calculation was performed bydegenerate Oppenheimer Fermi gas and described Vol by (C.24)-(C.26). Inof Fig. the 1 radius w [13] PoS(P2GC)004 ´ c (3.20) (3.19) (3.18) (3.16) (3.17) . 2 Neven Bili . It is important 0 ρ OV equations for a chosen .

lated, small elements of matter the maximal allowed NS mass? h similar assumptions about the termine which equation of state M arrison-Wheeler equation of state 2 NS mass based on rather general / is known. 1 quark level (preons, pre-preons etc.; 0 − ion ρ er collapse ending in a BH  mass limit, assuming general relativity n of state used below ts cannot be significantly altered. , 3 low the neutron drip. In fact, it turns out 0 − . . ble to expect that the equation of state will cts stop the collapse to a singularity by a bounce of ,

≥ 1 M M p gcm 2 6 ≤ ∂ . . 0 ∂ρ 14 2 s 5 3 ρ c 12 ≡ 10 ' ' 2 s × c 6 max max . 4 M M  2 . 3 ' max M gives 0 ρ maximizes the mass. Then, the numerical integration of the T 0 ρ that is, the speed ofwould sound spontaneously is collapse. real. If this condition were vio that is, the speed of sound is less than the speed of light. In their estimate, Rhodes and Ruffini took the the so-called H A semianalytic treatment of Nauenberg and Chapline [15] wit What happens if the mass of the collapsing star is larger than Rhodes and Ruffini performed a variational calculation to de It has been recently proposed that loop quantum gravity effe 2 1. The TOV equation determines the equilibrium structure. 2. The equation of state satisfies the local stability condit 3. The equation of state satisfies the causality condition 4. The equation of state below some “matching density" which accurately describes the nuclear matter densities be that the upper mass limit is not very sensitive to the equatio Abandoning the causality constraint stillto leads be to valid. a One severe finds (see, e.g., Hartle and Sabbadini [16]) equation of state gives still satisfy the above conditions and hence, the above limi 3.3 Black Holes to note that even if asee, new e.g., physics a exists black-hole at sceptical subnuclear paper and [17]), sub it is reasona In such a case, there is nothing to prevent the star from furth above equation of state below Black-Hole Phenomenology NS. It turns outassumptions that (Rhodes it and Ruffini is [14]): possible to set an upper limit to the the infalling matter [18] PoS(P2GC)004 ´ c M (3.26) (3.25) (3.23) (3.21) (3.22) (3.24) with zero Neven Bili . This may is timelike, (X-ray bina- has a zero at ∞ . µ , and electric max )

ξ R 2 R bed completely → M aM φ t = d = θ 1 for gravitationally R 2 gy per unit mass J < nonrotating BH) to sin , E 2 + / , 1 . 2 2 − ˙ t θ  ) tually, an astrophysical BH is 2 2 # d ( 2 as a function of E E . 2 ing from several ˙ t  R + dt + e surface follows a radial timelike asymptotically as  / 1 avitating spherically symmetric ball 1 dt − dR erized by just three numbers: mass rized by measuring just two parame- 2 M will usually be rapidly neutralized by R − M − dR  2 2 his is valid outside the star but also, by dt 1 R M R tion 2.2.1) implies that the metric outside − / # − = 2 2  1 M R   2  M 2 ( R . on the surface, we have 2 1 dt  = τ dR − ) dt ˙ d t t ) −  M ( R 13 R 00 1 1 2 R = / g −  ˙ t − = M  = − 1 2 r M µ 0, so R   τ − 2 d 2 0 for timelike and null geodesics as long as dx 1 is constant along the geodesics and > 1 M R . We consider the collapse to begin at ( E − µ 2 2 > E 1 ξ M τ = dR 2 − d E  = 2 1 min = = −  max E  R 2 R R  " Z dt dR ds M = R E  2 1 = − t (in galactic nuclei). Being so massive, these BHs are descri then decreases and approaches . The quantity 0 and 1

R , defined such that the angular momentum of the BH is  M = a M 2 " 5 φ . is a Killing vector, by Proposition 1 in appendix A.5 the ener 9 > = d t 2 R 10 ∂ , of which the latter is constrained to lie in the range from 0 ( = / a ds − θ ∂ and a minimum at 6 d [3, 4, 19] (see also J. Zanelli’s lectures at this School). Ac = and Q max ξ M R A useful toy model that illustrates the collapse is a self-gr Astrophysical BHs are macroscopic objects with masses rang = , spin parameter continuity of the metric, on the surface. If (maximally-rotating BH). 3.3.1 Spherical Collapse of dust (i.e., zero pressure fluid).the Birkhoff’s star theorem is (sec Schwarzschild (section 2.2, equation (2.15)). T be seen by integrating charge velocity. The radius not likely to have anysurrounding significant plasma. electric Therefore, charge the because BH it ters, can be fully characte R bound particles. Using (3.24) in (3.22) gives by classical general relativity.M As such, each BH is charact ries) to 10 We plot this function in Fig. 2. The surface radial velocity Since i.e., as long as Black-Hole Phenomenology Zero pressure and spherical symmetrygeodesic, imply that a point on th where is a constant of motion. Note that PoS(P2GC)004 ´ c (3.27) (3.28) Neven Bili R ...... R ...... In fact, it reaches the ......  ...... 2 . . 1 ...... face (figure from [7])...... f the star, the relevant time ...... E ...... − ...... M ...... 2 − ...... 1 . R . . max ...... R ...... max ...... τ ...... = ...... d ......  ...... R . d ...... ) ...... 2 ......  . max ...... E ...... R ...... finite proper time . . . . M ...... R ...... − . . 2 ...... in ...... 1 ...... − ...... M . .. . 1 ...... = ( . . So an observer “sees” the star contract at most ...... 2 . . . . for a spherically symmetric collapse of a ball of dust ......  ...... M .. . . .  R ...... = . . . . . 14 .. . . 1 ...... E . . 2 .. . 2 ...... R ...... E ...... = ...... → ...... + ...... τ ...... d .. . 1 ...... d ...... min ...... ˙ .. . . t ...... − . . 1 . . . R ...... M .. . . • • ...... through ...... 2 ...... M . . as = ...... R ...... 2 ...... M .. .. ∞ ...... d ...... max . . . dt ......  ...... 2 ...... R ...... → ...... = ...... t ...... = ...... 2 ...... R ......  ...... 2 ...... τ ...... d . .. .  dR ......  ...... dt dR ...... 0  ...... 2 ...... Obviously  The surface radial velocity as seen by the observer on the sur M τ ...... 2 d dR > but no further.  The surface radial velocity squared versus M min 2 R Figure 3: However, from the point of view of an observer on the surface o = R The star surface falls from variable is the proper time along a radial geodesic, so we use to where to rewrite (3.25) as Black-Hole Phenomenology Figure 2: (figure from [7]). PoS(P2GC)004 , ´ r c ∞ θ − , by r and v (3.33) (3.34) (3.30) (3.29) (3.31) , v v Neven Bili [20] 0. Because of ranges from between > ∗ ) r r r , ( ∗ ∞ r to + t M . 2 2 = 0 in the Schwarzschild metric Ω ld metric at the Schwarzschild t of either the metric tensor or Ω v d d = 2 , emovable, then it is irremovable, 2 r r r 2 coordinate system. A simple two- able. Such singularities are called ) − ic or its inverse diverge. However, a ngoing radial null coordinate) ∗ − ). ∞ 

s. Such singularities, usually referred ild spacetime 1 dr 2 lapses to a singularity in a finite proper − or which some scalar constructed from ( ∗ < M ranges from 2 2 . v M drdv dr ≡ r 2 2 2 0 (3.32) − 2 / < − 3 r − 2 ) dr

∞ 2 2 M E )= . As dt ln ) − ∗ π r dv  r − 15 / M 1   ± 300000 km s 2 , ( because the relation M t / ∗ 1 M M ( r r 2 r + . The singularity at = 2 2 d r M ingoing Eddington-Finkelstein coordinates − , which suggests that we investigate the spacetime near + τ 2 3km t − − 1 = M ( 1 1 > ∗ 2 = = r   r = c v = / 2 R = = dt Sch R 2 , are removable by a coordinate transformation. , but it can now be analytically continued to all ∼ ds true singularities M s ( 2 tortoise radial coordinate 5 or − > or r in coordinates adapted to infalling observers. Regge-Wheeler M : coordinate singularities Nothing special happens at A singularity of the metric is a point at which the determinan If no coordinate system exists for which the singularity is r We now show that the apparent singularity of the Schwarzschi 2 0 singularity in the proper time . Thus φ ∞ = = is defined only for and rewrite the Schwarzschild metric in This metric is initially defined for to on radial null geodesics. Now, define a new time coordinate (i where of its inverse vanishes, or atsingularity which of some the elements metric of may thedimensional be metr example simply is due the to origin ato in as failure plane of polar the coordinate 3.3.2 Eddington-Finkelstein Coordinates So, for an observer following radialtime of geodesics, the the order star of col 10 R and Black-Hole Phenomenology R is the i.e., a genuine singularity of spacetime. Any singularity f curvature singularities the curvature tensor blows up as it is approached, is irremov radius is removable. On radial null geodesics in Schwarzsch is an example. PoS(P2GC)004 ´ c ' 2 Pl (4.3) (4.1) (4.2) (3.35) m . This / M M 2 to prevent Neven Bili ≤ ∼ M r 2 = r , . 2 from 0 0 for future-directed θ , so the singularity in d M > ≥ . The horizon occurs at 2 Σ M 2 2 from the unstable photon 0 dv − > aM ds 2 r = = once the surface has passed = . r dr ∆ J θ Σ ∆ 2 ng to a black hole never appears e Kerr metric expressed in Boyer- reach llows that it takes an infinite time h an outside observer sees the star − ht travel time effect. Space can be 2 0 when cos t be said that the star collapses to a xactly at the horizon and are pointed 2 φ ≤ a d . φ f light at the horizon, and exceeding the 0 (i.e., ingoing radial null geodesics at 2 hrough space at the speed of light being . The star does actually collapse: it just ed to get to the outside world! θ 2 s from view within a time + . For the external observer, the surface 2 a ity. There is nothing at Ω 2 d of the Universe? = d r dt d − 2 sin Ω θ r 2 = d 2 ! − M Σ , θ 0. The null coordinate 2 sin 2 M p Σ 16 ≤ 2 , the redshift of light leaving the surface increases black hole dv , ± 2 sin M  a = 2 aMr 2 1 M 4 a r + drdv ∆ − = → + . Σ − 2 ± r Mr M r , we have r M 2 2 2 dt
2 M 2 −  2  a = 2 − r ≤ r + Σ 2 Mr r 2 , but as = 2 r ds M ∆ − 2 −

1 are related to the total angular momentum = =  − 0, i.e., at the roots of the quadratic equation r M = = 0 with equality when 2 ∆ drdv ≤ and 2 ds . The star has collapsed to a a . dr M M 2 3 = cross-term, the metric in EF coordinates is nonsingular at s . The late time appearance is dominated by photons escaping = r 6 r ). − M From the point of view of an outside observer, a star collapsi The star does in fact collapse inside the horizon, even thoug The spacetime around a rotating black hole is described by th Thus, no signal from the star’s surface can escape to infinity However, no future-directed timelike or null worldline can 2 10 drdv × = to collapse, but rather freezessingularity, at if it the never horizon. appears to How collapse then even till can the i en freezing at the horizon.regarded The as freezing falling into can the be black hole,speed regarded reaching of as the light a speed inside o lig theradially horizon upwards [21]. stay The there photons for ever, that their are outward e motion t for light to travel from the horizon to the external observer Lindquist coordinates [3, 4, 5] canceled by the inward flow of space at the speedtakes of an light. infinite time It for fo the information that it has collaps 4. Rotating Black Holes orbit at 5 through r may be seen as follows. When never actually reaches exponentially fast [19] and the star effectively disappear those points where worldlines, so Schwarzschild coordinates is really a coordinate singular the star from collapsing through Hence, for all timelike or null worldlines, with The parameters Black-Hole Phenomenology the PoS(P2GC)004 ´ c L ψ (4.8) (4.4) (4.5) (4.6) (4.9) (4.7) and 2, one / ξ π Neven Bili = θ , y. One may use ˙ 0 without loss of φ , one would have θ ≥ M 2 . a 2 ) sin (for details, see [4, 22]), ! aE θ ph , − 2 r ˙ and an angular momentum φ 0, which requires (for details, see Wald [5] and L exceeded ( > sin θ E 3 = a 2 , and substituting the results into 2 r r M r a L gy ∆ sin − egrate the geodesic equation (A.12) Σ ere exist two Killing vectors al plane reduces to solving a problem problem by symmetry considerations. Σ −  ot within an event horizon. A naked and 2 2 2 . for all
Mar r a E 0 2 n effective potential. The calculations are ) 2 , so we may choose , a r , + a ur where ˙ 0 = ( + κ ˜ 1 + L ˙ t r 2 V  = r ∂  )= ∂ ) r ν 2 and ˙ ( x 17 E

Σ Mr is called a maximally rotating black hole. µ ) V ˙ 2 r . In the case of equatorial geodesics, x in terms of , − + ( r + M ˙ 0 ˙ t ˜ φ ˙ µν − κ r E ( θ g = 1 2 1 cosmic censor conjecture = 2 1 2 |  a and V | + sin ˙ t = Σ 2 2 for a black hole to exist. If µ r L 2 ξ Mar µ M 2 + u − r = M = κ E . In addition we have µ − τ 1 for timelike, null, and spacelike geodesics, respectivel ψ effectively changes the sign of d = µ − / φ u µ V − dx 0, the Kerr solution reduces to the Schwarzschild solution. = →− = = L is the radius of the photon circular orbit. must be less than φ µ a | u 1, 0, and ph a r | = = 0, equations (4.9) have solutions > µ κ r x > generality. Taking When κ • • (per unit mass) along geodesics 4.1 Geodesic Motion A straightforward approach todirectly. finding However, all it is geodesics often is moreThe to economic Kerr to int spacetime simplify is the stationary and axially symmetric so th a gravitational field withsingularity a is “naked" forbidden singularity, by i.e., thereferences one therein). so n A called black hole with Remarks Thus, the problem of obtaining theof geodesics ordinary, nonrelativistic, in one-dimensional the motion equatori inrelatively a simple for circular orbits. Circular orbits occ where where ˙ where For which by Proposition 1 (appendix A.5) yield a conserved ener Black-Hole Phenomenology Note that finds (4.6) one obtains a differential equation for ˙ equations (4.4) and (4.5) to solve for where PoS(P2GC)004 ´ c (4.11) (4.17) (4.18) (4.16) (4.12) (4.14) (4.15) (4.13) Neven Bili (counterrotating orbit). M 4 . = . The requirements (4.9) for null ph  r ) use, in addition to (4.9), stability ph bits for particles, i.e., for timelike r M ii / ts, it may be easily shown (exercise) a 0 (4.10) . . , ∓ 2 2 . for equatorial circular orbits. In this case, ( = / φ a 1 1 . 0 00 0 φ . u − g g u ˙ ˙ + r 2 t r φ . Mr M / λ λ aM 3 − 0 1 2 √ 2 ph cos √ = r + + ± M a = ≥ (corotating) or 3 3 2 ≥ φ 2 0 φ 2 18 0 . /  φφ 2 2 u u L E q V 3 g M g r ± 2 r E E − ∂ = ∂ − cos = M ± − ± 2 = Ω 1 + = ph = = r 1 λ − Ω , L 2 Ω  r M M 2 = a = ph r Keplerian frequency , while for M 3 0). 0 in (4.8)) yield a cubic equation in is called the = > = Ω ph r κ κ 0, equations (4.9) have a solution 0, can be expressed as = = The angular velocity of an orbiting particle is defined as Photon circular orbits are possible only for particular rad It is obvious that not all circular orbits will be stable beca Ω a κ geodesics ( For For From (4.8) we obtain the quantity This is the general-relativistic form of Kepler’s third law The photon orbit isgeodesics the ( innermost boundary of the circular or where 4.1.2 Angular Velocity Black-Hole Phenomenology 4.1.1 Photon Circular Orbit For equatorial circular orbits one finds [22] Using the parameterization (B.9) forthat the velocity componen with the solution [22] 4.1.3 Innermost Stable Circular Orbit requires PoS(P2GC)004 ´ c 27 / (4.19) (4.20) (4.22) (4.21) (4.24) (4.23) 25 for the (ISCO) r p 9 (a=0) to . √ Neven Bili − / ph 8 r p ≥ − is r , # 3 9 (a=0) to 1 / 1 / , as the maximum binding 8 i  2 2 and in Schwarzschild space- η 2 p / 1 a M ) ∞ , − 2 cy BH accretion disk as an energy . −  Z 2 2 increases from 1 1 le trajectories (i.e., timelike geode- / θ gy for a maximally rotating BH is (counterrotating). Obviously, com- , in Kerr spacetime it need not be leased by matter spiraling in toward 1 2  , . + η .  be negative only if the time translation bits. A negligible amount of energy is M 1 θ θ 0 innermost stable circular orbit + 9 is 2 2 ircular orbit satisfies Z cos 3 M r > 2 / Mr 2 3 = + 1 a 2 cos cos 3 is  2 2 − ( Mr + r 2 a a 2 2 2 1 2 / a r 1 M  − − − ) 2 2 1 − − + θ 19 Z 1 M M 2 1 1 [22]  −  p p = is normally timelike at 3 cos " = ( is 2 + − 3 . ξ ˜ / is the radius of the (corotating) a E 2 into the BH. ∓ 1 00 / M M is 2 g 1 + is −  r M r Z 2 2 >  < 1 2 = r 2 1 r r = a + M µ Z ≡ is 3 ξ r h − + η µ , for timelike geodesics, we obtain a quartic equation in 1 2 ξ 2 M ) M  a M r = ( 3 = + ˜ is E 1  a r = = marginally stable orbit 1 2 ; for Z Z M is spacelike. The vector 6 ξ = is r 3 (a=M) for corotating orbits, while it decreases from 1 0, / 1 = is timelike only if A quantity of great interest for the potential efficiency of a A curious property of rotating BHs is that there exist partic a p ξ − source is the binding energy of the ISCO. Defining the efficien with 4.2 Ergosphere sics) with negative energies. The energy definedKilling by vector (4.4) can pared with the photon circular orbit, the innermost stable c For so Plugging in the solution (4.19), one finds that the efficiency time it is timeliketimelike everywhere everywhere outside outside the the horizon horizon. because However Black-Hole Phenomenology Substituting the solution energy per unit rest mass, from (4.18) (with =) one finds (a=M) for counterrotating orbits.42.3% The of maximum the rest-mass! binding This ener isthe the BH amount through of a energy succession that ofreleased is almost during re circular the equatorial final or plunge from 1 limiting case of equality. The solution also referred to as the This implies PoS(P2GC)004 , , ´ c in E E > (4.27) (4.28) (4.29) (4.25) (4.26) out E Neven Bili urface metric 29% of the mass (irreducible mass) ≈ irr M . The maximum energy 2 , can become spacelike in a a is its 4-momentum, we can  ξ n be understood as follows. 2 . Hence, a particle trajectory − µ requires particular conditions a 2 q a mechanism for the extraction . ising mechanism for extracting − M e area theorem of BH mechanics, ve a mass 2  rgoregion, we may have . If 2 √ andford [25, 26],) because magnetic M ). Hence, the maximally extracted . Thus, reases [24]. The area of the horizon a h falls into the hole with energy o regions farther out. Recent general- + duces to zero, i.e., if the BH becomes θ θ hat it penetrates the ergosphere. Now as it is beyond the horizon. The outer M p . This represents − a magnetized plasma in the ergosphere 2 ) ergoregion 2 M 2 + = M . cos = √ a M 2 / in p  . The ergosphere intersects the event horizon + a 1 E µ r . + ξ M − − − µ = 2 π 1 M q E 8 r ( 20  M is not necessarily positive in the ergoregion because = M = = 0, . By conservation of energy µ M p E ξ φ π = out out = + µ in d 8 E E q irr θ M = dr M = g d = 2 irr = in − r M E det dt M π p is 16 stationary limit surface , but the horizon for other values of Z with energy E M = ∞ < A or the out is the determinant of the metric on the horizon surface. The s 2 for a maximally rotating BH ( outside E / φφ M g 0, so = θθ g > 2 irr ergosphere in = M , but it lies E g π , 0 The energy extraction by the Penrose process is limited by th In 1969 Penrose exploited this property of Kerr BHs to design The Penrose process for extracting energy from a rotating BH = energy has been extracted from the black hole θ may be spacelike there. Thus, if the decay takes place in the e Normally, while the other escapes to which states that the surface areais of the BH horizon never dec so which satisfies where det extracted by the Penrose process isSchwarzschild. obtained if A the Schwarzschild BH BH spin re with the same area will ha This gives of energy from a BHSuppose that [23]. a particle The approaches mechanism a proposed Kerr by BH along Penrose a ca geodesic ξ region outside the event horizon.inside This the region ergosphere is may called have the negative energy! 4.2.1 Penrose Process is obtained from (4.1) by setting Black-Hole Phenomenology The inner boundary of thisboundary region , is i.e.,the not hypersurface physically relevant is called the identify the constant of motion as its energy (see sectionsuppose A.5). that The the trajectory particle is decays chosen into so two t others, one of whic energy from a BH of mass energy of the BH. [22] which are veryenergy difficult from a to rotating BH realize is via infields magnetic are nature. fields (Znajek capable and of Bl connecting A regions more veryrelativistic close magneto-hydrodynamics prom to (MHD) the simulations BH of t at PoS(P2GC)004 - ´ c 3 − 10 ∼ B Neven Bili . The pulse d matter onto the magnetic ersion of the kinetic energy ses is not easily distingushed usually a normal star. Most of of the neutron star, of the order specific nature of the compact systems, with optical primaries binary X-ray pulsars 3 n star or a black hole, is an ex- dates includes all three kinds of stellar mass BH candidates have e they are dark and very compact. lies a small emitting region. , de Villiers et al. [31], McKinney al potential energy into X-ray radi- these facts [4]: ld lines are azimuthally twisted and along the rotation axis. The process he spin axis, then the compact object he beam (or the beams, according to d of the companion optical star) into reting gas from the companion star. etic field ocess of collapse from a “normal” star ( . 1000 s. Those with short periods of about 1 s are 21 variations are called < ∼ P < ∼ periodic because of a close analogy between this mechanism [29] and X-ray binaries 5 s 10 km) . ∼ R km) to a neutron star ( MHD Penrose process 6 T. The magnetic field of the compact object drives the accrete 10 9 ∼ R ation. orbiting optically invisible secondaries. tremely efficient means of converting released gravitation –10 7 T, Obtained from conservation of the magnetic flux during the pr Astrophysical observation of NSs and BHs is difficult becaus An X-ray binary is an X-ray source with an optical companion, This interpretation of the observational data follows from The reasons why WD and BHs cannot be pulsars are: The standard model explains X-ray emission as due to the conv Binary X-ray sources displaying In general, the list of possible Galactic X-ray source candi 3 2 1. The variability of X-ray emission on short2. timescales imp Many of the sources are positively confirmed to be in binary 3. Mass accretion onto a compact object, especially a neutro − 10 Black-Hole Phenomenology of a rotating BH (Koide etand al. Gammie [32]) [27, show 28, that, 29], nearthe Semenov the twist et equatorial then al. plane, propagates [30] the outward fie is and dubbed transports the the energy Besides, a BH the massfrom of a which NS is because of ofbeen the their identified order only similar of as properties. a the so-called few So solar far, mas NSs and Penrose’s original idea [23]. 5. Stellar Mass BHs versus NSs 5.1 X-ray Binaries the Galactic X-ray sources are probably compact objects acc of 10 polar caps, and if the magneticacts field as axis a is “lighthouse”, not giving alignedthe rise with geometry) t to crosses pulsed our emission line of when sight. t of the accreting matter (comingradiation, from because of the the intense interactions with stellar the win strong magn normally identified with rotating NS. compact objects: WDs, NSs,object and can be BHs. identified. But in special cases,5.1.1 the Binary X-ray Pulsars periods are observed in the range 0 PoS(P2GC)004 ´ c 2 X f a and and s, if / + (5.4) (5.3) (5.2) (5.5) (5.1) ) O 1 f O orb a P M = ( f Neven Bili a = O will be Doppler f 2 M . Clearly, 2 ble quantities a ively. The ratio density too low to explain and 1 retion would not be periodic , then the object is very likely are via Kepler’s Third Law. lack holes could produce the a

and have no structure on which ariations in the time of arrival of M identifying BH candidates [8]: 3 . ic star such as a NS of the order he mass functions 3 2 is the semimajor axis of the ellipse. can be measured and hence one gets v G along the line of sight: . BH from a NS are their masses. The out 2 a 2 π 2 v orb 2 t travel time across the projected orbit – 3  P M se / , , 1 π i orb = and 2 of  2 P a 2 2 X O 2 3 ) sin v f f orb  ) 2 2 i P M a 1  = M + M sin ) 22 π = + orb 1 2 1 2 P 1 M X O M M M ( = M M in a circular orbit about their center of mass (CM). The (see section 5.3) ( 3 + = 2 a 2 1 a v ≡ ≡ M M ) q ( 1 the inclination of the orbital plane to the line of sight. Thu G M i and ( f 1 and their distances from the CM are M a shows periodic variations, then by the definition of the CM. Any spectra emitted from, e.g., 1 2 . M quasi periodic oscillations a is called the “mass function" and depends only on the observa c 2 / f for both the optical companion and the X-ray source, respect i ). M i ) is the orbital period and X = sin sin 2 1 M (see section. 3.2) allows the following simple criteria for orb 2 a a ( P . Alternatively, for X-ray pulses one can measure periodic v a 1 such short periods. Rotating BH are excluded because they areto axially symmetric attach a periodic emitter.to Any the mechanism depending observed on precision. acc so-called However, accretion disks round b Rotating WD are excluded because they are too large and their f i

Now, Kepler’s Third Law states For several X-ray binaries, it has been possible to measure t The most reliable means of determining astronomical masses M • • M = (or sin 3 2 X 2 f shifted, depending on the orbital velocity projection and pulses. The amplitude of these variationsthat is is, simply the ligh a gives th mass ratio the spectrum of separation of the two masses is v a black hole. Consider two spherical masses The quantity If a compact astrophysical object has the mass larger than ab we obtain Note that this is valid also for elliptical orbits in which ca Black-Hole Phenomenology 5.1.2 Mass Measurements The first important criteria thatfact may be that used there to exists distinguish∼ a a maximum mass of a compact relativist Using this and where PoS(P2GC)004 . ´ c ) and M ( (5.7) (5.8) (5.6) 2 f v > tal of 20 es Neven Bili M . i , with error bars such 300 km and establish itself larger than or of

launched by NASA off ardless of uncertainties ) < ∼ M M R ( f ording to (5.6), al supergiant star with a well- and Murdin [34], Bolton [35]) Uhuru about the mass of O and using en obtained for six eclipsing X- e order famous Hercules X-1 neutron star w that Cyg X-1 periodically cycles yg X-1 is variable on all timescales llite, arge masses makes them strong can- ly for those for which the inclination km, one obtains the mass function of e of the radius of O [37]. The absence from 1 to 2.3 nothing 6 cal constraints on sin . 10 he anniversary of Kenya’s independence. 2 . [4]. From the measurements of the orbital X-ray states [36]. The most dramatic vari- ) i are the binary X-ray sources with observed ×

q 3

, ) . In practice, the X-ray eclipse duration and/or M i + a M R low 3 sin 08 5 . 1 . . ( 3 ≥ 0 8 23 X i f and > ± ∼ ∼ , which is expected on the basis of current theoretical 1 in (5.6), one obtains a minimum value for the mass cos = X O 82

. X = M M 5 BH binaries hard M i M = ( i sin . Most of the 20 X-ray binaries have O M a . Setting sin

with nonperiodic time variability. As of today there are a to , which depends only on the two accurately measured quantiti M )

) still depends on knowing sin M M X ( f 010 4 . M 6 days and . 0 5 ± . Therefore, these systems are excellent BH candidates, reg =

252 orb M . P 0 is well known, which is not always the case. Fortunately, acc i = ( It is possible to set a convincing lower limit assuming The optical companion star (O) in the Cyg X-1 system is a typic Cygnus X-1 is a typical BH binary discovered in 1972 (Webster Black hole X-ray binaries, or short, In this way a complete description of the binary system has be “Uhuru" means “freedom" in Swahili; the launch occurred on t X , is a strict lower bound on 4 f orb only the absence of a prominent X-rayof eclipse eclipse and implies the estimat X, elements, known spectrum and the mass of at least the object to be highly compact. as a first stellar-mass BH candidate.varying The from X-ray ms source to (X) months of andthrough C years. two Recent accretion observations or sho spectralability states: is the 1-ms bursts, which set a maximum size for X of th in their inclinations and companion star masses. 5.2.1 Cygnus X-1: A Black-Hole Candidate the order 3 P The mass function scenarios for NS formation.discovered One in of 1972 these [33] X-ray in pulsarsthe the is coast data the of of Kenya. the first astronomy sate 5.2 Black-Hole Binaries masses larger than 3 confirmed BH binaries (Remillard and McClintock [1])didates Their for l BHs. However, the massangle estimates are reliable on that all the data fit in the range 1.2 - 1.6 ray pulsars with optical companions [4]. Their masses range variation in the optical light curve are used to set geometri Black-Hole Phenomenology and then from (5.4) we can write A unique value for PoS(P2GC)004 ´ c give (5.9) (5.11) (5.10) (5.12) (5.13) X f 5 kpc with Neven Bili . 2 . In contrast , and is very solid. a d '

, [1]. d R M

d luminosity com- 4 . . Various other less M

, we need test particles M a .13) of 3 me parameter ve been proposed for estimating bsorption. tion vs. distance curve calibrated ls in through a sequence of nearly he surrounding objects. Only for in of a BH or of any other rotating . instability [38] and magnetoviscous X-1 is 6.8-13.3 . le circular orbits are available, so the . t measurements of ted distance for O is . main source of instability and loss of 2 ISCO) (see section 4.1.3). Such test km ) O

i i he ISCO serves effectively as the inner , and so 2 . )  M M truncated disk model [40] R 3 2 refore, to measure sin 2 sin 2 ) X + i f √ X R O  d X 3 a X M ( 1 kpc f ( sin M / d 3 i  O ≥ 2 kpc 6 a √ 24 i ( 5 kpc increases this to 5.3 3 sin R 2 .  2 10 is the separation of the two objects. Using (5.3), we 2 O 4 . a × ≥ cos O 3 ' i a X ≥ 62 d ≥ . one finds i M + sin 6 i X X X M = a cos sin M R = O a a and X f has a maximum value of 2 2 x cos x to Cyg X-1 is determined using two methods: a) from the assume from luminosity and the effective temperature, one finds [4] d R is the radius of O and R of the accretion disk. A variety of observational methods ha Measuring the BH spin amounts to measuring the Kerr-spaceti 5 The gas in an accretion disk starts from large radii and spira We can summarize the situation as follows: the lower limit (5 The inner edge may not be very pronounced; compare, e.g., the 5 5.3 Spin Estimates to BH mass estimates whereastrophysical Newtonian gravity object applies, does the not sp relativistic have orbits does any spin have Newtonian measurable effect effects. The on t edge orbiting very close toparticles the are provided innermost by stable the accretion circular disk. orbit ( circular orbits as it viscously losesangular angular momentum momentum. are MHD The effects,instability e.g., [39]. magnetorotational When the gas reachesgas the accelerates ISCO, no radially more and stab free-falls into the BH. Thus, t even larger values. The currently accepted mass range of Cyg rigorous but more realistic arguments, as well as more recen Adopting the more reasonable value of an absolute minimum of 2 kpc in order to produce the observed a can write this as Estimating The function sin Hence, Black-Hole Phenomenology where From (5.12) with known The distance pared with the apparent luminosityfrom of a O large sample and of b) stars from in the the same absorp direction. The estima PoS(P2GC)004 , , ´ c 4 Ω acc / M 1 L ] 99. . , then σ (5.14) 0 is / emitted r ) ) ' r r ( > Neven Bili a ( (essentially F r [ oximately as F 2 d ≡ / i ) r ( cos is the distance to the eff 2 in T r d are available, and thus an blackbody at each radius, , one can use this method he Keplerian frequency l cause the spectrum of an is M r . . By comparing this quantity ions shows one or two peaks e treated with caution. a each BH binary corresponds to ν = alysis reveals that the emission d Hz. The peaks are relatively and . Next, we briefly describe some d in is ν d r r , L ature profile frequency at any radius i om Sgr A*, the supermassive BH in cillations but rather to quasiperiodic at teristic radius; it is plausible that this hod requires accurate estimates of ]. ions (4.16) and (4.19), one can express e BH horizon and they find ity η s of lculate the flux of radiation , y ) n tentative evidence for QPOs with a period M / a , corresponding to an accretion luminosity ( ˙ , one then obtains F M is 25 1 r is the inclination angle and M . Assuming that i = M / with is is available. The method has been applied to a few BH a Ω in r M = x 5 [8]. A number of QPO frequencies in the X-ray flares from . 0 > a received at Earth, one obtains an estimate of (see section 6.3.1), the accreting gas tends to radiate appr ν Edd d ν L F is obtained. Identifying (spectral fitting method; relativistic iron line method), t provided an estimate of in is a known function of in a r r ) x is the Stefan-Boltzmann constant. If the disk emits as a true ( . Therefore, spin estimates obtained by this method should b F σ d For some BH binaries, the power spectrum of intensity variat One possibility is that the QPO with the highest frequency in A major weakness of this method is that a number of effects wil When a BH has a large mass accretion rate , and by the accretion disk, and hence obtain the effective temper Relativistic Iron line source. In a few BH binaries, sufficiently reliable estimate the circular Keplerian frequency of gasradius corresponds blobs to at the some inner charac edge ofthe the Keplerian disk. frequency Using of equat the ISCO as i Quasiperiodic Oscillations (more like bumps inbroad, some indicating cases) that at they frequencies dooscillations of (QPOs). not a correspond few to hundre coherent os accretion disk to deviate from a blackbody. Besides, the met estimate of a blackbody. In this spectral state one can theoretically ca the BH must be rotating with Sgr A* have been identifiedfrom by the inner Aschenbach parts et of al. the accretion [42]. disk is Their quite close an to th above a few per cent of of these methods. For more details and for references, see [8 Spectral Fitting the projected solid angle of the disk), where to estimate where it is a simple matter to calculate the total spectral luminos Black-Hole Phenomenology the radius (quasiperiodic oscillations method), or the binding energ with the spectral flux binaries (for references, see [8]). Recently, thereof has bee 17 minutes in thethe infrared Galactic emission Center (Genzel . et If al. the QPOs [41]) correspond fr to the Keplerian where PoS(P2GC)004 ´ c emission r edge of Neven Bili α 8 (Miller et . 0 93. Among BH > . 0 a , and the emissivity > i , a a using the same line data from t it requires no knowledge of comes from a relativistic disk diating gas follows Keplerian llenging to make fundamental i he radius range over which the , motivated by the standard disk s is the presence of a large mass β ability could provide interesting fts as well as to the gravitational ve galactic nucleus (AGN) MCG- ems to indicate clei are powered by accretion onto − r f a rapidly spinning BH. Assuming s increases substantially. This gives seen in a few other AGN and X-ray ng mostly the following three meth- . [45] estimate ion of a rapidly spinning BH. onal redshift. The detection of such the observed line extends from about line as fluorescent iron K f not all, galaxies [52, 53]. In several nation rmost radius of the disk which in turn ] (movies courtesy P. Armitage). us, ears there has been increasing evidence g the motion of stars orbiting around the s that they are supermassive black holes, 26 solar masses. Comprehensive lists of about 30 supermassive 10 to 10 6 is particularly dramatic. As the BH spin increases, the inne ). is a r = , one can fit the shape of the line profile by adjusting is in r r ≥ r (since a is estimated. a It is now widely accepted that quasars and active galactic nu The main criterion for finding candidates for such black hole The variability of the line with time means that it will be cha In spite of some weaknesses, the method has the advantage tha In the case of MCG-6-30-15, the data confirm that the emission Given a system with a broad iron line, and assuming that the ra A strong broad spectral line in the X-ray spectrum of the acti The line width and its shape, among other factors, depend on t massive black holes [49, 50, 51]. Further,that over massive the dark last few objects y may residecases, at the the best centers explanation for of the most, nature i of these objects i with masses ranging from 10 tests of gravity withopportunities this to study method. disk dynamics and On turbulence [47, the 48 other hand, the vari 6. Supermassive Black Holes BHs at galactic centers may be found in [2, 54, 55,6.1 56]. Masses and Radii within a small region. The massods: and gas the spectroscopy, maser size interferometry, are and estimated measurin usi galactic nucleus. which al. [46]). the BH mass or of the distance, and it solves for the disk incli binaries, the source GX 339-4 shows a broad iron line which se the disk comes closer to thea horizon wider and range the of velocity Doppler ofextreme the levels shifts, of ga as broadening well may be as taken a as larger a strong gravitati indicat and at least some ofthat the there data is sets no can emission from be within interpreted the in ISCO, terms Reynolds et o al function; the latter is usually modeled as a power law in radi orbits with radii Black-Hole Phenomenology 6-30-15 was recently discovered [43].from cool They gas interpreted in the thebinaries. accretion disk. Whereas the Similar rest broad energy lines4 of were to the iron 7 line keV. is Thisredshift. 6.4 broadening keV, is due to Doppler blue- and red-shi emission occurs and in particular ondepends on the position of the inne model [44]. The effect of PoS(P2GC)004 ´ 1 c

0, − M = 7 (6.2) (6.3) (6.1)

10 2 × dt / Neven Bili 5 I . 2 d

km, shows that the gas resolution of 0.1 kms e structure of accreting , and no form of matter , so the total mass of the

14 m rium, then the analysis of the Hubble M 10 city of the gas measured by been discovered in the radio 9 × l theorem. It uses the measured 10 6 th a velocity difference of about n the galaxy’s central region are the most massive black hole ever re of ionized gas in the innermost ject with a mass of 3 × ≈ erian dynamics (Bragg et al. [62]). ]. 3 stars of equal mass, we find ich provides an angular resolution as , the gas follows circular orbits with a ill look the same (in statistical sense) ∼ i N , p. 1007). For simplicity, we restrict U h . . i = N U 4 Mpc) has been studied in detail [59, 60, 61]. i U . h stars, each of mass = K 6 , see Fig. 1 in [8]). Furthermore, the acceleration h 2 i = 2 N v 2 ' / 27 i 1 d − K − ∑ h r  with 2 m N I 2 ∝ 2 − R − v dt d km of the center of NGC 4258. The case for this dark mass  1 2 12 10 × 4 = 0.13 pc . ∼ Nm as (microarcseconds) at a wavelength of 1.3 cm and a spectral . This leads to a mass of the central object of O masers orbiting compact supermassive central objects. Th µ = 1 2 − is the region’s moment of inertia. If the galaxy is in equilib M I A simple method to obtain a mass estimate is based on the viria The time-averaged kinetic and potential energies of stars i A clear and compelling evidence for black holes has recently An example of measurements via gas spectrography is given by A spectral analysis shows the presence of a disklike structu velocity dispersion of starsattention in to the a central spherical region cluster (ref. of radius [6] being a BH is quite strong. 6.1.3 Virial Mass related by the equation (ref. [6], p. 54-56) confined within bulge is material around the nearby galaxy NGC 4258 ( regime: H can occupy such a smallobserved. region except for a black hole. This is 6.1.2 Maser Interferometry fine as 200 with the aid of very long baseline interferometry (VLBI), wh of the gas has been measured and it too is consistent with Kepl From these measurements it is inferred that there is a dark ob or less, radio interferometry measurementsnearly have perfect shown Keplerian velocity that profile ( recedes from us on one920 side, km and s approaches us on the other, wi resulting in the usual statement of the virial theorem Furthermore, for a large number ofat stars, any time, the and central the bulge time w averaging can be dropped. So for Black-Hole Phenomenology 6.1.1 Gas Spectroscopy Space Telescope observations of the radio galaxy M 87 [57, 58 few arc seconds inspectroscopy the at vicinity a distance of from the the nucleus nucleus of of the M order 20 87. pc The velo where PoS(P2GC)004 - ´ c x (6.8) (6.5) (6.6) (6.7) (6.4) 8 pc. This . Neven Bili 0 ' R , 2 r σ 3 rom the center of our own =

tral BH of M31 (Andromeda). measures the spread in the r estimates for the mass of the ow time-elapsed images of the 2 r s ution in the special case when ears by two independent groups . The existence of a dark massive v x ∗

σ 3 roximately 240 km/s within 0.2 as , otions of stars and gas in its vicinity. 2 as corresponds to . = . N 2 / ,

1 ; Ghez et al. [66, 67]). The movies (1st a spherical distribution of the total mass . s made it possible to follow the orbits of 2 φ eSgrA several stars. #

v ,

, 2 ix

M 2 r v 2 M 7 7 σ + i G R ∑ R 10

GM 5 " 2 θ virial mass × v 3 5 to 10 1 28 N

6 = 6 − + = =

= x 2 r virial σ v U M

virial M =

2 v

= 2 i v ∑ 1 N —————————————————————————————– − ∗ 0. = is the dispersion in the radial velocity. i x r v σ ——————————————————————————————————— It is equal toh the standard deviation of the velocity distrib component of the peculiar velocities of stars and is defined a For a region in space with a large number of stars This is, of course, just an order of magnitude estimate. Othe Perhaps the most convincing evidence for a black hole comes f This equation can be used to estimate a virial mass for the cen Galactic Center region revealing the (eccentric) orbits of individual stars around this objectmovie (Schödel et courtesy al. R. [63, Genzel; 64] 2nd movie courtesy A. Ghez [68]) sh object at the center of theThe Galaxy has motions been of inferred the from the stars[63, m were 64, observed 65, and 66, recorded 67]. for High-resolution many infrared y observation within a sphere of radius 0.8 pc. supermassive BH of M31 range from about 10 galaxy that coincides with the enigmatic strong radiosourc 6.2 Sagittarius A gives a total mass of roughly Using the (approximate) gravitational potential energy(exercise) of equations (6.2) and (6.3) yield where the mass obtained in this way is called the Velocity dispersion Black-Hole Phenomenology The average velocity squared is The central radial velocity(arcseconds). dispersion Given the is distance measured to Andromeda to of be 770 kpc, app 0. where PoS(P2GC)004 ´ c v = and (6.9) ∗ SCH has an e names R ∗ Neven Bili , where with a velocity ∗ SCH (AGN). To this class R 6000 km/s at a distance ∼ v of 15.2 yr around Sgr A axy be emitted from a volume f gravitational potential energy otential around Sgr A P t few years by Ghez et al. [67] us amounts of energy from their . with luminosity (in particular the ed during the last decade [63, 64] ts an upper limit to the size of the

es egions of active galactic nuclei are etween April and May 2002, S2(S0- riod elocity . hows that for matter falling straight 0 AU from Sgr A M ve galactic nuclei would need to be an provide. black hole [49, 69]. Even using an 6 han one ten-millionth the size of the 6 ar U-turn, crossing the point of closest ordinary galaxies by several orders of ly one has survived close scrutiny: the s. gly suggesting that the object is a BH. , are often used to refer to both types of objects ei fluctuate on very short time scales – 10 × active galactic nuclei 7 . This implies that an enormous mass of the . quasar quasars ∗ 3 = and 29 3 2 a 2 QSO π GP 4 (QSO) and . Quasars are radio-loud whereas QSOs are radio-quiet . Thes = . ∗

neutron star, about 21% of the rest mass is released. This is M SgrA

6 M M of the order of 4.62 mpc. 10 a × 6 . 2 = quasistellar objects M quasistellar radio source form, for radii larger than 60 AU, corresponding to 1169 1 − r 0.051 AU for Many galaxies possess extremely luminous central regions, The most efficient way of generating energy is by the release o What makes these galaxies “active” is the emission of enormo How could a luminosity hundreds of times that of an entire gal The salient feature of the new adaptive optics data is that, b Another star, S0-16 (S14), which was observed during the las The projected positions of the star S2(S0-2) that was observ Quasar is short for Then, neglecting the star mass, Kepler’s third law (5.2) giv = 6 9000 km/s. Ghez et al. thus conclude that the gravitational p M 6.3 Supermassive BHs in Active Galactic Nuclei luminosity of X-ray radiation) exceedingmagnitude. the luminosity These of luminous central regions are called almost 30 times larger than the energy that hydrogen fusion c 2 belong the so-called billions of times smaller? Ofrelease all of proposed explanations, gravitational on energyenergy by source matter as falling efficient towardssupermassive as in a gravity, order the to produce black the holes luminosities of in quasar acti 6.3.1 Eddington Limit through mass accretion. Fordown example, onto a the simple surface calculation of s a 1.4 nuclei. Moreover, the luminosities of active galactic nucl emitting region. For thisonly reason, light-minutes we or know light-days thatgalaxy across, the in making emitting which them they r sit. less t within days or sometimes even minutes. The time variation se approximately ∼ 2) apparently sped past the point of closest approach with a v of about 17 light-hours [63] or 123 AU from Sgr A with the Keck telescope in Hawaii, recentlyapproach made a at spectacul an even smaller distance of 8.32 light-hours or 6 central object is concentrated in a very small volume, stron Black-Hole Phenomenology suggest that S2(S0-2) is moving on a Keplerian orbit with a pe the estimated semimajor axis [6]. are sometimes confused in the literature. Both terms, PoS(P2GC)004 r ´ γ c ε N = r (6.15) (6.11) (6.12) (6.13) (6.14) (6.10) (6.16) p Neven Bili per electron 423, with the . . Equating the r 0 < η < interaction of radiating 0572 ergy of the innermost stable , the number of photons . ough an accretion disk of a L . surface of a compact object of umber of collisions per electron 2 . 1 otating BH, respectively. cm − ty is 4.1.3, 0 s and there is a limit to the luminosity 25 er unit time, so we multiply by drogen gas is the scattering of photons − tional attraction, exceeds gravity and . erg s electron (acting via the proton) . 10 2 r p  p σ sets an upper bound to accretion luminosity , × π , L

m 2 m 2 is the enclosed mass at radius 4 ˙ σ r M r 65 M M . π M M M = L η 6  4 r r π − ' p = ε 30 = 38 4 γ (i.e., the radiation energy released per unit of time) 2 = 10 N = = M  disk γ σ e × L 2 is grav N e Edd m 3 F = r . L 1  F , through the disk may be written as ) with the cross section π = 3 ˙ 8 M Edd = L T σ is the photon-electron scattering cross section. The force σ accretion luminosity , where γ Thomson scattering N σ Eddington limit is the radiative efficiency of the disc equal to the binding en is the mean energy transferred radially per collision. The n . The upward force on the infalling matter is mainly due to the r η ε M The efficiency may be even larger if the matter is accreted thr However, the radiation is interacting with the accreting ga Consider a fully ionized hydrogen plasma accreting near the circular orbit per unit rest mass. As we have shown in section radiation force (6.12) with the gravitational forceknown (6.13) as the photons with electrons in the plasma. If the photon luminosi This gives where In order for accretion to occur, the gravitational force per must exceed the radiation force (6.12). Here, is just the rate at which the momentum is deposited radially p The dominant electron-photon process in a highlyby ionized free hy electrons ( per unit time is to obtain crossing unit area per unit time at radius Black-Hole Phenomenology rotating black hole. The where generated by a mass accretion rate, lower and upper bounds for a nonrotating and for a maximally r above which the radiationthereby pressure, stops the acting accretion. against gravita mass PoS(P2GC)004 . ´ c dr and + n eglect (6.22) (6.21) (6.18) (6.20) (6.23) (6.19) (6.17) p r m Neven Bili = and s in the shell r ρ of a BH the Schwarz- .

M 8 M er if the general relativistic 8 . 10 2 , 10 / al period of 1 hr. This sets the × 1 . × 7 − 2 . dr dp ) / his is an incredibly small size. 3 r s. (Carroll and Ostlie [6], p. 1181) 1 2 ng gravitational field is larger by a 2 ust be less than the Eddington limit, 14 ill nonrelativistic, i.e., gravitational force is no longer valid. / / . . / − . 1 1 2 ) ) . The force acting on the shell element estimated above. This fact supports the r

− )= r ng the relativistic expression (6.20), we M km orce per electron (acting via the proton) ) / = / 2 G dr R M r 9 8 2 / is equivalent to more than 500 galaxies of ! − ( M M 10 / 1 2 10 1 2 M ( 0 − BH R × 2 − × − 1 M M . 1 = 3 1 − 2 M . 2 ( 1 ( r p 31 3 1 erg s ( BH − m V = M = 46 d M p − 0 2 2 0 σ BH m 10 = with thickness M = drdS M M π r 7 AU × 4 4 > is the difference between radial pressures at = ' N + = dF M d V dSdp R 1 dp d = = Edd s L

0 dF grav F M by the right-hand side of the TOV equation (2.18) in which we n by assumption). According to (2.11), the number of particle > dr ρ . Then, the gravitational force per particle is / M  V dp p is quite close to the mass is nd 0 thus obtained with the photon force (6.12), we find = M dS | N grav d F | , provides a lower limit for the mass of the central object . Edd ρ The estimated value of the lower mass limit is somewhat small The typical quasar luminosity of 5 The luminosity of a typical quasar varies in time with a typic Consider a spherical shell of radius In a strong gravitational field the expression (6.13) for the L  < p the pressure (since schild radius of which is equal to the radius of the quasar idea that supermassive BHs are responsible for powering AGN correction to the Eddington limit isfind taken into account. Usi Next, we substitute The mass limit the Milky Way size! Now, theL constraint that the luminosity m Considering that AGNs are the most luminous objects known, t is the proper volume element and upper limit to the radius of about Hence, compared with (6.14),relativistic the correction Eddington factor. limit in the stro 6.3.2 Radius and Mass Estimate Equating where Black-Hole Phenomenology Relativistic Eddington Limit Here we derive the general relativistic expression for the f element is under the assumption that the energy density of the fluid is st of surface area PoS(P2GC)004 . ´ c ?

a

M M 4 ) or the 10

(provided − Neven Bili M 2 disc 20 L − 5 . emaine [71]). Such 3 ion. ∼ M equals the binding energy ( nstrate that it possesses an . η observations of high redshift a advection-dominated accretion tion with very low radiative ef- ve BHs per unit volume of the er retion luminosity at the intermediate mass BHs do specific BH. . Assuming that supermassive BHs he population of AGN as a whole, rate estimate of the mass accretion e current data suggest an efficiency hypothesis), one can divide the two in nearby galaxies, one can estimate n 4.1.3, termediate mass BHs [73]. However, ynamical mass measurements would of an accretion disk is defined as the , 76] emoving the angular momentum from X-ray sources have been detected [72]. η a binary companion to provide a robust e produced by stellar mass BHs, e.g, by Poutanen et al. [74]). Hence, it is at present 32 or more exceed the Eddington limit (6.16) of 10 40 for an individual AGN with the precision needed to estimate ). Are there BHs in the intermediate mass range 10 η

M 6 10 > M are possible only if supermassive BHs have significant rotat no reason for such BHs not to exist. η 15 for supermassive BHs on average (Elvis et al [70], Yu and Tr . 0 a priori − 1 , so one cannot calculate . ˙ 0 In a typical accretion system, one can easily measure the acc To prove that a BH-like object is indeed a BH, one needs to demo Advection dominated accretion flows [77, 78] describe accre There is a tentative evidence based on the Eddington limit th So far we have been concerned with either the stellar mass BHs It should be noted that this is only a statistical result for t According to equation (6.10), the radiative efficiency M ∼ the distance is known), butrate one practically never has an accu However, a rough estimate canAGN, be made one for can an estimate averageuniverse. the AGN. Similarly, From mean by energy taking a radiated samplethe by mean of mass supermassi supermassive in BHs BHs per unitacquire volume most of the of universe their at present massquantities via to accretion obtain (a the not mean unreasonable radiative efficiency of AGN. Th large values of BHs, these objects are argued tolarge be apparent good candidates luminosity for may, under the in speciala conditions, supercritical b accretion (for details and references, see unclear what exactly the ultraluminous X-ray sources are. D settle the issue but, unfortunately, nonemass of estimate. the sources has 8. Observational Evidence for the Horizon event horizon. The majorflows tests (ADAF), ii) for X-ray a bursts, and BH iii) horizon direct are imaging [8, based 75 on8.1 i) ADAF ficiency in which the energy released by viscosity friction r supermassive BHs ( exist. In several nearby galaxies aAs number of their ultraluminous luminosities of the order of 10 There is and the method does not say anything about the rotation of any 7. Intermediate Mass BHs η Black-Hole Phenomenology 6.3.3 Estimate of the AGN Efficiency energy it radiates per unitof accreted gas mass. at the As ISCO, shown which in in sectio turn depends on the BH spin paramet PoS(P2GC)004 ´ c and ˙ ersect M Neven Bili and accretion ˙ M te with high . Because of strong R 02 mas which is not beyond . radius 0 h cases. imple. At present, it is widely c) giant elliptical galaxy M87, onstruct an image of the region ∼ ely such a comparison could be hose compact object is a BH. For surface. Therefore, the argument n to the quiescent X-ray luminos- candidates are in a class of X-ray ry. Narayan and Heyl [80] argued es" are dimmer. They came to the . Material cannot accumulate on an s or a few tens of seconds. Remark- 01 mas, is another object of interest t and is compressed by the object’s . ies can be explainedf in X-ray terms binaries of supposed to an contain y low accretion rate w. If an ADAF forms around a BH, n. The angular size of the boundary, ly the Eddington limit in less than a t horizons. However, this conclusion 0 BH candidates. n, whereas if the accreting object is a hat, in accordance with the prediction tion of accreted material [80, 81]. Ac- server will see an apparent boundary of ). ∼ ed by a variable mass accretion rate, and es unstable thermonuclear burning, which 75] 33 must have a surface and at a distance of 8 kpc, is

M 6 2 [82]. Rays with impact parameters inside this boundary int 10 / × R and an expected angular size of 4 3

√ ∼ M 9 M 10 × with . Only occasionally do they go into outburst, when they accre ∗ acc L A key point is that the compact object The explanation for such an absence of X-ray bursts is quite s In some X-ray binaries, X-ray bursts are observed in additio The most promising line of search for a direct evidence isFor to definiteness, c consider a nonrotating BH with a horizon at Narayan, Garcia, and McClintock [79] suggested that precis accepted that X-ray bursts arise fromcreted thermonuclear material detona builds up ongravity. the After sufficient surface material of accumulates, the itwe undergo compact observe objec as an X-ray burst. ity. In a typicalsecond, X-ray and burst the flux luminosity then increases declines overably, up a no to period X-ray of near bursts a of few second thisthat kind the were lack observed of in bursts any is BH a bina strong evidence for the horizon in become bright. Spectral observationsADAF. Narayan of et quiescent al. BH [79] compared binar BHs quiescent with luminosities those o of neutron-star X-rayof binaries the and ADAF realized model, t systemsconclusion containing that black-hole they found “candidat evidence forhas been the challenged presence (for of details even and references, see [8, 75] 8.2 X-ray Bursts event horizon, and so no burstsmore can details, come from references, an and X-ray critical binary discussion, w see [8, 8.3 Direct Imaging near the event horizon using interferometry. gravitational lensing in the vicinity of the BH, a distant ob the BH at a radius of 3 luminosity Black-Hole Phenomenology the accreting matter is notthe radiated stored away energy but will stored be inNS, lost the this forever flo under energy the must event begoes, horizo radiated BHs away should once be matter dimmer than lands NSs on if its an ADAF is presentdone in using bot X-ray binaries. Mostbinaries of called X-ray the novae. known These stellar-mass systems BH tend are characteriz to spend most of their time in a quiescent state with a ver reach. The supermassive BHwith in a the mass nucleus of 3 of the nearby (15 Mp the horizon, while rays outsidee.g., the for boundary Sgr miss A the horizo [83]. PoS(P2GC)004 ´ 2 c e

− M m 6 (9.1) is the OV Neven Bili would have GM 2

M = f the radio-galaxy 9 . 53, 56]. What still Sch 10 R Sch × R 3 = with no singularities in the ino stars at the OV limit with OV jects ranging from a few 10 nterferometry, where angles less M ating ball of degenerate fermionic for this scenario is dard astrophysical scenario in the re, it should be possible to operate Hs with the Schwarzschild (or Kerr) , . r brown dwarfs) although not entirely odel proposed for supermassive com- low the OV limit (3.14), with an ate" or become a BH in much less time objects built out of more or less exotic 2 4 s a radius of 3 egeneracy pressure of fermions due to (or neutralino) stars, 2) Boson stars, 3) 91] eutrino star at the OV limit and a BH, in in the case of NGC 4258 and our Galaxy, nos in the keV mass range have recently = = egenerate fermion gas composed of, e.g., nos or supersymmetric fermionic partners g g 52 light-days, where . 1 = 34 km . m 10 14 keV for 12 keV for as a generic name for any degenerate fermion star com- 10 ' ' × m m 1 mm and with baselines as large as the diameter of the Earth < 9396 . λ 3 . Thus, at a distance of a few Schwarzschild radii away from th = OV neutrino stars M Sch R , such as the supermassive compact dark object at the center o 4466 .

, the existence of black-hole-like objects is beyond doubt [ 4 M

9 = M 9 10 OV × R km. For details, see Falcke et al. [82]. 3 4 ' Let us assume that the most massive objects are sterile neutr Given the accumulated evidence for supermassive compact ob Here we use the term A number of alternatives to classical BHs have been proposed The best angular resolution achievable today is with radio i As we have seen in the example of a neutron star, a self-gravit 10 OV ∼ supermassive object, there is little difference between a n particular since the last stable orbit around a BH already ha Schwarzschild radius for a radius M87 [57]. Using (3.14) we find that the neutrino mass required functional dependence on the fermion mass M 9. Alternatives to Supermassive BHs to a few 10 interior. The three representative modelsDark are: energy stars 1) Neutrino 9.1 Neutrino Stars remains an issue is whethermetric these describing supermassive the objects physics are of B thesubstance interior, but or some with other aform regular of behavior a compact in cluster theexcluded, of is dark interior. quite stars unlikely. (e.g., It A has neutron been stan stars demonstrated o that, than the age of the Galaxy [84]. such a cluster would be short-lived and would either “evapor From (3.14) and (3.15) it follows that a neutrino star of mass Black-Hole Phenomenology than 1 mas are routinelyinterferometers at resolved. wavelengths In the not too distant futu b pact objects at the galacticheavy centers sterile is neutrinos a [85, self-gravitating 86, d 87,been extensively 88, discussed 89]. as Sterile dark-matter candidates neutri [90, posed of neutral weakly interacting fermions,such e.g., as neutri neutralinos, gravitinos, and axinos. The simplest m matter is supported against gravitationalthe collapse Pauli by exclusion the principle, d provided the total mass is be PoS(P2GC)004 ´ c 0 max OV → (9.3) (9.2) M = 63.9 T ≤ min . M Neven Bili m AU to be compared 3 , as observed in /

o star. 1 M  AU , as given by the OV 3

M > M ∼ is usually referred to as 5200  M 3 ' below a certain limit of the / 2 R theory, there are cases where . M  cularly appealing. However, it 4 ∗ evidence for intermediate mass 4 g e mass-radius range of compact φ lower than the OV limit above, we . Hence, a neutrino star made of  ∗ he Bose-Einstein (BE) condensate. 3 3 io can be extrapolated to explain all review and a comprehensive list of sate is prevented by a repulsive self- / nal BH scenario (in which BHs are . 8 ions (or spin 3/2 Majorana fermions), rid” scenario [92] in which all super- with or without self-interaction) com- ndensate may be regarded as a erived from the Lane-Emden equation , we find e and hence difficult to detect.  e, the intermediate mass neutrino stars

ized in nature, with the co-existence of of a condensed cloud of charged bosons

M M s, such as 8 6 m are black holes, while those with 10 10 , the supermassive compact dark object at the = 2 describes either spin 1/2 fermions (without 12 keV ∗ × × max g OV  6 5 . M 35 2 083 10 . > 60 AU for Sgr A 1 = × M ≤ = M R 2. We find . That makes it astrophysically interesting as its maximal is by about a factor of 10 1406 max OV / . 4 , and stellar-mass BHs with mass | ∗ 3 7 M

Φ | = = as a neutrino star, we need a minimal fermion mass M λ 3 3 n ∗ / 2 < ∼  2 g M  3 / 1 2 Pl M 3 , hence comparable with the mass of a neutron star or a neutrin m / 2 B 8 m m / = 4. The maximal mass of a neutrino star made of these fermions , is a stable spherically symmetric configuration [94] which 3 Pl 51 g m . = 4. Using the canonical value m , interacting only gravitationally and having a total mass 15 keV neutrinos is ruled out as an explanation for Sgr A / 4 g m 2 Pl ∼ − for = M 2 R 12 c At first sight, such a hybrid scenario does not seem to be parti An indirect support for this scenario is the absence of clear Boson stars are static configurations of self-gravitating ( The gravitational collapse of a self-interacting BE conden Clearly, a neutrino star scenario which would cover the whol To be able to explain Sgr A Now, we wish to investigate the possibility that this scenar = are neutrino stars, is not excluded. neutron stars with masses is important to note that a similar scenario is actually real binary systems in the Galaxy. BH candidates, which isall difficult baryonic). to explain If in the the hybridwould exist conventio scenario but were having realized very large in radii natur would be rather9.2 dilut Boson Stars plex scalar fields. In self-interacting scalar field theorie mini-soliton star [95] orreferences, mini-boson see star Schunck and [96]. Mielke [97]. For a recent interaction, e.g., in the form of order of mass limit of a self-gravitating boson gas [93]. The ground state supermassive galactic centers is ruledmassive out. compact dark However, objects a with “hyb masses keV/ limit (3.14), is [92] m with the observational upper limit [67] Here, 1 mpc = 206.265 AU. The degeneracy factor antifermions) or spin 1/2 Majorana fermions.we For have Dirac ferm a homogeneous condensate is a stableHence, ground a state, boson known star as being t basically a self-gravitating BE co Black-Hole Phenomenology observed supermassive BH, and in particular Sgr A Galactic center. As the mass of Sgr A can use the Newtonian mass-radius scaling(2.28) relation with (2.32) the d polytropic index mass is PoS(P2GC)004 ´ c (9.4) (9.8) (9.9) (9.6) (9.5) (9.7) (9.10) in order is deter- 2 Pl ω Neven Bili m / . On the other 0 M R 2 2 = ed by Heisenberg’s ' Sch m / R

π ,. > 2 . M 0 ) 2 = KG reads  R | , med through a dissipationless Φ L Φ  Φ | λ m ( µν 1 eV ld KG , g U uid form (B.1, we can still identify equation of state, the matter here is  L . − − field, where the frequency ) 10 t 2 + , s estimates: For a boson to be confined | ∗ ime. Unlike in the case of a fermion ild radius, − ω i have recently attracted some attention as 0 rrelevant to astrophysical context, unless π Φ Φ − | R , respectively. 10 ν . 2 e = 0 16 pse. In this way we obtain an estimate ∂ . 0 ) × m m r T Kaup limit Φ ∞ )( − ( / −  Φ  2 Pl ϕ 0 has a nontrivial ground-state solution, called ) → µ 2 2 g and m | 4639 ∂ 1 r Φ = 36 . r 2 √ r π Φ ν 8 | dU T det ∂ U d < = − )( − ) + ( 0 as )= ∗ + t Φ M m p , Φ ν → / r one obtains the Klein–Gordon equation x 2 µ ( ∂ ) with 4 2 Pl , one has to solve a coupled system of Klein-Gordon and ∂ r  Φ Φ d )( ( ( yields Einstein’s equations (A.24) in which m ρ U ∗ ϕ Z µν Φ µν g 633 µ g . = ∂ 0 = BS = = ( S KG , the Compton wavelength has to satisfy L µν 0 max T R M and the density p is ridiculously small. . The gravitational collapse of a mini-boson star is prevent m Given the interaction potential Seidel and Suen have demonstrated that boson stars may be for The action of the gravitationally coupled complex scalar fie It turns out that even the simplest case mini-boson star to avoid instability against complete gravitational colla Einstein equations in spherically symmetric static spacet star, where the information aboutdescribed matter by was the provided KG by equation the (9.6). The stationarity ansatz hand, the star’s radius should be larger than the Schwarzsch with the Klein-Gordon Lagrangian describes a spherically symmetric bound state of the scalar mechanism, called gravitational cooling [98]. Boson stars they may well be candidates for nonbaryonic dark matter [99] Black-Hole Phenomenology the boson mass obtained numerically [96]. Clearly, mini-boson stars are i This upper bound is slightly larger than the so-called and the variation with respect to By varying the action with respect to Although this energy-momentum tensor isthe not radial in pressure the perfect fl a uncertainty principle. This provides us alsowithin with the star crude of mas radius mined by the asymptotic condition PoS(P2GC)004 ´ c (see 3 (9.15) (9.14) (9.13) (9.12) (9.11) / us of a 1 M Neven Bili ∼ 2 R m / 2 Pl m m / λ Pl 60 AU, and even less it m √ m ) / . < 1 π

36 . 8 R km galactic centers? To answer 3 M ∼ , √ Radius 2 2 ∗ ( R Effective =   / π acting bosons with a quartic self- OV d out as a unique explanation for , ity and in a good approximation is R = m gain, it is not excluded that some of m , fit the radii of all supermassive BHs. , for the supermassive compact dark assume that the most massive objects f SgrA 10 1GeV

1GeV s with the star mass as eff the same radius. This behavior is quite adius of an interacting boson star in the R M   10 9 303 AU λ λ × 2 . √ 3 10 √ = 4 m 1 | . × / = 0 Φ 2 3 km 3 Pl | 2 m 5096 m / . = λ m 37  10 ' / 1 λ 2 PL 2 3 Pl = 10 3 PL = m √ m m m max ) m U × 2 3 Pl λ M 1GeV π 633 m m . 53 √ 384 8 .  . 0 3 λ 4 Mass 0 √ λ π ( √ = 1 8 = Maximum √ / = π 1 √ R 8 π OV Kaup = √ = of the neutrino star discussed in section 9.1. Since the radi M M = max max OV eff M R M R Mini-Boson Star Boson Star Object Fermion Star Hence, boson stars with a quartic self-interaction are rule The results are summarized in the table Could boson stars fit the whole range of compact supermassive This result was later extended by Colpi et al. [100] for inter compact supermassive objects at galactic centersthe and galactic AGN. centers A harbor a boson star. fits the radius of a typical AGN of 7 AU (section 6.3.2). equation (9.2)). object at the center M87. From this we find boson star does not depend onHowever, its it mass, obviously this does value not should fit also the observed radius limit o quite close to the radius which also fixes the radius given by (9.13) this question, we proceed as inare the boson case of stars neutrino with stars. maximal We mass, i.e., Black-Hole Phenomenology interaction Hence, boson stars with adifferent quartic from self-interaction the all fermion have star case, where the radius scale quite similar to the fermion starNewtonian mass regime limit. turns The out effective to r equal be to independent of central dens In this case, the maximal mass is obtained as [100, 101] PoS(P2GC)004 ´ c 1. (9.17) (9.16) ≥− w vanished 2 Neven Bili / may depend 00 1 g . Chapline et w = ξ , where gravastar de (actually slightly further ρ w M r a . As this required negative 2 with a surface density and a = = avastars and Chaplygin gravas- de ilar to the the Einstein-de Sitter or an accelerated cosmology. In the horizon with a shell of stiff GM p terior at a spherical boundary of dS It is worth noting that DE could 2 , or ‘gravastar’, was taken up by R itably chosen interaction potential equation of state satisfies < logies to condensed matter systems gian. This scalar field theory would omenon. Specifically, assuming the 3. In addition, one also expects that tter than the usual DE. f-interaction. r ondensate to be described by de Sitter / structure, e.g., in the form of spherical properties [106]. In this way, DE stars onal instability, small inhomogeneities re the lapse function f state 1 , . n-shell formalism [114]. 2 2 − / / 1 1 < − as spherical solutions to Einstein’s equation  0 ρ w 2 2 dS r π 3 R 1) 8 38 − = r 1 a =  being the vacuum energy density. The lapse function is = dS increasing again at . 0 R ξ up to the de Sitter radius ρ . Visser and Wiltshire [112] and recently Carter [113] also ρ ξ 0 − ρ dark-energy stars is homogeneous and it is normally assumed that DE does not 1 dubbed “phantom energy" [102, 103] has recently received GM = = 2 − p de ρ ρ = < but at present ( r w a , it is necessary to put a thin spherical shell at the boundary R In cosmological context, In order to join the interior solutions to a Schwarzschild ex Subsequently, the idea of gravitational vacuum condensate Dark energy (DE) is a substance of negative pressure needed f Given the mass M, the interior is described by a solution, sim Here, we briefly discuss two examples of DE stars: de Sitter gr The simplest example of DE star is a gravitating vacuum star o Nevertheless, matter with attention as it might fit the most recent SN 1a data slightly be al. [107, 108] putwhere forth an the interesting effective general proposal relativitySchwarzschild based exterior, was on they an suggested ana that emergent the phen sphere whe DE satisfies the dominant energy condition, in which case the marked a quantum phase transition, out). Hence, the interior is de Sitter, radius Mazur and Mottola [109, 110]matter astride and the Dymnikova surface [111], at replacing given by with Black-Hole Phenomenology 9.3 Dark-Energy Stars cluster. However, it is not(analogous excluded to that dark-matter owing inhomogeneities) grow to and gravitati configurations. build Hence, we define order to achieve acceleration, DE must satisfy an equation o universe, with constant density examined the stability of the gravastar using the Israel thi with matter described bybe the DE described equation by of aand/or state scalar with [104, field a 105]. theory noncanonical (quintessence) kineticcorrespond energy with to term a an in effective su the equation Lagran could of be state regarded with as desirable boson stars but with a rather unusual sel tars. 9.3.1 De Sitter Gravastars on the cosmological scale pressure, the authors of [108] assumedspace the with interior the vacuum equation c of state PoS(P2GC)004 ´ c 1) − (9.20) (9.18) (9.21) (9.19) < w . Neven Bili A √ . A √ . The solutions depend A √ , with the de Sitter gravastar, / 0

ρ one has M , ion of state , has attracted some attention as 9 A ntify the most massive black-hole  r mass gravastar is much less than cally, any BH-like object observed 10 emely efficient cooling mechanism ocated at a radius just slightly larger 3 s cosmological constant problem is n the one hand, the quantum phase e black-hole candidates, it must vary √ met with a cool reception. Certainly, ve scenario for compact massive ob- on of the shell. ) × rior of a gravastar so much larger than Ar ≥ 3 values wanted for cosmology. Another ating Chaplygin gas and lookonfigurations for in static the phantom ( roaches the value π constant is a fundamental parameter; on yields the de Sitter gravastar solution in M 4 ρ 4 = 2 ) . is larger, smaller, or equal to − 114]. As the pressure does not vanish at the ρ − eV 2 . max 0 . r . Hence, there is practically no observational 3 r ( ρ A ρ A M M − r Pl π l ρ 4 √ 2 . In the following we summarize the properties of − 10 39 ( A = < ∼ =  √ for selected values of ε ρ p 2 , M A ) dr ρ ε r d ( − + ρ 1 M  2 relative to = = 0 0 ρ ρ R , to be contrasted with the 4 ) 7keV . 9 = ( 0 ρ Consider a particular form of DE with a rather peculiar equat Combining (9.18) with the TOV equations (2.18), and (2.19), In Fig. 4 we exhibit the resulting In spite of these attractive features the gravastar has been The gravastar has no horizon or singularity! Its surface is l then question is how does athe gravastar entropy form? of The an entropy ordinarybefore of star gravastars a could and stella form this during would stellar collapse require [75]. an extr 9.3.2 Chaplygin Gravastars essentially on the magnitude of three classes of solutions corresponding to whether the assumption of a detransition would Sitter suggest interior that presents the associated athe cosmological other quandary: hand, to o accommodate the massover range of some supermassiv six orders ofreversed for magnitude. gravastars: why In is addition, thethe vacuum the energy observed notoriou in vacuum the energy inte densitycandidate in observed the at the universe? center of If M87, we with mass ide a dark-energy candidate [106, 115].solutions. Consider In a particular, self-gravit we look for static Chaplygin gas c than the Schwarzschild radius way to distinguish a Schwarzschildin BH nature from may a be gravastar. a gravastar. Basi Equation (9.18) describes the Chaplygin gas which, for Black-Hole Phenomenology surface tension satisfying Israel’s junction conditionsboundary, [ it must be compensated by a negative surface tensi We show that these configurationsjects could at provide galactic an centers alternati [116].the Moreover, limit equation when (9.18) the central density of the static solution app regime, i.e., when PoS(P2GC)004 ´ c = , a ∞ dS at the (9.22) R except → 7 0 upermas- iii) ρ Neven Bili starting from r is always less than . r M . In the limit )= bh r R where they vanish. At that ( 0 decreases with is represented by the dot-dashed M R )= r up to the de Sitter radius ξ bh A erior solutions to a Schwarzschild R gravastar case. ere 2 ( √ , respectively, converge to 2 (short dashed), 0.98 (dotted), 0.9 (long phantom energy rather than in terms . t massive such object is described by M 1 ii) ying Israel’s junction conditions [114]. 3 G = / or 1 A ) e continuous without the presence of a thin shell for i) 2 √ / 0 up to the radius Gr A ρ behavior at small r 7, 118]. 7 π , where 2 3 / 40 18 bh 2 ( − R , it is necessary to put a thin spherical shell r ' R joins the Schwarzschild solution outside continuously, ) r ( iii) ρ owing to (9.18). The enclosed mass remains constant equal to decrease with ∞ increases and the lapse function ρ ξ − ρ , is of particular interest as we would like to interpret the s and iii) ρ , the density blows up to , both it happens discontinuously. A , the density A p A from above or from below, solutions ii) √ √ √ A = < √ 0 > and 0 , together with ρ 0 ρ i) → ρ ii) Density profile of the Chaplygin gravastar for 0 For ρ For For , never reaching the black-hole horizon, i.e., the radius wh As As in the case of a de Sitter gravastar, in order to join our int Case iii) ii) i) It has recently been demonstrated that the joining can be mad ) 7 . Hence, we recover the de Sitter solution precisely as in the 2 ( M / sive compact dark objects atof the a galactic classical centers black in hole. terms It of is natural to assume that the mos exterior at a spherical boundary of radius at the endpoint. The lapse function in boundary with a surface density and a surface tension satisf whereas in 2 point the pressure near the origin. and the de Sitter gravastar by the solid line. Figure 4: Black-Hole Phenomenology limiting solution exists with a singular behavior a gravastar made of the fluid with an anisotropic pressure [11 the origin up to the black-hole horizon radius dashed). The limiting singular solution with the r PoS(P2GC)004 - ´ c , can (A.1) (A.2) (A.3) (A.4) (9.23) (9.24) 21 lhr, ii) . 8 . Indeed, enters it is

= Neven Bili , so that the M p 9 km 3 , would have a 9 r 10

10 M ×  6 × 3 plerian nature of the 10 M = below the distances of 86 . × 8 3 , as in the usual Newtonian max = ugh the Riemann curvature , r = M γδ ρ ] max to Γ GC M , µν

oximation M ρβ α 2 g , which is similar to the Newtonian A Γ M 0 λ A = 6 π ρ ∂ ion, one cannot neglect the pressure . d at the center of M87, is a Chaplygin − 2 √ s at the galactic centers, as Chaplygin- 10 − = 17 lhr = 123 AU, [63, 64]) and SO-16 , max βδ ρ elationship for these solutions. This may = ,  = R Γ ν 2 0 µλ with respect to 0 , with the solution min g R α ργ σβγ ρ α dx ν . Clearly, solutions belonging to class R min constant Ar Γ ∂ µ R M π M ; + = max + 4 τα dx g  A M δγ α 41 = µν νλ 2 0 2 2 Γ 0 = g g r R π β µ ρ 15 ∂ = ∂ − 16 [ 2 − 1 τσβγ (depicted by the solid line in Fig. 4). The radius of this λσ = ds  R g βδ α 0 5 0 which phantom gravastars obey, offers the prospect of unify 1 2 Γ iii) M ρ R γ 5 0 ∂ R = = is well established. = ρ ∝ µν σ ∗ which the stars SO-2 ( Γ M ∗ δβγ α R . However, for the phenomenology of supermassive galactic c max M < M equal to the Schwarzschild radius 2 = 2.06 lhr if the scaling law (9.24) holds. This radius is well is the metric tensor. The spacetime curvature is defined thro dS R GC µν R g = 8.32 lhr = 60 AU [66, 67]) recently had and beyond which the Ke The geometry of spacetime is described by the metric min R the closest approach to Sgr A radius ( gravitational potential of Sgr A A. Basic General Relativity A.1 Geometry the compact dark object at the center of our Galaxy, with mass pressure term becomes dominant. Next, neglecting 2 assuming that the most massive compact darkphantom object, gravastar near observe the de Sitter gravastar limit, with approximation, equation (9.20) simplifies to important to find, at least approximately, thebe mass-radius done r in the low central density approximation, i.e., ing the description of all supermassivegas compact phantom dark gravastars object with masses ranging from fit all masses which gives a mass-radius relation tensor where defined by where approximation but, in contrast toterm the in Newtonian (2.18). approximat Moreover, as may be easily shown, in this appr Black-Hole Phenomenology the de Sitter gravastar, i.e.,object is solution The mass-radius relationship PoS(P2GC)004 ´ c + and µ x (A.8) (A.9) (A.5) (A.6) (A.7) µ ( (A.11) (A.12) (A.10) Y dx = + Neven Bili . µ x dY µσ A nce ν σρ Γ + . . ρ at the point σν consists of two parts: one  V A denotes an ordinary partial ν Y µ ρν ∂ µ σρ DY µ Γ , Γ µν Ricci tensor + urved geometry. Given a covari- + gg ν , ρ , µ , det µ . For a scalar, vector, and second-rank V µν . Hence, ves the Y A − being its velocity. A massless particle µ dx = e parallel displacement is related to the , is null. µ may be a tensor of any rank) is defined by µ ν ; = p ; dx . µ u . The latter part is basically related to the Y µ  , satisfies the equation µ ρ . Y ; scalar curvature µ u + ϕ µ 0 λανβ αβ µ ≡ V x ∂ µ µν R u R to be a curve whose tangent vector is parallel = µ x = g A , µ λν ν αβ ; dx ; ρ to 42 g g ) µ ϕ det , V 1 µ u Y = x = − µ µν ν ρ µσ u Γ (in general, ∇ √ A αβ R geodesic − R Y σ νρ = = ( ν Γ , ν µ ∇ − V DY µ = σν ∇ A ν ; of a field µν µ µρ g σ µ V Γ ∇ , we define a ≡ µ − ∇ ρ 2 , µν due to the parallel displacement and the other is the differe A parallelly displaced from denotes the covariant derivative. ) Y = µ µ ρ x ; µν due to the functional dependence on µ ( A V ) ϕ Y µν µ is an infinitesimal difference between the value of the field A x ( Y DY vector tensor scalar − Geodesics are the “shortest possible lines” one can draw in c Here we have used the usual convention in which a subscript A covariant derivative ) • • • µ A.3 Geodesics ant derivative operator moves along a null geodesic in which case the vector derivative and ; Black-Hole Phenomenology are the Christoffel symbols. A contraction of two indices gi and a further contraction of the Ricci tensor gives the A.2 Covariant Derivative where the quantity dx ordinary partial differentiation. Thecurvature of difference spacetime due and depends to on th the tensor nature of tensor one finds: [119] A massive particle moves along a timelike geodesic, The covariant d’Alembertian is given by is the change of propagated along itself, i.e., a curve whose tangent PoS(P2GC)004 ´ c µ k (A.15) (A.17) (A.18) (A.14) (A.20) (A.13) (A.16) (A.19) . Then the Neven Bili and µ µ 4 δ parameter group = µ k . A Killing vector , k . 0 = ν . ; . , Killing vector x µ 0 second term vanishes by the geodesic µφ . u µ g 0 ν g . u be a geodesic with tangent u ants of motion. ed a = 0 = , µ = γ . Suppose the metric components are inde- 0 χ µ µ ) µ ; = . ν ξ ψ φ = + , k , µ ν ν x µν x ; ∂ ; θ ∂ . In spherical coordinates + ) g µ as a differential operator , ; ∂ ; µ ∂ r t µ ν k µ ∂ µ χ , 0 ; φ 43 µ . In such a coordinate system, ∂ u t ν ∂φ δ µ δ = k x ( / µ = u / k = k ∂ µ χ = = ) k ∂ is independent of ≡ ( u ≡ ν ; µ µ 4 = ν ) = µ x µν u ξ = ( ν ψ ; ξ g k ψ µ ν ; ( ) k µ u . µ γ χ ( ν u is Killing if µ k be a Killing vector field and let that generates one parameter group of isometries, i.e., one µ 2 µ χ k , local coordinates can be found such that . Then there exist two Killing vectors: k φ Let is constant along µ and u t µ χ is one of these coordinates, e.g., x the generator of time translations the generator of axial isometries : We have A vector field Consider the spherical coordinate system • • where Proposition 1. A.5 Constants of Motion The following proposition relates Killing vectors to const quantity Proof For a vector field satisfies the Killing equation Black-Hole Phenomenology A.4 Isometries and Killing Vectors of transformations that leave the metric invariant, is call as was to be proved. It is convenient to represent the Killing vector Hence, one can say that The first term vanishes by the Killingequation equation (A.12) (A.13) . and the Hence pendent of Examples PoS(P2GC)004 ´ c (B.1) (B.2) (B.5) (B.3) (A.24) (A.21) (A.23) (A.22) Neven Bili the velocity, ies along the σ . , and n ∞ ,

φ ρ q , . p θ . Hence, in this convention, , ∞ 2 , ) ∞ µ

→ . sin 0 u . 0 r 2 mple discussed in appendix A.4 q 0 r ter through Einstein’s field equa- uity equation −−− = as = , = , e by p )= µ (+ ν µ µν µ q . µν ∂ ) ropy density of the fluid. The energy- q 1 µ ν T θ pg 0 tic flatness, i.e., u φ µ π 2 g = gnu Euler’s equation [120] µ g 8 − u ν − = ν sin − − 0 (B.4) , corresponding to time translation and axial u u 2 µ + √ µ r µ = = ( = q u p ∂φ u µ µ − µ ν µ / ) ; , ∂ 44 ξ R µν q ∂ ∂ 2 ρ g g µ r µν µν = − + − ψ 1 = T g − = ν µ p , ; − 1 2 √ . ξ µ ψ µ 1 µ u u = − − = µ = ( ν , µ u u µ mu 1 and ; µν ) ( ψ µν ) t R ρ = T µ µ ∂ + E / nu diag mu p ( ∂ ( − → = = ξ µν L g moving along a timelike geodesic, the two conserved quantit m energy-momentum tensor is the is the metric tensor with the Lorentzian signature µν µν T g the z-component of the angular momentum the energy Consider a stationary axially symmetric space, as in the exa The particle number conservation is described by the contin General relativity relates the geometry of spacetime to mat Consider a perfect gravitating relativistic fluid. We denot • • where applied to (B.1) yields the relativistic generalization of pressure, energy density, particle numbermomentum tensor density, of and a ent perfect fluid is given by where Black-Hole Phenomenology Examples with two Killing vectors, we have For a particle of mass A.6 Einstein’s Equations tions B. Basic Fluid Dynamics The energy-momentum conservation geodesics are isometries, respectively. In addition, assume the asympto PoS(P2GC)004 ´ c 1. < (B.9) (B.6) (B.7) (B.8) 2 (B.14) (B.12) (B.10) (B.13) (B.11) v ≤ Neven Bili : µ t is the unit vector µ t 1 and hence 0 . ≥ 0 00 p γ , where µ g µ , g ) 1 ∂ drostatic equilibrium. We can √ 0 ~ − − ) 1; = . 2 , which projects a vector into the id in terms of three-velocity com- v 3 = . ν . , ν , t y takes the form 2 , ξ = ( 2 , − 00 / µ , ν 0 µ ν 00 00 1  t g 1 µ  i 1 µ i g u ξ ξ g µ pendent of time, and the mixed compo- v ξ ) v g − µ ; ∂ √ ν = r j p = ( ∂ t − j 00 2 v µ µν 2 j / 00 t , = g = γ g 1 i . 0 i 00 g 1 2 µ 0 − 00 g − g µ µ ) g t u g u p ) √ ; − , µν µ i ; ρ t i j g 45 + v i 00 g 00 ; ; + v = ρ g 1 g + ( − p γ 00 00 √ = √ µ ( j µ µ 0 0 g g = ( t 0 2 δ δ   − γ g v 0 00 √ √ i γ γ u g 0 = = 0 g = = = = u p , µ j 0 µ µ µ ν µ u = µ 0 v u u u ξ ∂ i j Γ i j ν µ γ ) γ ξ ξ p = p i + v ρ = ( in two parts: one parallel with and the other orthogonal to µ µ t u are timelike unit vectors, a consequence of (B.7) is that µ t are zero. Equation (B.5) then gives and i µ 0 u g From Euler’s equation (B.5) we can derive the condition of hy From (B.7) with (B.6) we find (see, e.g., appendix of [121]) It is convenient to parameterize the four-velocity of the flu B.2 Hydrostatic Equilibrium use the comoving frame of reference in which the fluid velocit where subspace orthogonal to the time-translation Killing vecto We split up the vector where In equilibrium the metric is static; all components are inde nents Black-Hole Phenomenology B.1 Fluid Velocity ponents. To do this, we use the projection operator with the induced three-dimensional spatial metric: or Since PoS(P2GC)004 ´ c (C.8) (C.2) (C.6) (C.7) (C.1) (C.3) (C.5) (C.9) (C.4) Neven Bili ty is obtained using the e in equilibrium for which the rostatic equilibrium (B.14) [10, . gy is defined as [122, 123]. ate of entropy change with particle 2 / ) 1 ld vanish [124] , mber of particles ) . µ . µ ) Σ ξ 1 n T dn , µ d . N µ d µ ξ µ T are metric-dependent local quantities. Their , is related to the velocity of the fluid. µ T ρ = − µ µ − − − µ σξ ρ = ( const ξ particles in equilibrium at nonzero temperature. ρ Σ T = that contains the fluid is orthogonal to the time- µ ξ const T + = d d 46 Σ N ρ p µ Z Σ 1 ρ

= T nd (

n ; − 1 nu T n µ T = = ∂ µ ∂σ Σ ∂ ∂σ M u Z p σ = T ξ d = d σ = F µ ξ . From (C.4) and (C.5) we obtain . In equilibrium µ ρ ξ and the chemical potential T is the total mass as measured from infinity. The entropy densi M It may be shown that those and only those configurations will b Consider a nonrotating fluid consisting of The temperature and hence free energy assumes a minimum [122]. The canonical free ener which may be derived fromnumber the at physical fixed requirement energy that density the should r be constant, i.e., spacetime dependence may be120] derived and from the the thermodynamic identity equation (Gibbs-Duhem of relation hyd where ‘const’ is independent of should be fixed. Thetranslation Killing spacelike vector hypersurface field where standard thermodynamic relation The crucial condition is that the heat flow and diffusion shou A canonical ensemble is subject to the constraint that the nu Black-Hole Phenomenology C. Basic Thermodynamics C.1 Tolman Equations PoS(P2GC)004 . ´ c µ Σ (C.12) (C.14) (C.11) (C.13) (C.10) (C.15) Neven Bili 2 (spin up and = g he chemical potential take cal objects is almost always eful thermodynamic identity , ) d side of (C.11)-(C.14) with , T using the well-known momentum , , together with (B.14) and (C.5), , / T contained within a hypersurface / µ , T T µ / + / 0 ial integration, the last equation may m − µ T µ µ / T , − 14) one should also add the antiparticle − / 1 E T T = E / − / dp 1 E e may be chosen arbitrarily. In a canonical 2 . There, the numbers of neutrinos and antineutrinos e E E 1 n / e e 0 00 1 + + g µ 1 + + 1 )+ µ ( 1 1 2 E n q σ ln 3 3 3 47 and ( ) ) q q 3 3 ; 3 0 3 π ) π ) q q 0 T d 2 d Td 2 3 3 π π T ( ( because of the constraint (C.1) that the total number of d d 2 2 = ( ( ∞ ∞ = ξ 0 0 ∞ ∞ 2 Z Z 0 / 0 dw 00 1 are the local temperature and the local chemical potential, g g Z Z fermions with the mass g = µ = Tg gT N = n ρ = p and p T and 2 being the specific enthalpy. q n + are constants equal, respectively, to the temperature and t / 2 ) 0 . However, the contribution of antiparticles in astrophysi ρ m µ denotes the spin degeneracy factor. In most applications we µ is an implicit functional of + 0 p g − . p µ and 8 = 0 = ( E T w Equations (C.4) and (C.8) may be combined to yield another us Consider a gas consisting of One exception is neutrino stars made of Dirac type neutrinos 8 term which is of the same form as the corresponding right-han spin down). Strictly speaking, in each equation (C.11)-(C. The integer where where negligible replaced by respectively, defined by Tolman’s equations (C.10). By part be written as Black-Hole Phenomenology from which (C.6) immediatelyimplies follows. the well-known Tolman equations Next, equation (C.6) particles should be fixed. C.2 Fermi Distribution are equal and separately conserved, hence g=4 at infinity. In a grand-canonical ensemble, ensemble, The equation of state mayintegrals be over represented the Fermi in distribution a function parametric [125] form with PoS(P2GC)004 ´ c (C.17) (C.23) (C.26) (C.25) (C.24) (C.18) (C.22) (C.19) (C.21) (C.20) (C.16) 2, we find Neven Bili otational. = g nction that yields . in the form (C.22) ,  i µ X . With X X wu Arsh Arsh + entity (C.15), equation (B.5) − 1 1 + + 2 2 , X he quantity 3 functions of defined as X p i momentum X . . ativistic analogue of potential flow in ) on to an isentropic flow. A flow is said p 3 . µν 0 1 ) 0 m , ω 1 or in spacetime into its component in the (C.19) implies [126] ] mX 2 = − ν = σ , + ; 2 1 π u µ . = ; ρ 2 3 ϕ w µ [ X 0 ) , 2 µ µ u u X ν 3 2 0 ∂ ∂ = 2 σ ν m such that ( − ( h − = )= 3 wu X − − µ µ ϕ ρ ) X ( 48 ν q  n is constant, i.e.,when ν 2 σ h h 3 = δ ; π µν 4 ( ) − 4 d µ n 2 µ µ = ω µ ( m = / ν m ; ∂ 2 p F ) 2 σ µ ν q wu 1 µν π wu µ h 1 0 π ( = 8 ω 8 Z ν F wu 2 u = q ( = = 3 3 ) q ) n q 3 π 3 π d 2 d 2 ( ( 2 E E q 3 F . A flow with vanishing vorticity, i.e., when q F 0 µ q . In the following we assume that the flow is isentropic and irr Z 0 u Z 2 0, the Fermi distribution in (C.11)-(C.13) becomes a step fu 2 = = → ρ p T when the specific entropy irrotational isentropic As a consequence of equation (C.16) and the thermodynamic id There are two important limits: In the limit Euler’s equation is simplified if one restricts considerati A flow may in general have a nonvanishing vorticity simplifies to where The equation of state can be expressed in terms of elementary satisfies equation (C.21). Solutionsnonrelativistic of fluid this dynamics [120]. form are the rel C.4 Degenerate Fermi Gas is the projection operator which projectssubspace an orthogonal arbitrary to vect where the minus sign is chosen for convenience. Obviously, t Black-Hole Phenomenology C.3 Isentropic Fluid to be is said to be In this case, we may introduce a scalar function Furthermore, for an isentropic irrotational flow, equation an elementary integral with the upper limit equal to the Ferm PoS(P2GC)004 ´ c (C.32) (C.35) (C.31) (C.29) (C.33) (C.34) (C.28) (C.27) (C.30) . For an n Neven Bili , , . ,  mperature.    ...... l-known equation of state for an 9 ... manuscript. This work was sup- 7 (C.28) we find lic of Croatia under Contract No. X spitality. I am grateful to Andreas X X X tate in both the nonrelativistic and (C.30) yields 5 3 24 ln2 56 . . ln2 3 3 3 2 + / / 1 2 − . 4 5 7 5 Γ + n n − X n 2 , 3 3 X 2 may also be expressed in terms of / / 1 X , 5 4 2 3 ρ 3 14 X 10 Γ σ π π − m 4 − n K 3 3 + 5 − = 4 Γ / / + 4 5 1 2 p 3 if the equation of state may be written in the form X K 49 3 3 X + X X   =  4 = =  4 4 mn p 4 m p p . Using this equation and the thermodynamic identity m m 2 m = m 2 2 2 π 1 1 π π ρ 1 π 1 4 12 polytropic 3 . 15 = ρ = = = ρ p 1 ρ  p p  is called and the entropy density X m ρ is a constant that depends on K Ultrarelativistic limit which also holds for massless Bose and Fermi gases at finite te Nonrelativistic limit, Nonrelativistic limit. Retaining only the dominant term in Ultrarelativistic limit. In this case, the dominant term in Obviously, in this limit, Retaining the dominant terms, these equationsultrarelativistic yield gas the wel I thank the organizers of the school for the invitation and ho • • • • isentropic flow, it follows (C.15) the energy density where Müller for his valuable commentsported which by helped the me Ministry improve the of0098002. Science and Technology of the Repub Acknowledgment Black-Hole Phenomenology C.5 Polytropic Gases A gas of particles of mass A degenerate Fermi gas approachesthe a extreme-relativistic limits: polytropic equation of s PoS(P2GC)004 ´ c , . Neven Bili ]. and Future eral-relativistic . tational collapse , Wiley & Sons, NY , (1939) 374. tml 55 astro-ph/0603342 , (1958) 965. , Addison-Wesley, Reading gr-qc/0506078 110 Phys. Rev. gr-qc/0411060 (1972) 347. ]. , , 178 , W. H. Freeman and Company, NY 1973. , Clarendon, Oxford 1934; Dover, New York (2005) 199 [ Phys. Rev. ", 10 - 14 July 2006, Santa Fe, New Mexico; ]. 7 (1973) 277. (1977) 831. Gravitation (1974) 324. 50 Astrophys. J. 179 213 (1977) 433. 32 gr-qc/9707012 , hep-ph/9809563 , Wiley & Sons, NY 1972. New J. Phys. 179 , X-ray Properties of Black-Hole Binaries On Massive Neutron Cores gr-qc/0503041 (1931) 81. (1971) 1344. Black Holes, White Dwarfs and Neutron Stars The river model of black holes Astrophys. J. Astrophys. J. 74 MNRAS 26 (1969) 252. (1999) 173 [ An Introduction to Modern Astrophysics , University of Chicago, Chicago 1984. (1924) 192; D. Finkelstein, 1 Phys. Rev. Lett. (2002) 1688. 11 Gravitational phase transition of fermionic matter in a gen Supermassive Black Holes in Galactic Nuclei: Past, Present C 113 . 295 (1976) 270. A black hole mass threshold from non-singular quantum gravi (2005) 091302. [ Astrophys. J. 262 Nature Phis. Rev. Lett. Black Holes -Lecture notes 95 Science Gravitation and Cosmology Relativity Thermodynamics and Cosmology Black Holes in Astrophysics General relativity Riv. Nuovo Cim. A hierarchy of cosmic compact objects - without black holes Eur. Phys. J. Nature , astro-ph/0411247 , astro-ph/0606352 Research 1983 MA 1996. http://qso.lanl.gov/meetings/meet2006/participate.h 1987. framework Phys. Rev. Lett. [1] R. A. Remillard and J. E. McClintock, [2] L. Ferrarese and H. Ford, [3] C. W. Misner, K. S. Thorne, and[4] J. A. S.L. Wheeler, Shapiro and S.A. Teukolsky, [5] R.M. Wald, [6] B.W. Carroll and D.A. Ostlie, [7] P.K. Townsend, [8] R. Narayan, [9] “Physics and Astrophysics of Supermassive Black-Holes Black-Hole Phenomenology References [10] R.C. Tolman, [11] S. Chandrasekhar, [12] J.R. Oppenheimer andG.M. Volkoff, [13] N. Bilic and R.D. Viollier, [22] J.M. Bardeen, W.H. Press and S.A. Teukolski, [23] R. Penrose, [14] C.E. Rhodes and R. Ruffini, [24] S. W. Hawking, [15] M. Nauenberg and G. Chapline, [16] J.B. Hartle and A.G. Sabbadini, [17] J. Hansson, [25] R. Znajek, [18] M. Bojowald et al., [26] R. D. Blandford and R. Znajek, [27] S. Koide et al., [19] S. Weinberg, [20] A. S. Eddington, [21] A. J. S. Hamilton and J. P. Lisle, PoS(P2GC)004 ´ c (1991) 322 s from Science ]. , (2005) 328 376 Neven Bili , in T. e effect of 633 MNRAS , f Jet Formation with ]; movies at 10 - 14 July 2006, ]. Astrophys. J. , ay states Astrophys. J. ]. , (2004) 71 astro-ph/0309832 ]. 417 (2003) 1238. (1973) 337. (2004) 861 [ astro-ph/0302271 astro-ph/0410116 X-ray flares reveal mass and angular 599 24 Black Hole Spin in AGN and GBHCs 413 astro-ph/0403032 Magnetically Driven Accretion Flows in the Kerr 51 Astron. Astrophys. , (1972) L143. (2005) 71 [ Simulations of jets driven by black hole rotation astro-ph/0404512 (2003) 1041 [ (2004) L131. ]. From X-ray Binaries to Quasars: Black Hole Accretion on Astrophys. J. 174 , 300 (1974) 161. (2005) 60. [ 341 (1972) 37 . 606 A Measurement of the Electromagnetic Luminosity of a Kerr ]. The variability of accretion onto Schwarzschild black hole 34 Astron. Astrophys. 625 235 Astron. Astrophys. Relativistic emission lines from accreting black holes - Th , (2004) 977 [ (1969) 690. (2003) 934. (1995) 659. MNRAS The Dynamics of the Magnetoviscous Instability , (2003) 104010. 611 ]. ]. Nature (1972) 271. 223 425 375 The Dynamics of the Magnetoviscous Instability 67 Physics and Astrophysics of Supermassive Black-Holes A General Relativistic Magnetohydrodynamics Simulation o Astrophys. J. Lett. D 235 Astrophys. J. Astrophys. J. Lett. , Astrophys. Space Sci. astro-ph/0408371 Nature Nature Nature , Astron. Astrophys. Powerful jets from black hole X-ray binaries in Low/Hard X-r Astrophys. J. astro-ph/0008447 , Phys. Rev. (2004) 978 [ astro-ph/0504666 astro-ph/0401589 turbulent magnetized discs http://jilawww.colorado.edu/~pja/black_hole.html Santa Fe, New Mexico (see ref. [9]). Maccarone, R. Fender, and L. Ho (eds.), All Mass Scales [ disk truncation on line profiles momentum of the Galactic Center black hole [ 214. (2001) 31 [ 305 Metric I: Models and Overall Structure Black Hole a State Transition [48] C. S. Reynolds, talk at [49] D. Lynden-Bell, [46] J.M. Miller et al., [40] A. Müller and M. Camenzind, [41] R. Genzel et al., [47] P. J. Armitage and C. S. Reynolds, [42] B. Aschenbach, N. Grosso, D. Porquet and P. Predehl, [43] Y. Tanaka et al., [44] N. I. Shakura and R. A. Sunyaev, [45] C. S. Reynolds ,L. W. Brenneman and D. Garofalo, [39] T. Islam and S. Balbus, [37] B. Paczynski, [38] S. Balbus and J. F. Hawley, [31] J. P. De Villiers, J. F. Hawley and J. H. Krolik, [32] J. C. McKinney and C. F. Gammie, Black-Hole Phenomenology [28] K. I. Nishikawa et al. [33] H. Tananbaum et al., [36] R. P. Fender, [34] B.L. Webster,P. Murdin, [35] C.T. Bolton, Nature [29] S. Koide, [30] V. Semenov, S. Dyadechkin and B. Punsly, PoS(P2GC)004 ´ c , . ]. . , in L.C. ]. (2004) 27 . Neven Bili 155 (2000) 73 . idge University Press, 535 astro-ph/0105230 in Galactic Nuclei , astro-ph/9802264 (2002) 41 astro-ph/0009249 , Black holes and relativistic Theory of Black Hole astro-ph/0002085 15N6 , in astro-ph/0107134 , Astrophys. J. , (2002) L75. (1998) L69 [ Prog. Theor. Phys. Suppl. , in the Proceedings of the M87 , 565 497 Supermassive black holes: Their formation, (1999) 165 [ astro-ph/9803178 Phys. World rence, 4 - 8 October 2002, Kalua-Kona , ]. 20 astro-ph/9912346 (2002) 793. , Cambridge Univ. Press, Cambridge 2004 , Observational Evidence for Massive Black Holes in , in R.M. Wald (ed.), (2003) L127. 52 384 (2003) L121. 586 ]. Astrophys. J. Lett. Astrophys. J. Lett. 597 The Nature of Accreting Black Holes in Nearby Galaxy Nuclei , (1997) 579. (2002) 965. 489 (1998) 678. 335 J. Astrophys. Astron. Supermassive black holes in galactic nuclei , (2002) 694. (2002) 917. astro-ph/9901023 509 Relationship of black holes to bulges Supermassive black holes Astron. Astrophys. 419 331 . ]. ]. ]. Astrophys. J. Lett. MNRAS VLBA Continuum Observations of NGC 4258: Constraints on an Morphology of the nuclear disk in M87 Astrophys. J. Lett. Astrophys. J. http://www.astro.ucla.edu/~ghezgroup/gc/pictures/ Accelerations of Water Masers in NGC4258 astro-ph/9903141 (1999) 89 [ Nature 437 (2005) 1105. Black holes in active galactic nuclei: observations MNRAS , eds. M.A. Abramowicz, G. Bjvrnsson and J.E. Pringle, Cambr Astrophys. J. Ultra-Luminous Sources in Nearby Galaxies The Stellar-Dynamical Search for Supermassive Black Holes Evidence for massive black holes in nearby galactic nuclei 519 Astrophysical evidence for black holes Coevolution of Black Holes and Galaxies Nature , Univ. of Chicago Press, Chicago 1996; M.J. Rees, astro-ph/0306353 astro-ph/0001543 astro-ph/0206222 astro-ph/0411040 Accretion Discs Cambridge 1998 [ stars and their prospects as probes of relativistic gravity Ho (ed.), Advection-Dominated Accretion Flow [ the Centers of Active Galaxies Workshop, Ringberg castle, Germany, 15-19 Sep 1997, [ Astrophys. J. (Hawaii); A. Ghez et al, [ [ Black-Hole Phenomenology [50] G.M. Madejski, [51] T. de Zeeuw, [52] M.J. Rees, [54] D. Merritt and L. Ferrarese, [55] J. Kormendy and K. Gebhardt, [56] J. Kormendy, [53] F. Melia, [57] F. Macchetto et al., [60] J. M. Moran, L. J. Greenhill and J. R. Herrnstein, [59] J. R. Herrnstein et al, [58] Z.I. Tsvetanov et al, [61] W. Pieych and A. M. Read, [62] A. E. Bragg et al., [63] R. Schödel et al., [64] F. Eisenhauer et al., [65] A. Eckart et al., [66] A. Ghez et al.: [67] A. Ghez et al., invited talk at the Galactic Center confe [68] A. Ghez, movie at [69] L. Ferrarese and D. Merritt, [70] M. Elvis, G. Risaliti, and G. Zamorani, [71] Q. Yu and S. Tremaine, [72] R. Mushotzky, [73] E. J. M. Colbert and R. F. Mushotzky, PoS(P2GC)004 ´ c 74 . D 428 , ]. (2000) Neven Bili (2005) L006 590 (1999) 744. B , in J. Trampetic (1995) 623 526 Phys. Rev. , Astrophys. J. , 374 C0507252 . (2001) 105. ]. Nucl. Phys. es In Galactic Nuclei , astro-ph/0607072 Nature . (1997) L79. eConf , , , B515 astro-ph/0310294 ter 478 (1993) 79. (1998) L105; ibid. ]. On the nature of ultraluminous X-ray n 2004 [ 509 Phys. Lett. v, (1999) 024003. B306 (2003) R301. D59 , Proceedings of the 9th Adriatic Meeting, (1987) 3640. 20 astro-ph/0608096 (2000) L13. astro-ph/0207270 , (2000) 185. No observational proof of the black-hole 35 Astrophys. J. Lett. D 528 astro-ph/0609274 53 Phys. Lett. (1994) 2516. 564 (2002) L139. Astrophys. J. , (2003) 419. B 72 Phys. Rev. 574 Sterile Neutrino Dark Matter in the Galaxy (1994) 51. 599 ]. (2002) L31 [ Constraints on sterile neutrino dark matter 32 Advection Dominated Accretion Model For Sagittarius A: astro-ph/9710309 ]. Phys. Rev. 396 Astrophys. J. Class. Quant. Grav. Nucl. Phys. Astrophys. J. Phys. Rev. Lett. (1968) 1331. (1992) 163. ]. ]. Self-gravitating bosons at nonzero temperature Advection dominated accretion: A Selfsimilar solution Astrophys. J. Lett. Sub-milliarcsecond Imaging of SgrA* and M87 (1998) L181 [ ´ c, 172 ]. 220 Particle Physics and the Universe 494 astro-ph/0605271 Astron. Astrophys. Trust but verify: The case for astrophysical black holes Prog. Part. Nucl. Phys. , Detecting sterile dark matter in space astro-ph/9403052 Phys. Rev. Dynamical Constraints On Alternatives To Massive Black Hol Phys. Rep. ´ ´ ´ c and H. Nikoli c, F. Munyaneza and R.D. Viollier, c, R.L. Lindebaum, G.B. Tupper and R.D. Viollier, gr-qc/0006065 hep-ph/0511217 astro-ph/9411060 sources, or what a black hole should look like event-horizon [ (1994) L13 [ Evidence For A 10**6 Solar Mass Black Hole At The Galactic Cen [ Astrophys. J. Lett. (2006) 023527 [ and J. Wess (eds.), Dubrovnik, 2003, Springer Proceedings in Physics 98, Berli 575 [ Black-Hole Phenomenology [74] J. Poutanen, S. Fabrika, A. G. Butkevich and P. Abolmaso [75] M. A. Abramowicz, W. Kluzniak and J. P. Lasota, [76] S. A. Hughes, [77] R. Narayan and I. s. Yi, [78] R. Narayan, I. Yi and R. Mahadevan, [79] R. Narayan, M.R. Garcia, and J.E. McClintock, [80] R. Narayan, J.S. Heyl, [81] R. Narayan and J. S. Heyl, [82] H. Falcke, F. Melia, and E. Agol, [83] T. P. Krichbaum et al., [84] E. Maoz, [91] A. Kusenko, [85] R.D. Viollier, D. Trautmann and G.B. Tupper, [92] N. Bilic, G.B. Tupper, and R.D. Viollier, [86] R.D. Viollier, [93] N. Bili [87] N. Bili [88] N. Bili [94] D.J. Kaup, [89] F.Munyaneza, D. Tsiklauri and R.D. Viollier, [90] K. Abazajian and S. M. Koushiappas, [95] R. Friedberg, T.D. Lee and Y. Pang, [96] P. Jetzer, [97] F.E. Schunck and E.W. Mielke, [98] E. Seidel and W.-M. Suen, [99] E.W. Mielke and F.E. Schunck, PoS(P2GC)004 ´ c ]. g ]. 9545 , Class. ]. , 111 , (1999) Neven Bili les (2006) 013 ]. 31 (2005) 4551 22 0602 n of state Int. J. Mod. Phys. Class. Quant. Grav. , , , gr-qc/9908002 gr-qc/0508115 , JCAP center Gen. Rel. Grav. Proc. Nat. Acad. Sci. , , (1978) 301. astro-ph/9908168 (2001) 265. Relativity, Astrophysics and (1999) 3953. [ 10 (2006) 1525 [ astro-ph/0503200 511 Class. Quant. Grav. 16 (1967) 463. , 23 B (2002) 17. 48 ]. , Pergamon, Oxford 1994. Phantom Energy and Cosmic Doomsday (2002) 23 [ 535 (2004) 1135. B 545 21 B ]. (2004) 0205 [ Phys. Lett. , in W. Israel (ed.), 54 , Pergamon, Oxford, 1993. Boson stars: Gravitational equilibria of self–interactin (2003) 3587. Born-Infeld phantom gravastars Ann. Rev. Fluid Mech. Class. Quant. Grav. Phys. Lett. , Class. Quant. Grav. , C041213 18 , Phys. Lett. ]. , A Gravastars must have anisotropic pressures (2001) 235. ]. Action Integrals and Partition Function in Quantum Gravity astro-ph/0302506 Boson stars: Early history and recent prospects eConf 81 (1986) 2485. , Class. Quant. Grav. gr-qc/0511097 B Gravitational vacuum condensate stars Fluid Mechanics The classical theory of fields 57 S10 (1966) 1; Erratum-ibid B Gravitational Condensate Stars: An Alternative to Black Ho General-relativistic Thomas-Fermi model ]. (1976) 310. 44 ]. ]. B Gravastar solutions with continuous pressures and equatio 100 ]. gr-qc/0304110 Phil. Mag. Int. J. Mod. Phys. . . (2003) 071301 [ Stable gravastars with generalised exteriors A Phantom Menace? gr-qc/0505137 (2006) 2303. [ (1977) 2752. Spherically symmetric spacetime with the regular de Sitter 91 Stable dark energy stars Dark energy stars 23 Phys. Rev. Lett. 55 Relativistic Fluid Mechanics , Nuovo Cim. Ann. Phys. , D. Reidel Publishing Company, Dordrecht/Boston 1973. Survey of General Relativity Theory D Relativistic Acoustic Geometry ´ ´ ´ ´ gr-qc/0407075 c, G.B. Tupper and R.D. Viollier, c, G. B. Tupper and R. D. Viollier, c, c and R. D. Viollier, gr-qc/9903034 , 1015 (2003) [ (2005) 4189 [ 12 gr-qc/0509087 astro-ph/0503427 gr-qc/0109035 scalar fields (2004) [ gr-qc/9801063 D Phys. Rev. Lett. [ [ 22 Quant. Grav. 1105 [ Phys. Rev. Cosmology [110] P. O. Mazur and E. Mottola, [111] I. Dymnikova, [102] R. R. Caldwell, [101] E. W. Mielke and F. E. Schunck, [112] M. Visser and D.L. Wiltshire, [113] B. M. N. Carter, [106] N. Bili [103] R. R. Caldwell, M. Kamionkowski and N. N. Weinberg, [107] C. Chapline et al., [114] W. Israel, [108] C. Chapline et al., [104] G. Chapline, [115] A. Kamenshchik, U. Moschella and V. Pasquier, [109] P.O. Mazur and E. Mottola, [116] N. Bili Black-Hole Phenomenology [100] M. Colpi, S. L. Shapiro and I. Wasserman, [105] F. S. N. Lobo, [117] C. Cattoen, T. Faber and M. Visser, [118] A. DeBenedictis et al, [120] L.D. Landau, E.M. Lifshitz, [121] N. Bili [119] L.D. Landau, E.M. Lifshitz, [122] N. Bili [123] G.W. Gibbons and S.W. Hawking, [124] W. Israel, [125] J. Ehlers, [126] A.H. Taub,