Black Holes: From Speculations to Observations Thomas Baumgarte Bowdoin College
Mitchell and Laplace (late 1700 s) Escape velocity (G = c = 1) 2M v esc = R independent of mass m of test particle Early Speculations = If this applies to light, then stars with v esc > c are dark = This happens when R < 2M 2
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General Relativity Einstein s field equations [(1915)] G ab = 8πT ab Spherically symmetric vacuum solution ( ds 2 = 1 2M ) ( dt 2 + 1 2M ) 1 dr 2 + r 2 dω 2 r r [Schwarzschild, 1916] A. Einstein: I had not expected that one could formulate the exact solution of the problem in such a simple way. Schwarzschild himself considered solution physically meaningless 4
Three Strikes against Schwarzschild ds 2 = ( 1 2M r ) dt 2 + ( 1 2M r ) 1 dr 2 + r 2 dω 2 Singularity at Schwarzschild radius makes physical interpretation unclear r SS = 2M Analytical example of gravitational collapse considered too idealized (dust, no rotation, spherical symmetry) No observational need for compact objects (white dwarfs exotic enough...) = Astrophysical significance of Schwarzschild solution unappreciated until... 5
...the Golden Age of black hole physics (1960 s) Observational evidence for gravitationally collapsed objects Better understanding of the Schwarzschild solution Better understanding of gravitational collapse = Relativistic Astrophysics as a new field John Wheeler coins the term black hole (1967) 6
Observational evidence Identification of radio sources with cosmological objects [March 16, 1963 issue of Nature] = huge energies required: gravitational implosion to relativity limit? [Hoyle & Fowler (1963)] Radio image of 3C273 (MERLIN) 7
Identification of Schwarzschild radius r SS = 2M as harmless coordinate singularity [Kruskal (1960)] Schwarzschild radius is event horizon a one-way membrane through which not even light can leave collapsed region encompasses true spacetime singularity at r = 0 Schwarzschild black holes 8
Other theoretical advances Black holes can carry angular momentum J/M 2 < 1: Kerr solution [Kerr (1963)] Gravitational collapse to form spacetime singularity is generic after formation of trapped surface (a surface from which light cannot escape outwards) [Penrose (1965)] Uniqueness theorems suggest that (uncharged) black holes are uniquely determined by mass and angular momentum: have no hair [Israel (1967); Carter (1971); Hawking (1971)...] [Chandrasekhar (1987)] This is the only instance we have of an exact description of a macroscopic object... They are, thus, almost by definition, the most perfect macroscopic objects there are in the universe 9
Cygnus X-1 Very short time variation in X-ray signal (< 10ms) = Cyg X-1 is compact object Cyg X-1 is unseen companion to HDE 226868 From mass function can estimate mass M x > M NS max = Cyg X-1 a stellar-mass black hole 10
Infrared observations of galactic center: Star S2 follows Keplerian orbit with P = 15.2 yrs and a = 4.62 mpc = Enclosed mass is M = 4π2 a 3 GP = 3.7 2 106 M Pericenter distance 124AU 2100 r SS Sagittarius A http://www.mpe.mpg.de/ir/gc/ = Sag A a super-massive black hole [Schödel et.al.(2002), Ghez et.al.(2003)] 11
Black Hole Populations They are all Kerr black holes, differing only in mass and angular momentum... Stellar mass black holes (e.g. Cyg X-1) M 5 20M formed in collapse of massive stars Currently about a dozen very good candidates, but there may be millions in our own galaxy Supermassive black holes (e.g. Sag A ) M 10 6 10 9 M Formation not quite clear... (Hint: must have formed early; close relation between galaxy characteristics and black hole mass) Probably at core of most galaxies Some evidence for intermediate mass black holes Possibly primordial black holes (no observational evidence so far) 12
Where do black hole observations stand? So far: huge mass in small volume = black hole most conservative explanation Some evidence for absence of stellar surface Can we observe black hole spin? = effect on innermost stable circular orbit 13
The innermost stable circular orbit (ISCO) circular orbit extremizes binding energy E of test mass m at const. angular momentum L Newtonian point mass E m = M r + L2 2r 2 Schwarzschild black hole (( E m = 1 2M ) )) 1/2 (1 + L2 1 r r 2 14
Characteristics of accretion disk depend on ISCO Angular momentum J of black hole affects location of ISCO maximum speed of accreting material maximum Doppler shifts observed spectra = X-ray observations suggest that GX 339-4 and XTE J1650-500 are rapidly rotating J/M 2 > 0.8 [Miller et.al.(2004)] (NASA, M. Weiss; J. Miller) 15
Gravitational wave astronomy Planned space-based interferometer: LISA Supermassive black holes... Ground-based interferometers: VIRGO, GEO, TAMA... Stellar-mass black holes... LIGO, Courtesy NASA/JPL-Caltech 16
1e-16 1e-17 1e-18 Strain Sensitivities for the LIGO Interferometers H1 Performance Comparison: S1 through post S3 LIGO-G040439-00-E LHO 4km (2002.09.09) - S1 LHO 4km (2003.04.08) - S2 LHO 4km (2004.01.04) - S3 - Inspiral Range for 1.4/1.4 Msun: 6.5 Mpc LHO 4km (2004.08.15) - Inspiral Range for 1.4/1.4 Msun: 8 Mpc LIGO I SRD Goal, 4km h[f], 1/Sqrt[Hz] 1e-19 1e-20 1e-21 1e-22 1e-23 1e-24 10 100 1000 10000 Frequency [Hz] Courtesy D. Shoemaker (LIGO) 17
1e-16 1e-17 Strain Sensitivities for the LIGO Interferometers Best Performance for S4 LIGO-G050230-02-E LHO 2km (2005.02.26) - S4: Binary Inspiral Range (1.4/1.4 Msun) = 3.5 Mpc LLO 4km (2005.03.11) - S4: Binary Inspiral Range (1.4/1.4 Msun) = 7.3 Mpc LHO 4km (2005.02.26) - S4: Binary Inspiral Range (1.4/1.4 Msun) = 8.4 Mpc LIGO I SRD Goal, 4km 1e-18 h[f], 1/Sqrt[Hz] 1e-19 1e-20 1e-21 1e-22 1e-23 1e-24 10 100 1000 10000 Frequency [Hz] Courtesy D. Shoemaker 18 (LIGO)
Source Simulations Need theoretical models of promising sources to enhance likelihood of detection aid in interpretation of astrophysical signals Among most promising sources: inspiral and coalescence of binary black holes 19
Numerical Relativity Solve Einstein s equations G ab = 8πT ab numerically Similarly to Maxwell s equations, Einstein s equations split into Constraint equations constrain the gravitational fields at each instant of time ( div equations ) Evolution equations govern evolution of gravitational fields from one time to next ( curl equations ) Construct solution in two steps Initial data describing snap shot of initial state Dynamical simulation of subsequent time evolution = distinct mathematical, computational and conceptional problems [e. g. Baumgarte & Shapiro (2003)] 20
Binary black hole initial data Solve elliptic constraint equations = two black holes at separation l and with angular momentum J = construct circular orbits [Cook (1994)] Conceptional issue: some fields freely specifiable how to encode gravitational wave pattern? promising approach: conformal thinsandwich formalism [Gourgoulhon et.al.(2002); Yo et.al.(2004); Cook & Pfeiffer (2005)] [Baumgarte (2000)] [Cook & Pfeiffer (2005)] 21
Simulation of binary black hole coalescence Integrate hyperbolic evolution equations Many issues: formulation, coordinate conditions, boundary conditions, numerical stability... [Pretorius (2005, private communication)] 22
Black holes Perhaps most exotic consequence of Einstein s relativity Only macroscopic object for which exact description exists very good observational evidence Will discuss detailed gravitational wave observations at centennial of Einstein s general relativity! 23