Download Black Holes

General Relativity and
Applications
4. Black Holes
Edmund Bertschinger
MIT Department of Physics and
Kavli Institute for Astrophysics and
Space Research
“It was like a hole in the sky…”
– Iman Farid
Above: The moon occulting
the sun (29 March 2006,
APOD 31-Mar-06).
Left: A black hole occulting
Saturn (simulated).
(Cover photograph,
Exploring Black Holes)
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In my entire scientific life, extending over forty-five
years, the most shattering experience has been the
realization that an exact solution of Einstein's
equations of general relativity provides the
absolutely exact representation
of untold numbers of black holes that populate the
universe.
Subrahmanyan Chandrasekhar,
The Nora and Edward Ryerson Lecture, 22 April 1975
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What is a black hole?
A massive spacetime curvature singularity,
(a point or ring of infinite density and tidal acceleration)
Surrounded by an event horizon
(a spacetime boundary between causally disconnected
regions of the universe)
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Outline
The Astrophysics of Black Holes
 Two varieties (small, super-large)
 How do we recognize a black hole?
The Physics of Black Holes
The Fun of Black Holes
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Black Holes in the Universe
Most basic property: mass
Galactic Nuclei: 106 to 109 Msun
Dead stars: 3 to 15 Msun
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The biggest nearby black hole:
Sgr A*
Center of our galaxy: radio
source Sgr A*
Distance: 8 kpc = 8000 pc,
1pc = 3.3 lt-yr
Highly obscured in optical
Dense central star cluster
visible in infrared
Photo/illustration from A. Tanner, UCLA
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Stellar Dynamics of Sgr A*
From R. Genzel et al.,
Max-Planck-Institut für
extraterrestrische Physik
M = combined mass (BH + star)
a = semimajor axis of stellar orbit
P = orbital period
Star S0-2 has a=920 AU, P=14.5yr
M
= 3.7x106 solar masses
(20%)
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How do we know it’s a black hole?
No firm proof yet: Black holes are
indistinguishable from Newtonian
bodies at large distances r:
Lack of alternative: Any plausible
alternative would lead quickly to
gravitational collapse
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Pinning down a black hole:
Observe it close in
Accretion disk
Angular momentum
and dissipation
forces gas falling
toward the BH to
orbit in a disk
Friction causes the
gas to slowly spiral
in toward the BH,
and makes the gas
very hot so it glows.
The problem:
Relativistic disks
are much too
distant to resolve!
Figure from STScI
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The Anatomy of Black Holes
Indirectly
looking
close in:
Emission
from
inner
accretion
disk or
jets
Figure
from
NASA/
GSFC
Imagine
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Black Hole vs. Neutron Star:
Surface or Event Horizon?
Neutron stars are brighter
and have X-ray bursts from
thermonuclear explosions on
their surfaces. BH do not.
Garcia et al. 2001
Chandra X-Ray Observatory
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Relativistic Jets
From NRAO/AUI: R. Perley, C. Carilli,
J. Dreher
We don’t know how the
jets form!
Magnetic fields? BH spin?
From NRAO/AUI and STScI: D. Merritt
and R. Ekers
Did jets change direction
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when black holes collided?
Gamma-Ray Bursts
GRB 990123
20-50 keV
50-100 keV
Long-duration (> 2s)
GRBs caused by
relativistic jets of a
newly born BH,
pointing toward earth
100-300 keV
>300 keV
0
50
100 seconds
Figure from NASA CGRO/BATSE
Figure from NASA/
SkyWorks Digital
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Outline
The Astrophysics of Black Holes
The Physics of Black Holes
 Orbits
 Light cones and Penrose diagram
 Event horizon
The Fun of Black Holes
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Gravity as both an effect of
curvature and as a force.
In flat spacetime, freely-falling bodies follow
paths of maximal proper time.
Einstein equivalence principle: the same is
true in curved spacetime (geometry).
Geometry/force duality: Gravity is a force
and torque in local Lorentz frames
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Orbits in Newtonian Physics
f
Conserved quantities:
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Orbital dynamics from E, L
In Lecture 3, Veff(R) was introduced to deduce
the expansion R(t) of the universe!
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Orbital periods
Veff
r
E/m
For bound Kepler orbits, both r and f are
periodic functions of t with identical periods.
Consequence:
Bound orbits are
Closed ellipses!
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Numerical Integration of Orbits
JKerrOrbits was developed by S. Tuleja, T. Jezo, and J. Hanc
based on earlier Schwarzschild integrator of A. Riess and E. Taylor
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Orbits around a non-spinning BH:
Curved spacetime modifies E, L, vr
general relativity
special relativity
Taylor expand
v/c << 1,
r >> GM/c2
t=
proper
time
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Effective Potential for Non-spinning BH
Veff
E/m
2GM/c2
r general relativity
For details see Lecture Notes 3 and
Exploring Black Holes Chapters 3, 4
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Perihelion precession
Orbits around a
point mass in GR:
Two different
periods! Pr  Pf
Orbits do not close!
Mercury:
43”/century
PSR 1913+16:
8o/year
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Light Cone
ct
Minkowski
diagram
without a
black hole
My worldline (trajectory)
If you pass inside the cone, I can
communicate with you!
Also, my worldline can never
y move outside these cones!
The cone moves along the
trajectory with me!
x
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Penrose Diagrams: Spacetime Map
Map infinity to boundary of a compact domain:
Light cones preserved
(conformal transformation).
r=0
time
Minkowski diagram becomes a Penrose Diagram.
Purpose is to display “light cones” – worldlines of
all massive particles must remain inside the cone.
h
r=
Angular coordinates (q,f) not
shown in the diagram.
Severe distortion near
boundary.
space
x
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M.C. Escher, Circle Limit IV
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Spherical, Uncharged Black Hole
Oppenheimer and Snyder, 1939
Penrose diagram:
Global geometry
Topology
Causal structure
Event horizon:
Once crossed,
there is no
return!
Event
Horizon
r=0
Central
r=0
Singularity
r=
Outwardsdirected light
rays remain
stuck to the
event horizon!
Worldline of a
collapsing
star’s surface
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The Schwarzschild singularity is a
time, not a place
Across the horizon,all of space is
crushed, like a big bang singularity in
reverse.
Spatial geometry of a
black hole,
From Andrew Hamilton,
U. Colorado
(Google him to learn
about his fantastic
simulated BH voyage!)
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What happens at the event horizon?

Classically, nothing.


Event horizon is determined by global, not local,
structure
Quantum mechanically, Hawking
radiation



Particle vacuum state fluctuates with
creation/annihilation of virtual pairs of particles
Negative energy particles fall in, positive energy ones
escape, black hole loses mass as blackbody radiation
Completely negligible for astrophysical black holes
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How can matter escape from a BH?
Quantum tunneling produces a
Particle-Antiparticle pair from the
vacuum!
http://superstringtheory.com/
blackh/blackh3.html
Warning: this diagram is
misleading because quantum
particles are in fact waves!
Hawking radiation is
extremely weak and may
never be observed. Yet it
raises profound questions
of quantum gravity that
perhaps only Superstring
Theory will be able to
answer!
Note: Hawking radiation
also includes antiparticles!
(The particle can fall into
the black hole.)
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Outline
The Astrophysics of Black Holes
The Physics of Black Holes
The Fun of Black Holes
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Interstellar travel by BH?
Penrose diagram of a spherical, charged BH
(half of Reissner-Nordstrom spacetime).
Gravitational repulsion of electric fields
causes freely-falling observer to bounce
back to increasing r, crossing into another
universe! Similar to dark energy!
Problem: instability of the second horizon
destroys the connections (Poisson and
Israel “mass inflation”).
Cool diagram from Andrew Hamilton, U. Colorado
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Interstellar travel by BH?
Penrose diagram of a spherical, charged BH
(half of Reissner-Nordstrom spacetime).
Gravitational repulsion of electric fields
causes freely-falling observer to bounce
back to increasing r, crossing into another
universe! Similar to dark energy!
Problem: instability of the second horizon
destroys the connections (Poisson and
Israel “mass inflation”).
Cool diagram from Andrew Hamilton, U. Colorado
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Nature of the Singularity

Classical gravity: spacetime geometry near the
singularity not fully understood
Oscillatory curvature singularity (Belinsky, Khalatnikov, Lifshitz):
“spaghettification” or “eggbeater”?
 Spacelike, timelike, or null?
 Horizon structure? Singularities without horizons?



Quantum gravity: singularity resolved, but is
very small (10-35 m)
Quantum coherence and probability: whence
information? Superstring states?
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Outstanding Questions



Are the BH of nature exactly the BH
predicted by Einstein’s theory?
How do supermassive BH form? What
role do they play in galaxy formation?
(G. Kauffmann lectures)
How and why do relativistic jets form?
(R. Blandford’s lectures)
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More Questions




What is the nature of the singularity?
What is the quantum mechanical
wavefunction of a black hole spacetime?
Are four spacetime dimensions enough
to properly describe a black hole?
???
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