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Black Holes, Gravity to the Max
By Dr. Harold Williams
of Montgomery College Planetarium
http://montgomerycollege.edu/Departments/planet/
Given in the planetarium Saturday 19 November 2011
Black Hole in front of the Milky Way, out galaxy with
10 Solar Masses and viewed from 600km away
Black Holes
Just like white dwarfs (Chandrasekhar limit: 1.4 Msun),
there is a mass limit for neutron stars:
Neutron stars can not exist
with masses > 3 Msun
We know of no mechanism to halt the collapse
of a compact object with > 3 Msun.
It will collapse into a surface – an Events Horizon:
But only at the end of time relative to an outside
=> A black hole!
observer.
Escape Velocity
Velocity needed to
escape Earth’s gravity
from the surface: vesc
≈ 11.6 km/s.
Now, gravitational force
decreases with distance (~
1/d2) => Starting out high
above the surface =>
lower escape velocity.
If you could compress
Earth to a smaller radius
=> higher escape velocity
from the surface.
vesc
vesc
vesc
Escape Velocity Equation
•
•
•
•
Newtonian gravity
Ves=√(2GM/R)
Ves, escape velocity in m/s
G, Newtonian universal gravitational
constant, 6.67259x10-11m3/(kg s2)
• M, mass of object in kg
• R, radius of object in m
The Schwarzschild Radius
=> There is a limiting radius
where the escape velocity
reaches the speed of light, c:
____
Rs = 2GM
c2
G = gravitational constant
M = mass; c=speed of light
in a vacuum
Rs is called the
Schwarzschild radius.
Vesc = c
General Relativity
• Extension of special relativity to accelerations
• Free-fall is the “natural” state of motion
• Space+time (spacetime) is warped by gravity
Black Holes
• John Michell, 1783:
would most massive
things be dark?
• Modern view based on
general relativity
• Event horizon: surface
of no return
• Near BH, strong
distortions of spacetime
Schwarzschild Radius and
Event Horizon
No object can travel faster
than the speed of light
=> nothing (not even light)
can escape from inside
the Schwarzschild radius
 We have no way of
finding out what’s
happening inside the
Schwarzschild radius.
 “Event horizon”
“Black Holes Have No Hair”
Matter forming a black hole is losing
almost all of its properties.
black holes are completely
determined by 3 quantities:
mass
angular momentum
(electric charge)
The electric charge is
most likely near zero
Gravitational
Potential
The Gravitational Field of
a Black Hole
Distance from
central mass
The gravitational potential
(and gravitational attraction
force) at the Schwarzschild
radius of a black hole
becomes infinite.
General Relativity Effects
Near Black Holes
An astronaut descending down
towards the event horizon of
the black hole will be stretched
vertically (tidal effects) and
squeezed laterally unless the
black hole is very large like
thousands of solar masses, so
the multi-million solar mass
black hole in the center of the
galaxy is safe from turning a
traveler into spaghetti .
General Relativity Effects
Near Black Holes
Time dilation
Clocks starting at
12:00 at each point.
After 3 hours (for an
observer far away
from the black hole):
Clocks closer to the black
hole run more slowly.
Time dilation
becomes infinite at
the event horizon.
Event horizon
Observing Black Holes
No light can escape a black hole
=> Black holes can not be observed directly.
If an invisible compact
object is part of a binary,
we can estimate its
mass from the orbital
period and radial
velocity. Newton’s
version of Kepler’s third
Law.
Mass > 3 Msun
=> Black hole!
Detecting Black Holes
• Problem: what goes down doesn’t come
back up
• Need to detect effect on surrounding stuff
Hot gas in accretion disks
Orbiting stars
Maybe gravitational lensing
Compact object with
> 3 Msun must be a
black hole!
Stellar-Mass Black Holes
• To be convincing, must
show that invisible thing
is more massive than NS
• First example: Cyg X-1
• Now more than 17 clear
cases, around 2009.
• Still a small number!
• Scientist witness apparent black hole birth,
Washington Post, Tuesday, November 16,
2010.
http://chandra.harvard.edu/photo/2010/sn1979
c/
SN 1979C
Jets of Energy from
Compact Objects
Some X-ray binaries
show jets perpendicular
to the accretion disk
Model of the X-Ray Binary SS 433
Optical spectrum shows
spectral lines from material
in the jet.
Two sets of lines:
one blue-shifted,
one red-shifted
Line systems shift
back and forth across
each other due to jet
precession
Black Hole X-Ray Binaries
Accretion disks around black holes
Strong X-ray sources
Rapidly, erratically variable (with flickering on
time scales of less than a second)
Sometimes: Quasi-periodic oscillations (QPOs)
Sometimes: Radio-emitting jets
Gamma-Ray Bursts (GRBs)
Short (~ a few s), bright bursts of gamma-rays
GRB of May 10, 1999:
1 day after the GRB
2 days after the GRB
Later discovered with X-ray and optical
afterglows lasting several hours – a few days
Many have now been associated with host
galaxies at large (cosmological) distances.
Probably related to the deaths of very
massive (> 25 Msun) stars.
Black-Hole vs. Neutron-Star Binaries
Black Holes: Accreted matter
disappears beyond the event
horizon without a trace.
Neutron Stars: Accreted
matter produces an X-ray
flash as it impacts on the
neutron star surface.
Stars at the Galactic Center
Gamma Ray Bubble in Milky Way
Spectrum
Black Holes and their Galaxies
Gravitational Waves
• Back to rubber sheet
• Moving objects
produce ripples in
spacetime
• Close binary BH or
NS are examples
• Very weak!
Gravitational Wave Detectors
Numerical Relativity
• For colliding BH, equations can’t be solved
analytically
Coupled, nonlinear, second-order PDE!
• Even numerically, extremely challenging
Major breakthroughs in last 3 years
• Now many groups have stable, accurate
codes
• Can compute waveforms and even kicks
Colliding BH on a Computer: From
NASA/Goddard Group
What Lies Ahead
• Numerical relativity continues to develop
Compare with post-Newtonian analyses
• Initial LIGO is complete and taking data
• Detections expected with next generation,
in less than a decade
• In space: LISA, focusing on bigger BH
Assembly of structure in early universe?
Mass – Inertial vs. Gravitational
• Mass has a gravitational attraction for other
masses
Fg  G
mg mg'
r2
• Mass has an inertial property that resists
acceleration
Fi = mi a
• The value of G was chosen to make the values of
mg and mi equal
Einstein’s Reasoning Concerning
Mass
• That mg and mi were directly proportional
was evidence for a basic connection
between them
• No mechanical experiment could
distinguish between the two
• He extended the idea to no experiment of
any type could distinguish the two masses
Postulates of General Relativity
• All laws of nature must have the same form for
observers in any frame of reference, whether
accelerated or not
• In the vicinity of any given point, a gravitational
field is equivalent to an accelerated frame of
reference without a gravitational field
– This is the principle of equivalence
Implications of General
Relativity
• Gravitational mass and inertial mass are not just
proportional, but completely equivalent
• A clock in the presence of gravity runs more
slowly than one where gravity is negligible
• The frequencies of radiation emitted by atoms in a
strong gravitational field are shifted to lower
frequencies
– This has been detected in the spectral lines emitted by
atoms in massive stars
More Implications of General
Relativity
• A gravitational field may be “transformed away”
at any point if we choose an appropriate
accelerated frame of reference – a freely falling
frame
• Einstein specified a certain quantity, the curvature
of spacetime, that describes the gravitational effect
at every point
Curvature of Spacetime
• There is no such thing as a gravitational
force
– According to Einstein
• Instead, the presence of a mass causes a
curvature of spacetime in the vicinity of the
mass
– This curvature dictates the path that all freely
moving objects must follow
General Relativity Summary
• Mass one tells spacetime how to curve; curved
spacetime tells mass two how to move
– John Wheeler’s summary, 1979
• The equation of general relativity is roughly a
proportion:
Average curvature of spacetime a energy density
– The actual equation can be solved for the metric which
can be used to measure lengths and compute
trajectories
Testing General Relativity
• General Relativity predicts that a light ray passing near the
Sun should be deflected by the curved spacetime created
by the Sun’s mass
• The prediction was confirmed by astronomers during a
total solar eclipse
Other Verifications of General
Relativity
• Explanation of Mercury’s orbit
– Explained the discrepancy between observation
and Newton’s theory
• Time delay of radar bounced off Venus
• Gradual lengthening of the period of binary
pulsars (a neutron star) due to emission of
gravitational radiation
Black Holes
• If the concentration of mass becomes great
enough, a black hole is believed to be
formed
• In a black hole, the curvature of space-time
is so great that, within a certain distance
from its center (whose radius, r, is defined
as its circumference, C, divided by 2π,
r=C/2π), all light and matter become
trapped on the surface until the end of time.
Black Holes, cont
• The radius is called the Schwarzschild radius
– Also called the event horizon
– It would be about 3 km for a star the size of our Sun
• At the center of the black hole is a singularity
– It is a point of infinite density and curvature where
space-time comes to an end (not in our universe!)
Penrose Diagram of Spherical
Black Hole
All Real Black Holes will be
Rotating, Kerr Solution
• Andrew J. S. Hamiton & Jason P. Lisle (2008)
“The river model of black holes” Am. J. Phys. 76
519-532, gr-qc/0411060
• Roy P. Kerr (1963) “Gravitational field of a
spinning mass as an example of algebraically
special metrics” Phys. Rev. Lett. 11 237--238
• Brandon Carter (1968) “Global structure of the
Kerr family of gravitational fields” Phys. Rev. 174
1559-1571