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Transcript
Black Holes and Curved Space-time
• When a massive star collapses at the end of its
life, it can become a black hole
• A black is an object that is so massive that light
cannot escape from it
• The theory that describes gravity is called general
relativity

Put forward by Einstein in 1916
• Based in simple principles but makes interesting
predictions

Light does not always travel in a straight line
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
1
The Principle of Equivalence
• The principle of
equivalence states that
the force of gravity is
equivalent to force from
acceleration
• You cannot tell if you
are floating in space or
falling freely in a
gravitational field
• The laws of physics are
the same in every
inertial reference frame
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
2
Paths of Light and Matter
• Let’s try a thought
experiment
(Einstein’s favorite
technique)
• Suppose the space shuttle is in orbit and one of
the astronauts shines a laser from the back of the
shuttle to the front
• While the light is traveling, the shuttle is “falling”
out of its straight path
• The light would strike B’ instead of B if light were
not bent by gravity
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
3
Implications of Gravity Bending Light
• The space shuttle has mass while light does not
• Einstein postulated that it was actually the fabric
of space-time that was distorted by gravity
• The light still travels in a straight line in spacetime by that space was warped by the presence of
a massive object
• Mass tells space-time how to curve and the
curvature of space tells matter how to move
• The amount of distortion depends on the mass of
the object

Everyday objects do not have enough mass to distort
space-time
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
4
Space-time Examples
• Let’s imagine we are driving to
•
•
Minneapolis
We stop for lunch on the way
We can plot the space-time for this
trip using 1 dimension
• Imagine an ant walks across a
rubber sheet
• Now put a heavy weight on the
rubber sheet and let the ant walk
across again
• The ant always walks in a straight
line but his path through spacetime is curved
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
5
Facts from Relativity
• One’s perception of space and time depends on
one’s reference frame
• For these effects to be noticeable, the relative
speed must be an appreciable fraction of the speed
of light
• Some interesting facts



The clock in a moving frame runs slow from your
point of view
A fast-moving object appears to be shortened
If you are moving fast, everything around you seems
to speed up
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
6
Relativistic Effects on Mercury’s Orbit
• The orbit of Mercury is
•
•
Differences exaggerated
for pedagogical purposes
very eccentric
Normally the point at
which Mercury is
farthest from the Sun
would always be the
same
However, this point
moves around the Sun

Precession of the
perihelion
• The motion of the other planets causes part of this precession


0.531 seconds of arc is predicted
0.574 seconds of arc is measured
• The difference is caused by the warping of space-time by the Sun
• When Mercury is close to the Sun, there is a small additional push
due to the warping of space
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
7
Deflection of Light
• A second prediction of general relativity was that light would be
bent by gravity
• Einstein suggested that starlight bent by the Sun could be observed
•
•
during a solar eclipse
Indeed was observed by Eddington in 1919 but not to great
accuracy (20%)
Recent measurements with radio waves confirm Einstein’s
predictions to 1%
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
8
• Massive
Gravitational Lensing
invisible
objects can
create images
of distant
objects
HST picture showing
gravitational lensing
QuickTime™ and a
Cinepak decompressor
are needed to see this picture.
ISP 205 - Astronomy Gary D. Westfall
• On the left is an
animation of
gravitational lensing
by a spiral galaxy
Lecture 21
9
Time in General Relativity
• General relativity predicts that time runs more
slowly in a strong gravitational field
• Experiments with atomic clocks have shown that
indeed time runs more slowly in higher
gravitational fields
• In 1976, Viking sent a radio pulse to Earth from
the other side of the Sun
• The timing
of the pulse
agreed with
relativity to
within 0.1%
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
10
Black Holes
• Astronomer had long
speculated that stars
might exist that have
escape velocities faster
than the speed of light
• Relativity states that gravity is a
•
•
ISP 205 - Astronomy Gary D. Westfall
curvature in space-time
A sufficiently large mass could
distort space-time so that nothing,
even light, could escape
If an object gets within the event
horizon of a black hole, the object
will disappear from our view
Lecture 21
11
Event Horizon
• Karl Schwarzchild calculated mathematically using
•
•
relativity that an event horizon would exist around a
black hole in 1916
The event horizon does not get smaller as a collapsing
star compresses
The Schwarzchild radius depends only on the mass in the
black hole



About 3 km for 1 Msun
Feed the black hole, and the radius will grow
Collapse a globular cluster to a black hole and it will have a radius of
300,000 km
• The gravity from black holes is just like normal star as long as you
are not too close to it
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
12
A Trip into a Black Hole
• We cannot see into a black hole but that does not prevent
•
us from trying to calculate what it must be like inside a
black hole
In the center there is a singularity in space-time

The laws of physics as we know them break down
• Suppose a spaceship decides to enter a black hole and
•
•
•
send out a light pulse every second
The light pulses would get farther apart and have longer
wavelength as the ship went in until no more pulse would
arrive
The ship would not return
Tidal forces would destroy the ship
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
13
Approaching a Black Hole
QuickTime™ and a
decompressor
are needed to see this picture.
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
14
Circling a Black Hole
QuickTime™ and a
decompressor
are needed to see this picture.
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
15
Evidence for Black Holes
• One way to study black holes is to look at binary
star systems where one partner has become a
black hole
• Matter will be sucked from the normal star to the
black hole giving off visible radiation

Accretion disk
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
16
Gravity of a Black Hole
• As long as you are a reasonable distance from the
black hole, the black hole acts as is it mass is
concentrated at the center (and it really is!)
• Newton’s laws are in effect
• The event horizon is small so ordinary distances
(like 1 AU) are safe from a black hole of 1 to a
few 10s of solar masses
• Only if you approach to with a few solar
diameters will the tidal effects and relativistic
effects become apparent

Einstein’s law come into effect
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
17
The Milky Way Galaxy
• The Milky Way Galaxy can be seen in the night
sky as a streak of dim white light stretching from
horizon to horizon
• All the stars we can see in the night sky are in the
Milky Way
• The Milky Way is a
flat, spiral galaxy
containing 200
billion stars
• The picture on the
right is the center of
the Milky Way
taken with infrared
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
18
The Architecture of the Milky Way
• In 1785 William Herschel discovered that the Sun belongs to a
•
group of stars on the shape of a disk
He observed that the Sun was near the center of this system and the
disk was about 6000 LY across
• We now know that dust blocked Herschel’s view and the
Milky Way is much larger than Herschel thought


The full disk is about 100,00 LY across
The Sun is not at the center of the Galaxy
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
19
Globular Clusters and the Center of the Galaxy
• The center of the Milky Way is shrouded with dust so that one
•
•
•
cannot look far along the plane of the Galaxy
However, one can see much farther if you look up through the dust
instead of along the disk
By looking out of the plane of the Galaxy, Shapley observed
globular clusters in 1917 and their distances were measured using
Cepheids and RR Lyrae stars
Shapley found that the globular clusters were distributed
spherically and their center was far from the Sun
• Shapley postulated that these
globular clusters defined the
galaxy size
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
20
Overview of the Galaxy
• The Galaxy has been studied using infrared radiation
Picture
toward the
center of
the Galaxy
Schematic
Diagram of
the Galaxy
• Most of the stars and dust of the Galaxy are found in the
•
•
thin disk of the Galaxy
The Galaxy disk is embedded in a spherical halo of faint
stars that extends to a distance of 50,000 LY
Close to the center, the stars are no longer confined to the
disk but form a nuclear bulge consisting of old stars
Picture of the Galaxy
taken in near infrared
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
21
Interstellar Matter in the Galaxy
• Radio observations shows that the Galaxy’s cold
atomic hydrogen is confined to an extremely flat
layer that is about 400 LY thick
• In the plane of the Galaxy, this cold hydrogen
extends out 80,000 LY from the center
• Dust is also confined to the plane of the Galaxy
being about the same thickness as the hydrogen
gas but more concentrated in the spiral arms and
toward the Galactic center
• Large molecular are found in the spiral arms along
with many young stars
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
22
The Spiral Arms of the Galaxy
• The Milky Way has four spiral arms



Sagittarius-Carina, Perseus, Cygnus arms
We are located in a sub-arm called the Orion arm
There is a fourth, unnamed arm difficult to observe
• The spiral arms are thought to have arisen from differential galactic
rotation
• The spiral arm structure is thought to get frozen in by a
•
spiral density wave that occurs when material encounters
the material in the spiral arms
Other mechanisms such as chain reaction formation of
stars may also be responsible
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
23
Two Kinds of Stars
• The stars in our Galaxy can be divided into two
groups

Population I stars
Bright blue stars in the spiral arms
 Found only in the Galactic disk
 Follow nearly circular orbits around the Galactic center
 The Sun is population I
 This group includes many young stars


Population II stars
Stars in the halo, nuclear bulge, and globular clusters
 No correlation with the spiral arms
 Found throughout the Galaxy
 Can be found in elliptic orbits out of the plane of the disk
 This group consists entirely of old stars 12 billion years old

ISP 205 - Astronomy Gary D. Westfall
Lecture 21
24
The Mass of the Galaxy
• The Sun orbits the Galactic center every 225 million
•
years
We can use Kepler’s laws to calculate the mass of the
Galaxy inside the Sun’s orbit
3
9
1.6x10
a3
11
M Galaxy  M Sun  2 

10
M Sun
2
P
225x10 6




• Most of the luminous matter of the
•
Galaxy is within 30,000 LY of the
center
However, there is a large amount of
invisible matter which causes distant
objects to orbit faster
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
25
Dark Matter
• The mass of the galaxy is 10 times larger than the mass
•
of the observable objects
This mass is not in the form of



stars (we could see them)
gas (we could observe the neutral or ionized gas)
dust (would obscure major parts of the Galaxy)
• This mass cannot be in the form of black holes, white
•
•
dwarfs, or neutron stars
Could be brown dwarfs or massive planets or huge black
hole
Could be unknown elementary particles

WIMPs
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
26
The Nucleus of the Galaxy
• The center of the Galaxy is a
•
•
crowded and complicated place
The first radio source found was the
Galactic center
There is strong evidence that a
massive black hole is at the center of
the Galaxy
• The VLA has shown that the radio source at the Galactic
•
center, Sagittarius Ao, has a radius of 10 AU
Star orbit measurements show that the Galactic center is
a million times denser than any known star cluster
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
27
Finding the Source
• There could have been a black
•
hole formed at the center of
the Galaxy early in the history
of the Galaxy
Matter is falling into the center
at a rate of 1 Msun per 1000
years
• At this rate, a black hole with a mass several million
•
times the mass of the Sun could have been formed
Our galaxy is not the only one with a black hole in the
center

Supermassive black holes have been observed in the center of
other galaxies
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
28
The Protogalactic Cloud
• Because the oldest stars are distributed in a sphere
•
•
centered around the nucleus of the Galaxy, it is logical to
assume that a protogalactic cloud was roughly spherical
Like in star
formation, the
cloud collapsed
and formed a thin,
rotating disk
Gravity caused the
disk to clump into
star clusters
• There are problems with this idea in the form of
“backwards clusters” that are several billion years older
than the Galaxy
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
29
Collisions of Galaxies
• In 1994 astronomers discovered a satellite galaxy of the
•
•
Milky Way, the Sagittarius dwarf galaxy
It is approaching the Milky Way and is being torn apart
by tidal forces
The Large and Small Magellanic Clouds are satellite
galaxies that are spiraling closer and closer to the Milky
Way

Tidal forces have pulled out dust trails in front and behind these
galaxies
• There are 8 other nearby galaxies that appear to have split
•
off from the Magellanic Clouds
The Galaxy formed in two stages


The process described from a protogalactic cloud
Other stars and clusters were captured, retrograde motion
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
30
Galaxies
• In 1925 Edwin
Hubble
announced that
the Andromeda
Galaxy was a
separate galaxy
from the Milky
Way
• Previously most
astronomers
believed that the
Milky Way was
the only galaxy
• Now we know
that there
millions and
maybe billions
of other
galaxies
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
View of very distant
galaxies by the HST
31
Spiral Galaxies
• Spiral galaxies (like the Milky Way
•
and Andromeda) consist of a
nucleus, a halo, and spiral arms
Interstellar matter is spread
throughout the disks and bright
nebula and hot young stars are
present
• The disks are often dusty
which shows when the
galaxies are viewed edge-on
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
32
Types of Spiral Galaxies
• Spiral galaxies rotate such that their arms appear to trail
• A third of spiral galaxies have bars running through their nuclei and
•
•
another third have faint bars
There are many different shapes ranging from mostly nucleus to
mostly disk
Spiral galaxies contain many young stars and lots of gas and dust
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
33
Elliptical Galaxies
• Elliptical galaxies contain
•
•
•
mainly old stars and are
shaped like spheres or
ellipsoids
Their light is dominated by
old reddish, population II
stars
Dust and gas are not
conspicuous in elliptical
clusters
The stars do not all orbit the
center in the same direction
like spirals
The giant elliptical galaxy M87 with
globular clusters surrounding it
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
34
Irregular Galaxies
• Everything else besides spiral and elliptical is
called irregular
• The best known irregular galaxies are the Large
and Small Magellanic Clouds
Large Magellanic Cloud
ISP 205 - Astronomy Gary D. Westfall
Small Magellanic Cloud
Lecture 21
35
Masses of Galaxies
• The masses of spiral galaxies can be measured as
the mass of the Milky Way was measured

Kepler’s law, period of star orbits
• Elliptic galaxies do not rotate so Kepler cannot
help us
• We can look at the broadening of the absorption
lines and calculate the average speed of all the
stars
• Elliptic galaxies are the most massive and the
least massive
• Irregular galaxies have less mass than spirals
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
36
Mass-to-Light Ratio
• We can characterize galaxies by their mass to light ratio
•
•
•
•
•
with the Sun mass-to-light ratio being 1
Because galaxies have many small stars, their mass-tolight ratio is generally greater than 1
Young galaxies have a mass-to-light ratio of 1 to 10
Older galaxies have a mass-to-light ratio of 10 to 20
As much as 90% of the mass of galaxies is not visible in
any electromagnetic wavelength
Dark matter may have a mass-to-light ratio as high as 100
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
37
Distance Scale for Galaxies
• The distance to stars was measured using variable stars
• Locating variable stars in galaxies is difficult
• There are two kinds of Cepheids and they were mixed up
•
•
•
and the distance scale to galaxies had to be increased by a
factor of 2 in the 1950s
Variable stars are only visible in nearby galaxies so we
need a new method
The method is to notice that galaxies come in groups and
clusters with average characteristics
This method of the “standard bulb” is always used with
caution
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
38
New Techniques
• These new distance measuring techniques have
been calibrated with nearby galaxies
• The first method is to exploit the relationship
between rotational velocity and luminosity for
spiral galaxies

Use 21 cm radiation to measure rotational velocity
• The second method involves the measurement of
the bumpiness of the apparent surface of an
elliptical galaxy

The less bumpy, the farther away
• The two methods are complementary
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
39
The Expanding Universe
• We know that the universe is expanding
• The discovery of this expansion began with the
search for Martians and other solar systems
• Vesto Sipher, working for Lowell, was asked to
measure the spectra from spiral nebula to search
for chemical compositions expected for newly
forming planets
• The spiral nebulae are very dim and exposures of
20 to 40 hours were required
• It took 20 years to measure the spectra from 40
nebulae
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
40
Redshifts
• What Sipher found was that these spiral nebulae
showed astounding redshifts


The lines in the spectra were all shifted toward longer
wavelengths
They were all moving away from us at high speed
• Only a few spirals such as M31, which we now
know to be a close neighbor, were moving
towards us
• These measurements were announced in 1914,
years before Hubble found that these spiral nebula
were galaxies and before anyone knew how far
away they were
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
41
The Hubble Law
• In the 1920s, Edwin Hubble found a way to estimate the distance
•
to the galaxies
He found, along with Humason, that there was a direct relationship
between the distance to the galaxy and it velocity of recession


v = Hd, Hubble Law
H is the Hubble constant
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
42
Hubble’s Law
• Every galaxy measured showed the relationship of
velocity and distance

Implies that the universe is expanding
• Having this relationship allows astronomers to
measure the distance the galaxies just by
measuring their redshift

Redshifts have been measured up to 90% of the speed
of light
• This method allows astronomers to measure the
distance of galaxies far beyond previous methods
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
43
Implications of the Hubble Constant
• The best measurement of H was done in 1999
using the Hubble Space Telescope viewing the
spiral galaxy NGC 4603
• The distance was
measured using Cepheid
variable stars
• The value of H was
measured to be:


70  7 km per second per megaparsec
20  2 km per second per million light years
• 1/H is the age of the universe
• 15  1.5 billion years
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
44
Models for an Expanding Universe
• At first one might think that an expanding
universe in all directions means we are at the
center of the universe
• However, in a uniformly expanding universe, we
and all other observers, must see the same
expansion
ISP 205 - Astronomy Gary D. Westfall
Lecture 21
45