Re-cap from last time (General Relativity):
• Gravitation & Acceleration are equivalent
• Straightest path is path taken by light and in general it is curved
• Time runs slowly and lengths are contracted in a high gravity environment
• Light escaping from a high gravity environment is redshifted
• Light falling into a high gravity environment is blueshifted
• Is gravity really a force as Newton envisoned it?
Outline - Feb. 18, 2010
• Is gravity a classical force (like Newton said)?
• Warped “spacetime”
• Tests of Special and General Relativity (Ch. S3, 449-465)
• Black Hole basics (pg. 595-601)
• Should you believe in black holes? How would you go about
finding one?
Gravity: Newton vs. Einstein
“Spooky” Action at Distance (Newton):
Sun “tugs” on the planets and pulls them around in their orbits, like
a string tied to a whirling kitty toy. But how?? Where’s the string??
If the sun disappears, the planets should instantly “fly off” into
space on straight line trajectories. But information can’t travel
faster than c, so how can they know “instantly” that the sun is
gone?
General Relativity (Einstein):
Matter/mass tells space how to CURVE
Curvature of space tells things how to MOVE
Information about changes comes from gravity waves (ripples of
curvature in spacetime) that travel at the speed of light
2-dimensional, rubber sheet analogy to General Relativity
(Note, in reality this is 4-dimensional in the universe.)
If you start a marble rolling across the rubber sheet in a straight line,
what happens?
Einstein’s view of orbits
Planets move along their natural curves in space, caused by
the mass of the sun “warping” space. Now what happens if
you pluck the sun out of the center of the solar system?
Who’s right? Einstein or Newton?
Classic tests of General Relativity:
1. Precession of the perihelion of Mercury
2. Gravitational lensing
Einstein’s theory of gravity gives the same answers as Newton’s theory
in the limit of extremely weak gravity. They only differ where gravity is
particularly strong (e.g., nearby very massive objects such as stars,
centers of galaxies, galaxy clusters)
Precession of the Perihelion of Mercury
This effect is noticeable for the
innermost planets.
Every century the total change in
the location of “periheilon” for
Mercury is 43 arcseconds (= 0.012
degrees). For Venus the change is
8.6 arcseconds (= 0.002 degrees),
and for Earth the change is 3.8
arcseconds (= 0.001 degrees)
Gravitational Lensing
Light has to follow the curved path of space around a massive object (like the sun).
The closer the light passes to the sun, the more it is “deflected” by the curved path.
Gravitational lensing by the sun first detected in 1919, validating General
Relativity over Netwonian gravity.
Gravitation lensing can create multiple images of the same object
We actually see this in Nature!!
“QUASAR” named QSO 0957+561
Two images of the same object, discovered in 1979
Einstein “Cross”; 4 images of the same object
Gravitational lenses make REALLY bad eyeglasses
The images are highly DISTORTED!
Black Hole Basics:
Warped Spacetime
Black holes are
so “compressed”
that spacetime
nearby a black
hole is warped to
the point of
being cut off
from the rest of
the universe.
Black Hole Basics:
Escape Speed
Black hole is a region of space where gravity is so strong that
not even light can escape.
Minimum speed of escape from the gravitational pull of an object
with mass M is:
Vesc = ( 2GM / R )1/2
Imagine compressing the earth.
How would the escape speed from the surface of the earth change?
Radius
R = Rearth = 6500km
R = 0.25 Rearth = 1600 km
Vesc
11 km/s (Saturn V rocket)
22 km/s
R = 1 km
890 km/s
R = 1 cm
300,000 km/s = c
Black Hole Basics:
Escape Speed, II
Compress the earth to a radius smaller than a grape and Vesc > c, so
not even light could escape off the surface of the planet!
Note: black holes are not necessarily infinitely dense with the
mass contained within a physical radius of zero size (a “singularity”),
they just need to be sufficiently dense that Vesc > c.
Note: the FARTHER you are from a black hole (the larger “R” is in the
equation), the LOWER is Vesc.
How close can you get to a black hole and still escape?
Black Hole Basics:
Schwarzschild Radius and Event Horizon
Schwarzschild radius is the
distance from the black hole
at which Vesc = c. If R < RSch,
Vesc > c. If R > RSch, Vesc < c.
black hole
RSch = 3 x ( Mbh / Msun ) km
Event Horizon is the effective
“surface” of a black hole. It is
a sphere with radius equal to
RSch that defines the region
of space within which no
event can be seen, heard,
or known by an outside
observer.
Black Hole Basics:
Don’t be frightened…
Black holes are NOT cosmic vacuum cleaners!
(but they are cosmic “heaters”…)
What happens as you travel toward a black hole?
Release a space probe near a black hole and watch it fall in.
The space probe has a clock on board and it sends out a radio signal
to a stationary observer.
As distance between probe and black hole decreases:
• gravitational pull increases, spacetime become more warped
• as seen by an outside observer, the clock on probe ticks more and
more slowly
• wavelength at which and outside observer receives the signal gets
longer (have to tune to lower and lower frequencies)
Travel to a Black Hole, II
As seen by an outside observer:
• as probe nears event horizon, time on probe comes to a stop
(takes an infinite amount of time to pass between events on board
the probe)
• everything on probe is frozen in time and it takes an infinite
amount of time for the probe to cross the event horizon
• black holes are the ultimate in length contraction, time dilation, and
gravitational redshift
Travel to a Black Hole, III
Suppose somebody was crazy enough to go for a ride on the space
probe…
Very close to the event horizon, the occupant of the space probe:
• would notice nothing strange as far as length and time on the
probe (physics appears normal within the probe as you approach the
black hole)
• would say time is running faster and faster outside the probe, and
all light is blueshifted (falling toward the black hole)
• can only see in the direction from which the probe came, not in
the direction of travel (the direction of the black hole)
Do Black Holes Always Chew their Food?
Will the space probe actually survive crossing the event horizon?
Maybe, maybe not.
We have to consider tidal gravitational forces acting across the
probe. Gravitational pull near the black hole increases so rapidly that
the nosecone may experience a greater pull than the booster rockets
and could be turn apart.
Properly, it’s the curvature of space (caused by the black hole)
compared to the size of the space probe that truly matters.
Surviving the initial fall into a black hole…
Two extreme scenarios:
1. Curvature of space at the event horizon is large compared to
the size of the probe (BH with M = 1 Msun has RSch = 3 km, so a
probe that is a few meters long is BIG compared to the effective
radius of the BH)
2. Curvature of space at the event horizon is small compared to
the size of the probe (BH with M = 106 Msun has RSch = 3x106 km,
so a probe that is a few meters long is SMALL compared to the
effective radius of the BH)
Case 2: probe (and any occupants) pass through the event
horizon and don’t notice anything strange occurring. Like a
boat pulled along by current, they fall toward center (won’t see
mass at center because all light is pulled inward). Eventually
will be torn apart as go deeper into the black hole.
Should you believe in black holes?
Is this science or fiction?
Suppose you wanted to find a black hole. What would you look for?
Remember, a black hole itself emits no light!!
But, material near to the black hole and outside the event
horizon may emit light that you could observe.
Key properties to black hole observations: large, “invisible”
amount of mass crammed into a very tiny space
Black Hole “Binary” System
A star is observed to be in orbit (by Doppler
shift of spectrum) around an “invisible” object.
Optical image of Cygnus X-1
Artist’s conception of black hole
binary system Cygnus X-1
If star is sufficiently large and is sufficiently
close to its unseen companion, matter from
the star may transfer over and build up in an
“accretion disk” around the black hole.
Gas spirals toward BH, is accelerated up to
high speeds by gravity, suffers violent
collisions and heats up (millions of degrees =
X-ray emission).
Note: stars are not strong X-ray sources
Constraining the Size of the Region that Contains the
Invisible Mass
If the X-ray light flickers (on/off) very rapidly, this places a direct constraint
on the size of the accretion disk (just outside the event horizon).
Time scale over which you observe the light to be flickering must be
smaller than the time it takes for light to travel across the accretion
disk, or you won’t notice the flickering - it will be smeared out!!
Example: sound waves (time delay of arrival of sound due to its
distance; e.g. thunder vs. lightning)
World’s Longest, Loudest Marching Band
All band members play one short, staccato note.
What do you hear?
Speed of sound = 343 m/s, so you don’t hear the back
row of the band until 10 seconds after the single
note is played
If band plays 1 staccato note every half second you
would hear continuous sound (no “quiet” or “off” time)
If band plays staccato notes more than 10 seconds
apart, then you will notice breaks in the sound
Band is 37.5 times the length
of a football field…
Time to traverse the length of the band has to be
shorter than the time between which the notes occur
in order for you to experience “off” time (same goes
for light)
Constraining the Size, II
Diameter of the emitting region has to be less than the distance light
could travel over a time equal to the time scale for flickering (t)
D < c t
If t < 1 second, D < 300,000 km (i.e., 20% of the diameter of the sun)
So, if you see flickering on a time scale less than about 4 or 5
seconds, the size of the emitting region (the accretion disk) is
smaller than a star, so the companion cannot possibly be a star!
Back to Cygnus X-1
Cygnus X-1 consists of a bright star with
mass = 18 Msun and an unseen
companion with mass = 10 Msun
Rapidly flickering of X-rays says
companion is much too small to be a star
Most theoretically conservative
conclusion: companion is a black hole
Many such X-ray binary systems exist in
our Galaxy, with black holes that have
masses between 4 Msun and 10 Msun
These black holes were formed when an
extremely massive star died in a
supernova explosion
“Supermassive” Black Holes
MBH > 106 Msun
If something (star, disk of gas) is orbiting about a black hole, the speed of
rotation should decrease with distance from the black hole:
V = (G MBH / R)1/2
If you can measure V and R, you can deduce MBH
Look for rapidly rotating disks at the very centers of big galaxies,
motions of stars near the very center of our own galaxy (Milky Way).
What do you find?
“Supermassive” Black Holes
MBH > 106 Msun
Rapidly rotating disk within only 16 light years of the center of
giant elliptical galaxy M87 gives MBH = 3 x 109 Msun
“Supermassive” Black Holes
MBH > 106 Msun
Rapidly rotating disk within only 0.64 light years of the center of
spiral galaxy NGC 4258 gives MBH = 4 x 107 Msun
“Supermassive” Black Holes
MBH > 106 Msun
Over course of about 20 years astronomers have followed the motions of
stars at the very center of the Milky Way, and have determined their orbits
with very high accuracy. From orbital speeds of stars within 0.03 light
years of the center of the Milky Way, MBH = 106 Msun
Last word on Black Holes
• Black holes really do exit
• Black holes with mass MBH < 10 Msun probably result from death
of massive star in a supernova explosion
• Probably all large galaxies (galaxies at least as big as our own)
harbor “supermassive” black holes at their centers (formation
mechanism not yet understood)
• You have nothing to fear from black holes, you just want to stay
far enough away that the maximum speed of your space ship
exceeds the local escape speed.