Download Lect18-4-19-10

Now we go on to discuss some of the aspects of Einstein’s theory
of gravity, which revises Newton’s theory a little.
General Relativity
Einstein’s Theory of Gravity
Paul Woodward
4/19/10
This is called the general theory of relativity.
The generalization here is that instead of only considering
observers in states of constant motion in straight lines, now
Einstein considers observers whose velocities are changing as
the result of acceleration.
In our everyday experience, we know two kinds of acceleration.
The first is the acceleration of the earth’s gravity, which we feel
when we stand on our feet, which then feel the weight of our
bodies, or when we sink into a chair.
We don’t feel accelerated in either case, because we do not perceive
that our velocity is changing. Nevertheless, we are definitely
being acted upon by a force, no question about it.
The second kind of acceleration we are familiar with seems more
deserving of the name.
This is the acceleration we experience when we push down on the
accelerator pedal in our cars.
So long as we are not wimpy about this, we not only see our
velocity changing by noting the increasingly more rapidly
passing scenery, but we also feel the acceleration by being
pressed
d bback
k against
i t the
th seat.
t
Einstein suggests that if we remove the evidence of the passing
scenery, we cannot tell the difference between these two very
different reasons for being pressed against a seat.
It is always possible by
performing an
experiment to discover
that you are in an
accelerated frame of
reference.
It is also possible to
know when you are
freely falling, because
you are weightless.
Think about an elevator (no scenery).
This Einstein called the principle of equivalence.
It is impossible to know
whether you are hovering
in a gravitational field or if
you are simply accelerating
far from any gravitating
masses.
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Unlike Maxwell’s assertion that the speed of light is the same to all
observers, Einstein’s principle of equivalence is really not weird
at all.
Einstein decided that the generalization of the concept of constant
velocity motion, without being subjected to forces, is motion of
observers in free-fall.
But Einstein, of course, did not stop there.
The purpose of this generalization is to account for the presence
and influence of masses in our world.
He went on to generalize the concept of a straight line.
Not a straight line in space, but a straight line in space-time.
I G
In
Galileo’s
lil ’ world,
ld andd also
l in
i the
th world
ld off special
i l relativity,
l ti it a
straight line is the path followed by an object moving at a
constant velocity and not acted upon by any force.
This definition, of course, is circular, since when turned around it is
also the definition of a constant velocity. At least for Galileo.
But then Galileo had no particular reason to think too deeply about
what is straight and what is not.
So now you and Jackie may still be in your space ships, but you are
no longer out in deep space far from any other objects.
The idea is that if you don’t turn on you rocket motors, both you
and Jackie will be freely falling and you will travel in “straight”
lines – not straight lines in space, but straight lines in spacetime.
The forces exerted on you by masses through their gravity will
show up through a redefinition of the concept of a straight line,
that is, through a redefinition of space itself.
Einstein decided that the
generalization of the
concept of constant
velocity motion, without
being subjected to forces,
is motion of observers in
free-fall.
Einstein decided that the situations illustrated here in (a) or in (b) are equivalent.
The people in cases (a) or (b) can perform no experiment from which they can
learn which version of (a) or (b) they are experiencing.
In Einstein’s view, masses distort space-time, and you, while freely
falling, just travel along straight lines in this distorted context.
This is, so far as we know, a perfectly OK way of thinking about
gravity, but other ways of thinking about it are probably OK too.
But no one else has thought as deeply about gravity as Einstein
using any other conceptual framework. (Others have tried, and
the thoughts, for example, of Richard Feinman, a Nobel prize
winner on this subject have recently been published,
winner,
published but they
did not get far enough to be called a theory.)
But we can get an idea by projecting pictures onto 2-D pieces of
paper, exploiting the fact that our brains know how to build 3-D
perceptions from 2-D perspective drawings.
So here’s representation #1.
We draw a flat space as a flat rubber trampoline, and we illustrate
the distortion of this flat space by masses by showing them
sinking down in it and making the surrounding regions of the
t
trampoline
li curve downward.
d
d
The distortion Einstein is talking about is in a 4-dimensional space,
and to “see” this distortion properly, we would have to “embed”
this 4-D space-time in a 5-dimensional space that was not
distorted.
No one has ever achieved such “sight,” or perhaps we should say
“insight.”
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In this analogy, bigger masses create greater distortions of the
surrounding space, and this is just what Einstein’s view of
gravity says they should do.
The analogy works quite well, even for orbits of other small
masses, as shown on the next slide.
Just as Kepler says for planets orbiting the sun, a marble rolling on
the surface of the distorted trampoline will roll faster when near
th masses that
the
th t distort
di t t its
it surface.
f
You can verify this at the Minnesota Science Museum, where a set
up just like on the next slide is on display.
The diagrams on the following slide illustrate this concept of the
distortion of space by a concentration of mass in a slightly
different way.
At the left, we see images similar to the one taken from your
textbook that appears on the previous slide.
To the right of each of these images is a view of the same situation
from a vantage point directly above the object in the third
dimension that we are using to make its distortion of the
surrounding space visible to us.
These views on the right give us the classic pictures that we would
plot by applying Newton’s theory of gravity, much as he did in
his lifetime.
This slide is to illustrate how Einstein was able to obtain precisely
the same result as Newton, in this simple case, but from a rather
more peculiar way of thinking about the problem.
In Einstein’s view, the large mass has distorted the space around it,
so that freely falling particles, particles on which no forces
additional to gravity are acting, follow paths that are curved by
the distortion of the space they are moving through.
For Einstein, if we can calculate the curvature of space-time
everywhere from a knowledge of the positions of the
gravitating masses in the space-time, then we can find the paths
of all freely falling particles. Such paths, for example, are the
orbits of the planets or of space craft in our solar system.
For Newton, we imagine that space is neatly ruled out with straight
lines, with fiducial markings, like on graph paper. From the
positions of the masses, we can use his gravitational force law
and add up all the gravitational tugs on a freely falling particle
and, with some calculus, then compute its path, or orbit.
This
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Both approaches get the same result in all circumstances except for
some rather exotic cases in which Einstein, so far, seems to be
right and Newton wrong.
Einstein says that it is possible to tell if the space around you is
distorted. All you need to do is to rule out in the best way you can
three straight lines that meet at the 3 corners of a triangle. Then
you get out your protractor and measure the three inner angles and
add them up. If space is flat, the sum will be 180°. If not, it will be
more or less than that. The distortion of space by the mass of the
earth is extremely small, so here in the classroom, you will get
180°, but right up close to a pulsar, for example, you would not.
This concept is not really so outrageously peculiar. Think about the
two-dimensional space that is the surface of the earth. If we draw
triangles on this surface using straight lines that, of course, remain
on the surface, then those straight lines will be “great circles.”
These give the shortest distances between two points on the surface
and are taken by airplanes (which is why the shortest path from here
to Tokyo will nearly take you up to Alaska). For a big triangle, like
Minneapolis to Tokyo to Cancun and back, we will not get 180°.
In the example using the earth’s surface as our space, we said the
straight lines would be great circles, which are the paths that
airplanes take. These are just what Einstein would choose, because,
if we get rid of the air resistance and get rid of all the mountains and
valleys, space ships would orbit (in free fall) precisely along these
paths just above the earth’s surface. So these are indeed Einstein’s
straight lines.
This trampoline analogy also works as a way to understand the
effects of compressing a given mass into smaller and smaller
volumes of space.
As, for example, we make the sun progressively more compact,
going from main sequence star to white dwarf, or even to
neutron star or black hole, the outer region is unaffected, but the
inner region becomes ever more greatly distorted.
You can take an especially simple example by building the triangle
using the north pole and two cities on the equator, say someplace in
Ecuador and someplace in equatorial Africa. The two angles at the
equatorial cities will be precisely 90° each, and the angle at the pole
will certainly be greater than zero, so Einstein’s way (actually
Riemann’s way, to be precise) of determining the curvature of the
space definitely works.
4
Let’s look again at the first diagram with the orbits, because it helps
to bring out another important point.
Notice that two of the orbits cross at two points.
If these orbits are “straight” lines, how can they intersect with each
other twice?
The resolution of this question comes from recognizing that in this
analogy
gy we are projecting
p j
g things
g from our 4-D space-time
p
onto a
2-D image.
In space-time, the orbits do not intersect more than once, since it
takes different amounts of time to go along the different orbits
between the 2 intersection points.
In space time, a circular orbit maps into something that we might
better visualize as a helix, although even then it is hard to think
of this as a straight line.
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It is possible to have many straight lines passing through any
particular point in space-time, and these would be the paths of
freely-falling objects that have different velocities (both speeds
and directions).
But the paths of light rays through this point are special, because
light goes at the fastest possible speed.
The trampoline analogy shows how even light will be deflected as
it passes by
b a condensed
d
d mass, like
lik the
th sun or like
lik a galaxy.
l
The following diagrams show that not only can light be deflected,
but it can reach an observer along multiple possible paths
simultaneously, producing what is called an Einstein ring.
Both the deflection of light by the sun and the phenomenon of the
Einstein ring have been observed, so this aspect of the theory
checks out.
Above are shown a couple of actual Einstein rings.
An entire cluster of
galaxies acts as a
gravitational lens. The
combined gravitational
field of the yellow
elliptical and spiral
galaxies has dramatically
distorted the image of a
more distant blue galaxy
into 5 separate parts. The
galaxy is about twice as
far away as the cluster
CL0024+1654. The
galaxy’s spiral shape has
been considerably
changed by the lensing
effect, but several details
are still visible. Its
clumpiness, for instance,
indicates its youth and
active star formation.
Colley et al. & NASA.
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A mysterious arc of light found behind a distant cluster of galaxies
has turned out to be the biggest, brightest, and hottest star-forming
region ever seen in space. The so-called Lynx arc is 1 million times
brighter than the well-known Orion Nebula, a nearby prototypical
star-birth region visible with small telescopes. The newly identified
super-cluster contains a million blue-white stars that are twice as
hot as similar stars in our Milky Way galaxy. It is a rarely seen
example of the early days of the universe where furious firestorms
off star birth
bi h blazed
bl d across the
h skies.
ki The
h spectacular
l cluster's
l
'
opulence is dimmed when seen from Earth only because it is 12
billion light-years away.
Text from Space Telescope Science Institute via European Space
Agency.
The Lynx Arc, shown here, is thought to be the brightest star cluster
ever observed, seen with the aid of this cosmic telescope.
This is the
artist’s
concept of
what the
Lynx Arc
really looks
like. Wow!
The idea is
that
h the
h fi
first
stars were
much, much
more
massive than
stars that
form today.
The distant
galaxy
certainly
doesn’t look
like much in
this picture.
An international team of astronomers may have set a new record in
discovering what is the most distant known galaxy in the universe.
Located an estimated 13 billion light-years away, the object is
being viewed at a time only 750 million years after the big bang,
when the universe was barely 5 percent of its current age. The
primeval galaxy was identified by combining the power of NASA’s
Hubble Space Telescope and CARA’s W. M. Keck Telescopes on
Mauna Kea in Hawaii. These great observatories got a boost from
the
h added
dd d magnification
ifi i off a naturall “cosmic
“
i gravitational
i i l lens”
l ” in
i
space that further amplifies the brightness of the distant object.
Text from Space Telescope Science Institute via European Space
Agency.
Now let’s think about a further consequence of the Principle of
Equivalence: clocks run slower in gravitational fields.
Where is that
artist who
d
draws
those
th
fabulous
concept’s
when you
need her?
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We talked about black holes before during our discussion of
binary star systems.
We will just recapitulate some of that discussion here.
Later, in discussing the expanding universe we will encounter
further ramifications of Einstein’s theory of gravity, his
general theory of relativity.
Because you are at the front of the space ship, the acceleration is always taking
you away from the point at which Jackie’s light flashes originate, thus the light
from each flash takes a little longer to reach you than it would in unaccelerated
motion. Therefore Jackie’s clock will seem to be running slower than yours.
This prediction that clocks in stronger gravitational fields run slower has been
tested, and it checks out.
We can think of a black hole as mass that is so concentrated that gravity
is so strong near it that even light is deflected so greatly that it orbits
the mass concentration rather like the way the planets orbit the Sun.
A black hole has what we call an event horizon, beyond which events
are unobservable to us.
Inside the event horizon, gravity is so strong that the escape velocity
exceeds the speed of light.
g cannot escape
p from within the event horizon.
Even light
Light loses energy upon working its way out from a strong gravitational
field. This energy loss takes the form of a redshift, and we call it the
gravitational redshift. It is related to the slower rate at which clocks
run in strong gravitational fields.
Fig. 17.12 (a) A 2-D representation of “flat” space-time.
The circumference of each circle is 2π times its radius.
radius
(b) A 2-D representation of the “curved” space-time around a black
hole. The black hole’s mass distorts space-time, making the
radial distance between two circles larger than it would be in a
“flat” space-time.
Light emitted directly outward from the location of the event horizon is
infinitely redshifted, that is, it reaches locations far from the black
hole redshifted to infinite wavelength (zero energy), and is thus
invisible.
Black holes may sound like very violent things, but they need not
be.
3 Solar mass
black hole with
photon sphere,
event horizon,
Schwarzschild
radius, incoming
light.
It has been popular for many years, although it is less popular
today, to think of our universe as closed. This would mean
that if we sent off a beam of light in any direction, it could not
escape from the universe, no matter how long it traveled. We
might fire off such a beam of light, it might go all the way
around the universe, and ultimately come back to us from the
opposite
i direction
di i to that
h in
i which
hi h we launched
l
h d it.
i Suchh a
closed universe would be very much like the inside of a black
hole. The round-about travel of the light beam is like the light
that orbits a black hole. In fact, we could live quite happily
inside a black hole, if it were large enough so that its tides did
not tear us apart and the orbits of its photons were so large that
they would appear nearly straight and infinite to our human
scale of perception.
8
Your textbook has a whole section about people in rocket ships
orbiting a black hole and probing it in various ways.
This is well worth reading, but there is no point to recap it here.
Fig. 17.13 Time runs more slowly on the clock nearer to the black
hole, and gravitational redshift makes its glowing blue numerals
appear red from your spaceship orbiting the black hole further out.
At the left is an image, taken in visual light, of the region of the
Galactic center, and at the right is a radio image, which is
enlarged on the next slide.
Most of the circular
features in this view
of the central region
of our galaxy are
supernova remnants.
The density of stars is
very high in this
region, so it is not
surprising that some
of them have
exploded recently
enough that their
remnants are still very
noticeable.
Radio close-up
of the central
region. We see
a “hypernova”
remnant toward
the left, and a
spiral of gas
centered on the
very center of
the Galaxy.
9
Still closer
view of the
central
region. We
see a
“hypernova”
remnant
toward the
left and a
left,
spiral of gas
centered on
the center.
Stars in the very central region orbit the center, and Kepler’s laws
tell us how massive the region they orbit must be:
about 2.6 million solar masses.
With high frequency radio observations, we can measure the size
of the central object, or at least a size that it cannot exceed. It
turns out to be no larger than the diameter of the earth’s orbit
about the sun.
Stars in the
very
central
region orbit
the center,
and
Kepler’s
laws tell us
how
massive the
region they
orbit must
be: about
2.6 million
solar
masses.
With high frequency radio observations, we can measure the size
of the central object, or at least a size that it cannot exceed.
Computation by Ed Seidel, et al.
The evolution of the collision of two equal mass black holes was
computed using a Cray C-90 supercomputer at the Pittsburgh
Supercomputing Center and Cray Y-MP at NCSA. The black
holes start initially at rest, and accelerate towards each other as a
result of their mutual attraction due to gravity. As the holes
collide, a large, distorted black hole is formed, which vibrates at
its characteristic frequency. The final oscillating hole emits
gravitational waves at this frequency as it settles down to its
quiet, spherical state.
The National Science Foundation has built two gravitational
wave observatories, one in Louisiana and one in Washington
state. They hope to “see” an occasional black hole collision.
10
The horizon history diagrams show the evolution of what are
known as the “apparent horizons” of the black holes. The
horizon marks the boundary of the black hole; light rays inside
this boundary are forever trapped inside the hole, while outgoing
light rays outside this boundary are able to expand away from the
hole. The left edge of the horizon history diagram shows the
shape of the initial configuration, corresponding to either two
separate holes (if they are far enough apart initially), or two holes
that are already merged into a single hole (if they are close
enough together). The color map on the surface shows the local
distortion in the hole’s shape. The right end of these diagrams
shows the shape of the final hole, and the region in between
shows the merging and oscillation of the horizons.
The gravitational wave diagrams show the waves leaving the
holes during and after the collision. The waves are shown as
blue and yellow color maps, as white contour lines, or as
surfaces, depending on the representation.
In this image we show a series of photons that are attempting to
get away from the system at the initial time. In fact, these
photons will all succeed in their flight away from the holes,
showing that for this initial data set there was no event horizon
surrounding both holes initially. (Initial data, mu=2.2 case)
Horizon history diagram at about t =1M for mu=1.2 .
Horizon history diagram at about t =50M for mu=1.2 .
Horizon history diagram at about t =100M for mu=1.2 .
Horizon history diagram at about t =1M for mu=2.2 .
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Horizon history diagram at about t =50M for mu=2.2 .
Horizon history diagram at about t =1M for mu=3.25 .
Horizon history diagram at about t =50M for mu=3.25 .
Horizon history diagram at about t =100M for mu=3.25 .
Gravitational waves emitted by a large black hole formed from
the collision of two smaller holes, shown at t=27M.
Gravitational waves emitted by a large black hole formed from
the collision of two smaller holes, shown at t=39M.
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Gravitational waves emitted by a large black hole formed from
the collision of two smaller holes, shown at t=68M.
The Chandra image of NGC 6240, a butterfly-shaped galaxy that
is the product of the collision of two smaller galaxies, revealed
that the central region of the galaxy (inset) contains not one, but
two active giant black holes.
An artist's conception shows a black hole surrounded by a disk of
hot gas, and a large doughnut or torus of cooler gas and dust. The
light blue ring on the back of the torus is due to the fluorescence
of iron atoms excited by X-rays from the hot gas disk.
VLA radio image (by Ron Ekers) of a jet, presumably caused by
accretion of material by a supermassive black hole at the center
of the galaxy NGC 326. It is supposed that only the merger of
this black hole with another one, as the result of a galaxy merger,
could have caused the visible realignment of the hole’s rotation.
The Chandra image of NGC 6240, a butterfly-shaped galaxy that
is the product of the collision of two smaller galaxies, revealed
that the central region of the galaxy (inset) contains not one, but
two active giant black holes.
A black hole merger movie.
The movie shows a merger of two galaxies (simulation) that
forms a single galaxy with two centrally located supermassive
black holes surrounded by disks of hot gas. The black holes orbit
each other for hundreds of millions of years before they merge to
form a single supermassive black hole that sends out intense
gravitational waves.
13
This sequence begins with a wide-field optical image of the galaxy
NGC 6240 and then zooms into the central region where it dissolves
to a Chandra X-ray image. The colors in the X-ray image show the
intensity of the low (red), medium (green) and high (blue) energy Xrays. This image then dissolves to show only the highest-energy Xrays detected by Chandra, which come from gas around the two
black holes in the center of the galaxy.
A 3D computer simulation by John Hawley produced this image
of a magnetized accretion torus about a black hole.
An artist's conception shows a black hole surrounded by a disk of
hot gas, and a large doughnut or torus of cooler gas and dust. The
light blue ring on the back of the torus is due to the fluorescence
of iron atoms excited by X-rays from the hot gas disk.
A 3D computer simulation by John Hawley produced this movie
of a magnetized accretion torus about a black hole.
Einstein’s view of gravity in terms of a distortion of space-time
that is experienced by any particles travelling through has
some interesting consequences.
First, it follows that if the masses in the universe move, the
distortion of space-time must change accordingly, but this
change cannot happen everywhere in the universe instantly.
This argument implies that a wave of space-time distortion
change must propagate outward (at the speed of light, of
course, what else?) from a moving mass.
A 3D computer simulation by John Hawley produced this movie
of azimuthally averaged density in a close-up view of a
magnetized accretion torus about a black hole.
These propagating modifications of the distortion of space-time
are called gravitational waves.
An apparently somewhat less than honest professor at the
University of Maryland claimed to have observed such waves
in the 1960s (what was he smoking?).
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However, since the 60s, several very careful, expensive, and
unsuccessful experiments have looked for gravitational
waves.
The latest is called LIGO, a set of gravitational wave
observatories in unlikely states (Washington and Louisiana)
costing the taxpayers some 60 million dollars.
Let’s hope that it’s too early for these observatories to have seen
any gravitational waves as yet. If and when they do, it will
be first page news.
The problem with gravitational waves is that they are weak.
Like, really weak.
You can sense their passage by observing a couple of incredibly
massive aluminum cylinders moving toward and away from
each other by a microscopic, laser sensed amount. The
trouble is that there are so many other reasons why such
cylinders might do this.
Indirect evidence of gravitational waves, however, does exist.
In the 1970s, a search for pulsars (rotating neutron stars) came
up with a binary pulsar. This is 2 pulsars in a close orbit
about each other.
The gravitational force between these 2 collapsed objects is so
strong that their orbiting must send out gravitational waves of
significant strength (of course we have not seen them
directly, they aren’t that strong).
The emission of the gravitational waves removes orbital energy
from the system, so that the pulsars spiral in toward each
other.
This orbital decay has been observed and checks out with the
predicted amount of gravitational wave energy loss. The guy
who worked this out won a prize for this.
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