General Relativity

General Relativity :
The second part of relativity is the theory of general relativity and lies on two empirical findings
that he elevated to the status of basic postulates. The first postulate is the relativity principle: local
physics is governed by the theory of special relativity. The second postulate is the equivalence
principle: there is no way for an observer to distinguish locally between gravity and acceleration.

Einstein discovered that there is a relationship between mass, gravity and spacetime. Mass distorts
spacetime, causing it to curve.

Gravity can be described as motion caused in curved spacetime .
Thus, the primary result from general relativity is that gravitation is a purely geometric
consequence of the properties of spacetime. Special relativity destroyed classical physics view of
absolute space and time, general relativity dismantles the idea that spacetime is described by
Euclidean or plane geometry. In this sense, general relativity is a field theory, relating Newton's
law of gravity to the field nature of spacetime, which can be curved.

Gravity in general relativity is described in terms of curved spacetime. The idea that spacetime is
distorted by motion, as in special relativity, is extended to gravity by the equivalence principle.
Gravity comes from matter, so the presence of matter causes distortions or warps in spacetime.
Matter tells spacetime how to curve, and spacetime tells matter how to move (orbits).
There were two classical test of general relativity, the first was that light should be deflected by
passing close to a massive body. The first opportunity occurred during a total eclipse of the Sun in
1919.

Measurements of stellar positions near the darkened solar limb proved Einstein was right. Direct
confirmation of gravitational lensing was obtained by the Hubble Space Telescope last year.
The second test is that general relativity predicts a time dilation in a gravitational field, so that,
relative to someone outside of the field, clocks (or atomic processes) go slowly. This was
confirmed with atomic clocks flying airplanes in the mid-1970's.
The general theory of relativity is constructed so that its results are approximately the same as those
of Newton's theories as long as the velocities of all bodies interacting with each other
gravitationally are small compared with the speed of light--i.e., as long as the gravitational fields
involved are weak. The latter requirement may be stated roughly in terms of the escape velocity. A
gravitational field is considered strong if the escape velocity approaches the speed of light, weak if
it is much smaller. All gravitational fields encountered in the solar system are weak in this sense.
Notice that at low speeds and weak gravitational fields, general and special relativity reduce to
Newtonian physics, i.e. everyday experience.

Black Holes

Black Holes:
The fact that light is bent by a gravitational field brings up the following thought experiment.
Imagine adding mass to a body. As the mass increases, so does the gravitational pull and objects
require more energy to reach escape velocity. When the mass is sufficiently high enough that the
velocity needed to escape is greater than the speed of light we say that a black hole has been
created.

Another way of defining a black hole is that for a given mass, there is a radius where if all the mass
is compress within this radius the curvature of spacetime becomes infinite and the object is
surrounded by an event horizon. This radius called the Schwarzschild radius and varys with the
mass of the object (large mass objects have large Schwarzschild radii, small mass objects have
small Schwarzschild radii).

The Schwarzschild radius marks the point where the event horizon forms, below this radius no light
escapes. The visual image of a black hole is one of a dark spot in space with no radiation emitted.
Any radiation falling on the black hole is not reflected but rather absorbed, and starlight from
behind the black hole is lensed.

Even though a black hole is invisible, it has properties and structure. The boundary surrounding the
black hole at the Schwarzschild radius is called the event horizon, events below this limit are not
observed. Since the forces of matter can not overcome the force of gravity, all the mass of a black
hole compresses to infinity at the very center, called the singularity.

A black hole can come in any size. Stellar mass black holes are thought to form from supernova
events, and have radii of 5 km. Galactic black hole in the cores of some galaxies, millions of solar
masses and the radius of a solar system, are built up over time by cannibalizing stars. Mini black
holes formed in the early Universe (due to tremendous pressures) down to masses of asteroids with
radii the size of a grain of sand.

Note that a black hole is the ultimate entropy sink since all information or objects that enter a black
hole never return. If an observer entered a black hole to look for the missing information, he/she
would be unable to communicate their findings outside the event horizon.

Galaxies

Galaxies:
A galaxy is a collect of stars, gas and dust bound together by their common gravitational pull.
Galaxies range from 10,000 to 200,000 light-years in size and between 109 and 1014 solar
luminosities in brightness.
The discovery of `nebula', fuzzy objects in the sky that were not planets, comets or stars, is
attributed to Charles Messier in the late 1700's. His collection of 103 objects is the first galaxy
catalog. Herschel (1792-1871) used a large reflecting telescope to produce the first General
Catalog of galaxies.

Before photographic plates, galaxies were drawn by hand by the astronomer.

Galaxies have certain features in common. Gravity holds the billions of stars together, and the
densest region is in the center, called a core or bulge. Some galaxies have spiral or pinwheel arms.
All galaxies have a faint outer region or envelope and a mysterious dark matter halo.

The contents of galaxies vary from galaxy type to galaxy type, and with time.
Almost all galaxy types can be found in groups or clusters. Many clusters of galaxies have a large,
supergiant galaxy at its center which has grow by cannibalizing its neighbors. Our solar system is
located in outer regions of a spiral galaxy we call the Milky Way. The nearest neighbor galaxy is
Andromeda Galaxy (M31).

Above is a 3D plot of most of the Local Group of galaxies, the population of galaxies within 1000
kpc if the Milky Way. Clustering of dwarf satellite galaxies around the great Milky Way and
Andromeda spirals can be seen.
Hubble sequence :
Almost all current systems of galaxy classification are outgrowths of the initial scheme proposed
by American astronomer Edwin Hubble in 1926. In Hubble's scheme, which is based on the
optical appearance of galaxy images on photographic plates, galaxies are divided into three
general classes: ellipticals, spirals, and irregulars.

Spiral galaxies

Spiral galaxies :
These galaxies are conspicuous for their spiral-shaped arms, which emanate from or near the
nucleus and gradually wind outward to the edge. There are usually two opposing arms arranged
symmetrically around the center. The nucleus of a spiral galaxy is a sharp-peaked area of smooth
texture, which can be quite small or, in some cases, can make up the bulk of the galaxy. The arms
are embedded in a thin disk of stars. Both the arms and the disk of a spiral system are blue in
color, whereas its central areas are red like an elliptical galaxy.
M100
Notice in the above picture of M100 from HST, that the center of the spiral is red/yellow and the
arms are blue. Hotter, younger stars are blue, older, cooler stars are red. Thus, the center of a spiral
is made of old stars, with young stars in the arms formed recently out of gas and dust.
NGC 4639
The bulge of NGC 4639 is quite distinct from the younger, bluer disk regions.
NGC 1365
NGC 1365 is a barred spiral galaxy. Note the distinct dark lanes of obscuring dust in the bar
pointing towards the bulge. A close-up of the spiral arms shows blue nebula, sites of current star
formation.
NGC 253 is a typical Sa type galaxy with very tight spiral arms. As spiral galaxies are seen
edge-on the large amount of gas and dust is visible as dark lanes and filaments crossing in front of
the bulge regions.
Irregular galaxies :
Most representatives of this class consist of grainy, highly irregular assemblages of luminous
areas. They have no noticeable symmetry nor obvious central nucleus, and they are generally bluer
in color than are the arms and disks of spiral galaxies.
NGC 2363
NGC 2363 is an example of a nearby irregular galaxy. There is no well defined shape to the
galaxy, nor are there spiral arms. A close-up of the bright region on the east side shows a cluster of
new stars embedded in the red glow of ionized hydrogen gas.

Galaxy Colors

Galaxy Colors:
The various colors in a galaxy (red bulge, blue disks) is due to the types of stars found in those
galaxy regions, called its stellar population. Big, massive stars burn their hydrogen fuel, by
thermonuclear fusion, extremely fast. Thus, they are bright and hot = blue. Low mass stars,
although more numerous, are cool in surface temperature (= red) and much fainter. All this is
displayed in a Hertzsprung-Russell Diagram of the young star cluster.

The hot blue stars use their core fuel much faster than the fainter, cooler red stars. Therefore, a
young stellar population has a mean color that is blue (the sum of the light from all the stars in the
stellar population) since most of the light is coming from the hot stars. An old stellar population is
red, since all the hot stars have died off (turned into red giant stars) leaving the faint cool stars.

The bottom line is that the red regions of a galaxy are old, with no hot stars. The blue portions of a
galaxy are young, meaning the stellar population that dominates this region is newly formed.

Star Formation

Star Formation :
The one feature that correlates with the shape, appearance and color of a galaxy is the amount of
current star formation. Stars form when giant clouds of hydrogen gas and dust collapse under their
own gravity. As the cloud collapses it fragments into many smaller pieces, each section continues
to collapse until thermonuclear fusion begins.

Initial conditions for a galaxy determines its rate of star formation. For example, elliptical galaxies
collapse early and form stars quickly. The gas is used up in its early years and today has the
appearance of a smooth, red object with no current star formation.

Spirals, on the other hand, form slower, with lower rates of star formation. The gas that `fuels' star
formation is used slower and, thus, there is plenty around today to continue to form stars within
the spiral arms.

Hubble's law

Hubble's law:
In the 1930's, Edwin Hubble discoveried that all galaxies have a positive redshift. In other words,
all galaxies were receding from the Milky Way. By the Copernican principle (we are not at a
special place in the Universe), we deduce that all galaxies are receding from each other, or we
live in a dynamic, expanding Universe.

The expansion of the Universe is described by a very simple equation called Hubble's law; the
velocity of the recession of a galaxy is equal to a constant times its distance (v=Hd). Where the
constant is called Hubble's constant and relates distance to velocity in units of light-years.

Distance Scale

Distance Scale:
The most important value for an astronomical object is its distance from the Earth. Since
cosmology deals with objects larger and brighter than our Sun or solar system, it is impossible to
have the correct frame of reference with respect to their size and luminosity as there is nothing to
compare extragalactic objects with.

Before the 1920's, it was thought that galaxies were in fact objects within our own Galaxy,
possibly regions forming individual stars. They were given the name ``nebula'', which we now
use to denote regions of gas and dust within galaxies.
At the turn of the century Cepheid variable stars, a special class of pulsating stars that exhibit a
particular period-luminosity relation, were discovered. In other words, it was found that their
intrinsic brightness was proportional to their period of variation and, hence, could be used for
measuring the distances to nearby galaxies.
In the late 1920's, Hubble discovered similar Cepheid stars in neighboring galaxies as was found
in our own Galaxy. Since they followed the same period-luminosity relation, and they were very
faint, then this implied that the neighboring galaxies were very far away. This proved that spiral
`nebula' were, in fact, external to our own Galaxy and sudden the Universe was vast in space and
time.
Although Hubble showed that spiral nebula were external to our Galaxy, his estimate of their
distances was off by a factor of 6. This was due to the fact that the calibration to Cepheids was
poor at the time, combined with the primitive telescopes Hubble used.
Modern efforts to obtain an estimate of Hubble's constant, the expansion rate of the Universe,
find it necessary to determine the distance and the velocities of a large sample of galaxies. The
hardest step in this process is the construct of the distance scale for galaxies, a method of
determining the true distance to a particular galaxy using some property or characteristic that is
visible over a range of galaxies types and distance.
The determination of the distance scale begins with the construction of ladder of primary,
secondary and tertiary calibrators in the search for a standard candle.

Primary Calibrators

Primary Calibrators:
The construction of the distance scale ladder is a process of building of a chain of objects with
well determined distance. The bottom of this chain is the determination of the scale of objects in
the Solar System. This is done through radar ranging, where a radio pulse is reflected off of the
various planets in the Solar System.

The most important value from solar system radar ranging is the exact distance of the Earth from
the Sun, determined by triangular measurement of the Earth and terrestrial worlds. This allows
an accurate value for what is called the Astronomical Unit (A.U.), i.e. the mean Earth-Sun
distance. The A.U. is the ``yardstick'' for measuring the distance to nearby stars by parallax.

The parallax system is only good for stars within 300 light-years of the Earth due to limitations
of measuring small changes in stellar position. Fortunately, there are hundreds of stars within this
volume of space, which become the calibrators for secondary distance indicators.

Secondary Calibrators

Secondary Calibrators:
Secondary calibrators of the distance scale depend on statistical measures of stellar properties,
such as the mean brightness of a class of stars. It has been known since the 1800's that stars
follow a particular color-luminosity relation known as the Hertzsprung-Russell Diagram.

The existence of the main sequence for stars, a relationship between luminosity and color due to
the stable, hydrogen-burning part of a star's life, allows for the use of spectroscopic parallax. A
stars temperature is determined by its spectrum (some elements become ions at certain
temperatures). With a known temperature, then an absolute luminosity can be read off the HR
diagram.

The distance to a star is simply the ratio of its apparent brightness and its true brightness
(imagine car headlights at a distance). The method allows us to measure the distances to
thousands of local stars and, in particular, to nearby star clusters which harbor variable stars.
A variable star is a star where the brightness of the star changes over time (usually a small
amount). This is traced by a light curve, a plot of brightness and time.

Particular variable stars, such as Cepheids, have a period-luminosity relationship. Meaning that
for a particular period of oscillation, they have a unique absolute brightness.

The result is that it is possible to measure the light curve of Cepheids in other galaxies and
determine their distances.

Tertiary Calibrators

Tertiary Calibrators:
The nearby region of the Universe, known as the Local Group and is located at the edge of what
is known as the the Virgo supercluster of galaxies. The use of Cepheid variables is limited to
within the volume of space outlined by Virgo system. Thus, the distances to nearby galaxies does
not measure the true Hubble flow of the expanding Universe, but rather the gravitational infall
into Virgo.
In order to determine Hubble's constant, we must measure the velocity of galaxies much farther
away then the Local Group or the Virgo supercluster. But, at these distances we cannot see
Cepheid stars, so we determine the total luminosity of the galaxy by the Tully-Fisher method, the
last leg of the distance scale ladder.

The Tully-Fisher relation is basically a plot of mass versus luminosity of a galaxy. Its not
surprising that luminosity and mass are correlated since stars make up basically most of a
galaxy's mass and all of the light. Missing mass would be in the form of gas, dust and dark
matter.
The key parameter for this last leg of the distance scale are the calibrating galaxies to the
Tully-Fisher relation, i.e. the galaxies where we know both the total luminosity from Cepheid
distances and the total luminosity from the Tully-Fisher relation.
There is currently a strong debate on the value of the Hubble's constant fueled by new data from
HST Cepheid studies of nearby galaxies. The community is divided into two schools of thought;
1) the old school which proposes a value for Hubble's constant around 50 to agree with the ages
of the oldest stars in our Galaxy, and 2) a newer, and larger school which finds a higher Hubble's
constant of 75. This higher value poses a problem for modern cosmology in that the age of the
Universe from Hubble's constant is less than the age of the oldest stars as determined by nuclear
physics.
So the dilemma is this, either something is wrong with nuclear physics or something is wrong
with our understanding of the geometry of the Universe. One possible solution is the introduction
of the cosmological constant, once rejected as unnecessary to cosmology, it has now grown in
importance due to the conflict of stellar ages and the age of the Universe.
Quasars:
Quasars are the most luminous objects in the Universe. The typical quasar emits 100 to 1000
times the amount of radiation as our own Milky Way galaxy. However, quasars are also variable
on the order of a few days, which means that the source of radiation must be contained in a
volume of space on a few light-days across. How such amounts of energy can be generated in
such small volumes is a challenge to our current physics.
Quasars were originally discovered in the radio region of the spectrum, even though they emit
most of their radiation in the high energy x-ray and gamma-ray regions. Optical spectra of the
first quasars in the 1960's showed them to be over two billion light-years away, meaning two
billion years into the past as well.

Over a thousand quasars have been discovered, most having redshifts greater than 10 billion
light-years away. The number density of quasars drops off very fast, such that they are objects
associated with a time when galaxies were young.
The large amount of radio and x-ray emission from quasars gives them similar properties to the
class of what are called active galaxies, such as Seyfert galaxies, originally recognized by the
American astronomer Carl K. Seyfert from optical spectra. Seyfert galaxies have very bright
nuclei with strong emission lines of hydrogen and other common elements, showing velocities of
hundreds or thousands of kilometers per second, where the high energy emission is probably due
to a Galactic mass black hole at the galaxies core (for example, NGC 4261 shown below). The
idea is that quasars are younger, and brighter, versions of Seyfert galaxies.
HST imaging showed that quasars are centered in the middle of host galaxies, giving more
support to the idea that the quasar phenomenon is associated with Galactic mass black holes in
the middle of the host galaxies. Since a majority of the host galaxies are disturbed in appearance,
the suspicion is that colliding galaxies cause stars and gas to be tidally pushed into the black hole
to fuel the quasar.
This process would explain the occurrence of quasars with redshift. In the far distant past there
were no galaxies, so no sites for quasars. In the early phases of galaxy formation, the galaxy
density was high, and there were many collisions producing many quasars. As time passed, the
number of collisions decreased as space expanded and the number of quasar also dropped.

Active Galaxies

Active Galaxies:
Most galaxies are `normal' in that most of their light is generated by stars or heated
gas. This energy is primarily radiated away as optical and infrared energy.
However, there exists a subclass of galaxies, known as active galaxies, which
radiate tremendous amounts of energy in the radio and x-ray regions of the
spectrum. These objects often emit hundreds to thousands of times the energy
emitted by our Galaxy and, because of this high luminosity, are visible to the edges
of the Universe.
Active galaxies usually fall in three types; Seyfert galaxies, radio galaxies and
quasars. Radio galaxies often have a double-lobe appearance, and the type of radio
emission suggests that the origin is synchrotron radiation.

Active galaxies emit large amounts of x-rays and gamma-rays, extremely high
energy forms of electromagnetic radiation. Strong magnetic fields (synchrotron
radiation in the radiio) plus gamma-rays implies very violent events in the cores of
active galaxies.
Although active galaxies different in their appearance, they are related in the
mechanism that produces their huge amounts of energy, a Galactic mass black hole
at the galaxy's center. The gas flowing towards the center of the galaxy forms a
thick circle of orbiting material called an accretion disk many hundreds of
light-years across.
Since the infalling gas retains the direction of orbital motion of the companion, the
stream of material forms a rotating disk.

Friction between the gas in neighboring orbits causes the material to spiral inward
until it hits the event horizon of the central black hole. As the spiraling gas moves
inward, gravitational energy is released as heat into the accretion disk. The release
of energy is greatest at the inner edge of the accretion disk where temperatures can
reach millions of degrees. It is from this region that the magnetic fields are
produced for the synchrotron radiation and the collision between atoms to emit
x-rays and gamma-rays.
Our own Galaxy core may harbor a small active nuclei similar to those found in
quasars. In fact, all galaxies may have dormant black holes, invisible because there
is no accretion. Seyfert, radio galaxies and quasars may simply be normal galaxies
in an active phase.
This hypothesis has been confirmed by HST imaging of distant QSO hosts which
show the bright quasar core is in the center of fairly normal looking galaxies.
Lookback Time:
The large size of the Universe, combined with the finite speed for light, produces
the phenomenon known as lookback time. Lookback time means that the farther
away an object is from the Earth, the longer it takes for its light to reach us. Thus,
we are looking back in time as we look farther away.

The galaxies we see at large distances are younger than the galaxies we see nearby.
This allows us to study galaxies as they evolve. Note that we don't see the
individuals evolve, but we can compare spirals nearby with spirals far away to see
how the typical spiral has changed with time.

Galaxy Evolution

Galaxy Evolution:
The phenomenon of lookback time allows us to actually observe the evolution of
galaxies. We are not seeing the same galaxies as today, but it is possible to trace
the behavior of galaxies types with distance/time.
It is known that galaxies form from large clouds of gas in the early Universe. The
gas collects under self-gravity and, at some point, the gas fragments into star
cluster sized elements where star formation begins. Thus, we have the expectation
that distant galaxies (i.e. younger galaxies) will be undergoing large amounts of
star formation and producing hot stars = blue stars. The study of this phenomenon
is called color evolution.

Computer simulations also indicate that the epoch right after galaxy formation is a
time filled with many encounters/collisions between young galaxies. Galaxies that
pass near each other can be captured in their mutual self-gravity and merge into a
new galaxy. Note that this is unlike cars, which after collisions are not new types of
cars, because galaxies are composed of many individual stars, not solid pieces of
matter. The evolution of galaxies by mergers and collisions is called number
evolution.

Thus, our picture of galaxy evolution, incorporating both these principles, looks
like the following:

Some types of galaxies are still forming stars at the present epoch (e.g. spiral and
irregular galaxies). However, the past was marked by a much higher rate of star
formation than the present-day average rate because there was more gas clouds in
the past. Galaxies, themselves, were built in the past from high, initial rates of star
formation.
The time of quasars is also during the time of first star formation in galaxies, so the
two phenomenon are related, the past was a time of rapid change and violent
activity in galaxies.
Space observations called the Hubble Deep Field produced images of faint galaxies
and distant galaxies at high redshift which confirmed, quantitatively, our estimates
of the style and amount of star formation. Nature lends a hand by providing images
of distant galaxies by gravitational lensing, as seen in this HST image of CL0024.
Interestingly enough, it is often easier to simulate the evolution of galaxies in a
computer, then use the simulations to solve for various cosmological constants,
such as Hubble's constant or the geometry of the Universe. The field of
extragalactic studies is just such a process of iteration on the fundamental constants
of the Universe and the behavior of galaxies with time (i.e. galaxy evolution).
Creation Event:
The debate about the origin of the Universe presupposes that there was an origin.
Instead of a beginning, the Universe may be experiencing an endless number of
cycles. Ancient Chinese believed that all events formed a periodic pattern driven
by two basic forces, Yin and Yang.

The Hindu cosmological system consisted of cycles within cycles of immense
duration (one lifecycle of Brahma is 311 trillion years). Cyclicity cosmologies, and
their associated fatalism, is also found in Babylonian, Egyptian and Mayan
cultures.
Judeo-Christian tradition was unique in its belief that God created the Universe at
some specfic moment in the past, and that events form an unfolding unidirectional
sequence. Key to this philosophy is that the Creator is entirely separate from and
independent of His creation. God brings order to a primordal chaos.

Belief that a divine being starts the Universe then `sits back' and watchs events
unfold, taking no direct part in affairs, is known as deism. Here God is considered a
cosmic engineer. In contrast, theism is the belief in a God who is creator of the
Universe and who also remains directly involved in the day-to-day running of the
world, especially the affairs of human beings. God maintains a personal and
guiding role. In both deism and theism, God is regarded as wholly other than, and
beyond, the physical Universe. In pantheism, no such separation is made between
God and the physical Universe. God is identified with Nature itself: everything is a
part of God and God is in everything.
A Creation event implies that everything came from nothing (creation ex nihilo)
since if there were something before Creation, than an earlier Creation is needed to
explain that something. God existed before Creation, and the definition is not
limited to work with pre-existing matter or pre-existing physical laws either. In
fact, the most obvious distinction between the Creator and the created Universe is
that the Creator is eternal and the created Universe had a beginning.
Hot Big Bang:
The discovery of an expanding Universe implies the obvious, that the Universe
must have had an initial starting point, an alpha point or Creation. In other words,
there existed a point in the past when the radius of the Universe was zero. Since all
the matter in the Universe must have been condensed in a small region, along with
all its energy, this moment of Creation is referred to as the Big Bang.
A common question that is asked when considering a Creation point in time is
``What is before the Big Bang?''. This type is question is meaningless or without
context since time was created with the Big Bang. It is similar to asking ``What is
north of the North Pole?''. The question itself can not be phrased in a meaningful
way.
The Big Bang theory has been supported by numerous observations and, regardless
of the details in our final theories of the Universe, remains the core element to our
understanding of the past. Note that an alpha point automatically implies two
things: 1) the Universe has a finite age (about 15 billion years) and 2) the Universe
has a finite size (its expanding at a finite speed in a finite time).

Geometry of the Universe

Geometry of the Universe:
Can the Universe be finite in size? If so, what is ``outside'' the Universe? The answer to both these
questions involves a discussion of the intrinsic geometry of the Universe.
There are basically three possible shapes to the Universe; a flat Universe (Euclidean or zero curvature), a
spherical or closed Universe (positive curvature) or a hyperbolic or open Universe (negative curvature).
Note that this curvature is similar to spacetime curvature due to stellar masses except that the entire mass
of the Universe determines the curvature. So a high mass Universe has positive curvature, a low mass
Universe has negative curvature.

All three geometries are classes of what is called Riemannian geometry, based on three possible states for
parallel lines
l never meeting (flat or Euclidean)
l must cross (spherical)
l always divergent (hyperbolic)
or one can think of triangles where for a flat Universe the angles of a triangle sum to 180 degrees, in a
closed Universe the sum must be greater than 180, in an open Universe the sum must be less than 180.
Standard cosmological observations do not say anything about how those volumes fit together to give the
universe its overall shape--its topology. The three plausible cosmic geometries are consistent with many
different topologies. For example, relativity would describe both a torus (a doughnutlike shape) and a
plane with the same equations, even though the torus is finite and the plane is infinite. Determining the
topology requires some physical understanding beyond relativity.

Like a hall of mirrors, the apparently endless universe might be deluding us. The cosmos could, in fact,
be finite. The illusion of infinity would come about as light wrapped all the way around space, perhaps
more than once--creating multiple images of each galaxy. A mirror box evokes a finite cosmos that looks
endless. The box contains only three balls, yet the mirrors that line its walls produce an infinite number of
images. Of course, in the real universe there is no boundary from which light can reflect. Instead a
multiplicity of images could arise as light rays wrap around the universe over and over again. From the
pattern of repeated images, one could deduce the universe's true size and shape.

Topology shows that a flat piece of spacetime can be folded into a torus when the edges touch. In a
similar manner, a flat strip of paper can be twisted to form a Moebius Strip.

The 3D version of a moebius strip is a Klein Bottle, where spacetime is distorted so there is no inside or
outside, only one surface.

The usual assumption is that the universe is, like a plane, "simply connected," which means there is only
one direct path for light to travel from a source to an observer. A simply connected Euclidean or
hyperbolic universe would indeed be infinite. But the universe might instead be "multiply connected,"
like a torus, in which case there are many different such paths. An observer would see multiple images of
each galaxy and could easily misinterpret them as distinct galaxies in an endless space, much as a visitor
to a mirrored room has the illusion of seeing a huge crowd.

One possible finite geometry is donutspace or more properly known as the Euclidean 2-torus, is a flat
square whose opposite sides are connected. Anything crossing one edge reenters from the opposite edge
(like a video game see 1 above). Although this surface cannot exist within our three-dimensional space, a
distorted version can be built by taping together top and bottom (see 2 above) and scrunching the
resulting cylinder into a ring (see 3 above). For observers in the pictured red galaxy, space seems infinite
because their line of sight never ends (below). Light from the yellow galaxy can reach them along several
different paths, so they see more than one image of it. A Euclidean 3-torus is built from a cube rather than
a square.

A finite hyperbolic space is formed by an octagon whose opposite sides are connected, so that anything
crossing one edge reenters from the opposite edge (top left). Topologically, the octagonal space is
equivalent to a two-holed pretzel (top right). Observers who lived on the surface would see an infinite
octagonal grid of galaxies. Such a grid can be drawn only on a hyperbolic manifold--a strange floppy
surface where every point has the geometry of a saddle (bottom).

Its important to remember that the above images are 2D shadows of 4D space, it is impossible to draw the
geometry of the Universe on a piece of paper (although we can come close with a hypercube), it can only
be described by mathematics. All possible Universes are finite since there is only a finite age and,
therefore, a limiting horizon. The geometry may be flat or open, and therefore infinite in possible size (it
continues to grow forever), but the amount of mass and time in our Universe is finite.

Density of the Universe

Density of the Universe:
The description of the various geometries of the Universe (open, closed, flat) also relate to their futures.
There are two possible futures for our Universe, continual expansion (open and flat), turn-around and
collapse (closed). Note that flat is the specific case of expansion to zero velocity.

The key factor that determines which history is correct is the amount of mass/gravity for the Universe as
a whole. If there is sufficient mass, then the expansion of the Universe will be slowed to the point of
stopping, then retraction to collapse. If there is not a sufficient amount of mass, then the Universe will
expand forever without stopping. The flat Universe is one where there is exactly the balance of mass to
slow the expansion to zero, but not for collapse.

The parameter that is used to measure the mass of the Universe is the critical density, Omega. Omega is
usually expressed as the ratio of the mean density observed to that of the density in a flat Universe.

Given all the range of values for the mean density of the Universe, it is strangely close to the density of a
flat Universe. And our theories of the early Universe (see inflation) strongly suggest the value of Omega
should be exactly equal to one. If so our measurements of the density by galaxy counts or dynamics are
grossly in error and remains one of the key problems for modern astrophysics.
Cosmological Constants:
The size, age and fate of the Universe are determined by two constants:

The measurement of these constants consumes major amounts of telescope time over all wavelengths.
Both constants remain uncertain to about 30%; however, within this decade we can expect to measure
highly accurate values for both due to the Hubble Space Telescope and the Keck twins.