 Accretion Disk
 In a binary system containing a star and a compact object (white
dwarf, neutron star, or black hole) gas may flow from the star to the
compact object. According to the theoretical model, the gas will
spiral in and fall to the surface of the compact object creating a
flow of matter in the shape of a disk. It is generally believed that
this model explains many features of Xray pulsars
 Apparent Horizon
 When matter falls inward to form a black hole it is not always
easy to see where the event horizon might be. It might appear at one time
that a
light ray is capable of escaping but infalling matter might eventually
prevent it from doing so. The apparent horizon is a surface on which
outgoing light rays are just trapped, and cannot expand outward. It is a
stronger condition than the event horizon, and the apparent horizon always
lies inside the event horizon, or coincides with it.
This situation is analgous to a man running through a corridor
filled with doors. He is trying to run outward, but the doors are
closing in sequence from the outside in. How many doors will he be
able to pass through before he is blocked by a closed door? The door
that is closest to him that is currently closed is analgous to the
apparent horizon. The door that he will actually reach before he
cannot travel further is analogous to the event horizon.
 Arc Second
 The size of a celestial object expressed in terms of the angle that it covers (or "subtends") when viewed
from Earth. For example, the moon subtends an angle of 1/2 a degree. One degree
of arc is defined as equivalent to 60 minutes of arc (or "arc minutes"). Arc minutes are further divided
into arc seconds, such that there 60 x 60 or 3600 arc seconds per degree. So the moon's apparent size can
also be expressed as 1/2 degree x 3600 = 1800 arc seconds. If the the distance to an object is also known, then
its angular size can be used to calculate its diameter in miles or kilometers.
 Axisymmetry
 An axisymmetric system looks the same if we change our point of
view by rotating our position about an axis. Since a symmetry in
physics is an operation that leaves our system unchanged, an object
that does not look different after a rotation about an axis has
"axissymmetry."
If we are looking at a soup can within our imaginary sphere and
put our imginary axis through the center of its two flat faces we will
find that it is axisymmetric if we take the label off of it, but is
not axisymmetric if we leave the label on.

Big Bang

The "fireball" of cosmic creation. Modern cosmology is founded on the "Big
Bang" model in which all the known universe is thought have have emerged
some 1320 billion years ago from an unimaginably hot, dense state born of
a singularity.
See also Naked Singularity.
 Black Hole
 A black hole is a region of spacetime enclosed by an event horizon. A black hole is formed by
the collapse of massive objects. If the heat and pressure supplied by
the fusion of the material within the star is less than the
gravitational pull inward, the object may collapse to form a white dwarf, a neutron star, or (if it is massive enough) a
black hole.
A black hole is termed "black" because nothing can escape
from within it, not even light. Everything that passes through the
event horizon is gone from the observable universe.

Binary Pulsar

A source that pulsates in the radio or xray spectrum is called a
"pulsar" and it is generally believed that a pulsar is a
neutron star (although some of the
pulsars with longer periods might be
white dwarfs). A binary pulsar
is a binary star system (a system where two stars orbit
each other), where one of the two is a pulsar.

Coupled Equations

An example of a coupled equation may be found in the example
of state and federal tax. The state (in our example) takes
20 percent of the part of your income (after the federal tax is deducted),
and the federal government takes 10 percent of your income (after the
state tax is deducted). Using the definitions
 s = state tax
 f = federal tax
 i = your income
We can write the coupled equations:
 s = 0.20 * (if)
 f = 0.10 * (is)
(Where * means "multiplied by.")

Coupled, HyperbolicElliptic, Nonlinear,
Partial Differential Equation

This is name is a sequence of adjectives, each with a specific
meaning, that tells us something about the type of differential equation we are trying to solve.
The terms "Hyperbolic" or "Elliptic" describe the
mathematical form of the differential equation. Hyperbolic equations
describe the propagation of waves moving at some speed, such as water
or gravitational waves. Hence a disturbance at one place (say a pebble
falling into a pond, or two black holes colliding) is only felt at another
place later in time, when the waves reach that point. Elliptic equations
generally describe a function,
like Newton's gravitational potential, whose effects are felt throughout
space at one instant in time. This is the mathematical description of
Newton's "actionatadistance." The term
coupled tells us that we have a set of
equations that must be solved all at once; the unknown quantities appear
all mixed together in each equation. The term nonlinear means that the unknowns appear as
squares or higher powers, and the solution is
likely to be more difficult to solve. With nonlinear equations it will be more
difficult to assess whether we have found a general solution or not. The Einstein equations
have all of these properties.

Cosmic Censorship

At the center of a mathematical description of a black hole
discovered by the German mathematician, Karl Schwarzschild, is a
singularity, a point at which the laws of physics break down and space
becomes infinitely curved. Fortunately, the Schwarzschild spacetime
has another feature, an event horizon, out through which nothing can pass
(although things CAN pass IN through the horizon).
Because the singularity is "clothed" by the event horizon it cannot
affect the exterior spacetime and we do not need to worry about the
presence of the singularity. If a "naked"
singularity were to exist, one without a surrounding event
horizon, it would present great difficulties for physicists (see singularity). Cosmic censorship is a
hypothesis which proposes that the natural laws do not permit a naked
singularity to form, that these laws will always work to modestly
clothe a singularity with an event horizon.

Critical Circumference

This is the circumference below which an object of given mass would
collapse to form a black hole. This circumference depends on the mass of
the object in question. For example, a collapsing star equal to 10 suns
will have a critical circumference of 198 kilometers or 118 miles.
See also Schwarschild Radius.

(Partial) Differential Equation

Solutions to algebraic equations, like x^2 = 2, are just numbers. The
solutions to differential, or partial differential equations are
functions. The term "differential" describes one aspect of the
equation familiar from Calculus (invented by Newton to describe his theory
of gravitation). This relates the slope of an unknown function to its
value in some way. The most obvious equation of this type asks the
question "what function has a slope equal to its value at each point"
and the answer is y=e^x.

Electromagnetic Field

An electromagnetic field consists of energy oscillations associated with
electric and magnetic fields initially caused by the motions of electric
charges. The resulting waves propagate through space at the speed of light.
The famous British Scientist, James Clark Maxwell (187179),
formulated the mathematical laws governing the propagation of
electromagnetic waves in space. These laws provided the underpinning for
classical "electrodynamics." Certain problems in the manifestation of these
laws prompted Einstein to formulate and publish his theory of Special
Relativity in 1905.


Electromagnetic Radiation

Although initially created by moving charges, electromagnetic radiation
electromagnetic fields propogates
freely through the vacuum requiring no further influence from matter
to sustain it. Such radiation is both generated by and indicative of a
wide range of phenomena in the universe (since visible light, radio,
xray and infrared are all manifestations of electromagnetic
radiatioan). By measuring the intensity and wavelength
of such radiation, scientists can gain insights into the underlying
chemistry and physics.

Event horizon

The event horizon defines the boundary of a black hole behind which
nothing, not even light, can escape. Consider an event (a given position at
a given time) in
spacetime. Now imagine that light rays shoot out in all directions
from this event. If none of them can escape to an infinite distance then
that event is inside the event horizon. If any can escape, that event is
outside the event horizon.

General Solution

See Types of Calcuations and their Solutions.

Gravitational Lensing

Consider the example given under
spacetime curvature where
we describe two dimensional creatures living in a bedsheet.
In that example we saw how parallel light rays could be
caused to meet by passing on either side of a massive object.
This tells us that the curvature of
spacetime can focus light rays. In effect,
the curvature of spacetime acts on light somewhat like a giant
convex lens extending around the massive object.
Because of this effect, sensitive devices sometimes see two
images of an astronomical object. A real image, formed by
light rays travelling without significant deflection to Earth,
and another image which is formed by light rays that pass
near a massive object and then toward the earth. Although
this effect is very small, its significance is magnified by
the great distances involved.

Gravitational Radiation

The energy that is emitted by strong sources of gravitational waves, for example, certain
collapsing or colliding stars.

Gravitational Waves

Think about the example described under
spacetime curvature
in which we have two dimensional creatures living on the
surface of a bedsheet.
Now imagine that a physics professor grabs one end of the
bedsheet and begins to shake it violently up and down. This
will cause ripples to travel through the fabric. The imaginary
creatures within the bedsheet will not be able to see what is
happening, but they they will be able to measure the time
variation in the geometry of their space. The wave travelling
on the bedsheet is analagous to a gravitational wave in our universe, the
difference being that our universe exhibits three spatial
dimensions not two! Gravitational waves have never been observed directly,
but scientists hope to detect them soon with extremely sensitive
instruments now under construction.

Gravity (as the warping of spacetime)

See Spacetime Curvature

Light Year

A light year is the distance light can travel in a year.
Light travels at 186,282 miles per second, so one can see
that this is truly a gigantic distance. Yet in some respects
it is still quite small. The nearest star to our
sun is over four light years away, and the galaxy itself is
about 100,000 light years across.

"Naked" Singularity

A singularity from which the universe is
"unshielded" because there is no event
horizon. The physical consequences of "naked" singularities are hotly
debated among physicists.
To quote the renowned mathematician and physicist Roger Penrose:
It is sometimes said that if naked singularities do occur, then this would be
disastrous for physics.
I do not share this view. We already have the example of the big bang
singularity in the remote
past, which seems not to be avoidable. The "disaster" to physics occured right
at the beginning.
Surely the presence of naked singularities arising occasionally in collapse
under much more "controlled" circumstances would be the very reverse of a
disaster. The effects of such singular occurences
could then be accessible now. Theories of singularities would be open
to observational test. The initial mystery of creation, therefore, would no
longer be able to hide in the obscurity afforded by its supposed
uniqueness.
From Annals New York Academy of Sciences, Vol. 224 (1973).
See also Cosmic Censorship

Neutron Star

If a star's mass is too great, its nuclear matter will be compressed
beyond the limits given by a white dwarf.
The electrons and protons of the star's matter will combine to form
neutrons, and the star will in some cases possess regions that are
more dense than an atomic nucleus. In a sense, Neutron Stars are like
giant atomic nuclei  although the physics of so large an object as a
neutron star will have many important differences.

Nonlinear Process

In linear processes the output is directly proportional to its input. For
example, the pressure of a gas in a fixed volume is directly proportional
to its temperature. In nonlinear processes this direct proportionality is
lost.
Two basic types of equations, linear and nonlinear, describe such
processes mathematically. Linear equations, like
y=mx+b (the formula for a straight line)
are generally easy to solve, whereas nonlinear equations, such as
xy^3 + x^2 + (yx)^2 = 6
are much harder to crack, and the solutions, if they can be found at all,
can behave in unexpected ways.
Linear processes can be accounted for by the sum of their parts and are
easy to predict. Not so for nonlinear processes: they tend to be complex;
their outcomes can be difficult to predict and often display socalled
chaotic behavior, and the mathematical equations describing them can in
some cases be very hard to solve. This is especially so in the case of
general relativity.
The Einstein Equations contain thousands of terms in many variables, not
just x and y, and these terms are nonlinear. For all but the simplest
spacetimes it's impossible to solve them precisely using traditional, analytical methods.
However, weak spacetime curvature, e.g. near the earth, is close to linear:
the tidal pulls of the sun and moon add up to result in the oceans' tides.
But near a
black hole, Einstein's Equations predict that the curvature of spacetime is
highly nonlinear. At the center of the black hole, distance and time become
infinitely stretched! Furthermore, the nonlinearity of the equations can
lead to strange effects such as the formation of black holes where none
exist initially. Nonlinearity complicates life, but makes it more
interesting!

Redshift

Redshift is the lengthening of the wavelength of electromagnetic radiation
(or, equivalently, the shortening of its frequency). There are three types
of redshift.
 The (Relativistic) Doppler Effect

Named after its discoverer, Christian Doppler (180353), this is the change
in frequency that results when the emitter is travelling away from
the viewer, or when the viewer is travelling away from the
emitter (in special relativity this is really the same thing). However, if
the emitter and observer are moving towards each other (relatively,
of course!), the observed radiation's frequency will be increased or
blueshifted.
 Gravitational Redshift

As a result of the slowing down of time within a gravitational
field, light emitted from the surface of a planet or near the
surface of a black hole is reduced in frequency, or redshifted. In effect,
the light's observed energy is diminished as it battles to escape the grip
of gravity.
 Cosmological Redshift

Due to the expansion of
the universe, light may also be
redshifted as viewed by an observer. This is a result of two effects, one
resulting from special relativity, the other from general relativity.
Imagine the source and the viewer of the
light to be two dimensional creatures living on the surface of a balloon.
As their universe, the balloon, begins to inflate, any
two points on the surface of the balloon will acquire a relative speed
and exhibit corresponding doppler shifts. The second effect is a direct
result upon the wavelength. Imagine two ants crawling at the same speed
across the surface of the balloon, each along the same path. These
ants represent the endpoints of a wave of light propagating through
the expanding universe. One can readily see that the expansion of the
balloon will cause the distance between the ants to separate, and
lightwaves to be "stretched" or redshifted.

Reference frame

Consider a astronaut travelling in a starship, and someone sitting on the earth.
According to the special theory of relativity, each perceives himself or herself
to be at rest while the other is perceived to be moving. Each of these
observers is said to be in his or her own "reference frame." Any two observers
travelling with the same velocity (the same speed and direction) are
said to be in the same reference frame. Two people travelling with
a different speed and direction are said to be in different reference
frames.

Singularity

In the center of the mathematical model of a
black hole is a singularity which has the shape of
a point (or a ring if the hole is rotating), at which
the curvature of spacetime becomes infinitely large.
A singularity represents a great difficulty for theoreticians
because it is impossible to predict how a singularity will
affect objects in its causal future. If
cosmic censorship is true, then this needn't cause any trouble
because they will only be found inside event
horizons.

Schwarzschild Radius

The Schwarzschild radius is the radius at which
the event horizon
of a Critical Circumference.

Spacetime

Space has three dimensions. However, the theory of relativity
predicts that time, like space, is a dimension. In order to
describe a four dimensional universe which has three spatial
dimensions and one time dimension the word "spacetime" was
coined. Each point in spacetime is called an event.

Spacetime Curvature

Imagine that the universe has two spatial dimensions instead
of three, and that there are flat creatures living on its surface.
Now imagine that the surface they are living on is subject to
deformations, something like a bedsheet. The creatures living
on the bedsheet can only see length and depth, they can only see
within the bedsheet. They cannot
even imagine the concept of height.
Now imagine that the bedsheet is draped over a basketball,
and the creatures are very small. If the creatures attempt
to travel along a straight line within the fabric of the
bedsheet they will be deflected by the presence of the
basketball. Although on their very small scale the bedsheet
appears to be flat, their path through it will be altered by
the presence of the basketball, distorting the geometry of
their world.
Because of this, if we have three of these creatures travelling
along parallel straight lines, and the middle creature's
path takes him across the top of the ball, the course of the
creatures to his right and left will be deflected inward.
Because of curvature effects, these three initially parallel
paths will meet. A similar effect can occur in spacetime.
If two light rays, initially parallel, pass on either side
of a black hole their paths will converge.
Gravity causes spacetime to curve, and this curvature in turn
affects the motion of objects in spacetime in much the same
way that the curvature of the bedsheet affects the paths of
motion of creatures wandering within it.
To understand how these distortions create gravity you need to
think of parallel worldlines. These
worldlines are not drawn on a flat page, but are drawn on a
curved surface. Because the surface is curved the intially
parallel lines can be drawn together. What we perceive as
gravity is the deflection in the path of worldlines caused
by their being traced on a curved surface.
 Speed of Light

Light travels at a speed of 186,282 miles per second in
vacuum from the point of view of a nearby observer. Because of the
effects of general relativity the speed of light near a massive object
will appear slower to a distant observer, and this effect has been
confirmed in experiments.
The speed of light is the theoretical limit to the speed of any
particle in the universe. More fundamentally, no cause can result in
an effect that requires travel faster than light. For example, I
cannot affect what is going on 3 light years
away but only 2 years in the future. I can, however, affect what is
going on 3 light years away but 4 light years in the future.


Stationary Black Bole

"Stationary" refers to a timeindependent mathematical description of
a black hole (not its rotation  a rotating black hole can still be
"stationary.") To understand
what this means, consider the physical system represented by a perfectly
symmetrical top spinning without friction. Every detail of this system
remains the same as time goes by, and thus we can say the system is
"stationary, " even though it is spinning.
By definition, a stationary black hole "sits" alone in space.
It interacts with no other matter or
gravitational radiation.

Supernova(e)

A supernova is an exploding star. Such an explosion occurs in our galaxy at
a rate of about one every 30 years. Its causes are not precisely
known, but the violent movement of matter within the star may
produce a significant amount of
gravitational radiation. It is
thought that supernovae produce
pulsars.


Types of Calculations and their Solutions

Take the simple equation x^2 = 2. The general
solution to this equation is x = +/ square root [2]. Both
x = +square root [2] and x = square root [2] are the analytic
solutions to this
equation, whereas x=1.414 is a numerical solution.
However, the types of equations in relativity are much more complex than
the example above; they deal with functions rather than merely numbers.
An equation which requires a function for its solution is called a
differential equation.

White Dwarf

When a star has burned most of its nuclear fuel it can no
longer provide the heat and pressure necessary to prevent its
gravitational collapse. However, there is still another effect
which can prevent the forming of a black hole.
The effect arises from quantum physics which tells us two things
about the electrons in the stellar material. The Pauli
Exclusion Principle tells us that no two electrons can exist in
the same place in space. Quantum mechanics restricts the
number of places that an electron can be in to a finite number
surrounding each atomic nucleus. As matter in a collapsing star
becomes more and more tightly packed these laws manifest as an outward
pressure that resists contraction due to gravity.
If the mass of a collapsing star exceeds a certain critical value it may
contract into a yet denser object, a neutron
star. And if the collapsing star's mass is sufficiently large, the
inward pull of gravity overcomes all outward pressures, causing the star to
collapse into a black hole.

Worldline

Imagine that time is like a spatial dimension, and it is plotted on
the yaxis of a sheet of graph paper before us. Let the xaxis be one
of the three spatial dimensions of our world. A ball travelling to
the right would be depicted as a line with a positive slope. A ball
at rest would be plotted as a straight vertical line. Because
physicists do think of time as a dimension similar to a spatial
dimension, they draw diagrams, like the one described above, to
illustrate the trajectory of a particle which they term that
particle's "worldline."
An important point to realize is that one can always rotate the paper
and redraw the axes so that one of the balls appears to be sitting
still (its worldline will simply be a vertical line). The choice of
axes is really somewhat arbitrary, but the fact that the balls are
moving apart is evident no matter how the coordinate axes are drawn.