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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 X-ray 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.


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 "axis-symmetry."

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 13-20 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 x-ray 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 * (i-f)
f = 0.10 * (i-s)
(Where * means "multiplied by.")

Coupled, Hyperbolic-Elliptic, 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 "action-at-a-distance." 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 (1871-79), 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, x-ray 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 non-linear, 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 + (y-x)^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 so-called 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 non-linear. 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 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 (1803-53), 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.


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.


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 time-independent 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.


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.


Imagine that time is like a spatial dimension, and it is plotted on the y-axis of a sheet of graph paper before us. Let the x-axis 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.

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NCSA. Last modified, May 28th, 1995.