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.
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.
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.
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.
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
- f = federal tax
- s = 0.20 * (i-f)
- f = 0.10 * (i-s)
(Where * means "multiplied by.") - f = 0.10 * (i-s)
Cosmic Censorship
See also Schwarschild Radius.
(Partial) Differential Equation
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.
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 Waves
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.
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
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!
- 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.
Singularity
Schwarzschild Radius
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.
By definition, a stationary black hole "sits" alone in space. It interacts with no other matter or gravitational radiation.
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.
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.
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.
