If the core remnant has a mass greater than 3 solar masses, then not even the super-compressed degenerate neutrons can hold the core up against its own gravity. Gravity finally wins and compresses everything to a mathematical point at the center. The point mass is a black hole. Only the most massive, very rare stars (greater than 10 solar masses) will form a black hole when they die. As the core implodes it briefly makes a neutron star for just long enough to produce the supernova explosion.
Since the mass M is in the top of the fraction, the escape velocity is greater for larger masses. The escape velocity is smaller for larger distances from the center because the distance is in the bottom of the fraction. The Earth's escape velocity at its surface is about 11 kilometers/second and the Sun's surface escape velocity is about 620 kilometers/second. White dwarfs and neutron stars have very large surface escape velocities because they have roughly the mass of the Sun packed into an incredibly small volume. A solar mass white dwarf has a radius of only 8,800 kilometers, so its surface escape velocity is about 5500 kilometers/second. A solar mass neutron star would have a radius of just 17 kilometers, so its surface escape velocity would be an incredible 125,000 kilometers/second! (Real neutron stars have masses above 1.4 solar masses and smaller radii, so their escape velocities are even larger!)
A black hole probably has no surface, so astronomers use the distance at which the escape velocity equals the speed of light for the size of the black hole. This distance is called the event horizon because no messages of events (via electromagnetic radiation or anything else) happening within that distance of the point mass make it to the outside. The region within the event horizon is black. Rearranging the formula above for the escape velocity and putting in the speed of light c for the escape velocity, you find the event horizon is at a distance of
Black holes have been portrayed as cosmic vacuum cleaners in Hollywood films, sucking up everything around them. Black holes are dangerous only if something gets too close to them. Because all of their mass is compressed to a point, it is possible to get very close to them and still be outside all of the mass. Recall that gravity is an inverse square law force, so at very small distances, the strength of gravity around a point mass becomes very large. But objects far enough away will not sense anything unusual. If the Sun were replaced by a black hole of the same mass, the orbits of the planets would remain unchanged (it would be much colder in the solar system, though!).
For very strong gravitational fields, Newton's description of gravity becomes inadequate. Einstein's theory of General Relativity must be used to describe the gravitational effects. Einstein found that gravity is not a force in the usual Newtonian description of force. Gravity is a result of a warping or distortion of space-time around a massive object. The stronger the gravity is, the more the space-time is warped or curved.
What is space-time? Time is not completely separate from and independent of space as you would ordinarily assume. In his Special Relativity theory, Einstein assumed that the speed of light will be measured to be the same by any two observers regardless of their velocity relative to each other---this assumption has now been shown to be correct in many experiments. To get the same value of the speed (= distance/time) of light, the two observers moving with respect to each other would not only disagree on the distance the light travelled as Newton said, they would also disagree on the time it took.
Time and space are connected to make four-dimensional space-time (three dimensions for space and one dimension for time). This is not that strange---we often define distances by the time it takes light to travel between two points. For example, one light year is the distance light will travel in a year. To talk about an event, you will usually tell where (in space) and when (in time) it happened. The event happened in space-time.
Another consequence of Special Relativity is that nothing can travel faster than the speed of light. Any object with mass moving near the speed of light would experience an increase in its mass. That mass would approach infinity as it reached light speed and would, therefore, require an infinite amount of energy to accelerate it to light speed. The fastest possible speed any form of information or force could operate is at the speed of light.
Newton's law of gravity seemed to imply that the force of gravity would instantly change between two objects if one was moved---Newton's gravity had infinite speed. Einstein proposed in his General Relativity theory that what is called gravity is really the result of curved space-time.
The Earth does not orbit the Sun because the Sun is pulling on it. The Earth is simply following the shortest path (a geodesic) in four-dimensional space-time.
If you have ever taken a long flight you probably already know that the shortest distance between two cities is not a straight line. Non-stop flights from the United States to Europe fly over parts of Greenland. On a flat map the plane's flight path looks curved, but on a globe, that path is the shortest one! Light travels along a geodesic path between two points in space-time. Far from any gravity source, the shortest distance is a straight line in three-dimensional space. Near a massive object, the shortest distance is curved in three-dimensional space. Stephen Hawking gives the nice analogy that what we see is like the curved motion of a shadow on the ground from a plane flying in a straight line over hilly terrain.
Within the event horizon space is so curved that any light emitted is bent back to the point mass. Karl Schwarzschild worked out the equations in General Relativity for a non-rotating black hole and found that the light rays within a certain distance of the point mass would be bent back to the point mass. The derived distance is the same as the event horizon value above. The event horizon is sometimes called the Schwarzschild radius in his honor.
If Einstein's theory of General Relavity is an accurate description of gravity, then there are some bizarre consequences. In this section the implications of General Relativity's claim that gravity is the warping of space-time will be explored in a prediction-observation format. A scientific theory must make testable predictions which are tested through observations and experiments.
Observation: During a solar eclipse we see that the stars along the same line of sight as the Sun are shifted ``outward''. This is because the light from the star behind the Sun is bent toward the Sun and toward the Earth. The light comes from a direction that is different from where the star really is. But wouldn't Newton's law of gravity and the regulst from Einstein's Special Relativity theory that E = mc2 predict light deflection too? Yes, but only half as much. General Relativity says that time is also stretched so the deflection is twice as great.
Observation: The light from quasars is observed to be bent by gravitational lenses produced by galaxies between us and the quasars. It is possible to see two or more identical images of the same background quasar. In some cases the light from background quasars or galaxies can be warped to form rings. Since the amount of warping depends on the mass of the foreground galaxy, you can estimate the total mass of the foreground galaxy.
The Einstein Cross is formed by a foreground galaxy lensing the light from a background quasar into 4 images.
Below is a picture from the Hubble Space Telescope showing the lensing of a background galaxy by a cluster of galaxies in front. The distorted blue arcs visible around the center of the picture are the lensed background galaxy. If you select the image, an enlarged version will appear (courtesy of the Space Telescope Science Institute).
Observation: a) Clocks on planes high above the ground run faster than those on the ground. The effect is small since the Earth's mass is small, so atomic clocks must be used to detect the difference. b) The Global Positioning Satellite (GPS) system must compensate for General Relativity effects or the positions it gives for locations would be significantly off.
This is a consequence of time dilation. Suppose person A on the massive object decides to send light of a specific frequency f to person B all of the time. So every second, f wave crests leave person A. The same wave crests are received by person B in an interval of time interval of (1+z) seconds. He receives the waves at a frequency of f/(1+z). Remember that the speed of light c = (the frequency f) × (the wavelength l). If the frequency is reduced by (1+z) times, the wavelength must INcrease by (1+z) times: lat B = (1+z) × lat A. In the doppler effect, this lengthening of the wavelength is called a redshift. For gravity, the effect is called a gravitational redshift.
Observation: spectral lines from the top layer of white dwarfs are significantly shifted by an amount predicted for compact solar-mass objects. The white dwarf must be in a binary system with a main sequence companion so how much of the shift is from the doppler effect and how much is from the gravitational redshift can be determined. Inside a black hole's event horizon, light is red-shifted to an infinitely long wavelength.
A well-done site that has further information about General Relativity and the tests is at the NCSA Online Expo.
To make rapidly varying X-rays, the unseen companion must be small! The fluctuation timescale gives us the maximum possible diameter of the object. Since the speed of light is finite, it takes a given amount of time for light to travel across the object. The time it takes for any interaction to occur is diameter/speed, where the speed can be up to the speed of light if the object could somehow brighten everywhere simultaneously. The diameter = (time interval) × speed. The maximum diameter possible is for a speed equal to the speed of light.
| accretion disk | degenerate gas | electron degeneracy pressure |
|---|---|---|
| event horizon | General Relativity | gravitational lens |
| gravitational redshift | lighthouse model | neutron degeneracy pressure |
| nova | pulsar | Schwarzschild radius |
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Stellar Nucleosynthesis and White Dwarf - Neutron Star sections
last update: 13 October 1998
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