BLACK HOLES

In 1916 while serving as an artillary officer on the Eastern (Russian) front for the German Army, Karl Schwartzschild received the joural in which Einstein's paper setting out General Relativity was published. He began to work on it, and published the first exact solution of the equations in 1916, a solution which is now called the Schwartzschild metric. Defining the radius as the circumference divided by 2 Pi, his solution had some strange features.

The time, as usual for Einstein's ideas, was somthing that depended on radius. In particular, the "distance" in time (ie that a body would see as its own proper time) depended on that radius as 1- 2G Mass/((vel light)^2 radius) of 1-2GM/(c^2 r). This is as expected if the second term is small, but he showed it was exact. But this means that at a non-zero value for the radius (for the circumference) this term went to zero. The amount of proper time for a given amount of "coordinate time" the proper time went to zero. This "singularity" as they called it, was mysterious. Of course in special relativity, one can have proper distances which are zero, without anything being weird. The length of a light ray is always zero. But they had a hard time thinking in this way. Until the end of his life, Einstein thought that this was unphysical, that there was something in nature that would prevent this "singularity" from appearing. After all Schwartzschild's soluton was for space without any matter. The presense of matter would alter the solution. If matter could not become too compact, then this singularity would never appear. The spatial distances were also strange. The radial distance increased by the same factor. Ie for a small difference in the circumference, the distance between those two circles would be bigger than the "radius" by a factor of 1 over the square root of the above factor (1-2M/c^2 r). Ie, the distance between two very nearby circles around the centre where the circumference differ by some small amount dr, is dr/sqrt( 1-2M/rc^2). Note that this becomes large as r gets close to (2M/c^2)

The key feature of a black hole is the horizon. The is that surface out of which nothing can travel. Thus the black hole really does represent a region disconnected from the outside world. While stuff can fall in, nothing can come out. It would have to travel faster than light to get out.

Do black holes form? Stars want to collapse. The gravity wants athe matter to travel into a more and more compact form. For an object like the earth, it is the electrons which hold it up. There is a quantum principle called the Pauli exclusion principle. This means that no two electrons can have exactly the same properties (which includes their location). This means that to have the same position, one of the electrons needs to have a higher energy than the other. The more electrons you try to squeeze into a space, the higher the energy you need. This means that to crowd them into a space (like the space taken up by the earth) you need to also give them more energy. If the earth tried to collapse the increase in energy would mean that you have to actively push it together. The gravitation is not sufficient to do so.

If we look at an object like the mass of the sun, if it is hot, that heat can help to hold it up. The pressure due to the heat can help hold it up, and is primary reason stars exist. Due to the fusion of the Hydrogen nuclei inside the stars into He, an immense amount of energy is released, which heats the star. That pressure due to the heat holds it up. However, eventually, that heat radiates away (the stars shine) the stars cool, and must get smaller. For a mass of the order of the sun, the electron pressure eventually becomes enough to hold up the matter, and one gets a white dwarf. These look very hot, but there is actually very little energy inside them to hold them up, and they will eventually cool down and just sit there. White dwarfs of the order of the mass of the sun have a radius of a few thousand kilometers (lower mass has a larger radius. There is less gravity, and thus the electron pressure can hold it out at a larger size.)

Subrahmanyan Chandrasekhar, on his boat trip to England to start his graduate studies, calculated the effect that special relativity would have on the pressure due to the electrons not being able to be in the same state. He showed that as the electron's velocity came close to the velocity of light, the pressure would be insufficient to hold up the mass of the star and the star would have to collapse. The maximum mass a star could have and still be held up by this so called electron degeneracy pressure (pressue due the electrons not being able to be in the same state) was about 1.4 times the mass of the sun.

For a larger size body, the electrons would in fact get forced into the protons, which would convert the protons into neutrons. Neutrons have the same behaviour as electrons, that they cannot be in the same state. Since they have a far greater mass, their velocity is not as near the velocity of light, and they can hold up the star for a little increase in mass. Between about 1.4 solar mass and 3 solar mass, one can have a neutron star-- a star made up of neutrons whose degeneracy pressure holds up the star. They have a radius of about 10km. Above about 3solar masses, the neutrons become relativistic, and again they cannot provide enough pressure. Since the radius of a black hole of 3 solar mass is about 9km, (3km for 1 solar mass), by the time the star is of the order of 3 solar mass, the star is of the same order as the black hole. Any larger and the neutron pressure cannot hold it up and it must form a black hole.

Since we know that stars of mass over 100 times the solar mass exist, their end must be a black hole. The immense pressure in the center causes the thermonuclear process to proceed very fast, burning up the fuel. Eventually the core of the star cools enough that it collapses. If it is small enough it might end up as a neutron star, but for a large star, it collapses into a black hole.

In the course of the collapse, the electrons being forced down the throat of the protons also releases neutrinos, tiny, very weakly interacting particles. But so may are created that much of the outer mass of the star is blown away. The blackholes formed by such stellar collapse thus seem to tend to have masses of about 10 solar mass or less. There should be lots around, but because they do not shine-- after all nothing can get out-- they are very hard to see. One tends to detect them either if all the mass blown away during their formation sticks around and forms an accretion disk around the black hole, and glows.

In the centers of galaxies, due to all of the gas that accumulated there in the formation of the galaxy, one also tends to get very large black holes ( 100,000 times to the mass of the sun to 10 billion times the mass of our sun-- the diameter of our black hole would be a bit smaller than the orbit of Mercury. The largest black hole found would be a bit larger than the whole the solar system. These can be seen due to the stars of the galaxy rotating around them. For example, over the past 30 years, radio and infrared observations of the center of our galaxy show that there are a bunch of stars there which are orbit something that is very massive (about 4 million times the mass of the sun) and is dark. Orbit of star about blackhole

One of the interesting features of the light deflection effects is that near a black hole the deflection can become very large-- even more 180 degrees, so that you could see yourself (rather distored) in the light that goes around the black hole.

If the light is aimed at the black hole toward a radius less than 5/2 of the radius of the black hole, the light ray will go into the black hole. If at exactly 5/2, it will orbit the black hole an infinite number of times. If outside that, it will be deflected by less.

This is a simulation of a black hole. The different coloured arrows show the outer image where the light is deflected by the smaller angle, and the other closer same coloured arrow points to the second image of that same star.

The deflection of the light rays for the first and second images of three different stars, one with almost 90 degree deflection.

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Copyright W Unruh