29:50 Modern Astronomy
Fall 1999
Lecture 19 ...October 8, 1999
Black Holes: What they are and whether they exist

Incipatur
Zeilek transparency of Earth, white dwarf, and neutron stars

The last time I ended up our description of stellar evolution with a categorization of the end products of stellar evolution. For ``low mass'' stars with about the mass of the Sun, the end of the line is a white dwarf star. A fantastic enough object with the mass of a star crammed into a sphere about the size of the Earth. The mean density is several hundred times that of water. According to theories, white dwarfs can only be 1.4 solar masses or smaller.

For a much more massive star (by massive I mean > 8 solar masses) the end involves a huge explosion called a supernova, the end product of which is a neutron star. A neutron star would seem to be the most fantastic thing one could imagine. A solar mass of material compressed to the density of an atomic nucleus times that of water. The radius of a typical neutron star is about 10 kilometers.

In spite of their incredible nature, they exist! There are now over 1000 cataloged radio pulsars, which are unambiguously identified as rotating neutron stars. There are about 100 other objects such as x-ray pulsars which are also securely known to be neutron stars. It is furthermore completely certain that we have only skimmed the surface of the total number of neutron stars in the galaxy. Neutron stars, despite their incredible properties, are common objects.

An obvious question to then ask is: are there objects more extreme than neutron stars? Are there conditions under which matter could be compressed (stepped on) to such a degree that something even stranger than a neutron star would be produced? The answer to this is yes, a sort of object called a Black Hole. They were theoretically predicted decades ago and have been the object of extensive astrophysical speculation. In the last couple of decades, and particularly the last 10 years, it has become completely clear that they exist. Furthermore, there are even two varieties of them: (1) BIG ONES. (2) little ones.

The point of todays lecture will be to first describe the physical nature of black holes, and then discuss the observational evidence of whether they exist or not.

There are two ways of describing what black holes are. The first is physically easy to grasp, and gives you the right equation for the conditions under which a black hole forms. However, it doesn't give you the really correct physical picture of what is going on. The physically correct picture of what is going on is a bit advanced for this class, but you can get some feel for what is involved.

Black Holes and the escape speed.
Let's talk about the speed of escape from an object. We have seen that a small object of mass m will orbit a large object of mass M at a distance r from the center of the large object at a speed given by

This formula would work perfectly well for an orbit right above the surface of the planet, with the radius of the planet being r. Obviously this would have to be an airless planet so air friction would not be important. On Earth there would also be problems from clotheslines, hang-gliders and high-jumping dogs.

Orbital and escape speeds from a planet.
Now what happen if you fire your rocket engines and speed up above the orbital speed. Newton's Laws say that you swing into a more elliptic orbit, with the same perigee (closest approach to the Earth), but a larger apogee. The faster you speed up, the more elliptical the orbit, and the furthest the apogee.

The Limiting Case is when the ellipticity and the apogee goes to infinity. In this case, you have escaped from the planet (or object) you were orbiting. Newtonian physics gives us an expression for this speed. The formula is very similar to that of circular orbit

so it is only a factor of greater than the circular orbit equation. This equation is verified every time we send a probe into deep space, such as the Mars Pathfinder spacecraft which is supposed to be launched today.

Now, as I have mentioned in class before, modern physics says that the ultimate speed limit for anything having mass or carrying energy is the speed of light c , 300,000 kilometers per second or 186,000 miles per second. If an object were so extreme as to have the speed of light as an escape speed. Any object which fell onto the surface would be unable to ever escape.

Let's work out the conditions for this to happen. Let's assume we have an object of mass M. How much would we have to compress it to make it a black hole? Well, we have from above

The condition for a black hole is that , so

Let's square things to make things easier to deal with

Now let's rearrange the equation so that the thing we want to know, the radius in which we have to cram the object, is on the left hand. We also use a new symbol for this, .

This radius is known in physics as the Schwarzschild Radius, after the German astronomer Karl Schwarzschild who first derived it.

Well, this is a nice formula, but let's see what it tells us. Let's say we wanted to make the Earth a Black Hole. How much would we have to compress the Earth. What would be the radius of the sphere we would have to put it in to make it a Black Hole?

Look up in the back of your book the mass of the Earth, which is kg. All of the rest is just physical constants.

This is a radius of 0.9 centimeters! This is the size of a marble! This gives an idea of the extreme conditions under which matter must be subjected to form a black hole. You might think that this would be totally impossible, but as we have seen, stellar evolution is capable of producing pretty strange objects.

General Relativistic Black Holes Before going on to a discussion of how one would search for Black Holes, it is worth talking about the second way of looking at Black Holes. This explanation is the one which modern physicists would say is the ``physically correct'' (another form of PC!) way of explaining these objects. Some go as far as saying that the description I have given you is not a correct description of things, and it is only a coincidence that it gives the right answer for the Schwarzschild radius.

The correct description for Black Holes comes from a theory of gravitation called General Relativity. General Relativity was developed by Albert Einstein early in this century, although the crux of the idea had been thought of by the famous German physicist and mathematician Karl Friedrich Gauss in the 19th century. The goal of General Relativity is to understand what causes gravity, rather than just being a mathematical description as in the description we have had up to now.

. By now you are used to the idea of objects moving in space, and that the time goes on as well. Einstein thought of generalizing all this to have objects moving in spacetime. You can think of spacetime as just a plot in which the spatial coordinate (x,y,z) are some of the axes, and time is the other dimension.

If an object moves in one dimension, it moves in a two dimensional spacetime.
Figure 20-22 from textbook.

If an object moves in two spatial dimensions (like the top of this table) it moves in a three dimensional spacetime.
Diagram of three dimensional spacetime

Obviously, generally motion of objects occurs in a four dimensional spacetime including the three spatial dimensions and the fourth dimension of time.

There are two ingredients of General Relativity. The first is that objects move between two points in spacetime A and B on the shortest path between the two points. The fancy mathematical term for this is a geodesic.
Question for Audience: What is the shortest path between A and B in this 3D spacetime?

So far so good, but where is the effect of gravity? That brings up the second main point of General Relativity, which is its most fascinating. The presence of mass warps or bends spacetime, so that the geodesics are no longer straight lines, but instead more complicated, curved paths. The larger the amount of mass in the smaller the region, the stronger the warping or bending of spacetime is. The central statement of General Relativity is the Einstein Field Equation which directly relates the curvature of spacetime to the presence of mass.

Motion along geodesics in curved spaces can look strange. The classical (and best illustration) involves motion along a two dimensional space embedded in a three dimensional one. This is characterized by a sheet of paper or the latex sheet here in the demonstration.
Demonstration of sheet.
Particularly note the overhead view which gives us a truly 2D view of things. If the sheet is is uncurved, or flat, the geodesics are straight lines and things are simple.

If however, the two dimensional surface has curvature, the geodesics are not straight lines in two dimensions. The most familiar illustration of this the paths taken by airplanes on transcontinental flights. Plotted on a two dimensional map, it looks they go way north out of their way. However, you can be sure the airlines are quite interested in flying the minimum distance (with the minimum fuel use) possible. They go on great circle routes which are geodesics on the surface of a sphere.

Let's do a little General Relativistic dynamics here with this experiment. As the ball moves across this sheet, it takes the path of minimum available distance from point A to point B. If there is no curvature, this is a straight path, and everything is simple.

If we now curve the space in which it is moving, it is still following a geodesic. However, as viewed things in a two dimensional space (as it appears on the TV screen) the trajectory appears more complicated and one would interpret its motion as due to a force between the large mass and the small one. I would recommend further reading in the book, beginning on p454, for a description of the curvature of spacetime and Black Holes.

Now above I said that the greater the mass concentrated in a smaller region, the larger the warping or curvature of spacetime. Einstein's theory of General Relativity predicts that for a sufficiently large concentration of mass in a sufficiently small volume, it is possible for the curvature to become infinite. Infinite curvature can be visualized as an infinitely deep hole, or a rip in the metric.

The conditions for this infinite curvature to occur are the same as for the Schwarzschild radius described above. Thus Black Holes can be viewed as a puncture or abyss in spacetime.

Next time, we will go on to a discussion of whether these remarkable objects exist in the sky.

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