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29:50 Modern Astronomy
Fall 2002
Lecture 22 ...October 30, 2002
Black Holes: What they are and whether they exist

Let's pick up where we left off last time. The last of the characteristics of massive star ``death'' was the phenomenon of Pollution of the Interstellar Medium. I can describe this as follows.

When the core collapse explosion occurs, it blows the massive star apart, leaving only the neutron star behind. The overlying layers of the star, which have been synthesized into heavy elements, are blown out into the interstellar medium. This is the material from which new stars form. We believe this has been going on for the full 13 billion year lifetime of the galaxy.

Question: What observations can you imagine that would verify (or falsify) this statement?

Interlude: A future spacecraft mission: the Interstellar Probe
Let's look at the expanded solar system Diagram of the "Heliosphere"

tex2html_wrap_inline85 In describing massive star evolution, I described the neutron star as a ``hard floor'' that stopped the core collapse. The idea is that a neutron fluid holds itself up by ``neutron degeneracy'', and that it is very stiff and resistant to compression. Nonetheless, very stiff does not mean that it can resist any force. We can repeat the calculation of Chandrasekhar for a massive object holding itself up by neutron degeneracy, and again get a mass-radius plot. This is given in Figure 20.16 of your textbook.
tex2html_wrap_inline87 Figure 20.16, Mass-radius relation for neutron stars.
Question: What is the implication of this diagram.

tex2html_wrap_inline85 This leads us to the 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.

tex2html_wrap_inline85 The point of today's 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.

tex2html_wrap_inline93 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

equation13

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.

tex2html_wrap_inline103 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 (the technical term is the eccentricity) tex2html_wrap_inline105 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

equation22

so it is only a factor of tex2html_wrap_inline107 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.

tex2html_wrap_inline85 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

equation22

The condition for a black hole is that tex2html_wrap_inline115 , so

equation36

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

equation40

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, tex2html_wrap_inline117 .

equation47

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 tex2html_wrap_inline119 kg. All of the rest is just physical constants.

equation54

equation61

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.

tex2html_wrap_inline93 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.

tex2html_wrap_inline85 . 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.
tex2html_wrap_inline125 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.
tex2html_wrap_inline125 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.
tex2html_wrap_inline125 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.

tex2html_wrap_inline85 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.

Visions of Blackholes
Artist's conception of Cygnus X-1
Artist's conception of the Vicinity of a Black Hole
Artist's conception of the Vicinity of a Black Hole




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Steve Spangler
Wed Oct 30 10:41:01 CST 2002