Relativity and Black Holes : The Beginning Becomes the End Becomes the Beginning : A study of cosmological birth and death

This essay will look at two parts of astronomical history; general relativity and the research into the phenomena of black holes. First will be a history of general relativity (GR) and a discussion of its impacts on scientific thought, then the history of black hole research and the specific interactions between GR theory and black holes.

History of General Relativity

Late in the sixteenth century, Galileo demonstrated that if one rolled two weighted balls down a declined plane, the balls would accelerate at the same rate, seemingly no matter what their weight. This fact allowed Newton to have a solid foundation for his laws of motion. In 1684 Newton was in contact with Edmund Halley who was interested in the problem of orbital motion. From this encounter, Newton went on to describe his theory of gravitation: every body attracts every other body with a force that is proportional to the mass of each body. Also, the further apart the bodies are, the weaker the gravitational force between them. This solved many problems for the cosmologists and astronomers, but created or highlighted others. For instance, Galileo and Newton didn't believe in a preferred state of rest as Aristotle did; Newton's first law of motion states that the real effect of a force is to change the speed of a body, not just to set it in motion. This assumes that all matter is constantly in motion, giving us a lack of an absolute standard of rest with which to make accurate measurements. It thus becomes impossible to determine whether two events that took place at different times occurred in the same position in space. Another problem was discovered with accurate measurements for observation made on objects traveling near or at the speed of light.

Newton also believed in absolute time, a concept widely accepted until the advent of instrumentation such as the atomic clock allowed the precision necessary to record the minute differences that pose a problem to this theory. However, all was reasonably well in the field of physics until they year 1887. In that year, the Michelson-Morley experiment (Fig. 1), named after the American physicist Albert Michelson and the American chemist Edward Williams Morley, was performed. It was an attempt to determine the rate of the motion of the earth through the ether, a hypothetical substance that was thought to transmit electromagnetic radiation, including light, and was assumed to permeate all space. The hypothesis was this: If the sun is at absolute rest in space, then the earth must have a constant velocity of 29 km/sec, caused by its revolution about the sun; if the sun and the entire solar system are moving through space, however, the constantly changing direction of the earth's orbital velocity will cause this value of the earth's motion to be added to the velocity of the sun at certain times of the year and subtracted from it at others. The result of the experiment was entirely unexpected and inexplicable; the apparent velocity of the earth through this hypothetical ether was zero at all times of the year. What the Michelson-Morley experiment actually measured was the velocity of light through space in two different directions. If a ray of light is moving through space at 300,000 km/sec, and an observer is moving in the same direction at 29 km/sec, then the light should move past the observer at the rate of 299,971 km/sec; if the observer is moving in the opposite direction, the light should move past the observer at 300,029 km/sec. It was this difference that the Michelson-Morley experiment failed to detect.

Figure 1: Michelson-Morley Experiment¹

Special Relativity

In 1905, Einstein published the first of two papers which dealt with the specific problems posed by the Michelson-Morley experiment. Einstein solved the problem of absolute motion by denying that it existed at all; he postulated that no particular object in the universe is suitable as an absolute frame of reference that is at rest with respect to space. Instead, any object chosen is a suitable frame of reference, and any motion can be measured in respect to that point. As Encarta Encyclopedia explains "...it is equally correct to say that a train moves past the station, or that the station moves past the train. This example is not as unreasonable as it seems at first sight, for the station is also moving, due to the motion of the earth on its axis and its revolution around the sun. All motion is relative, according to Einstein."². This was not necessarily a revolutionary idea, Newton had made a similar observation earlier, but failed to bring the idea to completion.

Einstein stated that the relative rate of motion between any observer and any ray of light is always the same, 300,000 km/sec, and thus two observers, moving relative to one another even at a speed of 160,000 km/sec, each measuring the velocity of the same ray of light, would both find it to be moving at 300,000 km/sec, and this apparently anomalous result was proved by the Michelson-Morley experiment. The fundamental hypothesis on which Einstein's theory was based was the nonexistence of absolute rest in the universe. Einstein postulated that two observers moving relative to one another at a constant velocity would observe identically the phenomena of nature. One of these observers, however, might record two events on distant stars as having occurred simultaneously, while the other observer would find that one had occurred before the other. In other words, it is not possible to specify uniquely the time when an event happens without reference to the place where it happens. Every particle or object in the universe is described by a "world line" that describes its position in time and space. If two or more world lines intersect, an event or occurrence takes place. The "distance" or "interval" between any two events can be accurately described by means of a combination of space and time, but not by either of these separately. The space-time of four dimensions (three for space and one for time) in which all events in the universe occur is called the space-time continuum.

General Relativity

In 1915 Einstein developed the theory of general relativity in which he considered the notion that objects accelerated with respect to one another. He developed this theory to explain apparent conflicts between the laws of relativity and the law of gravity. To resolve these conflicts he developed an entirely new approach to the concept of gravity, based on the principle of equivalence.

The principle of equivalence holds that forces produced by gravity are in every way equivalent to forces produced by acceleration, so that it is theoretically impossible to distinguish between gravitational and accelerational forces by experiment. In the theory of special relativity, Einstein had stated that a person in a closed car rolling on an absolutely smooth railroad track could not determine by any conceivable experiment whether he was at rest or in uniform motion. In general relativity he stated that if the car were speeded up or slowed down or driven around a curve, the occupant could not tell whether the forces so produced were due to gravitation or whether they were acceleration forces brought into play by pressure on the accelerator or on the brake or by turning the car sharply to the right or left.

Acceleration is defined as the rate of change of velocity. Consider an astronaut standing in a stationary rocket. Because of gravity his or her feet are pressed against the floor of the rocket with a force equal to the person's weight, w. If the same rocket is in outer space, far from any other object and not influenced by gravity, the astronaut is again being pressed against the floor if the rocket is accelerating, and if the acceleration is 9.8 m/sec2 (the acceleration of gravity at the surface of the earth), the force with which the astronaut is pressed against the floor is again equal to w. Without looking out of the window, the astronaut would have no way of telling whether the rocket was at rest on the earth or accelerating in outer space. The force due to acceleration is in no way distinguishable from the force due to gravity. According to Einstein's theory, Newton's law of gravitation is an unnecessary hypothesis; Einstein attributes all forces, both gravitational and those associated with acceleration, to the effects of acceleration. Thus, when the rocket is standing still on the surface of the earth, it is attracted toward the center of the earth. Einstein states that this phenomenon of attraction is attributable to an acceleration of the rocket. In three-dimensional space, the rocket is stationary and therefore is not accelerated; but in four-dimensional space-time, the rocket is in motion along its world line. According to Einstein, the world line is curved, because of the curvature of the continuum in the neighborhood of the earth.

Thus, Newton's hypothesis that every object attracts every other object in direct proportion to its mass is replaced by the relativistic hypothesis that the continuum is curved in the neighborhood of massive objects. Einstein's law of gravity states simply that the world line of every object is a geodesic in the continuum. A geodesic is the shortest distance between two points, but in curved space it is not generally a straight line. In the same way, geodesics on the surface of the earth are great circles, which are not straight lines on any ordinary map.

GR Theory's Impact on Astronomical Science

The impact the GR had on astronomy was profound and far-reaching. It did no less than give us a theory for the formation of the universe, and also reform our notion of space and time. For instance, Hubble observed a "red shift" in the light from far away galaxies, and postulated that this must be due to the "Doppler effect" which shifts the light spectrum according to the speed with which a body is moving away from an observer. This explanation was troublesome, because it belied all the ideas that physicists had about a static, unmoving universe. Einstein, when he formulated the theory of general relativity, was so sure that the universe was static that he modified his theory just to accommodate this belief. He termed this cosmological force "antigravity" and used it to explain how the universe had an inbuilt tendency to expand that would balance exactly with all the forces of attraction to give us a static universe.

One man did take GR at face value however, and Alexander Friedmann set out to explain GR's prediction of a non-static universe. In order to do this, he made two simple assumptions: that the universe appears identical in whichever direction we look, and that this would be true if we were observing the universe from anywhere else. These two hypotheses suggested that we should not expect the universe to be static. Friedmann then went on to form a model of how this could be possible, later expanded into three models by American physicist Howard Robertson, and British mathematician Arthur Walker. Graphs of the three models follow:

Figure 2: Friedmann's Expanding Universe Models³

Figure 2.1

Figure 2.2

Figure 2.3

In figure 2.1 (model discovered by Friedmann) the universe is expanding slowly enough that the gravitational attraction between the different galaxies causes the expansion to slow and eventually to stop. The galaxies then would move towards each other and the universe would contract. In 2.2 (post-Friedmann) the universe is expanding so rapidly that the gravitational forces can never stop it, although it may slow down slightly. Finally, 2.3 (post-Friedmann also) shows the universe expanding just fast enough to stave off recollapse, the separation starts at zero and increases infinitely, however, the speed of retreat gets slower and slower, but never reaches zero expansion.

Big Bang

The question of where, when and how our universe was formed was given a valuable analytical tool with the advent of the GR theory. As shown in the Friedmann models, all have a common feature; that at some point in the past (ten to twenty thousand million years ago) the distance between neighboring galaxies must have been zero. At that time, termed the "big bang", the density of the universe and the curvature of space-time would have been infinite. Since mathematics abhors infinite numbers, this "singularity" is the point where GR theory breaks down. Since the first possible moment of predictability (where GR works) is milliseconds after the "big bang", we may say that time began with the end of the singularity. An important step in furthering this research was undertaken by Roger Penrose in 1965. Using the behavior of light cones according to GR, together with the assumption that gravity is always attractive, he showed that a star collapsing under its own gravity is trapped in a region whose surface eventually shrinks to zero. And if the surface is zero, the total volume of the unit must be zero. Therefore, all the matter in the star must be compressed into a region of zero volume, so the density of matter and the curvature of space-time become infinite. The star becomes a singularity contained within a region of space-time known as a black hole.

Black Holes

The term black hole was coined in 1969 by American scientist John Wheeler. As far back as 1783, John Michell of Cambridge reported that because gravity can affect light, if there were a star with enough gravity, light may become trapped within the pull of the star. However, the first consistent theory of how gravity affects light is found in Einstein's theory of general relativity. Working from this foundation, in 1928 an Indian graduate student, Subrahmanyan Chandrasekhar worked out how big a star could be and still support itself against its own gravity after all its fuel was gone. Chandrasekhar realized that a cold star of more than approximately one and one half times the size of our sun would not be able to support itself against its own gravity. This mass is known as Chandrasekhar's limit.

The problem of what would happen to a star more massive that Chandrasekhar's limit was solved by Robert Oppenheimer in 1939. He postulated that there would be no observable consequences of these collapses that could be detected with contemporary telescopes. Oppenheimer's hypothesis is as follows: "The gravitational field of the star shifts the paths of light rays in space-time from what they would have been if the star were not present. As the star contracts, the gravitational field at its surface gets stronger and the light gets bent inward even more. This makes it more difficult for the light from the star to escape, and the light appears dimmer and redder to a distant observer. Eventually, when the star shrinks enough and acquires sufficient density, the gravitational field at the surface becomes so strong that the light is bent inwards so much that light can no longer escape. According to the theory of relativity, nothing can travel faster than light, so therefore nothing can escape the intense pull of this collapsing star. The boundary of such an event is the "event horizon" and it coincides with where the light rays just fail to escape from the star's pull.".⁴

The End?

What black holes potentially represent when studied in conjunction with GR theory is no less than the possibility of seeing the final end to the cosmos. If Friedmann's first mode of expansion (Fig 2.1) is correct and the universe will not expand infinitely then we are doomed to be swallowed by a mammoth black hole. This event is associated with the notion of a "collapsing universe". Kaufmann, in his book Black Holes and Warped Spacetime eloquently describes what happens after entering a black hole: "Nothing can survive passage through infinitely warped space-time. The most basic quantities in science become forever lost. The most fundamental numbers in physics that describe the most intimate details of matter and radiation become indeterminable. This is why we cannot intelligently ask what happened before the Big Bang roughly 20 billion years ago. Such regions of spacetime are completely cut off from us by the infinitely warped spacetime of the past singularity from which our universe was born. Even the basic structure of matter in the "previous universe" could have been vastly different from the atoms of which our present universe is composed. And the "next universe" - the universe that will be born from the condemned ashes of our cosmos- will be totally unrecognizable.".⁵

Conclusion

We have seen in this essay how present views of our universe in its creation, life and possible destruction have been built largely upon the theory of GR. It has been a tool of enlightenment and hope, but it carries within it the possibility of destruction. As such, it has become an integral part of our everyday scientific understanding of the processes of nature. Without GR, many astronomical events would be unintelligible and hidden from our observations behind a cloud of mistaken scientific acceptance. In other words, the way that we presently view our cosmos and its beginning are due in large part to Albert Einstein and his brilliance at manipulating theoretical possibilities into a vast and far-reaching equation: E=MC2.

Texts Consulted

Hawking, Stephen W., A Brief History of Time

Bantam Books, Toronto 1988

Kaufmann, William J. III, Black Holes and Warped Spacetime

W.H. Freeman & Co., San Francisco 1979

Leverington, David A History of Astronomy from 1890 to the Present

Springer Books, London 1995

Microsoft Encarta ‘95ä CDRom Encyclopedia

Van Flandern, Tom Dark Matter Missing Planets & New Comets

North Atlantic Books, Berkeley 1993

1 Microsoft Encarta ‘95ä CDRom Encyclopedia

2 Ibid.

3 Hawking, Stephen W., A Brief History of Time

Bantam Books, Toronto 1988

4 Ibid.

5 Kaufmann, William J. III, Black Holes and Warped Spacetime

W.H. Freeman & Co., San Francisco 1979