An Introduction to Black Holes: A Point-Form Summary. This section of the course notes, and the associated PowerPoint presentation, makes the following critical points: the word relativity refers to how the apparent motion of an object (or some other physical effect or phenomenon) depends on the state of motion of the observer. In a moving train car, you drop a coin 'straght down to the floor' whereas a person outside the train sees it move forward and down at slow speeds, we can merely add-and-subtract these motions in a 'common-sense' fashion light does not obey these simple rules. If you shine a flashlight in the forward direction while riding on a train, you can measure it to be moving away from you at "c", the speed of light. But exactly the same speed, "c", will be measured by an observer standing still beside the track, or indeed by any observer in any state of motion similar odd effects also apply for material objects too, but are only noticeable if the speeds are very high. In day-to-day life, in our low-speed world, the old common-sense methods work fine one consequence of the rigorous rules, however, is that in any frame of reference, no material object can be brought to a speed which exceeds that of light in vacuum the oddness of these effects is attributable to the fact that measurements of distances and time themselves depend on one's state of motion. In a sense, you can think of rulers 'stretching' as you travel at different speeds, and clocks running at different rates these are not just helpful visualisations: in a very real sense, this is exactly what happens! Indeed, the slowing down of clocks as a function of different rates of speed is absolutely demonstrable, and affects everything including all your subjective and physiological sensations of the passage of time

Associated Readings from the Text.

Please look at: Section S2, pages 433-450 - but a light reading only, please! (There is no need to develop a deep and rigorous understanding.) Section S3, pages 454-472 - similarly!

The Inevitability of Black Holes: An Introduction to Relativity.

Stars that are less massive than the Chandrasekhar mass can become stable white dwarfs, but many stars are more massive than this. Some massive stars can become stable neutron stars (and may throw off a fair fraction of their mass as an expanding supernova shell) but neutron stars also have upper limits. What if you have a remnant which is much too massive to be supported by the pressure of degenerate neutrons? The apparently inescapable answer is that it becomes a black hole. To understand these enigmatic objects, we first need to develop a better understanding of the way modern physics looks at gravity. The subject is deeply tied into Einstein's theory of relativity, which many non-scientists find to be baffling. My presentation will be qualitative and necessarily fairly superficial.

What Relativity Means: The Strange Behaviour of Light.

The word `relativity' refers to a description of how things look to observers who are in different states of motion (`relative motion'). As we saw in Physics 015, this concept was first discussed rigorously by Galileo, and indeed Galilean relativity seems to describe nothing more than everyday commonsense experience. Imagine the following situations: Picture yourself seated on a smoothly moving train and dropping a pencil to the floor. From your point of view, it falls `straight down' and lands by your feet. But if I am at the side of the track, I see the pencil fall forward and down. That is because you and the train (your whole frame of reference) are moving relative to me. (In Physics 016, we considered a similar situation when, along with Galileo, we speculated about the behaviour of an object which fell from a crow's-nest high on the mast of a smoothly sailing ship.) Now imagine sitting on the same train and shooting an arrow in the forward direction so that it leaves your bow at 100 km/hr. (Suppose the train itself is travelling at 150 km/hr.) From my perspective at the side of the track, the arrow will seem to be moving at 250 km/hr relative to me. This `makes sense': I see you sitting on the train, moving rapidly from my left to my right (say). The arrow which is streaking away from you must surely seem to be moving faster than the train is. Light behaves differently. Imagine being seated on the train again, and shining a flashlight in the forward direction. From your point of view, the photons race away at the speed of light, which is 300 000 km/sec. (This means that the photons cover a distance of about a foot in just a billionth of a second.) Now if the photons act like the arrow, a stationary person at the trackside should see the photons travelling even faster , covering more than 300 000 km every second. The remarkable thing is that they do not: if the trackside observer measures the apparent speed of the light, he will get exacly the same value as did the person on the train. And this would be true even if the train itself was travelling at 99% of the speed of light! Although this strange behaviour was first observed for light, it applies to all moving objects - not just photons, but electrons, atoms, bricks, and so on. Indeed, we encountered a manifestation of this in my discussion of the cyclotron a lecture or two ago, when we learned that no investment of energy, however much, could make an electron accelerate to the speed of light or beyond. Let us make this clear with another example. The bow-and-arrow on the train are moving at much too low a speed for relativistic effects to matter very much, and Galileo's old trick of merely `adding the speeds' would work well. But now imagine a situation rather like those shown on pages 442-443 of your text, one in which bodies do travel at speeds comparable to that of light: Visualise yourself on a rocket ship passing me at high speed. To be specific, suppose I measure your speed and conclude that you are travelling through my vicinity of space at 75% of the speed of light. Now suppose you launch a small missile in the forward direction, one which sets off at a speed which you measure as being 75% the speed of light relative to you. (You could even tell me by radio that you see the missile streaking away from you at this speed.) I will certainly see the missile streak off ahead of you, at high speed. But my measurements will show it to be travelling at less than the speed of light, rather than at one-and-a-half times the speed of light! (The mathematics of relativity theory allow us to work out the precise speed I will measure, by the way. I will not present the formulas here. But the theory is a fully-understood mathematical one, not merely hand-waving qualitative discussion. The relevant equations can be found on pages 444 if you are interested.) These examples seem to violate common sense, but are not only true but demonstrable in quite a number of modern experiments. Like it or not, this is the way the universe works! From everyone's perspective, photons of light always seem to travel at precisely the speed of light, no matter what the state of motion of the source or the observer; and no material object can be made to travel as fast as light itself does in vacuum.

Why Einsteinian Relativity `Works' the Way It Does.

Needless to say, Galileo (and many scientists before the twentieth century) would have found these notions utterly perplexing. It may seem to you that Einsteinian relativity complicates physics quite a bit, but in fact the reverse is true. Einsteinian relativity simplifies physics by implying that all physical laws have the same form for all observers, regardless of their different states of uniform motion. In essence, this is a statement of a `Principle of Mediocrity, ' a democratic statement that any one observer is just as good as any other from the point of view of understanding the laws of physics. But how does relativity produce its strange effects? To understand why every observer sees light travel at the same speed, think first about how we measure speed at all. Suppose we want to see how fast Donovan Bailey can run, for instance. There are two distinct kinds of measurements we have to make: First, we mark out a distance, like a 100-metre length of track. Next, we have Donovan Bailey cover that distance as fast as he can, recording the separate times at which he leaves the starting point and crosses the finish line. Then we calculate his speed by dividing the distance covered by the time elapsed. If he runs the race in ten seconds flat, his average speed over the distance is ten meters per second. In other words, in determining speeds we measure lengths and time intervals. The problem is that the measurements of things like distances and times are themselves influenced by the state of motion of the instruments doing the measurements, something which was unknown to Galileo and which is irrelevant and unobservable to us in day-to-day life (since we move around at such slow speeds). Thus, clocks run at different rates depending on their state of motion; yardsticks are not `the same length' as seen by observers moving in different ways. So the lapsed time and the distance covered by moving objects are not necessarily seen the same way by two different observers. Someone flying by in a rocket could watch Donovan run, and make the measurements, but would get different numbers and a different speed in his frame of reference.

Time Dilatation.

Most people are more comfortable with the notion of `stretchy' yardsticks than they are with the thought that the rate at which time passes depends on the state of motion of the observer. But Nature is the way it is whether we like it or not! It is incontrovertibly true that clocks moving at high speed appear to measure the passage of time more slowly, a phenomenon known as time dilatation. This means that two observers who synchronize their clocks and then set off on different journeys at different speeds will not have experienced the same passage of time when they are reunited later. Let me be quite specific about this, by referring to a real experiment. Suppose you take two very precise `atomic' clocks and synchronize them very carefully. Now load one onto an airplane and fly it around the world. When the clocks are brought back together, the one which flew around the world will be a little bit `behind' the other. This is not because the clock has been disturbed - say, because if got jiggled about on the aircraft - or because of the effect of high altitude. It is merely because it was moving at moderately high speed relative to us and the other clock. Please note that this does not apply only to mechanical clocks. All physical processes and phenomena are likewise slowed down. If you take a pair of twins, and fly one of them far out into space at high speed before turning the ship around and reuniting them, the moving twin will have `aged' less than the other - perhaps very much less (decades or more) depending on the distance covered and speed of travel. But here is an important qualifier: the twin who comes back still youthful does not enjoy the sensation of living for centuries in a body which miraculously does not age. Perhaps a hundred years pass on Earth, and our astronaut comes back to find her brother long since dead and buried, while she has aged a mere ten years. This means, however, that she will have had the sensation of exactly ten years of subjective lifetime - no more, no less. She has not enjoyed `extra time' to write the great American novel or compose all those symphonies!

A Helpful Visualisation.

As perplexing as all this sounds, do not let it stump you. Here is a helpful way of understanding the phenomenon. Suppose you and a friend leave your residence, each in a car, for a Sunday drive. Late that night you both return. The cars were together in the morning, and are now back together. You would certainly not be surprised to discover that they might have covered different numbers of miles. For instance, one of you might have gone for a long drive in the country, while the other drove downtown and parked near the lakeshore. What you now have to accept - since modern physics shows it to be demonstrably true! - is that the cars can likewise be reunited having experienced different time intervals as well, by virtue of having been in different states of motion. The dashboard clocks, carefully synchronized in the morning, will now disagree - but not because one of them is `wrong.' Each clock is right, because each one ran at the rate which was correct for its state of motion. On the positive side, these effects are utterly negligible, and the differences in the car clocks will be unimaginably and immeasurably tiny unless we consider very high (`relativistic') speeds or trips of enormously long duration, so that the tiny differences can accumulate. But as we will see next day, extremely strong gravitational fields impose something like the same effects, for related reasons: this is the realm of black holes. In day-to-day life, however, you needn't concern yourselves with relativistic effects - except as students of astronomy! Previous chapter:Next chapter


0: Physics 016: The Course Notes, spring 2005. 1: The Properties of the Sun: 2: What Is The Sun Doing? 3: An Introduction to Thermonuclear Fusion. 4: Probing the Deep Interior of the Sun. 5: The Sun in More Detail. 6: An Introduction to the Stars. 7: Stars and Their Distances: 8: The HR Diagram: 9: Questions Arising from the HR Diagram: 10: The Importance of Binary Stars: 11: Implications from Stellar Masses: 12: Late in the Life of the Sun: 13: The Importance of Star Clusters in Understanding Stellar Evolution: 14: The Chandrasekhar Limit: 15: Supernovae: The Deaths of Massive Stars, 16: Pulsars: 17: Novae: 18: An Introduction to Black Holes: 19: Gravity as Geometry: 20: Finishing Off Black Holes: 21: Star Formation: 22: Dust in the Interstellar Medium: 23: Gas in the ISM: 24: The Size and Shape of Our Galaxy: 25: The Discovery of External Galaxies: 26: Galaxies of All Kinds: 27: The Expanding Universe: 28: Quasars and Active Galaxies: 29: The Hot Big Bang: 30: The Geometry of the Universe: 31: Closing Thoughts:


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