Authors: Lawrence M. Krauss
Tags: #Science, #Energy, #Mechanics, #General, #Physics
Fear of Physics (9 page)
So, too, every physical process in the universe is multidimensional. It is only by realizing that we can understand each in a host of equivalent, but seemingly different, ways that we can appreciate most deeply the way the universe works. We cannot claim to understand nature when we see only one side of it. And for better or worse, it is a fact that only mathematical relations allow us to see the whole amid the parts. It is mathematics that allows us to say that the world
is
spherical cows.
is
spherical cows.
In one sense, then, mathematics does make the world more complex, by presenting for us the many different faces of reality. But in so doing, it really simplifies our understanding. We need not keep all the faces in our heads at the same time. With the aid of mathematics, we can go from one to the other at will. And if, as I will claim, it is the interconnectedness of physics that ultimately makes it most accessible, then mathematics makes physics accessible.
Moreover, the fact that mathematics allows us to rephrase the same phenomenon in many different guises offers us the continued
excitement of discovery. New visions of the same thing are always possible! Also, each new face of reality offers us the possibility of extending our understanding beyond the phenomena that may have led to our new insight. I would be remiss if I did not describe one well-known example of this, which I still find absolutely fascinating twenty-five years after I first learned it from Feynman.
excitement of discovery. New visions of the same thing are always possible! Also, each new face of reality offers us the possibility of extending our understanding beyond the phenomena that may have led to our new insight. I would be remiss if I did not describe one well-known example of this, which I still find absolutely fascinating twenty-five years after I first learned it from Feynman.
This example involves a familiar but puzzling phenomenon: a mirage. Anyone who has ever driven down a long, straight stretch of highway on a hot summer day will have had the experience of looking down the road and seeing it turn blue in the distance, as if it were wet and reflecting the sky above. This is the less exotic version of the same thing that happens to those poor souls wandering the desert looking for water and seeing it, only to have it disappear as they run toward their vision of salvation.
There is a simple, and standard, explanation of mirages that has to do with the commonly known fact that light bends when it traverses the boundary between two different media. This is why when you are standing in the water, you look shorter than you actually are. The light rays bend at the surface and trick you into thinking your feet are higher up than they are:
When light goes from a more dense to a less dense medium, as in the picture (going from your legs in the water to your eyes in the air), it always bends “outward.” Eventually, if it hits the surface at a big enough angle, it will bend so far that it reflects back into the water. Thus, the shark about to attack remains hidden from sight.
On a still, sultry day, the air right above a road surface gets very hot—much hotter than the temperature of the air higher up. What happens is that the air forms layers, with the most hot
and
the least dense at the bottom, progressing upward to ever cooler and more dense layers. When the light coming from the sky heads toward the road, it gets bent at each layer, until, if there are enough layers, it gets reflected completely until you see it as you look down the road from the car. Thus, the road appears to reflect the blue sky. If you look carefully next time you see a mirage, you will see that the layer of blue comes from slightly above the road’s surface.
and
the least dense at the bottom, progressing upward to ever cooler and more dense layers. When the light coming from the sky heads toward the road, it gets bent at each layer, until, if there are enough layers, it gets reflected completely until you see it as you look down the road from the car. Thus, the road appears to reflect the blue sky. If you look carefully next time you see a mirage, you will see that the layer of blue comes from slightly above the road’s surface.
Now this is the standard explanation, and it is satisfying, if not necessarily inspiring. There is another explanation of the same
phenomenon, however, which we now know is mathematically equivalent to this one but which presents a remarkably different picture of how the light gets to your eyes from the sky. It is based on the
principle of least time,
proposed by the French mathematician Pierre de Fermat in 1650, which states that light will always take the path that requires the least time to go from A to B.
phenomenon, however, which we now know is mathematically equivalent to this one but which presents a remarkably different picture of how the light gets to your eyes from the sky. It is based on the
principle of least time,
proposed by the French mathematician Pierre de Fermat in 1650, which states that light will always take the path that requires the least time to go from A to B.
This principle clearly is appropriate for the normal travel of light, which is in a straight line. How can it explain mirages, however? Well, light travels faster in a less dense medium (it travels fastest in empty space). Since the air near the road is hotter and less dense, the longer the light stays near the road, the faster it travels. Thus, imagine that a light ray wants to go from point A, to your eye, B. Which path will it take?
One way to do it would be to travel straight to your eye. In this case, however, the light, while traveling the smallest distance, will be spending most of its time in the dense air high above the road. Another way is to take the path shown in the illustration. In this case, the light travels a longer distance, but it spends more time in the less dense layers near the road, where it travels faster.
By balancing the distance traveled with the speed it travels with, you will find that the actual path it takes, the one that produces a mirage, is the one that minimizes the time.
By balancing the distance traveled with the speed it travels with, you will find that the actual path it takes, the one that produces a mirage, is the one that minimizes the time.
This is strange, if you think about it. How can light determine in advance when it is emitted, which path is the fastest? Does it “sniff” around all possible paths before finally choosing the right one? Clearly not. It just obeys the local laws of physics, which tell it what to do at each interface, and it just
happens,
mathematically, that this turns out always to be the path that takes the shortest time. There is something very satisfying about this finding. It seems more fundamental than the alternative description in terms of the bending of light at different layers in the atmosphere. And in some sense it is. We now understand that the laws of motion of all objects can be re-expressed in a form similar to Fermat’s principle for light. Moreover, this new form of expressing the classical Newtonian laws of motion led to a new method, developed by Feynman, for interpreting the laws of quantum mechanics.
happens,
mathematically, that this turns out always to be the path that takes the shortest time. There is something very satisfying about this finding. It seems more fundamental than the alternative description in terms of the bending of light at different layers in the atmosphere. And in some sense it is. We now understand that the laws of motion of all objects can be re-expressed in a form similar to Fermat’s principle for light. Moreover, this new form of expressing the classical Newtonian laws of motion led to a new method, developed by Feynman, for interpreting the laws of quantum mechanics.
By providing different but equivalent ways of picturing the world, mathematics leads to new ways of understanding nature. There is more than mere novelty at stake here. A new picture can allow us to avoid stumbling blocks that might get in the way of using the old picture. For example, the methods based on analogy to Fermat’s principle have allowed quantum mechanics to be applied to physical systems that had hitherto remained impenetrable, including the recent effort pioneered by Stephen Hawking to attempt to understand how quantum mechanics might affect Einstein’s theory of general relativity.
If mathematical connections help govern our understanding of nature by exposing new ways of picturing the world, inevitably this leads to the following issue I want to leave you with in this chapter. If our abstractions of nature are mathematical, in what
sense can we be said to understand the universe? For example, in what sense does Newton’s Law explain
why
things move? To turn to Feynman again:
sense can we be said to understand the universe? For example, in what sense does Newton’s Law explain
why
things move? To turn to Feynman again:
What do we mean by “understanding” something? Imagine that the world is something like a great chess game being played by the gods, and we are observers of the game. We do not know what the rules of the game are; all we are allowed to do is to watch the playing. Of course, if we watch long enough, we may eventually catch on to a few of the rules. The rules of the game are what we call fundamental physics. Even if we knew every rule, however, we might not be able to understand why a particular move is made in the game, merely because it is too complicated and our minds are limited. If you play chess you must know that it is easy to learn all the rules, and yet it is often very hard to select the best move or to understand why a player moves as he does. So it is in nature, only much more so.... We must limit ourselves to the more basic question of the rules of the game. If we know the rules, we consider that we “understand” the world.
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In the end, we may never go any further than explaining the rules, and may never know why they are as they are. But we have been wonderfully successful at discovering these rules, by abstracting out from complicated situations, where the rules are impossible to trace, to simpler ones, where the rules are self-evident—using the guidance of the tools I have described in this chapter and the last. And when we attempt to understand the
world, as physicists, that’s all we can hope to do. Nevertheless, if we try very hard and have luck on our side, we can at least have the pleasure of predicting how nature will respond in a situation that has never before been seen. In so doing, we can hope to observe directly the hidden connections in physics that mathematics may first expose and that, in turn, make the world so fascinating.
world, as physicists, that’s all we can hope to do. Nevertheless, if we try very hard and have luck on our side, we can at least have the pleasure of predicting how nature will respond in a situation that has never before been seen. In so doing, we can hope to observe directly the hidden connections in physics that mathematics may first expose and that, in turn, make the world so fascinating.
PART TWO
PROGRESS
THREE
CREATIVE PLAGIARISM
Plus ça change, plus c’est la même chose.
Popular Wisdom might have you believe that new discoveries in science always center on radically new ideas. In fact, most often the opposite is true. The old ideas not only survive but almost always remain seminal. While the universe is infinitely diverse in phenomena, it seems to be rather limited in principles. As a result, in physics there isn’t as much premium on new ideas as there is on ideas that work. Thus, one sees the same concepts, the same formalism, the same techniques, the same
pictures,
being twisted and molded and bent as far as possible to apply to a host of new situations, as long as they have worked before.
pictures,
being twisted and molded and bent as far as possible to apply to a host of new situations, as long as they have worked before.
This might seem to be a pretty timid, even uncreative approach to unlocking nature’s secrets, but it isn’t. If it takes great boldness to suppose that a slingshot might kill a giant, it is equally bold to suppose that the same methods that determine how far the shot will go might also determine the fate of the universe. It often requires great creativity, too, to see how existing
ideas might apply to new and unusual situations. In physics, less is more.
ideas might apply to new and unusual situations. In physics, less is more.
Transplanting old ideas has been successful so regularly that we have come to expect it to work. Even those rare new concepts that have worked their way into physics have been
forced
into existence by the framework of existing knowledge. It is this creative plagiarism that makes physics comprehensible, because it means that the fundamental ideas are limited in number.
forced
into existence by the framework of existing knowledge. It is this creative plagiarism that makes physics comprehensible, because it means that the fundamental ideas are limited in number.
Perhaps the greatest modern misconception about science is that scientific “revolutions” do away with all that has gone before. Thus, one might imagine that physics before Einstein is no longer correct. Not so. From here to eternity, the motion of a ball dropped from my hand will be described by Newton’s Laws. And while it is the stuff science fiction stories are made of, no new law of physics will ever make the ball fall up! One of the most satisfying aspects of physics is that new discoveries must agree with what is already known to be correct. So, too, the theories of the future will always continue to borrow heavily from those of the past.
This method of doing business complements the notion of approximating reality I discussed earlier. The same “Damn the torpedoes, full speed ahead” mentality suggests that one does not have to understand absolutely everything before moving on. We can probe unknown waters with the tools at our disposal without taking time out to build a new arsenal.
The precedent for this tradition was also created by Galileo. I spoke in the first chapter of Galileo’s discovery that concentration on the simplest aspect of motion and throwing out the irrelevant facts led to a profound reorganization of our picture of reality. What I didn’t explain was that he stated right up front that he didn’t care
why
things move; all he wanted to do, in his inimitably humble fashion, was investigate
how.
“My purpose is to set
forth a very new science dealing with a very ancient subject. There is, in nature, perhaps nothing older than motion, concerning which the books written by philosophers are neither few nor small; nevertheless I have discovered by experiment some properties of it which are worth knowing.”
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why
things move; all he wanted to do, in his inimitably humble fashion, was investigate
how.
“My purpose is to set
forth a very new science dealing with a very ancient subject. There is, in nature, perhaps nothing older than motion, concerning which the books written by philosophers are neither few nor small; nevertheless I have discovered by experiment some properties of it which are worth knowing.”
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