String Theory For Dummies. Andrew Zimmerman Jones

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СКАЧАТЬ said that force is equal to the change in momentum of an object. In calculus, this is the derivative of momentum with respect to time. Momentum is equal to mass times velocity. Because mass is assumed to be constant and the derivative of velocity with respect to time yields the acceleration, the popular F = ma equation is a simplified way of looking at this situation.

      

This equation can also be used to define inertial mass. If we take a force and divide it by the acceleration it causes on an object, we can determine the mass of the object. One question string theorists hope to answer is why some objects have mass and others (such as the photon) do not.

NEWTON MAKES SOME LAWS ABOUT MOTION

      Newton’s second law of motion, and the way it relates force, acceleration, and mass, is the only law of motion relevant to a string theory discussion. However, for true Newton-o-philes, here are the other two laws of motion, paraphrased for ease of understanding:

       Newton’s first law of motion: An object at rest remains at rest, or an object in motion remains in motion, unless acted upon by an external force. In other words, it takes a force to cause motion to change.

       Newton’s third law of motion: When two objects interact through a force, each object exerts a force on the other object that is equal and opposite. In other words, if we exert a force on a wall with a hand, the wall exerts an equal force back on the hand.

      Gravity: A great discovery

      With the laws of motion in hand, Newton was able to perform the action that would make him the greatest physicist of his age: explaining the motion of the heavens and Earth. His proposal was the law of universal gravitation, which defines a force acting between two objects based on their masses and the distance separating them.

      The more massive the objects are, the higher the gravitational force is. The relationship with distance is an inverse relationship, meaning that as the distance increases, the force drops off. (It actually drops off with the square of the distance — so it drops off very quickly as objects are separated.) The closer two objects are, the higher the gravitational force is.

      The strength of the gravitational force determines a value in Newton’s equation called the universal constant of gravitation or Newton’s constant. A striking property of the universal constant of gravitation is that it is … well, universal. This means that its value is the same in New York, Tokyo, Mars, Alpha Centauri, or the Andromeda galaxy. This value is obtained by performing laboratory experiments and astronomical observations, and calculating what the constant should be. One question still open to physics and string theory is why gravity is so weak compared to other forces.

Gravity seems fairly straightforward, but it actually causes quite a few problems for physicists because it won’t behave itself and get along with the other forces of the universe. Newton himself wasn’t comfortable with the idea of a force acting at a distance, without understanding the mechanism involved. But the equations, even without a thorough explanation for what causes gravity, worked. In fact, the equations worked well enough that for more than two centuries, until Einstein, no one could figure out what was missing from the theory. More on this in Chapter 6.

      Optics: Shedding light on light’s properties

      Newton also performed extensive work in understanding the properties of light, a field known as optics. Newton supported a view that light moves as tiny particles, as opposed to a theory that light travels as a wave. He performed all his work in optics assuming that light moves as tiny balls of energy flying through the air.

      For nearly a century, Newton’s view of light as particles dominated, until Thomas Young’s experiments in the early 1800s demonstrated that light exhibits the properties of waves — namely, the principle of superposition (see the earlier “Catching the wave” section for more on superposition and the later “Light as a wave: The ether theory” section for more on light waves).

      The understanding of light, which began with Newton, would lead to the revolutions in physics by Albert Einstein and, ultimately, to the ideas at the heart of string theory. In string theory, both gravity and light are caused by the behavior of strings.

      Calculus and mathematics: Enhancing scientific understanding

      To study the physical world, Newton had to develop new mathematical tools. One of the tools he developed was a type of math that we call calculus. At the same time Newton invented it, philosopher and mathematician Gottfried Leibniz also created calculus completely independently! Newton needed calculus to perform his analysis of the natural world. Leibniz, on the other hand, developed it mainly to explain certain geometric problems.

      

Think for a moment how amazing this really is. A purely mathematical construct, calculus, provided key insights into the physical systems that Newton explored. Alternately, the physical analysis that Newton performed led him to create calculus. In other words, this is a case where mathematics and physics seemed to help build upon each other! One of the major successes of string theory is that it has provided motivation for important mathematical developments that have gone on to be useful in other realms.

The Forces of Light: Electricity and Magnetism

      In the 19th century, the physical understanding of the nature of light changed completely. Experiments began to show strong cases where light acts like waves instead of particles, which contradicted Newton (see the “Optics: Shedding light on light’s properties” section for more on Newton’s findings). During the same time, experiments into electricity and magnetism began to reveal that these forces behave like light, except we can’t see them.

      By the end of the 19th century, it became clear that electricity and magnetism are different manifestations of the same force: electromagnetism. One of the goals of string theory is to develop a single theory that incorporates both electromagnetism and gravity.

      Light as a wave: The ether theory

      Newton had treated light as particles, but experiments in the 19th century began to show that light acts like a wave. The major problem with this was that waves require a medium. Something has to do the waving. Light seemed to travel through empty space, which contained no substance at all. What was the medium that light used to move through space? What was waving?

      To explain the problem, physicists proposed that space is filled with a substance. When looking for a name for this hypothetical substance, they turned back to Aristotle and named it luminous aether, or ether. (No relation to the gas used in old-timey surgeries as anesthetic.)

      Even with this hypothetical ether, though, there were problems. Newton’s СКАЧАТЬ