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1. Gravitation

Mass Attraction

Should something from the window fall

(and if it just the smallest be)

how jumps the law of gravity

as mighty as wind from the sea

at every ball or blueberry

and takes them to the core of all.

-Rainer Maria Rilke, *The Book of Hours*

Over large distances, the universe is governed by the gravitational force. In physics, the action of a force is the cause of motion or of any form of change. Complete rest is possible only if no net forces are acting. One scenario in which this can happen is the absence of any matter whatsoever-a state called vacuum. But matter quite obviously does exist, and just by its mass it causes gravitational forces on other masses. To realize motionless states of rest, at least approximately, all acting forces must compensate each other. In addition to gravity, there are the electric and magnetic forces to be considered, as well as two kinds of forces called weak and strong interactions, reigning in the realm of elementary particles.

While the electric force is easily compensated over large distances by the existence of positive and negative charges, mutually neutralizing each other, the forces that come into play in the interior of nuclei act only at extremely short range. What remains over long distances is gravity alone. It rules the general attraction of masses and energy distributions in space, and thus dictates the behavior of the universe itself. In contrast to electricity, there are no negative masses: Gravitational attraction cannot be fully compensated. Once massive objects such as stars or entire galaxies form, the resulting gravitational interaction dominates all that happens. The facets of this commonplace force, often ignored in recent research and yet-in cosmology and black holes-giving rise to a rich variety of exotic phenomena, are the topic of this book.

Newton's law of gravity:

Distant action and a fatal flaw

The first general law of gravity was formulated by Isaac Newton. As is typical for many important steps in gravitational research, this theoretical development required a unified view on well-known phenomena on Earth with a long list of intricate observations of objects in space: the moon and some planets. The latter was accomplished thanks to technologies that, for those times, were highly sophisticated; conversely, such research has spawned the development of new instruments. Combining fundamental questions and technological applications, in many areas of science and in gravitational research in particular, is a success story that continues into the present day.

Even before Newton, the initial untidy flood of data, as it was accumulated by astronomers such as Tycho Brahe, Johannes Kepler, and many others, was ordered into a model of the solar system. Since Nicolaus Copernicus and Kepler, this model has largely held the form we know today: Planets orbit around the sun along trajectories that, by a good approximation, can be considered as ellipses, or slightly oblong circles. But what is propelling the planets along their curved tracks? From common observations we know that a force is necessary to keep a body from moving stubbornly along a straight line. How can one describe or even explain the required force in the case of the planets?

Newton's groundbreaking insight-the existence of a universal force of gravity causing not only the motion of all planets around the sun, and of the moon around the earth, but also the everyday phenomena of falling objects-is impressive. It is an excellent example of the origin of scientific explanation: not an answer to a "why" question in the sense of an anthropomorphic motivation, but a plethora of complicated phenomena, unrelated at first sight, reduced to a single mechanism: a law of nature. Newton's mathematical description of the situation is very compact and highly efficient for predictions of new phenomena described by the same law. In the case of Newton's law of gravity, the unfathomable power of theoretical prediction has repeatedly been employed-for instance, to find new planets via small deviations imposed by their gravitational pull on the trajectories of other planets, or in planning modern satellite missions.

Such success stories, in which an elegant mathematical description explains and predicts a multitude of phenomena, can be found throughout physics; they are indeed the landmarks of its progress. Reliving such insights is often so gratifying that scientists employ the term "beauty"-a pragmatic kind of beauty whose core, the mathematical formulation, can be seen only by the initiated, but which in its concrete successes can also be appreciated by outsiders.1

Concretely, Newton's law of gravity describes the attractive force between two bodies caused by their masses. The force increases proportionally with the amounts of the masses: The attraction between two heavy bodies is larger than that between two light ones. It is also inversely proportional to the squared distance between the bodies; it weakens considerably when the bodies are farther apart. In addition to these proportionalities, the exact quantitative strength of the force is determined by a mathematical parameter, now called Newton's gravitational constant. In this value one can see the unification of earthly and heavenly phenomena. The gravitational constant can be derived from the tiny attraction of two masses on Earth, as was first accomplished in Henry Cavendish's laboratory in 1797 and '98; using the same value to calculate the force exerted by the sun on the planets shows exactly the right nudge required to hold the planets on their observed orbits.

In contrast to its clear dependence on distance, Newton's gravitational force is completely independent of time. Time independence sounds plausible, for a fundamental law of nature should, after all, be valid at all times in the same way. It is also consistent with the dominant understanding of space and time in Newton's age and long thereafter, not to mention our everyday conceptions of them. Although one can easily change the positions and distances of objects in space, space itself appears unchangeable. Also, time seems to pass simply and uniformly, without being influenced by physical processes or technical instruments. Since gravity, according to Newton, acts instantaneously-independently of how far apart the masses are-the force need be formulated only for the case of two masses not at the same place, but at the same time.

Despite its plausible form and celebrated successes, Newton's theory did have a flaw in its beauty. Like the beauty of the theory itself, this flaw, too, can be understood completely only with a sufficient amount of background knowledge. But even on the surface, it is a good example of the progress of theoretical physics. Newton himself had reportedly been uneasy about the "animalistic" tendencies of his law of gravitation: As an animal is attracted from far away by the expectation of food or companionship, a massive body appeared to move toward another one from a distance. This action at a distance, apparently without the more intuitive type of local interactions as realized for bodies pushing each other at close contact, was considered a serious conceptual weakness in spite of all concrete successes.

It is extremely difficult to correct this weak spot by constructing a theory only of local interactions that, of course, should otherwise remain compatible with the astronomical successes of Newton's theory. To start with, one will have to consider the time dimension, too, for such a local interaction must take some time to propagate from one body to the other. As it turned out, a consistent reformulation is possible only by radically changing Newton's-and our-intuitive conceptions of space and time. It requires much more highly sophisticated mathematical machineries and substantial efforts, but these efforts are rewarded by a theory of unprecedented beauty in the sense described above. All this required dedicated physical research and, not least, a strong mathematical grounding. The flaw in Newton's theory was to be corrected only long after Newton-by Albert Einstein.

Relativity of space and time:

Space-time transformers

All this took a long time, or a short time: for, strictly speaking, no time on earth exists for such things.

-friedrich nietzsche, Thus Spoke Zarathustra

In physics, as in all of science, it is important to distinguish between properties that depend on the person making an observation and properties independent of an observer. The mass of a particle refers only to the particle itself and will, if the particle remains unchanged, always be measured as the same value. Except for unavoidable experimental inaccuracies, it does not matter who is doing the measurement. A particle's velocity, on the other hand, appears different, and sometimes drastically so, depending on whether an observer is moving with respect to the particle. An observer moving along with the particle at exactly the same speed would perceive the particle as being at rest, well known from two cars cruising side by side along a straight stretch of highway. To the driver of one car, the other one seems not to be moving. Any other observer would see the car (or the particle) move and attribute to it a nonzero velocity. Relativity in general terms is the mathematical analysis of such relationships; it ultimately tells us what we can learn about nature in a fully objective, observer-independent way.

For many centuries, space and time were thought of as observer- independent. Distances between points and durations of temporal periods appeared absolute, no matter how an observer would be positioned or move. But the first fault lines in this worldview opened up toward the end of the nineteenth century, eventually leading to special relativity. In this new view, space and time cannot be seen in separation but are intertwined, interchangeable, and observer-dependent. Like the velocity of a particle, the values measured for them depend on the motion of an observer. In abstract terms, they describe different dimensions of a single physical object: space-time; and only space-time concepts, but not space or time themselves, are independent of the person making a measurement.

How can this be demonstrated by physical means? To answer this question and to explain the role of dimensions, we first consider space alone. Space has different dimensions, namely three: we can move sideways, back and forth, and up or down. Here, one might ask why these should be considered as three dimensions of a single space, rather than three completely independent directions: width, depth, height. The answer is simple. Width, depth, and height are not absolute and independent properties; they can be commuted into one another. We have only to turn around in space to make the height of a cube appear as its width, and in this sense height and width can be interchanged. This is not a transformation by a physical process, like a chemical reaction, but a much simpler one by means of changing our viewpoint. What we see as height, width, and depth depend on the place of an observer (or on conventions such as the use of Earth's surface along which to measure width and depth); they cannot be considered properties of space itself as a physical object. For this reason, one speaks of three-dimensional space, not of the existence of three independent one-dimensional directions.

Time is similar, although its transformation is harder. By simply turning around one can influence only one's view on space; the change of the angle of view (or, more precisely, the tangent of the angle as a mathematical function, which does not differ much from the angle when it is small) is expressed by the ratio of spatial extensions, such as the height before and after changing the viewpoint. By changing the angle, one can only transform spatial extensions into one another. If we want to transform space into time, we must vary a quantity given by a ratio of spatial and time extensions: a velocity. Traversing a certain distance in some period of time means that one moves at a velocity obtained as the ratio of that distance to the required time.

This consideration does, in fact, lead to the basic phenomenon of special relativity. If we are moving faster than a second observer while viewing a certain scene, spatial and time distances appear different to each of us. As changing the angle of view transforms spatial extensions, changing the velocity of an observer commutes spatial distances to timelike ones and vice versa. Distinguishing space and time extensions is thus dependent on the viewpoint (or the "viewtrack," if we are indeed moving); it cannot have a physical basis independent of observers' properties. Instead of separate space and time, there is only one joint object: space-time. Special relativity is the theory of these changing viewtracks (also called inertial observers) in the absence of the gravitational force.

As an illustration, these considerations are certainly no proof; not every ratio implies a transformation when it is changed. For instance, the birthrate of a country is the ratio of newborns to the total population, but a change in the birthrate does not mean that inhabitants are transformed into newborns. An important difference from the previous example is the role of observers: Changes are caused by observers taking different positions and states of motion; and since physical laws must be independent of the special private and personal properties of those making the observations, concepts distinguished only by viewpoints must be discarded. In special relativity, this "transformability" of space and time, forcing us to deny them separate meaning, has not only been substantiated mathematically; it has also been verified experimentally myriad times, especially in reactions of elementary particles. While the Newtonian concepts of a rigid space and an independent time would not agree with many measurements made in the last century, in a special relativistic view no inconsistencies arise.

Newton's view was able to enjoy great success for such a long time because noticeably transforming space and time requires very large observer velocities. Unless measurements are extremely refined and precise, in order to see an effect, speeds must be close to the immense velocity of light: roughly 300,000 kilometers per second. In everyday life, this makes the transformability of space and time imperceptible.2 For an observational verification, one needs either very high velocities or very precise time measurements in order to notice the tiny time changes at low velocities. Both methods have been developed in the past century.

Very precise time measurements are achieved by atomic clocks, making space-time transformations detectable even at the typical speeds of airplanes. (Since planes have to move at a certain height, additional effects arise due to a reduction of gravity acting on the clock farther away from the center of the earth. This general relativistic effect, depending on the gravitational force, is introduced below.)

At velocities close to that of light, space-time changes drastically: As an observer at rest would describe it, time is transformed almost completely into space, and thus passes ever more slowly. Once the speed of light is reached, which is possible only for massless objects such as light itself, all timelike distances vanish. Going beyond that speed limit is impossible, for all time has already been used up when we reach the speed of light. No signal can move faster than light. Delays in any transmission of information must always occur; they may be small, but they do become noticeable at large distances. (This maximum speed is that of light in a vacuum. In transparent media such as water, light usually moves more slowly than in a vacuum.

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