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has to replace masses with charges and Newton’s gravitational constant with an analogous one to quantify the electric force. (Moreover, one has to change the sign of the force, indicating its direction: Two charges of equal sign repel each other, while two—always positive—masses attract each other.) In particular, the dependence on the distance is the same in both cases, and in none of these formulas does a distance in time appear. For Coulomb’s law, an extension had already been found by Maxwell for different reasons, based on the relationship between electric and magnetic phenomena. This extension turned out to be consistent with the transformability of space and time; and it had been found well before Einstein, for whom it played a major role in his considerations leading to special relativity. Maxwell, however, did not recognize the connection between his extension of Coulomb’s law and the changes of space and time.

In 1905, when Einstein developed special relativity, uncovering the observer-dependent form of space and time, no reformulation of Newton’s law existed. The necessary extension of the theory, addressed afterward by Einstein himself, proved to be much more complicated than Maxwell’s generalization of Coulomb’s law. Another decade had to pass before Einstein came up with its final form—general relativity—in 1915. The payoff was not only a gravitational law reconciled with the principles of special relativity, but also a further radical change in our understanding of space and time as well as a mathematical foundation for cosmology. In the remainder of this chapter we will primarily be occupied with the structure of space and time, and much later come back to its role in the behavior of the whole universe, in the chapter on cosmogony (chapter 8).

GENERAL RELATIVITY:

SPACE AND TIME UNBOUND

Effortlessly swings he the world, by his knowing and willing alone.

—XENOPHANES OF KOLOPHON, Fragment

Special relativity does not include the gravitational force; in this sense it is not general enough. With general relativity a gravitational law is available that is consistent with special relativity. But it is not only an extended, more contrived form of Newton’s law; the theory of general relativity constitutes nothing less than the final promotion of space-time to an object of physical meaning. What is considered space and what is considered time is not only changeable, as in special relativity, depending on the viewpoint of an observer; it is itself dynamic and subject to physical processes: The form of space-time is determined by the matter it contains. Space-time is not a straight, flat, four-dimensional hypercube extending unchanged all the way to infinity. Like a piece of old rubber, it writhes under its own inner tensions into a curved structure. The inner tension of space-time is the gravitational force.

As velocities have to be very large to clearly bring out the effects of special relativity, changing space and time just by having a different viewpoint, the influence of gravity on space-time is usually weak. By currently available means it is impossible to exploit this technologically (even though we sometimes speculate about the construction of wormholes, warp drives, or mini–black holes). In astrophysics or cosmology, however, there are often objects so heavy that a precise description must take into account not only their matter content but also properties of space and time themselves. This has led to many tests of general relativity along with, as will be detailed later, new worldviews in cosmology.

In 1915, Einstein did not have such observations at his disposal, just as his earlier formulation of special relativity was driven by thought rather than experiments; his constructions were based solely, but solidly, on the possibility of a mathematically consistent realization of his overarching principles. The result is a theory whose elegance remains unsurpassed in physics. Drawing on general principles and a geometrical form of mathematics, which through a long and noble pedigree can be traced