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# 100 years General Relativity – The theory that made Einstein a genius

In Douglas Adams’ popular parody of intergalactic life “The Hitchhiker’s Guide to the Galaxy” at the beginning of the second book one finds that according to a certain theory, the world is something very bizarre and incomprehensible. And if anyone ever found out what the universe is all about, it would disappear instantly and be […]

In Douglas Adams’ popular parody of intergalactic life “The Hitchhiker’s Guide to the Galaxy” at the beginning of the second book one finds that according to a certain theory, the world is something very bizarre and incomprehensible. And if anyone ever found out what the universe is all about, it would disappear instantly and be replaced by something even more bizarre and inexplicable. And then there is yet another theory that states that this has already happened. These days marks the 100th anniversary of an event which appears to support the later of these two theories. On November 18th and 25th 1915, the 36-year-old Albert Einstein presented at a meeting of the Prussian Academy of Sciences his “generalized theory of gravitation”. The altogether four lectures were the conclusion of a nine-year intellectual odyssey which ultimately constituted Einstein’s reputation as the epitome of a genius and made him one of the most important men of the 20th century altogether. The thus presented theory which Einstein himself called “general theory of relativity” constitutes one of the greatest single contributions to science in history and a revolution in our thinking about the universe that has been unparalleled since.

Once asked where he actually had his laboratory Einstein replied “Here” and took his pen from his jacket pocket. In particular, general relativity spun off exclusively from Einstein’s own thoughts and mathematical reasoning. While nearly eleven years earlier specific unresolved issues and observations had urged him to develop new theories – the symmetry properties of the Maxwell’s equations and the Michelson-Morley experiment in the case of the theory of special relativity, black body radiation and the photoelectric effect in the case of the hypothesis of light quanta, which had the year 1905 already been awarded the name “miracle year” in Einstein’s biography, there was no indication that Newton’s theory of gravitation required any modification (besides maybe the movement of the perihelion of Mercury which slightly deviated from the law of Newton, but physicists had paid little attention to this; it was the problem Einstein solved first on November 18th 1915, before he published his final field equations on November 25th). Niels Bohr, next to Einstein the most outstanding physicist of his time, thus in 1914 called on Einstein to address the difficult outstanding problems of quantum physics rather than to deal with gravity, where everything was fine.

What was the background of Einstein’s thoughts? Even though his theory of gravitation would at last lead him to breathtaking heights of mathematical abstraction its beginning were of a simplicity and clarity so typical for Einstein. Shortly after the formulation of the special theory of relativity, he had succeeded in establishing a direct link between accelerated motion and the action of a gravitational force. Like the famous apple falling from the tree in 1666 is said to have led Newton to the basic principle of his theory of gravitation, a similar flash of inspiration can be found in Einstein’s biography representing the first step toward his theory. As Einstein himself later recalled:

I sat in my chair in the patent office in Bern. Suddenly I had an inspiration: If a person is in free fall, he will not feel his own weight. I was stunned. This simple thought experiment made a deep impression on me. It led me to a theory of gravitation.

Summarizing, the acceleration in free fall cancels off the effect of gravity. Einstein thus gave an answer to a millennium old question which already Aristotle had dealt with: whether and why all bodies fall equally fast (without disturbing influences such as friction). That they do so physicists had known since Galileo, as most of us will surely remember from our high school physics class. But even Galileo could not satisfactorily answer why this is so (and neither can quite a few high school physics teachers today). The answer lies in the equality of two very different concepts of mass: The inertial mass gives massive bodies resistance to changes in motion, whereas the gravitational mass is responsible for the gravity exerted on massive bodies. That these two masses are identical, is by no means directly obvious.

Another of Einstein’s consideration concerned the finite speed of light. According to Newton gravity exerts a force on bodies by means of their masses, analogous to charged particles in an electric or magnetic field. In the case of electromagnetism the force is based on fields emanating from electrical charge carriers and magnets. According to Maxwell’s equations the speed of propagation of these fields corresponds of the speed of light. And according to Einstein’s special theory of relativity from 1905 this constitutes a limit on any speed physically possible. Einstein therefore believed that this limit also applies to the propagation of gravitational forces. However this conclusion contradicted the idea of an instantly (timelessly) acting force in Newton’s theory. Therefore, Einstein saw the need to formulate a new theory for the gravitational field that analogous to Maxwell’s theory of electrodynamic satisfies the principles of special relativity. In abstract words, he wanted to include the gravitational force into the relativistic structure of space-time.

A final problem of the classical theory of gravitation for Einstein was the asymmetry between bodies and space. Although space acts on the bodies in the form of inertia, by giving them a resistance to any change in their motion, conversely bodies do not act on space. Physicists had regarded space as absolute, unalterable of bodies, movements, or forces. Einstein already knew from his work in 1905 that the absoluteness of space had to be abandoned. This led him to believe that the asymmetry between bodies and space can no longer be maintained, either. These three ideas of his, the identity of acceleration and gravitation, the finite speed of propagation of the gravitational forces, and the symmetry in effects between bodies and space, lead us directly to the heart of general relativity. They also demonstrate the powerful instinct which lead Einstein to the greatest single tour de force in 20th century physics.

But the path towards it proved to be much longer and more onerous than Einstein had imagined. The years from 1905 to 1916 were among the most stressful, difficult and self-sacrificing of his life. During these years, Einstein’s first marriage broke apart, he suffered two nervous breakdowns, and he spent his time in increasing isolation. After all, unlike his colleagues in atomic and quantum physics, he was all by himself in ascending the new summit of physics (only helped by some mathematicians). It was also the solitude in which Einstein mastered the summit, the existence of which most of his contemporaries were not even aware of, which eventually made him the epitome of a genius.

How can we summarize the main features of Einstein’s new “theory of general relativity”? By linking accelerated motion and gravity Einstein combined the concept of force with the structure of space and time, in which in turn forces act again. In the decisive step towards the general theory of relativity Einstein formulated that the effect of gravity is no longer given by a time-independent, spatially-acting force, but by an influence on the structure of space-time itself. In other words, gravity is now a consequence of masses causing changes in the geometric structure of four-dimensional space-time. Einstein thus combined what was sharply separated in classical physics: space and force, geometry and gravity. Masses no longer act on each other by forces, but they change the structure of space-time by tweaking or bending it, which in turn “gravitationally” influences the masses again. The classic flat, so-called “Euclidean geometry” of space ceases to be valid in Einstein’s theory and is replaced by a locally “bent” geometry, the curvature of which depends on the present mass distribution. In mathematical terms, Einstein’s equations provide a direct link between the mass distribution (represented by a mathematical object called “stress-energy tensor” on one side of the equation) and the geometric characteristics, the so-called “metric”, of space-time (represented by the “curvature tensor” on the other side of the equation). In physical terms: Space is not a container of the physical world, and the time is not an internal parameters of the movement, but both are an integrated object of physics with its own dynamics. Or as the physicist John Archibald Wheeler once summarized in simple terms: “Matter tells space how to curve. Space tells matter how to move.“

An analogy might help to illustrate those abstract ideas: A lead ball on a rubber mat causes a deformation at the point where it rests. This curvature in turn affects the motion of other balls on the mat. A second lead ball will (in the absence of frictional forces and depending on the angular momentum) revolve around the first ball. The balls on the rubber mat are thus not pulled towards each other due to a force but because they themselves change the shape of the space in which they are located. In analogy the deformation of the geometry of space-time caused by a massive body influences the movement of other mass bodies. Such geometrization of gravitation, however, introduces a complication which the example of the rubber mat is not capable of describing, since the deformation of the first two-dimensional rubber mat takes place into the well-known third dimension. The space that we consider in the gravitation however is already three-dimensional. Where should it thus be deformed into? We need a fourth dimension to describe the induced gravitational curvature of space. And this dimension is time. Thus, instead of describing solely the geometry of a three-dimensional space in which bodies move in an independent one-dimensional time, in a geometric description of gravity we must consider space and time combined into an inseparable four-dimensional world. This thought was not entirely new. Already in the special theory of relativity were space and time linked in a coherent four-dimensional “space-time continuum”, but still neither with a curvature shaped by the masses nor with its own dynamics.

The general theory of relativity is the first physical theory that is free of speculative ideas about the nature and essence of space and time. In it the structure of space and time possesses a dynamics described by a new physical theory. Therefore the general theory of relativity brought the philosopher a giant step closer to answering an age-old question: “What are space and time?” Einstein’s theory gives us an idea that space and time could be something very different than what our daily phenomenological experiences make us believe. Already Immanuel Kant said that we can form no idea of what the structure of space and time “in itself” is beyond our experience. At first glance this statement seems to be contradict Einstein’s theory, as this explicitly describes a physical structure of space and time, which lies beyond and even stands contrary to our direct experience, and puts this into the form of a mathematical law. Or maybe not? Readers who find it difficult to understand the abstractness of four-dimensional space-time geometry, its dependence on a mass dependent dynamics, and the complex mathematical structure of Einstein’s equation may judge for him- or herself whether Kant might have been right anyways and we cannot conceive all this. But they must be told: Without Einstein’s theory no GPS would function today, and cosmologists would lack a theoretical framework which fits so perfectly their observations.

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## 1 Comments

Free Fall

An elevator cabin is in free fall. On a plane, an observer is moving in a uniform linear motion horizontally. To him, the locus of the elevator is visible as a parabola.

On every atom of the former, inertial force is acting. On every atom of the latter, inertial force is not acting. Inertial force is not fictitious force. So, the two motions are not relative.

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Sorry, I cannot receive E-mail. I do not have PC.

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