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General Relativity

Einstein was not satisfied with special relativity for very long. He felt strongly that the principle of relativity for uniform motion ought to be generalized to arbitrary motion. In his popular book Relativity: The Special and the General Theory (1917) he used a charming analogy to explain what he found so objectionable about absolute motion. Consider two identical teakettles sitting on a stove. One is giving off steam, but the other is not. The observer is puzzled by this until realizing that the burner under the first kettle is turned on, but that the burner under the second is not. Compare this situation to that of two identical globes rotating with respect to one another around the line connecting their centers. Einstein discussed this situation in his first systematic exposition of general relativity in 1916. The situation of the two globes, like that of the two teakettles, at first seems to be completely symmetric. Observers on both globes will see the other globe rotating. Yet one globe bulges out at the equator, while the other does not. What is responsible for this difference? The Newtonian answer is: absolute space. What makes a globe bulge out is not its rotation with respect to the other globe, but its rotation with respect to absolute space. The special-relativistic answer is the same, except that Newton's absolute space needs to be replaced by Minkowski's absolute space-time. Einstein found this answer unsatisfactory, because neither Newton's absolute space nor Minkowski's absolute space-time can be directly observed. And without an observable cause for the difference between the two globes, we would have a violation of Gottfried Wilhelm von Leibniz's (1646–1716) Principle of Sufficient Reason. It would be as if the burners under both teakettles were turned on, but only one of them gave off steam.

There is a crucial difference, however, between the Newtonian and the special-relativistic answer as has been pointed out by Dorling). In special relativity, the situation of the two globes is not even symmetric at the purely descriptive level. Because of time dilation, one revolution of the other globe takes less time for an observer on the globe rotating in Minkowski space-time than it does for an observer on the other one. It therefore need not surprise us that the situation is not symmetric at the dynamic level either, that is, that only one globe bulges out. In terms of Einstein's analogy: if the two teakettles do not look the same, it need not surprise us that they do not behave the same. Absolute acceleration in special relativity thus does not violate the Principle of Sufficient Reason. Einstein failed to appreciate that special relativity had already solved what he himself identified as the problem of absolute motion.

Einstein's attempt to relativize acceleration.

In 1907 Einstein set out on the road that would lead to general relativity. In the 1919 article quoted in the preceding section, he vividly recalls the initial flash of insight. Immediately following the passage quoted earlier, he writes:

Then came to me the best idea [ die glücklichste Gedanke ] of my life … Like the electric field generated by electromagnetic induction, the gravitational field only has a relative existence. Because, for an observer freely falling from the roof of a house, no gravitational field exists while he is falling. The experimental fact that the acceleration due to gravity does not depend on the material is thus a powerful argument for extending the relativity postulate to systems in non-uniform relative motion. (Pais, p. 178; see Janssen, "COI Stories" for further discussion of this argument)

This extended relativity postulate, it turns out, is highly problematic. What it boils down to is that two observers accelerating with respect to one another can both claim to be at rest if they agree to disagree about whether a gravitational field is present or not. This curious principle is best illustrated with a couple of examples.

First, consider the unfortunate observer falling from the roof in Einstein's example. For a fleeting moment this person will feel like a modern astronaut orbiting the earth in a space shuttle. Einstein, watching his observer accelerate downward from the safety of his room in the Berne Patent Office where he was working at the time, is at rest in the gravitational field of the earth. For the falling person, however, there seems to be no gravitational field. He can, if he wishes, maintain that he is at rest and that Einstein and the Patent Office are accelerating upward.

As a second example, imagine two astronauts in rocket ships hovering side by side somewhere in outer space where the effects of gravity are negligible. One of the astronauts fires up the engines of her rocket ship. According to the other astronaut she accelerates, but she can, if she were so inclined, maintain that she is at rest in the gravitational field that suddenly came into being when her engines were switched on and that the other rocket ship and its crew are in free fall in this gravitational field. Einstein produced an account of the twin paradox along these lines as late as 1919.

Notice that the "relativity of acceleration" in these two examples is very different from the relativity of uniform motion in special relativity. Two observers in uniform relative motion are physically fully equivalent to one another; two observers in nonuniform relative motion are not. This is clear in both examples. (1) Free fall in a gravitational field (a) feels different from resisting the pull of gravity (b). (2) Hovering in outer space (a) feels different from accelerating in outer space (b). In both cases, the "relativity of acceleration" is relativity in name only. In fact, the pair (1a)–(2a) and the pair (1b)–(2b) feel the same.

Equivalence principle.

Einstein's excitement about "the best idea of his life" was nonetheless fully justified. Galileo's principle that all matter falls with the same acceleration in a given gravitational field cries out for an explanation. Newton incorporated the principle by giving two very different concepts of mass the same numerical value. He set inertial mass, a measure of a body's resistance to acceleration, equal to gravitational mass, a measure of a body's susceptibility to gravity. In Newton's theory this is an unexplained coincidence. Einstein correctly surmised that this equality points to an intimate connection between acceleration and gravity. He called this connection the equivalence principle. It was not until after he finished his general theory of relativity, however, that he was able to articulate what exactly the connection is.

That did not stop him from relying heavily on the equivalence principle in constructing his theory. Since accelerating in Minkowski space-time feels the same as resisting the pull of gravity, Einstein was able to glean some features of gravitational fields in general by studying acceleration in Minkowski space-time. In particular, he examined the situation of an observer on a rotating disk or a merry-go-round in Minkowski space-time. Appealing to the embryonic equivalence principle, the man on the disk can claim to be at rest and attribute the centrifugal forces due to his centripetal acceleration to a centrifugal gravitational field. Suppose the man on the disk and a woman standing next to it are both asked to determine the ratio of the disk's circumference to its radius. The woman will find 2, the answer given by Euclidean geometry. Because of length contraction, which affects the measuring rods placed along the circumference but not the ones along the radius, which move perpendicularly to their length, the man on the disk will find a value greater than 2. This means that the spatial geometry for an observer rotating in Minkowski space-time is non-Euclidean. The equivalence principle says that the spatial geometry for an observer in a gravitational field will then, in general, also be non-Euclidean. This simple consideration, in all likelihood, gave Einstein the idea to represent gravity by the curvature of space-time (Stachel and Howard, pp. 48–82).

Recognizing that gravity is part of the fabric of space-time made it possible to give a more precise formulation of the equivalence principle. The mature form of the equivalence principle is brought out very nicely by the analogy that Einstein sets up but never quite finishes in the 1919 passage quoted above. Special relativity made it clear that electric and magnetic fields are part of one entity, the electromagnetic field, which breaks down differently into electric and magnetic components for different observers. General relativity similarly made it clear that the inertial structure of space-time and the gravitational field are not two separate entities but components of one entity, now called the inertio-gravitational field. Inertial structure determines the trajectories of free particles. Gravity makes all particles deviate from these trajectories in identical fashion, regardless of their mass. These are the only marching orders that all particles have to obey. In general relativity, they are all issued by one and the same authority, the inertio-gravitational field, represented by curved space-time. Which marching orders are credited to inertial structure and which ones to gravity depends on the state of motion of the observer. The connection between acceleration (or, equivalently, inertia) and gravity is thus one of unification rather than one of reduction, as with the nominal "relativity of acceleration" discussed above, in which acceleration was being reduced to gravity.

How do the examples of nonuniform motion discussed above fit into the new scheme of things? Free fall in a gravitational field (1a) and hovering in outer space (2a) are both represented as motion along the straightest possible lines in what in general will be a curved space-time. Such lines are called geodesics. Resisting the pull of gravity (1b) and accelerating in outer space (2b) are both represented as motion along crooked lines, or nongeodesics. Since no change of perspective will transform a geodesic into a nongeodesic or vice versa, there is an absolute difference between (1a) and (1b) as well as between (2a) and (2b). Absolute acceleration survives in general relativity, as in special relativity, in the guise of an absolute distinction between geodesic and nongeodesic motion. This does not violate the Principle of Sufficient Reason, since geodesics and nongeodesics are already different at a purely descriptive or geometric level.

General covariance.

Einstein did not give up his crusade against absolute motion so easily. Once he had realized that gravity is space-time curvature, he quickly came up with a new (though once again flawed) strategy for extending the principle of relativity from uniform to arbitrary motion. To describe curved space-time, Einstein had turned to the theory of curved surfaces of the great nineteenth-century German mathematician Carl Friedrich Gauss (1777–1855). To describe such surfaces (think of the surface of Earth for instance) one needs a map, a grid that assigns unique coordinates to every point of the surface, and sets of numbers to convert coordinate distances (i.e., distances on a map) to real distances (i.e., distances on the actual surface). These sets of numbers are called the components of the metric tensor. In general they are different for different points. The conversion from coordinate distances to real distances is thus given by a field, called the metric field, which assigns the appropriate metric tensor to every point.

A simple example may help to better understand the concept of a metric tensor field. On a standard map of the earth, countries close to the equator look smaller than countries close to the poles. The conversion factors from coordinate distances to real distances are therefore larger near the equator than they are near the poles. The metric tensor field thus varies from point to point, just like an electromagnetic field. Furthermore, at one and the same point, the conversion factor for north-south distances may differ from the conversion factor for east-west distances. The metric tensor at one point thus has different components for different directions.

Gauss's theory of curved surfaces was generalized to spaces of higher dimension by another German mathematician, Bernhard Riemann (1826–1866). This Riemannian geometry can handle curved space-time as well. In the case of four-dimensional space(-time), the metric tensor has ten independent components. In Einstein's theory, the metric tensor field does double duty: it describes both the geometry of space-time and the gravitational field. Mass—or its equivalent, energy—is the source of gravitational fields. Which field is produced by a given source is determined by so-called field equations. To complete his theory Einstein thus had to find field equations for the metric field.

Einstein hoped to find field equations that would retain their form under arbitrary coordinate transformations. This property is called general covariance. The description of curved space-time outlined in the preceding paragraph clearly is generally covariant. One can choose any grid to assign coordinates to the points of space-time. Each choice will come with its own sets of conversion factors. In other words, the metric field encoding the geometry of space-time will be represented by different mathematical functions depending on which coordinates are used. Riemannian geometry is formulated in such a way that it works in arbitrary coordinates. It provides standard techniques for transforming the metric field from one coordinate system to another. If only Einstein could find field equations for the metric field that retain their form under arbitrary transformations, his whole theory would be generally covariant. In special relativity, Lorentz invariance expresses the relativity of uniform motion. Einstein—understandably perhaps, but mistakenly—thought that extending Lorentz invariance to invariance under arbitrary transformations would automatically extend the principle of relativity from uniform to arbitrary motion.

This line of thinking, however, conflates two completely different traditions in nineteenth-century geometry, as has been argued by John Norton. Minkowski's work with special relativity is in the tradition of projective geometry, associated with the so-called Erlangen Program of Felix Klein (1849–1925). Einstein's work with general relativity is in the tradition of differential geometry of Gauss and Riemann. The approaches of Klein and Riemann can be characterized as "sub-tractive" and "additive," respectively.

In the subtractive approach one starts from an exhaustive description of space-time with all the bells and whistles and then strips this description down to its bare essentials. The recipe for doing that is to assign reality only to elements that are invariant under the group of transformations that relate different perspectives on the space-time. This group of transformations is thus directly related to some relativity principle. The most famous application of this strategy in physics is Minkowski's geometrical formulation of special relativity. The group of transformations in this case is the group of Lorentz transformations.

In the additive approach one starts with the set of space-time points stripped of all their properties and then adds the minimum geometrical structure needed to define straight(est) lines and distances in space-time. To guarantee that the added structure describes only intrinsic features of space-time, the demand is made that the description be generally covariant, that is, that it not depend on the coordinates used. This procedure can obviously be applied to any space-time. Only in certain special cases, however, will there be symmetries such as Lorentz invariance in Minkowski space-time reflecting the physical equivalence of different frames of reference and thereby some relativity principle. In the generic case there will be no symmetries whatsoever and hence no principle of relativity at all. This shows that general covariance has nothing to do with general relativity. Comparing Lorentz invariance in special relativity and general covariance in general relativity is like comparing apples and oranges.

The hole argument and the point-coincidence argument.

It was not until 1918 that a German high school teacher by the name of Erich Kretschmann (1887–1937) set Einstein straight on this score. This was a few years after Einstein had finally found generally covariant field equations. These equations, first published in November 1915, formed the capstone of his general theory of relativity. For more than two years prior to that, Einstein had used field equations that are not generally covariant. He had even found a fallacious but ultimately profound argument purporting to show that the field equations for the metric field cannot be generally covariant. For reasons that need not be of concern here the argument is known as the "hole argument." The problem with generally covariant field equations, according to the hole argument, is that they allow one and the same source to produce what look like different metric fields, whereas the job of field equations is to determine uniquely what field is produced by a given source.

The escape from the hole argument is that on closer examination the different fields compatible with the same source turn out to be identical. The hole argument rests on the assumption that space-time points can be individuated and identified before any of their spatiotemporal properties are specified. Reject this assumption and the argument loses its force. The allegedly different metric fields only differ in that different featureless points take on the identity of the same space-time points. If space-time points cannot be individuated and identified independently of their spatiotemporal properties, this is no difference at all.

This comeback to the hole argument—a popular gloss on Einstein's so-called point-coincidence argument—amounts to a strong argument against the view that space-time is a substance, a container for the contents of space-time. The comeback shows that there are many ways to map spatiotemporal properties onto featureless points, all indistinguishable from one another. According to Leibniz's Principle of the Identity of Indiscernibles, all these indistinguishable ways must be physically identical. But then these points cannot be physically real, for that would make the indistinguishable ways of ascribing properties to them physically distinct.

The combination of the hole argument and the point-coincidence argument is thus seen to be a variant of an argument due to Leibniz himself against the Newtonian view that space is a substance. If space were a container, one version of the argument goes, God could have placed its contents a few feet to the left of where He actually placed it. But according to the Principle of the Identity of Indiscernibles, these two possible universes are identical and that leaves no room for the container, which could serve to distinguish the two. Einstein's fallacious argument against general covariance thus turned into an argument in support of a Leibnizian relational ontology of space-time.

Mach's principle.

During the period that Einstein accepted that the field equations of his theory were not generally covariant, he explored yet another strategy for eliminating absolute motion. This strategy was inspired by his reading of Ernst Mach's (1838–1916) response to Newton's famous bucket experiment. Set a bucket filled with water spinning. It will take the water some time to catch up with the rotation of the bucket. Just after the bucket starts rotating, the water will still be at rest and its surface will be flat. Once the water starts rotating, the water will climb up the sides of the bucket and its surface will become concave. Newton pointed out that this effect cannot be due to the relative rotation of the water with respect to the bucket. After all, the effect increases as the relative rotation decreases and is at its maximum when there is no relative motion at all because the water is rotating with the same angular velocity as the bucket. Newton concluded that the water surface becomes concave because of the water's rotation with respect to absolute space. Mach pointed out that there is another possibility: the effect could be due to the relative rotation of the water with respect to all other matter in the universe. Picture the earth, the bucket, and the water at the center of a giant spherical shell representing all other matter in the universe. If Mach were right, it would make no difference whether the bucket or the shell is set rotating: in both cases the water surface should become concave. According to Newton's theory, however, the rotating shell will have no effect whatsoever on the shape of the water's surface.

In 1913–1914, Einstein was convinced for a while that this was a problem not for Mach's analysis but for Newton's theory and that his own theory vindicated Mach's account of the bucket experiment. It only takes a cursory look at Einstein's calculations in support of this claim to see that this attempt to relativize rotation is a nonstarter. When Einstein calculated the metric field of a rotating shell at its center, he considered a shell rotating in Minkowski space-time. The rotating shell does produce a tiny deviation from the metric field of Minkowski space-time, but nothing on the order needed to make the water surface concave. What Einstein would have had to show to vindicate Mach is that the metric field produced by the rotating shell near its center mimics Minkowski space-time as seen from a rotating frame of reference. In that case the situation of the bucket at rest in this metric field would have been identical to that of the bucket rotating in the opposite direction in Minkowski space-time. But in order to calculate the metric field of a rotating shell, one needs to make some assumption about boundary conditions, that is, the values of the metric field as we go to spatial infinity. Rotation with respect to space-time rather than other matter thus creeps back in.

Einstein's flawed Machian account of Newton's bucket experiment receded into the background when he finally found generally covariant field equations for the metric field in November 1915. As is clear from Einstein's first systematic exposition of the theory in 1916, he still believed at this point that general covariance guarantees the relativity of arbitrary motion. The Dutch astronomer Willem De Sitter disabused him of this illusion in the fall of 1916 (see Stachel et al, vol. 18, pp. 351–357, for a summary of the debate that ensued between Einstein and De Sitter). De Sitter pointed out that Einstein used Minkowskian boundary conditions in his calculations of metric fields produced by various sources (such as the rotating shell discussed above) and thereby retained a remnant of absolute space-time. By early 1917, Einstein had worked out his response to De Sitter. He eliminated the need for boundary conditions at infinity simply by eliminating infinity. He proposed a model for the universe that is spatially closed. He chose the simplest model of this kind, which is static in addition to being closed. Such a static universe would collapse as a result of the gravitational attraction between its parts. Einstein therefore needed to add a term to his field equations that would provide the gravitational repulsion to neutralize this attraction. This term involved what has become known as the cosmological constant. In the late 1920s it became clear that the universe is expanding, in which case the gravitational attraction can be allowed to slow down the expansion and does not need to be compensated by a gravitational repulsion. In the wake of these developments, Einstein allegedly called the cosmological constant the biggest blunder of his life. In 1917, however, he felt he needed it to get rid of boundary conditions.

De Sitter quickly produced an alternative cosmological model that was allowed by Einstein's new field equations with the cosmological term. In this De Sitter world there is no matter at all. Absolute space-time thus returned with a vengeance. In reaction to De Sitter's model, Einstein formulated what came to be known as Mach's principle: the metric field is fully determined by matter and cannot exist without it. Einstein was convinced at this point that the addition of the cosmological term guaranteed that general relativity satisfies this principle, despite the apparent counterexample provided by the De Sitter solution.

Early in 1918, Einstein argued that the De Sitter world was not empty after all, but that hidden from view a vast amount of matter was tucked away in it. He concluded that general relativity satisfies Mach's principle and that this finally established complete relativity of arbitrary motion. All motion in general relativity is motion with respect to the metric field. But if the metric field can be reduced to matter, talk about such motion can be reinterpreted as a façon de parler about motion with respect to the matter generating the metric field. This certainly was a clever idea on Einstein's part, but by June 1918 it had become clear that the De Sitter world does not contain any hidden masses and is thus a genuine counterexample to Mach's principle. Another one of Einstein's attempts to relativize all motion had failed.

Einstein thereupon lost his enthusiasm for Mach's principle. He accepted that motion with respect to the metric field cannot always be translated into motion with respect to other matter. He also realized that motion with respect to the metric field or curved space-time is much more palatable than motion with respect to Newton's absolute space or Minkowski's absolute space-time anyway. The curved space-time of general relativity, unlike absolute space(-time), is a bona fide physical entity. It not only acts upon matter, like absolute space(-time), by telling matter how to move, but is also acted upon, as matter tells it how to curve (to borrow two slogans from Misner et al., p. 5). In his lectures in Princeton in May 1921, Einstein reformulated his objection against absolute space(-time) accordingly: it is something that acts but is not acted upon.

Einstein had a deeper reason to abandon Mach's principle. It was predicated on an antiquated nineteenth-century billiard-ball ontology. Einstein thought of matter as consisting of electromagnetic fields, in combination perhaps with gravitational fields. Mach's principle would thus amount to reducing one field to another. As can be inferred from a lecture delivered in Leiden in October 1920 (Einstein, 1983, pp. 1–24), Einstein came to accept that the metric field exists on a par with the electromagnetic field. Just as he had unified the electric and the magnetic field in special relativity and space-time and gravity in general relativity, he now embarked on the quest for a theory unifying the electromagnetic and the inertio-gravitational field. He would spend the rest of his life looking for such a theory.

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