# How are space and time interconnected

## Einstein's Spacetime

## Special Relativity

Physics at the end of the nineteenth century found itself in crisis: there were perfectly good theories of mechanics (Newton) and electromagnetism (Maxwell), but they did not seem to agree. Light was known to be an electromagnetic phenomenon, but it did not obey the same laws of mechanics as matter. Experiments by Albert A. Michelson (1852-1931) and others in the 1880s showed that it always traveled with the same velocity, regardless of the speed of its source. Older physicists struggled with this contradiction in various ways. In 1892 George F. FitzGerald (1851-1901) and Hendrik A. Lorentz (1853-1928) independently found that they could reconcile theory and experiment if they postulated that the detector apparatus was changing its size and shape in a characteristic way that depended on its state of motion. In 1898, J. Henri Poincaré (1854-1912) suggested that intervals of time, as well as length, might be observer-dependent, and he even speculated (in 1904) that the speed of light might be an "unsurpassable limit".

Einstein in 1905

None of these eminent physicists, however, put the whole story together. That was left to the young Albert Einstein (1879-1955), who already began approaching the problem in a new way at the age of sixteen (1895-6) when he wondered what it would be like to travel along with a light ray. By 1905 he had shown that FitzGerald and Lorentz's results followed from one simple but radical assumption: the laws of physics *and the speed of light* must be the same for all uniformly moving observers, regardless of their state of relative motion. For this to be true, space and time can no longer be independent. Rather, they are "converted" into each other in such a way as to keep the speed of light constant for all observers. (This is why moving objects appear to shrink, as suspected by FitzGerald and Lorentz, and why moving observers may measure time differently, as speculated by Poincaré.) Space and time are *relative* (i.e., they depend on the motion of the observer who measures them) — and light is more fundamental than either. This is the basis of Einstein's theory of special relativity ("special" refers to the restriction to uniform motion).

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## The Fourth Dimension

Minkowski

Lightcone diagram showing the worldline

of a moving observer

Einstein did not quite finish the job, however. Contrary to popular belief, he did *not* draw the conclusion that space and time could be seen as components of a single four-dimensional spacetime fabric. That insight came from Hermann Minkowski (1864-1909), who announced it in a 1908 colloquium with the dramatic words: "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality".

Four-dimensional Minkowski spacetime is often pictured in the form of a two-dimensional lightcone diagram, with the horizontal axes representing "space" *(x)* and the vertical axis "time" *(ct)*. The walls of the cone are defined by the evolution of a flash of light passing from the past (lower cone) to the future (upper cone) through the present (origin). All of physical reality is contained within this cone; the region outside ("elsewhere") is inaccessible because one would have to travel faster than light to reach it. The trajectories of all real objects lie along "worldlines" inside the cone (like the one shown here in red). The apparently static nature of this picture, in which history does not seem to "happen" but is rather "already there", has given writers and philosophers a new way to think about old issues involving determinism and free will.

Einstein initially dismissed Minkowski's four-dimensional interpretation of his theory as "superfluous learnedness" (Abraham Pais, *Subtle is the Lord...*, 1982). To his credit, however, he changed his mind quickly. The language of spacetime (known technically as tensor mathematics) proved to be essential in deriving his theory of general relativity.

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## The Equivalence Principle

Einstein's happiest thought (1907): "For an observer falling freely from

the roof of a house, the gravitational field does not exist" (left).

Conversely (right), an observer in a closed box—such as an elevator or

spaceship—cannot tell whether his weight is due to gravity or acceleration.

Soon after completing his special theory, Einstein had the "happiest thought of his life" (1907). It came while he was sitting in his chair at the patent office in Bern and wondering what it would be like to try to drop a ball while falling off the side of a building. Einstein realized that a person who accelerates downward along with the ball will not be able to detect the effects of gravity on it. An observer can "transform away" gravity (at least in the immediate neighborhood) simply by moving to this accelerated frame of reference — no matter what kind of object is dropped. Gravitation is (locally) equivalent to acceleration. This is the principle of equivalence.

To understand how remarkable the equivalence principle really is, imagine how it would be if gravity worked like other forces. If gravity were like electricity, for example, then balls with more charge would be attracted to the earth more strongly, and hence fall down more quickly than balls with less charge. (Balls whose charge was of the same sign as the earth's would even "fall" upwards.) There would be no way to transform away such effects by moving to the same accelerated frame of reference for all objects. But gravity is "matter-blind" — it affects all objects the same way. From this fact Einstein leapt to the spectacular inference that gravity does not depend on the properties of matter (as electricity, for example, depends on electric charge). Rather the phenomenon of *gravity must spring from some property of spacetime*.

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## Gravity as Curved Spacetime

Einstein eventually identified the property of spacetime which is responsible for gravity as its *curvature*. Space and time in Einstein's universe are no longer flat (as implicitly assumed by Newton) but can pushed and pulled, stretched and warped by matter. Gravity feels strongest where spacetime is most curved, and it vanishes where spacetime is flat. This is the core of Einstein's theory of general relativity, which is often summed up in words as follows: **"matter tells spacetime how to curve, and curved spacetime tells matter how to move"**. A standard way to illustrate this idea is to place a bowling ball (representing a massive object such as the sun) onto a stretched rubber sheet (representing spacetime). If a marble is placed onto the rubber sheet, it will roll toward the bowling ball, and may even be put into "orbit" around the bowling ball. This occurs, not because the smaller mass is "attracted" by a force emanating from the larger one, but because it is traveling along a surface which has been deformed by the presence of the larger mass. In the same way gravitation in Einstein's theory arises not as a force propagating *through* spacetime, but rather as a feature *of* spacetime itself. According to Einstein, your weight on earth is due to the fact that your body is traveling through warped spacetime!

Computer animation showing | Computer animation showing |

While intuitively appealing, however, the rubber-sheet picture has its limitations. Mostly, these have to do with the fact that it allows us to visualize the spatial aspect of Einstein's theory, but not the temporal one. To see this, we need only remember that Newtonian gravity must be *approximately* valid, whatever Einstein says, and Newton tells us that bodies move in straight lines unless acted upon by a force. Why, then, do the orbits of planets around the sun on the rubber sheet appear so far from straight, if there is no attracting force reaching out through spacetime to tug on them? The answer is that planetary trajectories *are* very nearly straight — *in spacetime*, not space. The worldline of the earth, for example, resembles a stretched-out spiral whose width in space is only one astronomical unit, but whose length in the time direction is measured in lightyears! Another way to appreciate the importance of the "time" in "spacetime" is to apply the equivalence principle and ask whether the fact that we experience a gravitational field on the earth's surface is "equivalent" to stating that the earth's surface is continually accelerating outward. Obviously not, for we do not observe the earth to grow larger! The trouble is that, in speaking of the earth's surface, we have again lapsed into thinking of acceleration in spatial terms. On earth, where speeds are small compared to the speed of light and the gravitational field is weak, it turns out that nearly all of our weight arises due to the warping of *time*, rather than space. What this means in practice is that gravity on earth is "equivalent" to acceleration mostly in the sense that clocks on the surface run more slowly than clocks in outer space.

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## General Relativity

General relativity is based physically on the equivalence principle, but the theory also has a second, more mathematical foundation. Known as the principle of general covariance, it is the requirement that the law of gravitation be the same for all observers — even accelerating ones — regardless of the coordinates in which it is described. (It is for this reason that Einstein named his new theory "general", as opposed to "special" relativity — he dropped the earlier restriction to uniformly moving observers.) This proved to be the most difficult challenge that Einstein ever faced. As he later said, to express physical laws without coordinates is like "describing thoughts without words". Einstein was obliged to master the abstract mathematics of surfaces and their description in terms of tensors, a field pioneered by Carl Friedrich Gauss (1777-1855) and generalized to higher dimensions and more abstract spaces by Georg Friedrich Bernard Riemann (1826-1866). In this labor he was aided above all by his friend the mathematician Marcel Grossmann (1878-1936). Another mathematician named David Hilbert (1862-1943) nearly beat him to his final equations.

Einstein in 1916

But general relativity is above all Einstein's achievement, and the phrase "Einstein's spacetime" is entirely appropriate. No theory of comparable significance before or since is more nearly due to the struggle of a single scientist. At the end of 1915 Einstein wrote to a friend that he had succeeded at last, and that he was "content but rather worn out". He later described this period as follows: "The years of searching in the dark for a truth that one feels but cannot express, the intense desire and the alternations of confidence and misgiving until one breaks through to clarity and understanding, are known only to those who have themselves experienced them".

## Relational or Absolute?

In 1918, Einstein described Mach's principle as a philosophical pillar of general relativity, along with the physical principle of equivalence and the mathematical pillar of general covariance. This characterization is now widely regarded as wishful thinking. Einstein was undoubtedly inspired by Mach's relational views, and he hoped that his new theory of gravitation would "secure the relativization of inertia" by binding spacetime so tightly to matter that one could not exist without the other. In fact, however, the equations of general relativity are perfectly consistent with spacetimes that contain no matter at all. Flat (Minkowski) spacetime is a trivial example, but empty spacetime can also be curved, as demonstrated by Willem de Sitter in 1916. There are even spacetimes whose distant reaches rotate endlessly around the sky relative to an observer's local inertial frame (as discovered by Kurt Gödel in 1949). The bare existence of such solutions in Einstein's theory shows that it cannot be Machian in the strict sense; matter and spacetime remain logically independent. The term "general relativity" is thus something of a misnomer, as pointed out by Hermann Minkowski and others. The theory does *not* make spacetime more relative than it was in special relativity. Just the opposite is true: the absolute space and time of Newton are retained. They are merely amalgamated and endowed with a more flexible mathematical skeleton (the metric tensor).

Nevertheless, Einstein's theory of gravity represents a major swing back toward the relational view of space and time, in that it answers the objection of the ancient Stoics. Space and time *do* act on matter, by guiding the way it moves. And matter *does* act back on spacetime, by producing the curvature that we feel as gravity. Beyond that, matter can act on spacetime in a manner that is very much in the spirit of Mach's principle. Calculations by Hans Thirring (1888-1979), Josef Lense (1890-1985) and others have shown that a large rotating mass will "drag" an observer's inertial reference frame around with it. This is the phenomenon of frame-dragging, whose existence Gravity Probe B is designed to detect. The same calculations suggest that, if the entire contents of the universe were to rotate, our local inertial frame would undergo "perfect dragging" — that is, we would not notice it, because we would be rotating too! In that sense, general relativity is indeed nearly as relational as Mach might have wished. Some physicists (such as Julian Barbour) have gone further and asserted that general relativity is in fact perfectly Machian. If one goes beyond classical physics and into modern quantum field theory, then questions of absolute versus relational spacetime are rendered anachronistic by the fact that even "empty space" is populated by matter in the form of virtual particles, zero-point fields and more. Within the context of Einstein's universe, however, the majority view is perhaps best summed up as follows: S*pacetime behaves relationally but exists absolutely*.

*James Overduin, November 2007*

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