Space-time distortions

Space and time are linked together, and the fabric of space-time can be distorted or warped by the presence of matter. Albert Einstein’s general theory of relativity reinterprets many diverse phenomena by replacing Newtonian gravity with distortions of space-time.

Overview

To the layperson, space is empty, but time is full of activity. Time is perceived as a flow, carrying consciousness from one present moment to the next. To a physicist, however, the terms space and time denote quite different concepts: space possesses physical properties and many levels of structure, and the flow of time is an illusion. Indeed, some theories hold that matter, rather than being located in space and time, is nothing more than disturbances in the fabric of space and time.

Although space and time can be perceived in radically different, separate ways, they are linked together by motion: average speed is defined as distance divided by time. The motion of material objects and light signals is described in relativity theory in terms of a single, unified reality called space-time. Anything that happens in some particular place at some particular time is called an event in space-time. Its location in space-time is specified by its three spatial coordinates and one time coordinate. The series of events that trace the where and when of an object is called its world line through space-time. Objects at rest or moving with a constant velocity follow straight world lines, while the world lines of accelerated objects are curved.

The structure of space can be described mathematically in terms of geometries. The geometry taught in most high schools is called Euclidean geometry. Developed (or at least described) by Euclid around 300 BCE, it deals with flat space. Some familiar features of flat space are that parallel lines never meet, even when extended infinitely far in both directions and that the sum of the interior angles of a triangle equals 180°. During the mid-nineteenth century, the Russian mathematician Nikolai Ivanovich Lobachevsky (1793-1856) and the German mathematician Georg Friedrich Riemann (1826-1866) developed self-consistent non-Euclidean geometries. In these new geometries, the sum of the angles of a triangle could be more or less than 180°, and locally parallel lines could converge or diverge over large distances. A physical space described by a non-Euclidean geometry is called a curved space, as opposed to a flat Euclidean space.

A familiar curved two-dimensional (2-D) space is the surface of a sphere, such as the Earth or a basketball. If one draws a triangle on the surface of a sphere, and that triangle is small (compared to the radius of the sphere), the angles of the triangle add up to 180° because the surface is nearly flat over small distances. However, if the triangle covers a large part of the surface, the angles add up to more than 180°. As an example, imagine doing this on the surface of the Earth. One side of the triangle is the 0° longitude line between the North Pole and the equator. The second side of the triangle is the 90° longitude line, also between the North Pole and the equator. The third side of the triangle is the equator from 0 to 90° longitude. Each of the three angles of this triangle is 90°, and thus, the sum of the angles is 270°. Similarly, lines of longitude are parallel where they cross the equator, but because they are located on a sphere, they eventually meet at both poles.

These results—the angles of a triangle adding up to more than 180°, and locally parallel lines converging—are characteristic of, and can be used as tests for, the geometry developed by Riemann for positively curved spaces. The opposite properties—the angles of a triangle adding up to less than 180°, and locally parallel lines diverging—are characteristic of, and can be used as tests for, the geometry developed by Lobachevsky for negatively curved spaces.

The mathematics of curved space geometries took on new physical significance in 1915 when Albert Einstein (1879-1955) put forth his general theory of relativity. One consequence of it is that four-dimensional (4-D) space-time, first introduced by Einstein in his 1905 special theory of relativity, is curved by the presence of matter. Gravity appears to exist only because mass warps space-time in its vicinity. Physicist John Wheeler, who contributed enormously to relativity theory, provided this succinct description of the relation between matter and space-time: matter tells space-time how to curve, and the curvature of space-time tells matter and light how to move. Where space-time is not flat, Euclidean geometry fails to provide an adequate description; non-Euclidean geometries must be used instead.

The fundamental postulate at the heart of general relativity is the principle of equivalence: the effects of a uniform gravitational field and the effects of a constant acceleration are indistinguishable. In a closed rocket accelerating through space at 9.8 meters per second per second (m/s²), there is no experiment that an astronaut could perform that would distinguish this situation from one in which the rocket is at rest on the Earth’s surface (where the acceleration of gravity is 9.8 m/s²). Einstein realized that if he could express this equivalence in a mathematical form, then he could relate both gravitation and acceleration to the curvature of space-time, thereby providing a geometric explanation of gravity. Non-Euclidean geometry provided the mathematical formalism to describe how the distribution of mass warps space-time. Einstein’s field equations of general relativity enable one to compute, in principle, the curvature or warpage in space and time due to a given mass distribution.

Naturally, the bending of space-time profoundly affects the world lines of objects. As space-time warps, the world lines warp as well since they are constrained to follow geodesics (locally straight lines) through a curved space. When an object moves on a world line that is curved, the object is being accelerated; that is, its motion is not uniform. From the perspective of general relativity, gravity is not to be considered as a force but rather as an effect of geometry—a distortion of space-time. The presence of mass will distort the fabric of space-time. What is experienced as gravitation is the warping of space-time because of mass.

Applications

The idea that matter distorts space-time led Einstein to make several remarkable predictions. In Euclidean geometry, the shortest distance between two points is a straight line. In a curved space, the shortest path is curved because space itself is curved. In order to gain a clearer understanding of this concept, consider that New York and Tokyo have similar latitudes but that the shortest distance between them, along the Earth’s surface, passes close to the North Pole. Although incomprehensible on a flat map, this fact is easily visualized on the curved surface of a globe.

Similarly, a light beam passing near a massive object will follow a curved path because the massive object distorts space-time in its vicinity. The bending of light in the presence of a massive object also is mandated by the equivalence principle. In a reference frame accelerating upward, a transverse light beam would appear to deflect toward the upwardly accelerating floor. Since the equivalence principle requires that there be no distinction between acceleration and gravitational fields, light has to be deflected toward a source of gravity—that is, a massive object that warps space-time around it.

In 1911, Einstein predicted that a ray of starlight passing near the Sun would be bent by the warpage of space-time around the Sun. (Stars can be seen near the Sun during total solar eclipses when the sky is darkened sufficiently.) Although the predicted deflection of starlight just grazing the Sun’s limb (edge) is only 1.75 arc seconds, it was first confirmed at the 1919 total solar eclipse, and later at many subsequent total solar eclipses since then. The effect has also been verified at radio wavelengths when the Sun occults (passes in front of) the radio source 3C279 every October 9. A similar effect, called gravitational lensing, has been observed when light from very distant galaxies is bent when it passes near massive, closer galaxies.

A previously known but unsatisfactorily explained effect—the advance of Mercury’s perihelion—is also explained by the warpage of space-time near the Sun. Mercury’s orbit around the Sun is slowly precessing (or pivoting) so that Mercury reaches perihelion (the point when it is closest to the Sun) slightly later with each orbit. Observations of Mercury’s motion were sufficiently accurate that its perihelion advance was known by the mid-nineteenth century. Almost the entire effect could be explained by the gravitational perturbations of the other known planets on Mercury, but a small residual was attributed to the gravitational perturbation of an as-yet undiscovered planet (given the name Vulcan) inside the orbit of Mercury. Vulcan was searched for, and some claims of discovery were even made, though none was confirmed. In 1915, Einstein explained how the warpage of space-time around the Sun causes Mercury’s orbit to pivot forward (or precess); the precession predicted by general relativity accounts almost precisely for the residual that once was attributed to Vulcan. This (along with the bending of starlight passing near the Sun) was one of the first predictions of general relativity to be confirmed observationally. Since then, residual perihelion advances have been measured for Venus, Earth, and the asteroid Icarus; all agree within their observational errors with the predicted values calculated from general relativity.

Since space and time are intrinsically bound together, the effect of mass on the geometry of space-time means that time will be warped as well as space. Einstein predicted that time would be slowed by the warped space-time in the vicinity of a mass. Thus, clocks at the Earth’s surface should run slightly slower than clocks at higher altitudes, which are farther from Earth’s mass, where the warpage is less. The effect is small (about one part in 10¹³ for each vertical kilometer of altitude), but this minuscule effect was first measured (indirectly) in 1960 by the physicists R. V. Pound and G. A. Rebka at Harvard, again verifying a prediction of general relativity.

Context

In 1827, the German mathematician Carl Friedrich Gauss published a paper in which he recorded his measurements of the interior angles of a large triangle formed by three mountain peaks. His measurement was an attempt to ascertain whether space was Euclidean. The experiment was inconclusive: he obtained 180° within the experimental accuracy of his measuring devices. Gauss’s experiment may have been the first attempt to test the long-established assumption that the universe is best described by Euclidean geometry.

By the middle of the nineteenth century, two other self-consistent geometries had been devised: the geometry of negatively curved space by the Russian mathematician Nikolai Ivanovich Lobachevsky and the geometry of positively curved space by the German mathematician Georg Friedrich Riemann. Until 1915, however, when Einstein published his general theory of relativity, most mathematicians and physicists assumed that non-Euclidean geometries were mathematical curiosities but had little to do with physical reality. In Einstein’s theory, space-time is curved by the presence of mass, and gravity exists only because mass gives space a non-Euclidean character.

Several months after Einstein’s theory was published, the German astrophysicist Karl Schwarzschild found rigorously exact solutions to Einstein’s field equations for two different cases: an ideal point mass and a finite spherical mass. The first case predicted that, at a relatively small radius from the mass point, some of the mathematical terms become infinite. This condition represents such an intense warping of space-time that any signal (whether matter or electromagnetic radiation) within this boundary would be unable to escape. This radius, called the Schwarzschild radius, defines a surface (called the event horizon) such that any mass or energy within this surface is forever trapped. Most physicists of the time, Einstein included, believed that it would be impossible for any real object to contract to such a small size that its mass would be contained within this surface. (For example, an object with the mass of the Sun would have to shrink within a radius of about three kilometers, and an object with the mass of the Earth would have to shrink within a radius of about nine millimeters.) In 1963, Roy Kerr developed a new exact solution to Einstein’s equations for a rotating mass, and the Schwarzschild solutions were seen to be special cases of Kerr’s solution. Since matter and energy could cross the event horizon in the inward direction but nothing could escape from inside the event horizon, the term black hole was coined.

By the 1970s, indirect evidence had begun to accumulate indicating the existence of just such black holes, and there now is compelling evidence for at least two classes of black holes. Stellar-mass black holes form when massive stars exhaust all the fuels they used for nuclear fusion reactions and finally explode as supernovae. Supermassive black holes have masses of millions of solar masses or more and are thought to exist at the centers of many galaxies, including our own Milky Way galaxy.

The application of the equations of general relativity to cosmology and the structure of the universe allows for the universe to be flat, negatively curved, or positively curved. Any curvature is related to how the average density of matter and energy in the universe compares to a value called the critical density, which is equivalent to about three hydrogen atoms per cubic meter.

In the simplest form of relativistic cosmology, any possible curvature of the universe is linked to its ultimate fate. If the average density of matter and energy equals the critical density, the universe is flat, and it will continue to expand but slow down in such a way that it just barely expands forever. In this case, the universe is said to be critically open. If the average density of matter and energy is less than the critical density, the universe is negatively curved, and it will easily expand forever, slowing just a little. In this case, it is said to be open. If the average density of matter and energy is greater than the critical density, the universe is positively curved. It will expand to some maximum size and then contract on itself in what some call the big crunch. In this case, it is said to be closed.

Various measures (mostly very indirect) indicate that the universe is extremely close to being flat, but the amount of matter and energy observed is substantially less than the critical density. This observation has led to the notion that most of the mass in the universe is in the form of dark matter, which has not yet been detected. Moreover, it seems as if only a small part of the unobserved dark matter can be in the form of conventional matter; most of it must be something exotic and as yet unknown.

A completely unexpected discovery in the mid-1990s was that the expansion of the universe seemed to be accelerating. This necessitates using a more general form of relativistic cosmology that includes an extra term involving a parameter called the cosmological constant (or some other modification similar to it). To account for the accelerating expansion of the universe, some unknown energy is required. Called dark energy, it may represent 70 percent or more of the total matter and energy in the universe.

General relativity, with its distortions of space-time, has thus become an important tool for understanding the origin, structure, and future of the universe and its contents. Although its basic structure has remained unaltered since 1915, it continues to find applications in such diverse areas as the precession of orbits, the bending of light, gravitational redshifts, black holes, cosmology, and gravitational waves, which were discovered in the twenty-first century to permeate the universe at low frequencies. It is truly amazing that an abstract theory concerning the warping of space-time has turned out to be so powerful and useful.

Bibliography

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