Understanding relativity

Summary: Albert Einstein’s theory of relativity is one of the most well-known theories in physics and helps describe the nature of the universe.

Albert Einstein’s theory of relativity forms one of the two pillars of modern physics, the other being quantum mechanics. It consists of two parts: the special theory of relativity from 1905, and the general theory of relativity from 1915, which both rely on significant mathematics.

The special theory of relativity describes how space and time are perceived by observers in different inertial systems. Albert Einstein derived this theory from a single physical principle of relativity. It was discovered in 1632 by Galileo Galilei that the laws of mechanics are the same in all inertial systems—a discovery, known as “Galileo’s principle of relativity,” that constituted a radical break with the prevailing Aristotelian physics. Einstein’s principle of relativity generalized this concept to all laws of nature, including Maxwell’s laws of electromagnetism, which govern the propagation of light. It thus follows from Einstein’s principle of relativity that the speed of light is the same in all inertial systems, a central result in the theory of relativity. Prior to Einstein, it was believed that light propagates through a luminiferous aether in the same way as sound propagated through air, but all attempts to measure the speed of the Earth relative to this aether, such as the Michelson–Morley experiment in 1887, failed. Special relativity explained the negative results of these experiments and made the aether hypothesis superfluous.

The general theory of relativity unifies special relativity with Isaac Newton’s law of universal gravity. Its basis is Einstein’s equivalence principle, according to which an accelerated system of reference (such as a so-called Einstein elevator) is indistinguishable from a system at rest in a gravitational field. Mathematically, Einstein’s field equations describe how the presence of mass, energy, and momentum gives rise to a curvature of space and time. Although this idea has little significance in weak gravitational fields, such as that of the Earth, general relativity is essential in the study of the universe as a whole. For example, Karl Schwarzschild in 1915 found an exact solution to Einstein’s equations that explains the existence of black holes.

The many surprising consequences of the theory of relativity have been described in numerous popularizations, most notably by George Gamow. Einstein’s theory must not be confused with the various relativist positions in philosophy, such as aesthetic, moral, cultural, or cognitive relativism.

Special Relativity

The Lorentz transformation forms the basis of the special theory of relativity. It is a set of equations describing how to translate suitably chosen coordinates of space and time between two inertial systems (S) and (S′) moving with the speed (v) relative to one another:

where c denotes the speed of light of 299,792,458 meters per second, and the dimensionless number

is the so-called Lorentz factor. In 1908, Hermann Minkowski gave a mathematical description of the Lorentz transformation as a rotation of the coordinate axes in four-dimensional space-time.

When v is much smaller than c, the Lorentz factor is close to 1, and the Lorentz transformation reduces to the classical Galilean transformation. When v approaches c, however, the Lorentz transformation has a number of consequences that radically contradict classical physics as well as common sense. For example, clocks in motion are slowed down (called “relativistic time dilation”), objects in motion are contracted in the direction of movement (called “relativistic length contraction”), and clocks in motion that are seen as synchronized by an observer moving with the clocks are seen as nonsynchronized by an observer at rest (called “relativity of simultaneity”).

It is another consequence of special relativity that no material objects—or signals of any kind—can travel faster than light. This “speed limit” exists because anything traveling faster than light relative to one observer would appear to be traveling backwards in time relative to another observer, thus leading to paradoxes regarding cause and effect. There is a quantum-mechanical phenomenon, the so-called Einstein–Podolsky–Rosen paradox, that seems to contradict this principle. According to quantum mechanics, the wave function of two entangled particles is affected by a measurement of the state of one of the particles, causing an instantaneous change to the state of the other, even if the two particles are located in different galaxies. But this phenomenon, which has since been verified experimentally, does not really contradict relativity since it cannot be used to transmit information from one galaxy to the other.

Special relativity dictates that mass and energy are connected by the equation E = mc², undoubtedly the most famous formula in all of physics. Any particle with mass m has a rest energy given by this equation. If the same particle is accelerated to the speed v, its energy is multiplied by the Lorentz factor , and its kinetic energy is found as the difference between total energy and rest energy, expressed algebraically as

The approximation, valid for v much smaller than c, equals the expression for kinetic energy in classical mechanics. This formula shows that it would require an infinite amount of energy to accelerate a particle with positive mass to the speed of light.

General Relativity

Einstein noted that special relativity implies that space appears to be curved, or “non-Euclidean,” to observers in accelerated systems (for example, on a rotating disc) and inferred from the equivalence principle that the same must be true in gravitational fields. However, after realizing this fundamental principle in 1907, it took him eight years to find the field equations that describe the exact curvature of space-time. The idea that physical space might be curved was not new. Already in 1823, Carl Friedrich Gauss investigated this question empirically by measuring the sum of angles of a triangle formed by three mountaintops but found no curvature. Bernhard Riemann further developed the mathematics of curved space in 1854 and this work would become an essential part of Einstein’s theory.

General relativity predicts that a body falling freely in a gravitational field, such as the Earth in its orbit around the sun, follows a “geodesic” in curved space-time. This geodesic is called the body’s “world-line.” In a curved space, geodesics are the least curved lines, in the same way as the equator is a least curved line on the surface of Earth. Although the predictions of general relativity are nearly the same as those of classical mechanics for bodies in weak gravitational fields, the interpretation of gravity is radically different: whereas classical mechanics explains the elliptical orbit of the Earth as a consequence of a gravitational force emanating from the sun, general relativity postulates that the mass of the sun gives rise to a curvature of space-time, and that the world-line of Earth is in fact a geodesic.

It is a consequence of general relativity that clocks in gravitational fields are slowed down. This effect is called “gravitational time dilation.” For a clock at rest in the gravitational field of Earth, the dilation factor is

where G is Newton’s gravitational constant, M is the mass of Earth, and r is the distance between the clock and the center of Earth.

Proofs and Applications of Relativity

Einstein showed in 1915 that general relativity explains the perihelion precession of the planet Mercury. This phenomenon, which had mystified astronomers since its discovery in 1859, is that the elliptical orbit of Mercury rotates around the sun with 43 arc seconds per century.

Also in 1915, Einstein predicted that light emitted from distant stars is deflected when passing through the gravitational field of the sun. Although this effect had previously been derived from Newtonian gravity alone, Einstein showed that the angle of deflection following from general relativity is twice the angle following from classical physics. Einstein’s prediction was confirmed dramatically by Arthur Eddington during the total solar eclipse of May 29, 1919.

Contrary to quantum mechanics, the technological implementations of which are ubiquitous, relativity has few practical applications. One notable exception is the global positioning system (GPS). GPS satellites revolve around the Earth twice per sidereal day at a height of about 20,000 kilometers (12,400 miles) and with a speed of about 4 kilometers (2.5 miles) per second. Because of the speed and altitude, the atomic clocks aboard the satellites are subject both to relativistic time dilation and to a reduced gravitational time dilation.

The first effect amounts to a loss of 7 microseconds per day, the second to a gain of 45 microseconds per day. In total, therefore, the atomic satellite clocks gain 38 microseconds per day relative to clocks on the ground. Failure to take these relativistic effects into account would render GPS useless since the resulting positional error would accumulate to 11 kilometers (6.8 miles) per day.

Bibliography

Einstein, Albert. Relativity: The Special and General Theory. New York: Henry Holt, 1920.

Feynman, Richard, Robert Leighton, and Matthew Sands. The Feynman Lectures on Physics. Reading, PA: Addison-Wesley, 1964.

Gamow, George. Mr. Tompkins in Wonderland. New York: Macmillan, 1946.

Grøn, Oyvind, and Sigbjorn Herv. Einstein’s General Theory of Relativity: With Modern Applications in Cosmology. New York: Springer Science+Business Media, 2007.

Møller, Christian. The Theory of Relativity. Oxford, Egland: Oxford University Press, 1952.

Russell, Bertrand. The ABC of Relativity. London: Kegan Paul, Trench, Trubner, 1925.

Ungar, Abraham. Analytic Hyperbolic Geometry and Albert Einstein’s Special Theory of Relativity. Singapore: World Scientific Publishing, 2008.