Two widely accepted theories proposed by Albert Einstein to account for departures from Newtonian mechanics. The special theory, of 1905, refers to nonaccelerated frames of reference, while the general theory, of 1915, extends to accelerated systems.

The special theory.

For Galileo and Newton, all uniformly moving frames of reference (Galilean frames) are equivalent for describing the dynamics of moving bodies. There is no experiment in dynamics that can distinguish between a stationary laboratory and a laboratory that is moving at uniform velocity. Einstein’s special theory of relativity takes this notion of equivalent frames one step further: he required all physical phenomena, not only those of dynamics, to be independent of the uniform motion of the laboratory.

When Einstein published his special theory, he realized that it could explain the apparent lack of experimental evidence for an ether, which was supposed to be the medium required for the propagation of electromagnetic waves (see Michelson–Morley experiment). Einstein recognized that the existence of an ether would render invalid any equivalent relativity principle for electromagnetic phenomena; i.e. uniform movement of the laboratory through the ether would lead to measurable differences in the propagation of electromagnetic waves in vacuo. Since no experimental evidence of the ether was forthcoming, Einstein was encouraged to continue his search for a relativity principle that encompassed all physical phenomena. Since light is a physical phenomenon, its propagation in vacuo could not be used to distinguish between uniformly moving frames of reference. Therefore in all such frames the measured speed of light in vacuo must be the same.

This conclusion has some important consequences for the nature of space and time. In his popular exposition of 1916, Einstein illustrated these consequences with thought experiments. In one such experiment, he invites the reader to imagine a very long train travelling along an embankment with a constant velocity ν‎ in a given direction (see diagram).

Relativity. Einstein’s thought experiment.

Observers on the train use it as a rigid reference body, regarding all events with reference to the train. Einstein posed a simple question: Are two events, which are simultaneous relative to the railway embankment, also simultaneous relative to the observer on the train? For example, two lightning strokes strike the embankment at points A and B simultaneously with respect to the embankment, so that an observer at M (the mid-point of the line AB) will record no time lapse between them. However, the events A and B also correspond to positions A and B on the train. M′ is the mid-point of the distance AB on the moving train. When the flashes occur, from the point of view of the embankment, M′ coincides with M. However, M moves with speed ν‎ towards the right and therefore hastens towards the beam of light coming from B, while moving on ahead of the beam from A. The observer at M′ would not agree with the observer at M on the simultaneity of the events A and B because the beam of light from B will be seen to be emitted before the beam of light at A.

At first sight there may seem to be a problem here. If the observer at M′ is ‘hastening towards the beam of light from B’, is this not equivalent to saying that the beam of light is travelling towards M′ at a combined speed of ν‎+c, where c is the speed of light in vacuo? The resolution of this problem is the basis of special relativity. According to Einstein, the moving observer at M′ must measure the speed of light in vacuo to be c, since there can be no experiment that distinguishes the train’s moving frame from any other Galilean frame. It is therefore the concept of time measurement that requires revision; that is, the time required for a particular event to occur with respect to the train cannot have the same duration as the same event when judged from the embankment.

This remarkable result also has implications for the measurement of spatial intervals. The measurement of a spatial interval requires the time coincidence of two points along a measuring rod. The relativity of simultaneity means that one cannot contend that an observer who traverses a distance x m per second in the train traverses the same distance x m also with respect to the embankment in each second. In trying to include the law of propagation of light into a relativity principle, Einstein questioned the way in which measurements of space and time in different Galilean frames are compared. Place and time measurements in two different Galilean frames must be related by a transformation preserving the relativity principle that every ray of light has a velocity of transmission c relative to observers in both frames.

Transformations that preserve the relativity principle are called Lorentz transformations. The form of these transformations looks complicated at first (see diagram). However, they arise from the simple requirement that there can be no experiment in dynamics or electromagnetism that will distinguish between two different Galilean frames of reference. These transformations suggest that observers in the two frames will not agree on measurements of length made in the y-direction. Indeed, the duration of intervals of time cannot be agreed upon in the two frames. This is exactly what was suggested in Einstein’s thought experiment on the train. Simple manipulations lead to the following formulae, which relate lengths and time intervals in the x′, y′, z′ frame to their equivalent quantities in the x, y, z frame:

Relativity. Lorentz transformations.

$$\Delta l=\Delta l^{\mathrm{\prime }}{(1-v^2/c^2)}^{1/2}$$

and

$$\Delta t=\Delta t^{\mathrm{\prime }}{(1-v^2/c^2)}^{-1/2}$$

where Δ‎l and Δ‎t are respectively intervals in space and time. Motion therefore leads to a length contraction of the x′, y′, z′ lengths with respect to the x, y, z lengths. Similarly, the equation relating the time intervals in the two frames leads to a time dilation in the x′, y′, z′ system compared to the x, y, z system. From these expressions it is clear that the velocity c plays the part of a limiting velocity, which can neither be reached nor exceeded by any material body.

The velocity c is often said to be the limiting velocity for the transfer of information in the universe. Faster-than-light signals violate causality when taken to their logical conclusions. The universe, therefore, according to the special theory of relativity, updates itself at the maximum speed of light c; that is, any local changes in the properties within a region of space are not communicated to the rest of the universe instantaneously. Rather, the universe is updated through a wave of reality, which emanates at speed c from the region in which the change took place.

The general theory.

In his special theory, Einstein updated the Galilean principle of relativity by including electromagnetic phenomena. Galileo and, later, Newton were well aware that no experiment in the dynamics of moving bodies could distinguish between frames of reference moving relative to each other at constant velocity (Galilean frames). If two Galilean frames move with respect to each other at uniform velocity, no experiment could determine which frame was in absolute motion and which frame was at absolute rest.

This is the basic principle of Einstein’s special theory of relativity. However, Einstein was not content with the apparent absolute status conferred to accelerating frames by the behaviour of bodies within them. Einstein sought a general principle of relativity that would require all frames of reference, whatever their relative state of motion, to be equivalent for the formulation of the general laws of nature. In his popular exposition of 1916, Einstein explains this by describing the experiences of an observer within a railway carriage that is decelerating. In his own words:

‘At all events it is clear that the Galilean law does not hold with respect to the nonuniformly moving carriage. Because of this, we feel compelled at the present juncture to grant a kind of absolute physical reality to nonuniform motion, in opposition to the general principle of relativity.’

Once again it was one of Galileo’s observations that provided the starting point for the formulation of Einstein’s ideas. Galileo observed that bodies moving under the sole influence of a gravitational field acquire an acceleration that does not depend upon the material or the physical state of the body. Einstein realized that this property of gravitational fields implied equivalence between gravity and accelerating frames of reference. This equivalence, which became the basis of his general theory of relativity, is well illustrated by one of Einstein’s thought experiments. Imagine an elevator, so far removed from stars and other large masses that there is no appreciable gravitational field. An observer inside the elevator is equipped with the appropriate apparatus and uses the elevator as a reference frame. Initially, if the elevator is a Galilean frame, the observer would feel weightless with only inferences from decorations inside the elevator to distinguish between ‘up’ and ‘down’. However, if a rope attached to the top of the elevator were to be pulled with a constant acceleration of 9.81 m s⁻², the observer would detect this acceleration as a force reaction on the floor of the elevator. The experiences of the observer in the elevator are equivalent to the experiences of an observer in an elevator in the earth’s gravitational field of strength 9.81 N kg⁻¹. Moreover, the force reaction at the feet of the observer in the accelerating frame is due to the observer’s inertial mass (the mass that represents the reluctance of the observer’s body to accelerate under the influence of a force).

An observer in the earth-bound elevator would feel the same force reaction at the floor of the elevator, but for this observer the force is due to the influence of the earth’s gravitational field on the observer’s gravitational mass. Guided by this example, Einstein realized that his extension of the principle of relativity to include accelerations implies the equality of inertial and gravitational mass, which had been established experimentally by Lólánd Eötvös (1848–1919) in 1888.

These considerations have significant implications for the nature of space and time under the influence of a gravitational field. Another of Einstein’s thought experiments illustrates these implications. Imagine a Galilean frame of reference K from which an observer A takes measurements of a non-Galilean frame K′, which is a rotating disk inhabited by an observer B. A notes that B is in circular motion and experiences a centrifugal acceleration. This acceleration is produced by a force, which may be interpreted as an effect of B’s inertial mass. However, on the basis of the general principle of relativity, B may contend that he is actually at rest but under the influence of a radially directed gravitational field.

A comparison of time-measuring devices placed at the centre and edge of the rotating disk would show a remarkable result. For although the devices would both be at rest with respect to K′, the motion of the disk with respect to K would lead to a time dilation at the edge with respect to measured time at the centre. It follows that the clock at the disk’s periphery runs at a permanently slower rate than that at the centre, i.e. as observed from K. The same effect would be noted by an observer who is sitting next to the clock at the centre of the disk. Thus, on the disk, or indeed in any gravitational field, a timing device will run at different rates depending on where it is situated.

The measurement of spatial intervals on the rotating disk will also incur a similar lack of definition. Standard measuring rods placed tangentially around the circumference C of the disk will all be contracted in length due to relativistic length contraction with respect to K. However, measuring rods will not experience shortening in length, as judged from K, if they are applied across a diameter D. Dividing the circumference by the diameter would produce a surprising result from K’s point of view. Normally such a quotient would have the value π‎=3.14159…, but in this situation the quotient is larger. Euclidean geometry does not seem to hold in an accelerating frame, or indeed by the principle of relativity, within a gravitational field. Spaces in which the propositions of Euclid are not valid are sometimes called curved spaces. For example, the sum of the internal angles of a triangle drawn on a flat sheet of paper will be 180°; however, a triangle drawn on the curved surface of a sphere will not follow this Euclidean rule.

Einstein fully expected to see this effect in gravitational fields, such was his belief in the general principle of relativity. In fact, it was the effect gravitational fields have on the propagation of light that was heralded as the major verification of his general relativity. Einstein realized that rays of light would be perceived as curving in an accelerating frame. This led him to conclude that, in general, rays of light are propagated curvilinearly in gravitational fields. Photographs taken of stars during the solar eclipse of 29 May 1919 confirmed the existence of the deflection of starlight around the sun’s mass by the amount that Einstein predicted.

The mathematics required to describe the curvature of space in the presence of gravitational fields existed before Einstein had need for it, and was used by Einstein to solve his general relativistic problems. In general relativity, material bodies follow lines of shortest distance, called geodesics. The line formed by stretching an elastic band over a curved surface would be a geodesic on the curved surface. Light follows geodesics called null-geodesics. The motions of material bodies are therefore determined by the curvature of the space in the region through which they pass. However, it is the mass of the bodies that causes the curvature of the space in the first place, which demonstrates the elegant self-consistency of Einstein’s general theory.