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Origins of the General Theory of Relativity
From: Cambridge University Press
| By:
Prabhakar Gondhalekar |
EDITOR'S INTRODUCTION |
How did Einstein discover his theory of general relativity? Did it hit him in a eureka moment or did he reach it through a deep understanding of the laws of nature?
 Taken from The Grip of Gravity (available through Fathom), Prabhakar Gondhalekar shows how Einstein's general theory of relativity is based on a mathematical extension of his previous theory of special relativity. The 'special' theory represented an idealisation of the world, where uniform bodies were in motion and gravity and acceleration were absent. Later he developed the 'general' theory, in which bodies move according to a gravity-bound universe. This theory takes on an entirely new interpretation of gravity, where the force behaves according to a distorted space-time continuum. |
 instein developed his theory of special relativity out of a deep conviction that in nature there were no preferred frames of reference. In this theory the concepts of inertial frame and Euclidean geometry are retained but the Galilean transformations are abandoned. But it was clear to Einstein that the special theory does not remove all restrictions on coordinate systems since it says nothing about the validity of laws of nature in non-inertial frames of reference (e.g. accelerated frames, rotating frames or frames in a gravitational field). Newton's law of gravitational attraction is not the same between inertial frames moving at high relative speed. This is because the distance and inertial mass, parameters used to express gravitational attraction between two bodies, have no absolute values but depend on the velocity of the observer and the bodies. The gravitational attraction between two approaching bodies, measured by an observer on one of the bodies, will be higher than the attraction between the bodies when the bodies are at relative rest. The increase in the mass and the decrease in the separation of the approaching body cause this difference in the gravitational attraction. Thus Newton's law of gravitational attraction does not conform to the invariance requirements of the special theory. The instantaneous action at a distance required by Newtonian gravity is also contrary to the basic principle of special theory, which asserts that a physical effect cannot be transmitted at speeds greater than the speed of light. |
To eliminate the flaws inherent in the special theory, Einstein introduced the principle of covariance: |
The laws of nature have the same mathematical form in all frames of reference. |
Compare this with his earlier principle--a law of nature must have the same form in all inertial frames--which he introduced in special relativity. By introducing the principle of covariance, Einstein insisted that all motions, uniform or accelerated, were relative. A relative accelerated motion appears contrary to our everyday experience. If we were in a smoothly flying plane we would be unaware of our motion, but if the plane banked or altered its speed we would immediately notice the accelerated motion. This would lead us to believe that accelerated motion was indeed absolute and different from uniform motion. But consider an astronaut standing on weighing scales in a rocket. When the rocket is launched it will accelerate away from Earth and the astronaut's weight will appear to increase because of the acceleration of the rocket. If the rocket motor is now switched off, the rocket will begin to fall, the astronaut will float freely in the rocket and, in that frame, the astronaut will appear to be weightless. The astronaut would be led to believe that there was no force acting on him or her. But to an observer at the launch pad the action of gravity is quite obvious. Such 'thought experiments' led Einstein to formulate his fundamental principle of equivalence. |
There is no way for an observer in a non-inertial frame (i.e. accelerated frame) to distinguish between gravitational force and inertial forces acting on bodies in that frame. |
This principle does not just assert that gravitational and inertial forces are equivalent, but insists that every effect produced (on any physical system) by acceleration or observed by an observer in an accelerated frame can be reproduced by an appropriate gravitational force and observed by an observer at rest in the gravitational field. To put it another way, the equivalence principle prevents an observer from detecting uniformly accelerated motion. Observed accelerated motion could be attributed either to acceleration in gravity-free space or to a gravitational field. Compare this with the principle of relativity--this principle prevents an observer from detecting uniform motion. The equivalence principle follows naturally from Galileo's so-called 'Leaning Tower of Pisa' experiment demonstrating the equality of gravitational and inertial mass. Galileo had shown that all bodies fall down to Earth with the same acceleration (neglecting the resistance due to air); that is, the acceleration of a body is independent of its mass or composition. From a relativistic point of view, it is entirely consistent to consider that it is the ground that has accelerated to the bodies, which are at rest. In this case the composition of the bodies is irrelevant, and all bodies should appear to be approached by the ground at the same rate of acceleration. Newton had tested the equality of gravitational and inertial mass with his pendulum experiments. In the late nineteenth century the Hungarian scientist Baron Eötvös performed precise experiments to show that the inertial and the gravitational mass were equal to an accuracy of one part in a billion. Einstein apparently did not know of these experiments but referred to them extensively after the work had been pointed out to him. Newton accepted the equality of inertial and gravitational mass as a phenomenological fact, and ascribed it no special importance, but to Einstein this equality was not a mere accident of nature: he saw a profound physical significance in it. The principle of equivalence forms a cornerstone of general relativity and its verification is crucially important. Increasingly precise experiments have been performed to test it and the current limit is about one part in 1012. |
Einstein began to formulate the general theory sometime in 1906 and it was to be eight long years before he was ready to present it in its full intellectual rigour. During those eight years Einstein moved between a number of universities in Europe, but 1912 appears to have been crucial to the mathematical development of the theory. In August that year Einstein returned from Prague to Zurich. By now he had convinced himself that time and light were affected by gravitation, but these ideas had to be put on a firm mathematical basis. In Zurich Einstein turned to his friend and former fellow student Marcel Grossmann for help. Grossmann was then a professor of geometry and the dean of the mathematics and physics section of ETH. He pointed out to Einstein that to solve the problem of gravitation he would need a space-time possessing the Riemannian geometry as opposed to the flat Euclidean geometry of special relativity. Einstein was blissfully unaware of Riemann and of his multi-dimensional geometry, and its significance to his work. The transition from Euclidean geometry to Riemannian geometry was the crucial step that led Einstein, initially with Grossmann's collaboration, to his ultimate formulation of post-Newtonian gravity. In 1914 Einstein moved to Berlin and on 25 November 1915 he presented to the Prussian Academy of Science his paper The Field Equations of Gravitation. |
The physical consequences of the gravitational field in the general theory of relativity formulated by Einstein can be summarised as follows: space-time is a four-dimensional non-Euclidean continuum, the curvature (or warping) of the continuum being a consequence of the local distribution of matter or energy. Particles and light rays travel along the geodesic (stationary distance) of this four-dimensional geometric world. |
In the general theory, space-time is a four-dimensional continuum, as in the special theory. But this is where the similarity ends. In the general theory the invariant intervals are defined only locally between events taking place close to each other. Only small regions of space-time resemble the continuum envisaged by Minkowski, just as small sections of a sphere appear nearly planar. Far from being rigid and homogeneous, the general-relativistic space-time continuum has geometric properties that vary from point to point and are affected by local mass or energy. General relativity thus makes geometry part of physics and properties (such as curvature) of the space-time defined by this geometry can be studied by means of scientific experiments. The basic idea in the theory of general relativity has been summarised thus: space-time tells mass how to move, and mass (or energy) tells space-time how to curve. There are two principal consequences of the geometric nature of gravitation: (1) the acceleration of bodies depends only on their mass and not on their chemical or nuclear constitution, and (2) the path of a body or light rays in the vicinity of a massive body is different from that predicted by Newtonian mechanics. |
Just five days before Einstein presented his paper to the Prussian Academy of Science, David Hilbert (1862-1943), one of the greatest mathematicians of all time, presented to the Royal Academy of Science in Göttingen the mathematical framework on which the general theory of relativity is based. Hilbert had followed Einstein's work and was fascinated by his ideas. In the summer of 1915 he invited Einstein to Göttingen to give seminars on the work he was doing. In the following months he pondered over what he had learned from Einstein's summer seminars. In the autumn, during a vacation, the key ideas fell into place and within a few weeks he had formulated the elegant form of Einstein's field equations, which describe how mass curves space-time. Einstein had arrived at the same result after a number of diversions down blind alleys and after months of frustrating trial and error. Einstein and Hilbert fell out over this, because Einstein felt that Hilbert had stolen his thunder, but they made up after a few months and no lasting rift was produced. The credit for the theory of general relativity is, rightfully, given to Einstein. He had the physical insight, such as the equivalence principle and the geometric form of the theory; the 'paradigm shift' was entirely due to Einstein. Hilbert had taken the last mathematical step, and although his was an intellectually elegant step it was, nevertheless, only the last step. Einstein (and Hilbert) had formulated the field equation of gravitation, a goal that had eluded Riemann 50 years earlier. Einstein's field equation does not describe the magnitude and direction of force in the vicinity of a gravitating body, as Maxwell's field equations do for a charged body. Rather, it describes the curvature of space-time in the vicinity of a body. |
In the theory of general relativity Einstein reasoned that gravity was linked to space-time and the linking agency was Riemannian geometry. To illustrate this, consider the motion of two bodies, one on a perfectly flat surface and the other on the surface of a perfect sphere. The body on the flat surface will continue to move in a straight line, but the body on the sphere will move along a curve determined by the surface of the sphere. According to Newton's First Law, there is no force acting on the body on the flat surface, but the body on the sphere is constrained to move along the curve by a force that is always directly towards the centre of the sphere. This was Newton's explanation for the near-circular orbits of planets, the inward directed force being gravity. Einstein asserted that this was an illusion, there was no force constraining the body on the sphere to move along a curve, but the gravitational field was a distortion of geometry of space-time from Euclidean to non-Euclidean form. If we accept this, then the motion of a body on a flat surface and a spherical surface are equivalent. It is now necessary to restate Newton's First Law--a free body moving in any frame of reference moves along a path that is the stationary distance between any two points on the path. In Euclidean geometry the 'stationary distance between two points' is a straight line, but in the non-Euclidean geometry this distance is a geodesic. This restatement of Newton's First Law may appear to contradict our everyday experience. If an object is thrown at any angle to the vertical it moves along an arc of a parabola that is certainly not the shortest geometrical distance between the point from which the object started and the point where it comes to rest. The shortest geometrical distance between these two points is a straight line. Why does the body not move along this straight line? But note that the straight line is in a three-dimensional space. The path of a body in motion has to be considered in a four-dimensional space-time and not just in a three-dimensional space. The parabola is the three-dimensional projection of the four-dimensional stationary path--a geodesic. |
The special theory of relativity synthesised the separate Newtonian concepts of three-dimensional space and one-dimensional time into a single four-dimensional Euclidean space-time continuum. The general theory retains the four-dimensional space-time continuum as the geometrical framework in which the laws of nature are to be stated, but the continuum is non-Euclidean. In general relativity the curvature of the space-time is determined by the local distribution of matter (or energy): the greater the density of matter in a region, the higher the curvature of the space-time in that region. It is worth noting that in his general relativity Einstein was not attempting to find a different interpretation of Newtonian gravity or to 'fix' special relativity to include gravity; he was attempting an entirely new interpretation of gravity. In the special theory of relativity space and time are combined, but space and time are still a fixed background in which events happen. It is possible to choose different paths through space-time, but the background of space-time is not modified. In general relativity, gravity is no longer a (Newtonian) universal force that operates in a fixed background of space-time. Instead it is a property (curvature or distortion) of space-time caused by local mass or energy. It is interesting to note that since gravity is a universal force there can never be a truly inertial frame and special relativity will always be an approximation. An important property of mass and energy is that they are always positive, and that is why gravity is always an attractive force. Repulsive gravity has never been detected. According to general relativity, this means that space-time is curved back on itself like the surface of a sphere or that the curvature of space-time is always positive. If mass (or energy) had been negative, or gravity repulsive, space-time would have had a negative curvature like the surface of a saddle. When Einstein applied general relativity to explain the universe, he regarded the positive curvature of space-time as a problem. |
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