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The principle of equivalenceEveryday experience indicates that in a given field of gravity, such as the field caused by the Earth, the greater the mass of a body the greater the force acting on it. That is to say, the more massive a body the more effectively will it tend to fall toward the Earth; in fact, in order to determine the mass of a body one weighs it--that is to say, one really measures the force by which it is attracted to the Earth, whereas the mass is properly defined as the body's resistance to acceleration. Newton noted that the ratio of the attractive force to a body's mass in a given field is the same for all bodies, irrespective of their chemical constitution and other characteristics, and that they all undergo the same acceleration in free fall; this common rate of acceleration on the surface of the Earth amounts to an increase in speed by approximately 32 feet (about 9.8 metres) per second every second. This common rate of gravitationally caused acceleration is illustrated dramatically in space travel during periods of coasting. The vehicle, the astronauts, and all other objects within the space capsule undergo the same acceleration, hence no acceleration relative to each other. The result is apparent weightlessness: no force holds the astronaut to the floor of his cabin or a liquid in an open container. To this extent, the behaviour of objects within the freely coasting space capsule is indistinguishable from the condition that would be encountered if the space capsule were outside all gravitational fields in interstellar space and moved in accordance with the law of inertia. Conversely, if a space capsule were to be accelerated upward by its rocket engines in the absence of gravitation, all objects inside would behave exactly as if the capsule were at rest but in a gravitational field. The principle of equivalence states formally the equivalence, in terms of local experiments, of gravitational forces and reactions to an accelerated noninertial frame of reference (e.g., the capsule while the rockets are being fired) and the equivalence between inertial frames of reference and local freely falling frames of reference. Of course, the principle of equivalence refers strictly to local effects: looking out of his window and performing navigational observations, the astronaut can tell how he is moving relative to the planets and moons of the solar system. Einstein argued, however, that in the presence of gravitational fields there is no unambiguous way to separate gravitational pull from the effects occasioned by the noninertial character of one's chosen frame of reference; hence one cannot identify an inertial frame of reference with complete precision. Thus the principle of equivalence renders the gravitational field fundamentally different from all other force fields encountered in nature. The new theory of gravitation, the general theory of relativity, adopts this characteristic of the gravitational field as its foundation.
Curved space-time
The principlesIn terms of Minkowski's space-time, inertial frames of reference are the analogues of rectilinear (straight-line) Cartesian coordinate systems in Euclidean geometry. In a plane these coordinate systems always exist, but they do not exist on the surface of a sphere: any attempt to cover a spherical surface with a grid of squares breaks down when the grid is extended over a significant fraction of the spherical surface. Thus a plane is a flat surface, whereas the surface of a sphere is curved. This distinction, based entirely on internal properties of the surface itself, classifies the surface of a cylinder as flat, as it can be rolled off on a plane and thus is capable of being covered by a grid of squares. Einstein conjectured that the presence of a gravitational field causes space-time to be curved (whereas in the absence of gravitation it is flat), and that this is the reason that inertial frames cannot be constructed. The curved trajectory of a particle in space and time resulting from the effects of gravitation would then represent not a straight line (which exists only in flat spaces and space-times) but the straightest curve possible in a curved space-time, a geodesic. Geodesics on a sphere (such as the surface of the Earth) are the great circles. (The plane of any great circle goes through the centre of the Earth.) They are the least curved lines one can construct on the surface of a sphere, and they are the shortest curves connecting any two points. The geodesics of space-time connect two events (or two instants in the history of one particle) with the greatest lapse of proper time, as was indicated in the earlier discussion of the twin paradox. If the presence of a gravitational field amounts to a curvature of space-time, then the description of the gravitational field in turn hinges on a mathematical elucidation of the curvature of four-dimensional space-time. Before Einstein, the German mathematician Bernhard Riemann (1826-66) had developed methods related directly to the failure of any attempt to construct square grids. If one were to construct within any small piece of (two-dimensional) surface a quadrilateral whose sides are geodesics, if the surface were flat, the sum of the angles at the four corners would be 360º. If the surface is not flat, the sum of the angles will not be 360º. The deviation of the actual sum of the angles from 360º will be proportional to the area of the quadrilateral; the amount of deviation per unit of surface will be a measure of the curvature of that surface. If the surface is imbedded in a higher dimensional continuum, then one can consider similarly unavoidable angles between vectors constructed as parallel as possible to each other at the four corners of the quadrilateral, and thus associate several distinct components of curvature with one surface. And, of course, there are several independent possible orientations of two-dimensional surfaces--for instance, six in a four-dimensional continuum, such as space-time. Altogether there are 20 distinct and independent components of curvature defined at each point of space-time; in mathematics these are referred to as the 20 components of Riemann's curvature tensor.
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