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Four-dimensional space-timeThe German mathematical physicist Hermann Minkowski pointed out that the invariant interval between two events has some of the properties of the distance in Euclidean geometry. Based on Euclidean geometry, the Cartesian coordinate system is designed to identify any point (event) in space by its reference to three mutually perpendicular lines or axes meeting at an arbitrary point of origin. The distance s between two events, in accordance with Pythagoras' theorem, in any Cartesian (rectilinear) coordinate system is obtained by taking the square root of the sum of the squares of coordinate distances, s2 = x2 + y2 + z2, and its value is independent of the choice of coordinate system, though the values of x, y, and z are not. The invariant interval, similarly, is the square root of a sum and difference of squares of intervals of both space and time. Accordingly, Minkowski suggested that space and time should be thought of as comprising a single four-dimensional continuum, space-time, often also referred to as the Minkowski universe. Events, localized as regards both space and time, are the natural analogues of points in ordinary three-dimensional geometry; in the history of one particle, its proper time resembles the arc length of a curve in three-space. In Minkowski's space-time the invariant interval may be either timelike or spacelike. If L2 - c2T2 for two events happens to be zero, the invariant interval is neither, but null, or lightlike, as a light signal emanating from the earlier of the two events may just pass the second as the latter occurs. By contrast, in ordinary geometry the distance between two points, s, vanishes only if the two points coincide. To this extent the analogy between space-time and ordinary space is imperfect. Minkowski's four-dimensional, geometric approach to relativity appears to add to the original physical concepts of relativity mostly a new terminology but not much else. Nevertheless, for the further conceptual development of relativity Minkowski's contribution has been of inestimable value.
The general theory of relativity
Physical originsThe general theory of relativity derives its origin from the need to extend the new space and time concepts of the special theory of relativity from the domain of electric and magnetic phenomena to all of physics and, particularly, to the theory of gravitation. As space and time relations underlie all physical phenomena, it is conceptually intolerable to have to use mutually contradictory notions of space and time in dealing with different kinds of interactions, particularly in view of the fact that the same particles may interact with each other in several different ways--electromagnetically, gravitationally, and by way of so-called nuclear forces. Newton's
explanation of gravitational interactions must be considered one
of the most successful physical theories of all time. It accounts
for the motions of all the constituents of the solar system with
uncanny accuracy, permitting, for instance, the prediction of eclipses
hundreds of years ahead. But Newton's theory visualizes the gravitational
pull that the Sun exerts on the planets and the pull that the planets
in turn exert on their moons and on each other as taking place instantaneously
over the vast distances of interplanetary space, whereas according
to relativistic notions of space and time any and all interactions
cannot spread faster than the speed of light. The difference may
be unimportant, for practical reasons, as all of the members of
the solar system move at relative speeds far less than Proceeding on the basis of the experience gained from Maxwell's theory of the electric field, Einstein postulated the existence of a gravitational field that propagates at the speed of light, c, and that will mediate an attraction as closely as possible equal to the attraction obtained from Newton's theory. From the outset it was clear that mathematically a field theory of gravitation would be more involved than that of electricity and magnetism. Whereas the sources of the electric field, the electric charges of particles, have values independent of the state of motion of the instruments by which these charges are measured, the source of the gravitational field, the mass of a particle, varies with the speed of the particle relative to the frame of reference in which it is determined and hence will have different values in different frames of reference. This complicating factor introduces into the task of constructing a relativistic theory of the gravitational field a measure of ambiguity, which Einstein resolved eventually by invoking the principle of equivalence.
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