![]() ![]() ![]() For the two worldlines with numbers, an observer moving with the rightmost would judge time to pass on the leftmost at half the speed. The inset numbers indicate proper time elapsed along the worldlines. The distances between the worldlines as measured along these hypersurfaces remain the same no matter how long the acceleration proceeds. The figure shows the hypersurfaces of simultaneity of the worldlines observers accelerating with any of the worldlines will agree on the one set. A small piece of the hyperbolic motion will roughly coincide with the earlier, Newtonian parabolic motion. It only approximates one in the early stages of acceleration. The trajectory overall is a hyperbola, not a parabola. Together, reflecting the relativistic length contraction. The worldlines of the uniformly accelerated points start initially around "0" time in a roughly parabolic trajectory then they approach the speed of light asymptotically. So we expect the motion initially be like the Newtonian parabolic trajectory but then, as speeds close to light are achieved, it should level off at something that approaches but never gains the speed of light.īoth these requirements are met by the hyperbolic Second, the acceleration cannot continue to add speed without limit or the speed of light will be exceeded. That means that different points of the body must follow trajectories with slightly different shapes, so that there is a convergence of worldlines. Matters are more complicated in special relativity for two reasons:įirst, the accelerating body must contract according to the familiar relativistic effect. In Newtonian theory, the uniform acceleration of the points forming a body is simply given by setting all the points on trajectories in spacetime with the same parabolic shape. Back to Einstein's Pathway to General Relativity Uniform Acceleration in a Minkowski Spacetimeĭepartment of History and Philosophy of Scienceįor a development of the mathematics underlying these figures, see " TechnicalĪppendix" to the chapter "Philosophical Significance of General Relativity: The Relativity of Accelerated Motion." ![]()
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