HPS 0410 Einstein for Everyone

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# Uniform Acceleration in a Minkowski Spacetime

John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh

For a development of the mathematics underlying these figures, see "Technical Appendix" to the chapter "Philosophical Significance of General Relativity: The Relativity of Accelerated Motion."

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. Matters are more complicated in special relativity for two reasons:

First, the accelerating body must contract according to the familiar relativistic effect. That means that different points of the body must follow trajectories with slightly different shapes, so that there is a convergence of worldlines.

Second, the acceleration cannot continue to add speed without limit or the speed of light will be exceeded. 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.

Both these requirements are met by the hyperbolic motion of the spacetime diagram. 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. As they do, the worldlines bunch together, reflecting the relativistic length contraction. The trajectory overall is a hyperbola, not a parabola. It only approximates one in the early stages of acceleration. A small piece of the hyperbolic motion will roughly coincide with the earlier, Newtonian parabolic motion.

The figure shows the hypersurfaces of simultaneity of the worldlines; observers accelerating with any of the worldlines will agree on the one set. The distances between the worldlines as measured along these hypersurfaces remain the same no matter how long the acceleration proceeds.

The inset numbers indicate proper time elapsed along the worldlines. For the two worldlines with numbers, an observer moving with the rightmost would judge time to pass on the leftmost at half the speed.

Unlike the Newtonian case, a uniformly accelerated motion in a Minkowski spacetime cannot cover the entire spacetime. It is restricted to the wedge of spacetime shown, bounded by two lightlike curves.

One feature of the diagram is misleading. It looks as if the hypersurface of simultaneity drawn horizontally in the diagram is special. That is merely an artefact of drawing. All the hypersurfaces are physically equivalent. That is, the spacetime geometry assigns no properties to the horizontally drawn hypersurface that it does not assign to all the rest.