Relativistic Chaos

A new study shows that general relativity, a theory in which observers in different reference frames measure time differently, is not incompatible with chaos theory, in which events unfold in absolute time. Chaos is an ordinary word with lots of meanings. In physics, however, the meaning is more precise: a system---a weather system, say---is chaotic if a very slight change in initial conditions sends the system off into a very different history. How different? The degree to which a system is chaotic can be encapsulated in a parameter called the Lyapunov exponent: when it is positive the system is chaotic and to some extent unpredictable; for a negative value, the system becomes nonchaotic---a small perturbation will not radically change its history. What has worried physicists for many years was the fear that a shift in a frame of reference might so alter the time parameter as to change the Lyapunov exponent from null or negative to positive or vice versa. In other words, the change of frame would seem to make a chaotic system nonchaotic or vice versa. Now, the work of Adilson Motter of the Max Planck Institute for Complex Systems in Dresden, Germany lays this matter to rest. He shows that over a wide range of conditions, a change of time parameter does not alter the Lyapunov exponent enough to change chaos in a system. Motter believes that this is good news since the equations of general relativity are nonlinear, as are those of chaotic systems, and many common situations described by general relativity, such as the motion of massive bodies near black holes or a nonuniform expansion of the universe at the time of the big bang ("mixmaster universe model," see PNU #158) are expected to be highly chaotic. (Physical Review Letters, 5 December 2003)