Time and causation in gödel's universe

Abstract
In 1949 the great logician Kurt Gödel constructed the first mathematical models of the universe in which travel into the past is, in theory at least, possible. Within the framework of Einstein’s general theory of relativity Gödel produced cosmological solutions to Einstein’s field equations which contain closed time-like curves, that is, curves in spacetime which, despite being closed, still represent possible paths of bodies. An object moving along such a path would travel back into its own past, to the very moment at which it “began” the journey. More generally, Gödel showed that, in his “universe”, for any two points P and Q on a body’s track through spacetime (its world line), such that P temporally precedes Q, there is a timelike curve linking P and Q on which Q temporally precedes P. This means that, in principle at least, one could board a “time machine” and travel to any point of the past. Gödel inferred, in consonance (as he observes) with the views of Parmenides, Kant and the modern idealists, that under these circumstances there could be no such thing as an objective lapse of time, that time or, more generally, change, is an illusion arising from our special mode of perception. For consider an observer initially at point P (with time coordinate t seconds as indicated by his own clock). At point Q (with time coordinate t′) he boards a time machine and travels back to point P, taking time t′′ to do so. In that case, according to his own clock, t′ – t + t′′ > 0 seconds have elapsed, and yet an identical clock left at P would show that 0 seconds have elapsed. In short, there has been no “objective” lapse of time at all. Gödel remarks that in his universe this situation is typical: for every possible definition of an “objective” time one could travel into regions which are past according to that definition. He continues.
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