Time thought experiments
[Written some years ago.]

Godel's theorem and a time travel paradox In How to Build a Time Machine (Viking 2001), the physicist Paul Davies gives the 'most baffling of all time paradoxes.' Writes Davies: 'A professor builds a time machine in 2005 and decides to go forward ... to 2010. When he arrives, he seeks out the university library and browses through the current journals. In the mathematics section he notices a splendid new theorem and jots down the details. Then he returns to 2005, summons a clever student, and outlines the theorem. The student goes away, tidies up the argument, writes a paper, and publishes it in a mathematics journal. It was, of course, in this very journal that the professor read the paper in 2010.'

Davies finds that, from a physics standpoint, such a 'self-consistent causal loop' is possible, but, 'where exactly did the theorem come from?... it's as if the information about the theorem just came out of thin air.' Davies says many worlds proponent David Deutsch, author of The Fabric of Reality and a time travel 'expert,' finds this paradox exceptionally disturbing, since information appears from nowhere, in apparent violation of the principle of entropy.

This paradox seems well suited to Godel's main incompleteness theorem, which says that a sufficiently rich formal system if consistent, must be incomplete. Suppose we assume that there is a formal system T -- a theory of physics -- in which a sentence S can be constructed describing the mentioned time travel paradox. If S strikes us as paradoxical, then we may regard S as the Godel sentence of T. Assuming that T is a consistent theory, we would then require that some extension of T be constructed. An extension might, for example, say that the theorem's origin is relative to the observer and include a censorship, as occurs in other light-related phenomena.

hat is, the professor might be required to forget where he got the ideas to feed his student. But, even if S is made consistent, there must then be some other sentence S', which is not derivable from T'. Of course, if T incorporates the many worlds view, S would likely be consistent and derivable from T.

However, assuming T is a sufficiently vigorous mathematical formalism, there must still be some other sentence V that may be viewed as paradoxical (inconsistent) if T is viewed as airtight. How old is a black hole? Certainly less than the age of the cosmos, you say. The black hole relativistic time problem illustrates that the age of the cosmos is determined by the yardstick used. Suppose we posit a pulsar pulsing at the rate T, and distance D from the event horizon of a black hole.

Our clock is timed to strike at T/2, so that pulse A has occurred at T=0. We now move the pulsar closer to the event horizon, again with our clock striking at what we'll call T'/2. Now because of the gravitational effect on observed time, the time between pulses is longer. That is T' > T, and hence T'=0 is farther in the past than T=0. Of course, as we push the pulsar closer to the event horizon, the relative time TN becomes asymptotic to infinity (eternity).

So, supposing the universe was born 15 billion years ago in the big bang, we can push our pulsar's pulse A back in time beyond 15 billion years ago by pushing the pulsar closer to the event horizon. No matter how old we make the universe, we may always obtain a pulse A that is older than the cosmos. Yes, you say, but a real pulsar would be ripped to shreds and such a happening is not observable.

Nevertherless, the general theory of relativity requires that we grant that time calculations can yield such contradictions. Anthropic issues A sense of awe often accompanies the observation: 'The conditions for human (or any) life are vastly improbable in the cosmic scheme of things.' This leads some to assert that the many worlds scenario answers that striking improbability, since in most other universes, life never arose and never will. I point out that the capacity for the human mind to examine the cosmos is perhaps 2.5 x 104 years old, against a cosmic time scale of 1.5 x 109. In other words, we have a ratio of 2.5(104)/1.5(109) = 1.6/105.

n other words, humanity is an almost invisible drop in the vast sea of cosmic events. Yet here we are! Isn't that amazing?! It seems as though the cosmos conspired to make our little culture just for us, so we could contemplate its vast mysteries. However, there is the problem of the constants of nature. Even slight differences in these constants would, it seems, lead to universes where complexity just doesn't happen. Suppose that these constants depend on initial cosmic conditions which have a built-in random variability.

In that case, the existence of a universe with just the right constants for life (in particular, humanity) to evolve is nothing short of miraculously improbable. Some hope a grand unified theory will resolve the issue. Others suggest that there is a host of bubble universes, most of which are not conducive to complexity, and hence the issue of improbability is removed (after all, we wouldn't be in one of the barren cosmoses). For more on this issue, see the physicist-writers John Barrow, Frank Tipler and Paul Davies.

At any rate, it doesn't seem likely that this drop will last long, in terms of cosmic scales, and the same holds for other such tiny drops elsewhere in the cosmos. Even granting faster-than-light 'tachyon radio,' the probability is very low that an alien civilization exists within communications range of our ephemeral race. That is, the chance of two such drops existing 'simultaneously' is rather low, despite the fond hopes of the SETI crowd. On the other hand, Tipler favors the idea that once intelligent life has evolved, it will find the means to continue on forever. Anyway, anthropomorphism does seem to enter into the picture when we consider quantum phenomena: a person's physical reality is influenced by his or her choices.