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Authors: Michael Heller

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Tipler’s Theorem

If in the general theory of relativity some dynamic questions may take a form so drastically different from their counterparts in classical mechanics, we should ask whether in the Einsteinian theory there is a counterpart of Poincaré’s theorem of recurrence, and if so, then what does it say. The answer to this question was found by Frank Tipler, who in 1980 proved the existence of a relativistic counterpart of Poincaré’s theorem.
Tipler’s theorem is expressed in highly technical language, and the proof calls for advanced mathematical tools; below I present just the basic idea of the theorem.

We want to learn whether the relativistic universe will one day return to a former state. By the state of the universe at any moment in time
we mean the set of all the events taking place in the universe at time
. In the technical language used by cosmologists this is called a space
section of the universe at time t
(or an instantaneous
section of the universe
). If the initial conditions determining the further development of the universe are determined on such an instantaneous section, it is called a
global Cauchy surface
. A cosmological model is
if it has two identical global Cauchy surfaces with the same initial conditions at two different moments of time.
Two Cauchy surfaces of this kind represent the same state of the universe. Thus a time-periodic model describes the universe returning to a former state. By analogy
we may describe the return of such a universe to a state close to a former state.

Tipler’s theorem states that if space-time
contains a closed
Cauchy surface such that the initial data for it determine the entire history of the universe;
gravitation is an attractive force;
and every history of a particle or photon experiences an attractive gravitational force at least once – then space-time
cannot be time-periodic.

In other words, if the conditions of this theorem are fulfilled, then the universe
return to a former state. The last two conditions are very tolerant and we should expect them to be met in the real universe. The condition of spatial closedness is essential, for if it is removed the theorem cannot be proved. Moreover, examples are known of spatially open worlds which are time-periodic, although all the other conditions of Tipler’s theorem are observed in them. One such model is an empty world, appropriately symmetrical, with just one, static star. Admittedly, it is not very realistic as a practical proposition for the description of our universe. However, it falsifies Tipler’s theorem for open models.

It is noteworthy that classical determinism was one of the salient assumptions in Poincaré’s theory of recurrence, whereas in the general theory of relativity this condition (1. in Tipler’s theorem) is one of the factors leading to the conclusion ruling out eternal returns.

On the basis of this result and other reflections in this chapter we may reach a conclusion that ideas intuitively drawn from classical physics should not be transferred uncritically to relativistic physics. On its largest scale the universe is relativistic, and hence global cosmological conclusions should be reached on the grounds of precise analysis rather than in a flash of intuition.


So does the concept of eternal return have a chance of fulfilling the function of an “ultimate solution” in contemporary cosmology? As we have seen, the fairly appealing, commonsensical idea that the history of the universe is made up of an infinite series of cycles comes up against a number of serious obstacles. We cannot say that the cosmological model corresponding to this idea has been abandoned altogether, but at present it is undoubtedly creating more conceptual problems than it is resolving.

Currently the most serious difficulty in this model seems to be the occurrence of singularities at the beginning and end of each cycle. Tolman and his contemporaries might have entertained the hope that the singularities were an artefact effected by the adoption of oversimplified assumptions for the construction of the model. Very often the assumption of a homogeneous and isotropic nature of the universe was suspected as responsible for this. Both Einstein and Tolman expressed such an opinion. However, already Lemaître’s early research
had shown that singularities still occurred in the cosmological model even if the assumption of isotropy was cast aside. On the contrary, the removal of this assumption increased the tendency of singularities to occur. In the 1960s Stephen Hawking, Roger Penrose, Robert Geroch and others
proved a series of theorems indicating that the occurrence of singularities in space-time theories like the general theory of relativity was the rule rather than the exception. Moreover, the initial and final singularities in the Friedman-Lemaître models belong to the class of strong curvature singularities and are characterised by a breakdown of the structure of space-time (in other words the concept of space-time becomes meaningless in them); and hence we may speak of only one cycle in them for the history of the universe, which starts with the initial singularity and ends with the final singularity. No solution can be prolonged beyond the singularities.

It should be stressed, however, that theorems of the occurrence of singularities apply only to “classical singularities,” that is analyses which do not take the quantum effects of gravity into account. This offers an escape route for the avoidance of such theorems. Perhaps the quantum effects of gravity will breach one of the conditions in the theorems for the occurrence of singularities, thereby facilitating a smooth transition from the contraction phase to the expansion phase. Many scientists have set their sights on this possibility, which looks appealing from the vantage-point of the interests of the search for ultimate explanations. However, the snag is that hitherto we have not yet worked out a generally acknowledged and experimentally confirmed quantum theory of gravitation, and the diverse trends in the research and the partial results obtained in the most popular approaches such as the theory of superstrings or Ashtekar’s loop, have not yielded an unambiguous answer in this respect. Nonetheless we may observe a distinct trend: authors have a clear preference for solutions in which there are no singularities at all, or else the singularities seem easy to remove. What is more, they tend to treat precisely these attributes of their model as the criteria making it appealing.

To conclude this chapter I would like to relate a certain episode from the history of science which should serve as a warning to all those who are guided in their choices in science by grounds other than mathematical consistency and experimental verification. In the nineteenth century, when the heat death hypothesis cast a shadow of doubt over the concept of an eternal universe, W.J.M. Rankine
put forward a conjecture that the energy dissipated in the universe (on the grounds of the second law of thermodynamics) would one day come up against a barrier in “the interstellar ether” situated at a finite distance away from the Earth, rebound from it, and once again accumulate in diverse “foci.” This process was supposed to be periodic, which would ensure the world of eternal existence. You can do it that way if you like, but reasoning on the strength of this strategy gives ultimate explanations which fade away into oblivion within a few years.

Michael Heller,
Ultimate Explanations of the Universe
, DOI: 10.1007/978-3-642-02103-9_4, © Springer-Verlag Berlin Heidelberg 2009
4. A Looped Cosmos

Michael Heller

ul. Powstańców Warszawy 13/94, 33-110 Tarnów, Poland
[email protected]
One of the more macabre ideas of how to eliminate the beginning from the history of the universe is the concept of closed time: a sequence of events recurring an infinite number of times; a history heading nowhere but only endlessly reiterating what has already occurred an infinite number of times; an unending chain of births, deaths, and renewed births; the hopelessness of the impossibility of wresting free from an inexorable loop. Nonetheless this idea has been resurging quite often, both in our none too strictly controlled imagination as well as in the history of human ideas.
Visions of Closed Time

One of the more macabre ideas of how to eliminate the beginning from the history of the universe is the concept of closed time: a sequence of events recurring an infinite number of times; a history heading nowhere but only endlessly reiterating what has already occurred an infinite number of times; an unending chain of births, deaths, and renewed births; the hopelessness of the impossibility of wresting free from an inexorable loop. Nonetheless this idea has been resurging quite often, both in our none too strictly controlled imagination as well as in the history of human ideas.

In Antiquity the Stoics reduced the idea of eternal returns to its logical extreme. The history of the world was cyclical: after each cycle the world returned to its original state (
), passing through a phase of destruction by fire and then starting a process of ordering itself anew (
). In each cycle exactly the same structure was reconstructed, down to its most minute detail: “After the passage of centuries the same Socrates will be teaching in the same Athens, and in the streets of the same cities the same people will be going through the same suffering.”

Every so often this doctrine is self-renewed and arises out of its own ashes. A contribution to its popularity in the modern period has come from Friedrich Nietzsche, who was very fond of it and treated it as a sort of religious message. He also tried to find a scholarly justification for it, albeit rather ineptly. In his opinion, the world should be envisaged as “a particular number of foci of force” and therefore “had to go through a calculable number of combinations, as if in a game of dice, in the grand game of existence.” Hence the world was a sequence of identical combinations “which had already been repeated an infinite number of times and which continued to play out their game
ad infinitum

You might think that today the idea of looped time could persist only in literary visions and science fiction. But the history of science turns out to be stranger than fiction.

Kurt GÖdel’s Universe

Quite out of the blue it turned out that the general theory of relativity lends fairly strong support to the concept of closed time. The first solution of Einstein’s equations involving closed time-like curves was discovered in 1924 by Cornelius Lanczos, who was later Einstein’s assistant.
It was rediscovered in 1937 by Willem Jacob van Stockum, a Dutchman who was killed in action during the Second World War, as a pilot fighting for the Allies.
This solution (now known as van Stockum’s dust) describes a space-time with a cylindrical symmetry, in which matter in the form of dust rotates around an axis of symmetry. This fact physically distinguishes the axis of symmetry, as a result of which the space-time is not isotropic. Van Stockum’s solution has one other feature, apart from closed time-like curves, for which it is hard to give a physical interpretation: the density of the dust particles increases with distance away from the axis of rotation.

In spite of their exotic properties, neither Lanczos’ nor van Stockum’s solution attracted much notice. It was not until Kurt Gödel’s discovery of another solution in 1949 that people’s attention was turned,
most probably thanks to the fact that Gödel was already a well-known personality and also because from the very start he promoted his solution as a cosmological model. Gödel’s solution entailed closed time-like curves, and understandably the possibility of a return to one’s own past stirred up a sensation. To reach a closed time-like curve in Gödel’s world you would have to have an unrealistically immense store of energy available to accelerate your spaceship appropriately, but what was that in view of the prospect of conquering time? Let’s take a closer look at Gödel’s solution.

Gödel’s universe is filled with matter consisting of dust with a constant density, just like the Friedman-Lemaître standard models. His space-time is flat and homogeneous (it is called
space) and has rotational symmetry around an axis. This axis may be identified as the trajectory of a particle with an initial velocity in the radial direction. Hence it may be said that the matter in Gödel’s model is rotating around this axis, or – equivalently – the axis is rotating with respect to the matter at rest. But this time the “axis of rotation” is not distinguished in any way at all. It may be transferred to any arbitrary point by a simple change of coordinates, such that the history of any arbitrarily chosen particle moving along a radial path may serve as Gödel’s axis.

The rotational symmetry of Gödel’s world is associated with a phenomenon which has already been mentioned – a closed time-like curve passes through every point in Gödel’s universe.
In other words in Gödel’s solution there is no cosmic time capable of “increasing at a uniform rate” with the history of any observer or particle (of non-zero rest mass).

There are no singularities – neither an initial nor a final singularity – in Gödel’s model,
and hence the “haunting prospect of a beginning” has been eliminated from it. But such a model is just a purely mathematical option, since it does not incorporate the effect of an expanding universe, in other words this model offers no explanation for the red shift observed in galactic spectra – a phenomenon which quite definitely exists in the real world.

There are many indications that Gödel started his search for a solution prompted by his own philosophy of time. Shortly after publishing his model he wrote a separate paper presenting his views on this subject.
He believed that time could be objective (real) only if there existed an infinite number of successive “layers of the present,” one following another. But the special theory of relativity rules out such a possibility. The situation seemed to be saved by the fact that in all the cosmological solutions to Einstein’s equations known at the time there exists a global time, which makes a succession of “layers” of the present possible. However, the solution discovered by Gödel shows that it is not a typical situation, but enforced by a symmetrical distribution of matter, thanks to which it was possible to apply a privileged system of coordinates extending over the whole of space-time in such a manner that one of the coordinates may be interpreted as global time. In the general case there was no such thing as global time nor an absolute “present moment.” Hence – according to Gödel – time may not be considered objective; it was only a figment of our imagination projected onto the universe.

However, the great logician seems to have made a mistake: he treated the absoluteness of time (and simultaneity) as identical with its objectivity. But the relative need not be subjective. Dependence on a system of reference may be, and often is, an objective fact. We are not obliged to agree with Gödel’s intuitions, but we should appreciate his difficulties. The theory of relativity had introduced a host of new concepts and a lot of time had to pass before physicists, philosophers and others who were interested in the issue could cope with all this and take stock of it intellectually.

There was an “existential” backdrop to Gödel’s grapple with time. After his death the notes he left revealed that for nearly two decades he had been searching for a theoretical possibility to overcome death by making use of closed time-like curves.

Gödel’s solution triggered an avalanche of papers in which more and more solutions to Einstein’s equations were found not only containing closed time-like curves but also exhibiting numerous temporal and causal pathologies.

Although Gödel’s solution does not offer a description of the real world, it has played an important role in the development of the mathematical methods applied in the general theory of relativity. From the very beginning physicists and philosophers, as well as many bystanders, have been intrigued by the problem of time in Einstein’s theory. Many believed that the time riddles would be “straightened out,” or at least grasped, if the theory were presented in the form of an axiomatic system. A few attempts were made to axiomatise it, undertaken by scientists like A.A. Robb, R. Carnap, H. Reichenbach, and H. Mehlberg.
Guided by their intuition or philosophical premises, they selected their axioms to give space-time the “most sensible” flow of time and other properties. But not until Gödel’s discovery of a solution with closed time-like curves did researchers realise that it was not worthwhile ruling out certain possibilities a priori; instead as many solutions to Einstein’s equations should be found as possible, and studied from the point of view of their global properties. This new trend helped to devise the global methods for the examination of space(-time) and launched a new style in differential geometry, different from the traditional practice. Once a series of particular solutions have been analysed it is possible to formulate general rules, and then to set about finding proofs for them. This is how many groundbreaking theorems have been arising. One of the first results of this approach was R.W. Bass and L. Witten’s proof of the theorem which says that every compact space-time contains a closed time-like curve.
This was followed by a tide of further results. Brandon Carter systematised them in an extensive paper.
The crowning achievement in this line of research was the proof of the celebrated theorems of the existence of singularities (see the previous chapter). Global methods have become well-established both in relativistic physics as well as in pure geometry. The fountainhead giving rise to this new style of thinking was Kurt Gödel’s work on a universe with closed time-like curves.

Gott and Li’s Suggestion

Gödel’s solution not only launched new research methods. It also provided an opportunity for reflections of an ideological character. As we saw in the introduction to this chapter, there was no dearth of ideas before, either, to raise up the ideology of closed time to the rank of “the ultimate solution,” but now an opportunity opened up to turn this ideology into a “self-explanatory” cosmological model. An example of work heading in this direction is the extensive paper by J. Richard Gott and Li-Xin Li under the suggestive title “Can the Universe Create Itself?”
Let’s take a closer look at their idea.

Gott and Li wanted to make use of the “remarkable” property of the general theory of relativity – its admission of solutions with closed time-like curves, but they were aware of the difficulties this property implied. Closed time meant problems with causality, often expressed in the question what would happen if someone who made use of the time-loop killed his father before his own birth. In physics this provocative question translates into computational problems connected, for example, with the expected behaviour of a solution to a differential equation given its initial conditions. Moreover, there are no experimental clues that in our universe time is a closed loop. On the contrary, the scientific reconstruction of its history, by now based on numerous observations, makes up a coherent cosmic history with a linear time-scale stretching back to the first moments following the Big Bang. Gott and Li are too experienced as cosmologists not to know of all these difficulties. That is why the model they proposed was far more sophisticated than the simple models with a closed history.

To obviate a beginning, Gott and Li assume that “the early universe contained a region with closed time-like curves.” Such a universe is neither eternal, nor has a beginning. Every event that happens in it has an event which preceded it, but the question which event was the earliest is as meaningless as asking which is the easternmost point of the Earth’s surface. But at a certain moment this spell of “dodged history” came to an end, and now history is proceeding in a one-way direction towards the future. However, it is not enough to juxtapose the “early” period with a closed history with the “later” period of linear history. To avoid a variety of pathologies with causation, the region with closed time-like curves must be separated off from the later one-way history of the universe. Relativistic cosmology offers such a possibility. Due to the maximum, finite velocity at which physical interactions can propagate in space-time, there may exist regions with which no communication whatsoever is possible. No physical interaction can “get through” from one such region to another, and we describe the situation by saying that they are separated from each other by a Cauchy horizon. Gott and Li applied this mechanism to save the later history of the universe from the causal anomalies generated by closed time-like curves in the early universe. The two periods are separated off from each other by a Cauchy horizon.

But that was not the end of the new model’s problems. Papers were published which showed that for Cauchy horizons of this type in space-times with closed time-like curves certain mathematical expressions describing the distribution of matter tended to infinity.
So it was possible to by-pass the pathologies connected with closed time only at the expense of bringing in other pathologies (the introduction of infinity). Gott and Li have challenged this finding. According to them in such situations it is possible to remove the “tendency to infinity” by finding a solution exactly reproducing conditions
which have already occurred before. A proposition which is enjoying considerable popularity nowadays are the so-called inflationary models, with the early universe expanding at a dramatically rapid rate – increasing its volume up to 10
times or more in a fraction of a second! Although there are no observations to confirm them, such models have been well received in contemporary cosmology, because they resolve several theoretical difficulties.
Gott and Li argue as follows: let’s assume that “in the beginning” there was an inflationary model; then the small volume of space-time was inflated to a gigantic size. If in that huge, inflated universe a small sub-region happened to occur with the same conditions as those in the initial, small volume, then closed time could have occurred without the need for infinity to come into play. “If that happened the universe could be its own mother.”

It has to be admitted that the construction Gott and Li have presented is intricate, but still incomplete. An accumulation of computations and particular examples is not yet a full cosmological model. To construct a full model it is necessary to examine its stability and determine the set of initial conditions (to form the space of all initial conditions) which yields such a solution. If the solution calls for highly specific initial conditions it is in need of explanation itself, rather than serving to explain. And above all there is the question of whether the given solution corresponds to reality, viz. can it be verified by observed facts. The model proposed by Gott and Li cannot claim an answer in the affirmative to this question.

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