rotation, universe, cosmology, general relativity, Einstein, Mach, Gödel, causality, inertial system, vorticity, centrifugal, space-time, curvature, large scale structure, time, light cone, closed timelike curve
What's a Rotating Universe?
Einstein's General Relativity
Gödel's Rotating Universe
Is our Universe Rotating?
Time Traveling and Newtonian Physics
Time Traveling and Special Relativity
Time Traveling and General Relativity
Time Traveling and Rotation of the Universe
Time Traveling and Human Mind
Is Time Traveling Possible in our Universe?
A title like 'Rotating Universes and Time Traveling' is certainly enough for some people to consider me a mad scientist or an incurable geek. Maybe they are right, I don't know. In any case, my Ph.D. thesis  deals with these subjects, particularly the rotation part. I believe there are some people on the Net who may be interested in this stuff. That's why I decided to write this article, largely based on my thesis, and put it online.
I should mention that I left the research in 1984, and there may be some development in this area that I'm not aware of. I would be grateful if some reader could notify me in that case.
The article begins with explaining what is meant by a rotating universe. Experience has shown that this is a very difficult concept. But as it is, Einstein's General Relativity theory does allow for rotating universes: There are such explicit mathematical solutions. Some of the rotating universes also turn out to have the shocking property that they allow (in theory) for traveling backwards in time, although the connection with rotation is not clear.
When I tell people about the possibility of a rotating universe, their reaction is usually either a silly smile, or the very well motivated question: With respect to what would the universe rotate? I viciously reply: With respect to something that does not rotate, that is, something that does not experience any centrifugal forces. OK, this is correct, but it needs some elaboration.
First of all, don't try to imagine the universe as rotating as a whole. That way of thinking is misleading. I'll come back to rotation as a whole later.
Second, don't think that this implies some center of rotation. According to the Copernican principle, all places in the universe are equivalent. This is a simplifying assumption adopted by most cosmologists; whether it holds in reality is an open question. On smaller scale the universe is badly inhomogeneous, but there is still hope that the large scale structure is homogeneous.
Third, study carefully the following attempt to visualize a rotating universe.
Imagine you are in a laboratory without windows floating around somewhere in the universe. If you and the other objects in the laboratory get pressed against the walls, you would say that the laboratory is rotating, and centrifugal forces are responsible for the effects. Now, the laboratory happens to be equipped with small engines that can be used to control the rotation. Use the engines until you have totally eliminated the centrifugal forces, and thereby the rotation. When done, drill some peepholes in the laboratory (but please make sure you don't lose your air supply). Observe the galaxies. If you find that the galaxies rotate around you, then the universe is said to be rotating.
You have actually only seen that the universe rotates around the point where you are, but if the Copernican principle holds, then it rotates around any point. That's a rotating universe.
So keep in mind that when I talk about a rotating universe, I mean that the matter of the universe rotates around the non-rotating observer. There is a better word for it: vorticity. In classical hydrodynamics, the vorticity w of a velocity field v is defined using the rot operator:
In general relativity, there is a similar definition. One expresses the vorticity four-vector in terms of matter four-velocity field (a four-vector is a vector with one 'time' and three 'space' components).
The Earth is rotating. Clearly. Centrifugal forces flatten the Earth a little. Also the Coriolis forces, having a strong impact on winds and missiles, indicate that the Earth is rotating. Now, answer my question: With respect to what is the Earth rotating?
If you are well familiar with Newtonian physics, your answer may be: With respect to an inertial reference system (non-rotating, non-accelerating system; a system where no inertial forces appear; a system where e.g. F=m·a holds). It is a postulate that there exists such a system.
If, and maybe even if not, you have heard about Mach's principle, you may say: With respect to distant galaxies (or fixed stars, or all masses in the Universe).
At least rough measuremens seem to indicate that the Machians are right in their prediction, and thus that the inertial system of the Newtonians in some magic way appears to be aligned with the masses of the Universe. Strange coincidence, isn't it, for the Newtonians, as in Newtonian theory there is no connection with the reference systems and the masses that move around in them. And Machians: Please explain exactly how the distant matter in the universe can create the centrifugal forces down here at the Earth (no reasoning; a consistent set of equations that can be checked against experiments).
If the Machians are right, then of course rotation of the universe is impossible.
I write about Mach's principle as it is pretty well known among scientists - but possibly it is best known among amateur scientists. Even Lenin had opinions about Mach's principle. However, Mach's principle has no place in modern science since it explains absolutely nothing. At most it can be seen as a guiding idea...
... and one of those who were inspired by Mach's principle is Albert Einstein. He formed the General Relativity Theory much along these lines. As Misner, Thorne, and Wheeler  put it: "Space-Time tells matter how to move. Matter tells space-time how to curve." The galaxies tell the space-time how to curve through the field equations. The space-time tells the matter how to move: 'geodesic' motion. In particular, space-time tells what is rotating and what is not. Sounds Machian, doesn't it?
The General Relativity is the best theory that we have today for the large scale structure of the universe. Some of its predictions have been verified (the orbit of Mercury etc.). No experiments are to my knowledge inconsistent with General Relativity (as long as we remain in its realm, the macroscopic world. Quantum effects is another story.) Remember, a theory can never be verified, only falsified.
In 1949 Kurt Gödel found a cosmological solution to Einstein's field equations with rotating matter . The lesson to be learnt from that is that the relation between galaxies and the inertial system is more intricate than common-sense believes. Mach's principle is only poorly incorporated in Einstein's theory.
Einstein himself has written: 'Von dem Mach'sen Prinzip sollte man eigentlich überhaupt nicht mehr sprechen.' (One really should not speak any more of Mach's principle.) If you are still interested in Mach's principle, consult .
Gödel's universe has another property that is perhaps even more shocking: It allows traveling backwards in time (although the practical difficulties would be more than overwhelming). You enter a rocket, and you take a journey in the universe along a certain path. Then you will - after many billions of years - return to the starting point before you started. I will say more about the possibilities of time traveling, but there is a question that should be answered first:
Whether our Universe is rotating is a question for measurements, not theory. The outcome of measurements can be either 'Yes, it's rotating', or they can put an upper limit to the rotation, if any. Measurements can never prove that the rotation is exactly zero. As it is, there is no certain evidence that our Universe rotates. Let's have a look at the upper limits then.
The best method we have for directly measuring the rotation is by observing the orbits of the planets . The positions of the planets are in practice always determined in relation to the background of the fixed stars. Then one adapts the parameters of a theoretical model (consisting of Kepler's laws plus some hundred correction terms) to the measured orbits. One of the parameters is the rotation of the background. It turns out that the latter parameter is zero within the experimental errors. The largest source of the errors is the limited resolution power of the telescopes.
Direct measurements: w < 0.1 arc seconds per century.
The microwave background radiation gives another means of determining the vorticity. It is known to be highly isotropic (equal in all directions), whereas rotation most likely would give some distortion (blueshift in some directions, redshift in others). Calculations along these lines  give 5-13 orders of magnitude lower limits on the vorticity. But they don't constitute any proof as they are based on certain cosmological models.
The rotation of Gödel's universe is of the order w = 0.01 arc seconds per century.
In conclusion, the direct observations are far too uncertain to detect even a rotation as large as that of the queer Gödel universe. But there are indirect indications that the rotation, if any, is far lower than that of Gödel's universe.
If we accept Newtonian physics as the basis for our existence, then there is no room for time traveling. It is simply taken for granted that there is a global time that runs equally for everybody under all circumstances. That's it. It's so evident, so perfectly in line with common sense, that it may not even be mentioned in Newtonian texts.
A warning: If you try to modify the time concept in the Newtonian framework, you'll have to modify all the other laws of nature accordingly. That's a huge undertaking. But that's what Einstein did in his Special Relativity Theory.
In special relativity the concept of a common global time is abandoned. Reference systems that move relative to each other must have different clocks. As a result of Special Relativity a traveler returning home will not have aged quite as much as those who stayed at home. But traveling back to an earlier point of time, as in the Gödel universe, is something entirely different. This has to do with the speed of light c, which is the limit for all information and all material particles.
Consider two events A and B in the universe, separated by time dt and by space dx. If dx² < c²dt², then the separation is said to be timelike, otherwise it's spacelike. If the separation is timelike, then it is possible for a material particle to visit both events. As a side remark, the classification of separations as timelike or spacelike is independent of the motion of the observer.
A light cone can be used to visualize which points can be reached and which points can not. The figure is drawn as if c=1.
Now, let's try to make a trip like that in Gödel's universe. We would start at A, and then make a loop to reach B which is at the same place but at an earlier time. We won't succeed as we would have to move out of the light cone - exceeding the speed of light. Sorry.
General Relativity is basically a local theory. The properties of space-time and its relation to matter is described locally by differential equations. The large scale effects will show up if one is able to integrate the differential equations. And the results may be quite surprising.
Except for special cases, in general relativity one can't choose a global time coordinate and draw global light cones. Each observer has its own small local light cone. He can move within the light cone, draw a new light cone from that position, etc. And, believe it or not, the light cones may be "tilted" so that a huge closed timelike curve is formed. The figure here may illustrate the idea; don't take it to literally as the meaning of the coordinate axes is not clear. A material object can travel from an event A, around the universe, and then back to event A. The curve can even be modified so that the object arrives before it was launched. If it then waits to see the launching, the timelike curve is closed.
Violation of causality is the scientific expression for time traveling.
Is it just a coincidence that both causality violation and rotation occur in the Gödel universe? That's hard to believe. In fact, the basic question behind by work is:
What is the connection, if any, between rotation of universe and violation of causality?
There does not seem to be an easy answer to the question. The reason may be that it is not posed in the spirit of general relativity. I would give one of its commandments in the following way:
Thou shalt not speak about the motion, relative to thee, of distant objects.
The rotation of the universe, i.e., the vorticity, is a local property. The closed time-like curve is a global feature. Accordingly the connection may be obscure. We don't have any concept of a universe rotating as a whole: The picture is complicated very much by the non-existence of a suitable global reference system; by the presence of shear and expansion; by the space-time curvature; and by the infinity of the universe.
There are rotating universes that do preserve causality. I am not aware of any non-rotating universe that violates causality.
In this section I give some personal, non-scientific, views on this type of causality violation.
What's so disturbing with causality violation? I haven't encountered any situation that can be described as a paradox or impossible as long as one keeps human mind and free will out of the reasoning.
Suppose a student would like to know the answers to an exam before doing the test. He decides to use a time machine. The following happens:
So far, so good. But what if, after having passed the test, the student does not bother to send the answers to the past? From where did he then receive them?
Assuming there is a free will, this is clearly a paradox. As far as I can see, free will is inconsistent with time traveling.
Other effects may be unexpected or unusual, e.g. that there would appear two copies of the same astronaut - one young and one old - at the launch pad. Although we have never experienced something like that, is it really in principle impossible?
Without the assumption of free will, I can't think of anything theoretically impossible with time traveling. If any reader can find out a paradox occurring in a universe inhabited by zombies, then please tell me.
In order to be able to travel to the past, three conditions should be fulfilled:
My personal views on these three points are:
There is however an easier way to violate causality if requirements 1 and 2 are met: Sending messages by radio. If the universe is inhabited by civilizations, you could send a message to one of them with instructions on how to forward the message to the next civilization, and so forth back to Earth. You don't have to wait for long to see whether the chain of transmissions has been successful; in fact, you should already have received the message...
 Sviestins, E. (1983). 'Rotating Relativistic Models of the Universe - Construction and Interpretation', Thesis, University of Stockholm.
 Misner, C.W., Thorne, K.S., and Wheeler, J.A. (1973). 'Gravitation' (Freeman, San Francisco).
 Gödel, K. (1949). 'An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation', Rev. Mod. Phys., Vol. 21, p. 447.
 Reinhardt, M. (1973). 'Mach's Principle - A Critical Review', Z. Naturforsch. Vol 28a, p. 529.
 Clemence, C.M. (1957). 'Astronomical Time', Rev. Mod. Phys. Vol. 29, p. 2.
 Hawking, S.W. (1969). 'On the Rotation of the Universe', Mon. Not. R. astr. Soc. Vol. 142, p. 529.
 Collins, C.B., and Hawking, S.W. (1973). 'The Rotation and Distortion of the Universe', Mon. Not. R. astr.Soc. Vol 162, p. 307.
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