General Relativity is one of the most beautiful theories in physics. Simply by changing the global concept of inertial frames to a local version, and connecting the curvature of the space-time manifold to the matter and energy distribution, we get a relativistic theory of gravity. The speed of light is inherited from special relativity, and Newton’s Gravitational constant is inherited by matching the low velocity, low field results with Newton’s theory. No new parameters are introduced into this theory making it highly vulnerable to attack by experimental counter-evidence. In spite of this, it has survived reasonably stringent experimental tests over the last 100 years. It is by far the most appealing theory of classical gravity.

Enter Kurt Godel. This fantastic mathematician caused many a great mathematicians to go mad with his famous incompletenss theorem. He has done a fair amount to perturb physicists as well. In 1949, he published a solution of general relativity, which represented a homogenous rotating universe. This universe was a very strange solution indeed. Firstly it was Anti-Machian. Contrary to the Machian philosophy that the inertial frames would be the set of frames in which the “fixed stars” do not rotate, in this solution the inertial frames see the matter of the universe rotating, and in the frame in which the matter is at rest, coriolis forces act on any moving object. Godel showed moreover, that in this solution, time doesn’t care enough about moving forward forever. He showed that you could have rockets propelled appropriately, that would go in a loop and come back to the same point in spacetime. Such a loop is referred to as a closed time loop (CTC). It is time-like because everywhere, the particle is travelling at a speed less than that of light, and yet it returns to the same point in spacetime. By the same token you could have propelled rockets that return to the same point in space before they left!

You may ask why bother, the solution doesn’t sound like our universe anyway … As far as we know, the matter is stationary in our inertial frames. Why is it surprising that such a strange solution has such strange properties. However, that is not the end of the story. A few years later closed time like loops (CTCs) were discovered in other solutions of general relativity, the Kerr solution, the Von Stockum solution, the Raychoudhari-Som solution and many more. One thing common to all these solutions – Rotation. (Besides rotating solutions, the wormhole class of solutions can also have CTCs). And some of these solutions are not far from reality. In particular when a massive star collapses to a blackhole, it would be described by a Kerr solution, because no matter how slowly it is rotates before collapse, it would gain significant angular velocity as it collapses (conservation of angular momentum).

In the 70s, Stephen Hawkings proposed a chronology preservation conjecture to rule out solutions that permitted “time travel”. However, this proposition did not have any physical grounds, as explicit physical paradoxes could not be constructed with these solutions to rule them out. Some people started proposing designs for time machines based on these solutions, primarily the Von Stockum solution for the rotating cylinder and the Kerr solution for the rotating blackhole. However, people still don’t understand the implications of CTCs in these solutions well enough to conclude without doubt that time travel would indeed be possible in such a case.

A few groups of general relativists are still studying these solutions to understand what exactly a CTC means, and whether their presence would imply a contradiction of physical laws. In particular, they are trying to study what would happen if a thermodynamic system were moved along a CTC. Would its entropy keep increasing as it loops around, or would the entropy increase upto a point, and then necessarily decrease, so that the CTC is closed with respect to entropy too, i.e the entropy returns to the same value every time it goes around the loop. This would suggest that the CTC should be interpreted as time itself being cyclic, rather than a phenomenon of time travel in a universe with time propagating forever forward.

Does general relativity permit time travel? No one yet knows for sure. One thing is clear – the meaning of time in general relativity is not clear.

**Posted by:** Ravishankar S

P.S : A preprint of the first part of our work can be accessed here http://arxiv.org/abs/gr-qc/0611093

What do you mean by Anti-Machian. The fact that matter far away in the universe influences the inertial properties of objects in the vicinity is Machian irrespective of whether the far away bodies are at rest or some bizarre motion .

Anyway, interesting read but you may have supplemented it with a few equations (the metric atleast!! ) for those who have some familiarity with the theory.

Venkateshan

Ravi,

Can you just include a background para at the beginning and yeah give a link to the Raychoudhari-Som solution if you have one . I couldn’t find a good link for that one.

And the following is the classical paper by Hawking in which he tries to exclude such solutions with Closed Timelike Curves. I am yet to sit down and work it out properly – for now, I am very confused about the physical import of the various assumptions he is making in the paper.

Here is another more recent paper trying to make sense out of Closed Time-like curves.

Ok, I´ve given a reference to Som-Raichaudhuri solution. But, It is a very old pdf file and I would still like to know if there are any better references.

There are some other papers I came across, which I thought might be useful…

and also see this paper.

More references on “time-travel”

Now, I’ve always wondered before what does existence of CTCs mean to quantum mechanical theory. Today, I found some interesting papers on this issue.

and then there are papers like the following one which mix computational complexity with CTCs.

The following paper by Scott Aaronson also has a section on this. Anyway, even otherwise it is a fun paper to read 🙂 (especially look at the section titled “anthropic computing” 😉 )