## Rotation and Time Travel Saturday, Apr 1 2006

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

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## Symmetry breaking Saturday, Apr 1 2006

Necessary Background: Not much as the article starts out at a sort of popular science level, but those of us doing Particle Physics now, may appreciate this a bit more.

Comment: This blog is quite interesting by the way, with a lot of interesting discussions by a few High energy and Astrophysics people, so check it out when you have the time at http://www.cosmicvariance.com/ or http://cosmicvariance.com/category/science/

Posted by: Shanth

## Identical Particles Saturday, Apr 1 2006

This came up during a Particle Physics discussion :

Here is a proof that the wavefunction of n identical particles has to be either completely symmetric or anti-symmetric.

Wavefunction ψ=ψ(a1,a2,……..,an)

Let Pij be the operator which flips particles i and j

Now indistinguishability requires

Pijψ=c ψ

Since flipping twice results in the same wavefunction

Pij Pij ψ = ψ

Implies c=1 or -1

Now consider this relation

Pij=Pik Pkj Pki

One can easily veryify that this true by looking at the action of Pij on the ordrerd set (i,j,k).

Since Pik Pik=1

Pij=Pkj

Extending this simlarly one gets

Pij = Pik = Pkl for any k,l

Thus Pfg = c for all f,g

Thus the wavefunction has to be completely symmetric or antisymmetric.

Venkateshan

PS: I took the liberty of tidying up the notation, hope I didn't make any mistake/ change your argument. –Shanth

## Does Position operator generate momentum Saturday, Apr 1 2006

$\langle x|p \rangle =e^{i\mathbf{p.x}}$

Consider

$U = e^{i\mathbf{X.p}}$ (Capitals for Operators)

$\langle x|e^{i\mathbf{X.p'}}|p\rangle= \int dx' \langle x|e^{i\mathbf{X.p'}}|x'\rangle \langle x'|p\rangle$

$= e^{i\mathbf{(p+p').x}}$ (skipped the delta function integ. step)

does this imply that position operator generates momentum?

Posted By : Venkateshan

## How to Talk to a Physicist: Groups, Symmetry, and Topology Saturday, Apr 1 2006

Necessary Background : Mathematical Physics, Quantum mechanics, probably quantum field theory/Particle Physics for the later parts. But, more than anything else, interest in symmetry groups and topology as they enter into physics.
Link : from http://physics.harvard.edu/~dtlarson/tutorial05/

## Groups, Symmetry, and Topology

Comment : This is a collection lecture notes written in a kind of no-nonsense approach. Contains a short intro to Lie Algebra, its representations and stuff. Do look at the list of books provided at the site linked above !

Posted By : Loganayagam.R.(Tom)

## Half-integral spins and Pauli exclusion Saturday, Apr 1 2006

Necessary Background : Basics of quantum statistics(bosons fermions and all that) , a bit of topology(basically knowing what the word homotopy means)and some geometric imagination.

# half-integral spins and Pauli exclusion

Comment :
It is a kind of popular science explanation on why fermions obey Pauli’s principle

Posted By : Loganayagam.R.(Tom)

## MIT OpenCourseWare – Physics Saturday, Apr 1 2006

Necessary Background :This is for Everyone

MIT OpenCourseWare » Physics

Comment :
This site lists the links to the various course sites at MIT. Many of these sites have lecture notes/assignments which you might find useful.

Posted By : Loganayagam.R.(Tom)

## Do We Really Understand Quantum Mechanics ? Saturday, Apr 1 2006

Necessary Background : Basic ideas in quantum mechanics, Bra-Ket Notation and yeah interest in Interpretation of quantum mechanics,EPR, Wave-function collapse and stuff like that . 🙂

Link : From http://arxiv.org/abs/quant-ph/0209123

Do we really understand quantum mechanics?
Authors:
Franck Laloe (LKB – Lhomond)
Journal-ref:
American Journal of Physics 69 (2001) 655 – 701

Comment :
This is an article which appeared on AJP quite sometime ago. I think it does a good job of talking about various weird things which crop up in quantum theory. It’s quite loooong, but I guess depending on what your background and interests are, you can safely skip a lot of it.

So, tell me what do you think of this article. You can post any interesting link related to this subject in the comments below. By the way, for those who are not familiar to blogs, you can click the link below with “comments” written on it and leave your comments there.

And well, if you worry so much about your privacy that you don’t want to leave your name here (Horrors !) , then comment with name as anonymous , but do leave a valid e-mail(Say the iitk email) so that we among ourselves can figure out who is commenting. The email ID will not appear here – to see the email ids of the commenters one has to login using username+Passwd. We will soon try to come up with an About page How to post in this blog page with all such info.

And again, Comments please !

Posted By : Loganayagam.R.(Tom)