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Imagine we live on cylinder(we are 2d creature), put a rope around that cylinder and start pulling both ends of the rope against each another. Will the space get deformed? I guess it will, I have to put some energy to the rope and energy deforms the space. Is it possible to calculate shape of the space based on the force I put in the rope?

In order to calculate this I guess you have to make 3d analogy and use laws of general relativity.

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  • $\begingroup$ What could the rope, or anything that causes the effect of the rope, be from a physics point of view? $\endgroup$
    – Dilaton
    Jul 25 '13 at 11:13
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I suspect you're thinking that pulling a string wrapped round a cylinder would compress it in the same way that pulling on a string wrapped round a coke can would crush it. If so, this won't happen because the tension in the string can't produce any force out of the 2+1D manifold that it lives in.

However you are quite correct that the tension in the string will contribute to the stress-energy tensor, and therefore act as a source of gravity. I have to confess I don't know what the effect would be in 2+1D, however in 3+1D the effect would be similar to a cosmic string. Assuming the mass of the string is negligable, so the tension is the only contribution to the stress-energy tensor, the spacetime around the string is flat but has an angular deficit. That means you don't feel any gravitational field from the string, but if you travel in a circle around the string you'd find you had rotated by less than 360º.

I suppose the equivalent to your 1D object in the 2+1D universe would be a 2D object in our 3+1D universe i.e. instead of a stretched string you'd have a stretched membrane. An example of this would be a domain wall. This does produce a gravitational field, and in fact the field is repulsive i.e. you would be accelerated away from the wall in a direction normal to it. There is a description of the field produced in this book.

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