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However, the lower dimensionality severely reduces the number of independent components of the curvature tensor.

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JesseM

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Here's a pretty detailed discussion of the analogue of GR in a universe with 2 spatial dimensions:

http://gregegan.customer.netspace.net.au/SCIENCE/GR2plus1/GR2plus1.html

Basically, it seems you can't actually have spacetime curvature at any finite distance from a massive object in 2D, although you can have point particles which have the property that when you travel around them the angle is less than 360, like traveling around the tip of a cone (which is also a surface that's flat everywhere in differential geometry terms, since you can 'unroll' it into a flat surface with a pie-shaped wedge cut out, see http://www.allendesigns.com/AreaofCone/AreaofCone.htm [Broken]). There's a somewhat less technical discussion here:

http://www.theory.caltech.edu/people/patricia/ttif.html [Broken]

(see especially the section on 'What is the spacetime geometry?')

http://gregegan.customer.netspace.net.au/SCIENCE/GR2plus1/GR2plus1.html

Basically, it seems you can't actually have spacetime curvature at any finite distance from a massive object in 2D, although you can have point particles which have the property that when you travel around them the angle is less than 360, like traveling around the tip of a cone (which is also a surface that's flat everywhere in differential geometry terms, since you can 'unroll' it into a flat surface with a pie-shaped wedge cut out, see http://www.allendesigns.com/AreaofCone/AreaofCone.htm [Broken]). There's a somewhat less technical discussion here:

http://www.theory.caltech.edu/people/patricia/ttif.html [Broken]

(see especially the section on 'What is the spacetime geometry?')

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