Geodesics in (2 + 1)-dimensions
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Date
2009-08-14
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Abstract
The geodesics in (2+1) dimensional spacetime should be the same as the geodesics in (3+1) dimensional spacetime, since the two dimensional surface on which these geodesics lie should be embeddable 3-dimensional co-ordinate space. In the present thesis, we show precisely that .We demonstrate that the plane, the spherical surface, the ellipsoidal surface and the surface of a saddle on which these geodesics lie are embeddable in 3-dimensional co-ordinate space. We then go ahead and fine these geodesics. And clearly, the determination of the geodesics in (2+1) dimensional spacetime should be easier than in 3-dimensional spacetime since the number of equations involved in (2+1) dimensional spacetime is much smaller than in (3+1) dimensional spacetime. We talk about (2+1) and (3+1) dimensional spacetime because it is easier and more elegant to use the techniques of general relativity in the determination of these curves.
After showing that the surfaces indicated above are embeddable in 3-dimensional co-ordinate space, we go ahead and construct the 3-dimensional equivalent of the Robertson-Walker metric. The equation for the geodesics in general relativity is well known, and using our 3-dimensional metric, we compute all the geodesics on these surfaces which turn out to be the surfaces of zero, positive and negative curvatures
Not surprisingly, the geodesics in the plane and spherical surface were found to be straight line and great circles respectively. What can apparently be considered to be new results are the geodesics on ellipsoidal surface and the surface of a saddle which can really be described as 2-dimensional hyperbolic space. The geodesic on these last two surfaces were found to ellipses and hyperbolae. But it should be emphasized that in the relativistic language, curves are the geodesics in curved space and it is perhaps worth nothing that these curves are the trajectories of bodies attracted or repelled in force fields of the inverse square law
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A Thesis submitted to the Department of Mathematics
Kwame Nkrumah University of Science and Technology in partial fulfillment of the requirements for the degree of Master of Science.