The D to 2 limit of general relativity

A method for taking the D → 2 limit of D-dimensional general relativity is constructed, yielding a two-dimensional theory which couples gravitation to conserved stress-energy. We show how this theory is related to those obtained via an alternative dimensional reduction approach. The study of gravity in two spacetime dimensions has been of considerable interest for several years [1, 2, 3, 4, 5, 6, 7, 8]. Such theories can have quite a rich and interesting structure which reproduces qualitatively much that is found in general relativity, (e.g. black holes, FRW-type cosmologies, gravitational collapse) even though they are mathematically much simpler. This simplicity makes them both a useful pedagogical tool and an interesting arena for the study of quantum gravitational effects. The topological character of the Einstein-Hilbert action in D = 2 dimensions has led theorists to construct such gravitational theories in a manner which circumvents the Einstein equations [1, 7, 9]. Indeed, the triviality of Einstein’s equations in two spacetime dimensions seemingly indicates that D = 2 general relativity does not make sense. We wish to point out here that this is not the case. Specifically, we show that (at least formally), one can take the D → 2 limit of Einstein’s equations. This yields a 2D theory of gravity previously considered in the literature [2, 3, 4, 5] in which gravitation is generated by stress-energy and stress-energy is in turn acted upon by gravitation, just as in D> 2 general relativity. We then elucidate via a dimensional-reduction argument the connections between this approach and other 2D theories [1, 7]. We begin with the gravitational action in D dimensions where L (D) M S = 1 κD d D x √ −gR + L (D)

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