ar X iv : h ep - t h / 05 12 16 8 v 7 4 J ul 2 00 6 The Einstein - Hilbert Lagrangian Density in a 2 - dimensional Spacetime is an Exact Differential ∗

Recently Kiriushcheva and Kuzmin [1] claimed to have shown that the Einstein-Hilbert Lagrangian density cannot be written in any coordinate gauge as an exact differential in a 2-dimensional spacetime. Since this is contrary to other statements on on the subject found in the literature, as e.g., by Deser [3], Deser and Jackiw [4], Jackiw [5] and Grumiller, Kummer and Vassilevich [6] it is necessary to do decide who has reason. This is done in this paper in a very simply way using the Clifford bundle formalism. In this version we added Section 18 which discusses a recent comment on our paper just posted by Kiriushcheva and Kuzmin [2]. In [1] authors claim to have shown that: 'if general covariance is to be preserved (that is, a coordinate system is not fixed) the well known triviality of the Einstein field equations in two dimensions is not a sufficient condition for the Einstein-Hilbert action to be a total divergence'. This statement is contrary to well known statements, as, e.g., in [3, 4, 5, 6]). So, we need to decide who is correct. In what follows we explain that even if at first (and even second) sight the arguments of [1] seems to be correct, they are not complete and indeed the Einstein-Hilbert Lagrangian in a 2-dimensional spacetime can always be written in any coordinate gauge as an exact differential.