Nuclear Physics B (Proc. Suppl.) 6 (1989) 405-407 405 North-Holland, Amsterdam NEW SUPERI:~TENTIALS IN GENERAL RELATIVITY Mauro FRANCAVIGLIA * and Marco FERRARIS** • Istituto di Fisica Matematica "J. L. Lagrange", University of Torino, Via Carlo Alberto 10, 10123 Torino, Italy. • * Dipartimento di Matematica, Universityof Cagliari, Via Ospedale 72, 09100 Cagliari, Italy. In the remarkable 1982 paper 1 about the notion of mass and angular momentum in General Relativity, R. Penrose wrote: "It is perhaps ironic that cnergy conservation, a paradigmatic physical concept arising initially from Galileo's (1638) studies of the motion of bodies under gravity, [.... ] should nevertheless have found no universally applicable formulation, within Einstein's theory, incorporating the energy of gravity itselF" The definition of energy, and, more generally, of conserved quantities, is in fact, despite the large literature on the subject, a still intriguingproblem in General Relativity. One of the reasons which generate difficulties in this matter is the fairly well known fact that, in order to define the energy of the gravitational field starting from a variational formulation of Einstein's field equations, one usually works with expressions having validity only in a local frame or under particular restrictive hypotheses 2. According to a general procedure in Physics, which dates back to the work of E. N6ther 3, the energy and the other conserved quantities corresponding to a (Lagrangian) field theory should be obtained as the generators of the field dynamics along some preferred vector field in space-time. As is well known, this requirement is 0920-5632/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division) implemented in Classical Mechanics and first-order field theories by relying on a suitable l-form, called the Poincar6-Cartan form 4, living either in the tangent bundle of configuration space or in the appropriate phase space (possibly via reduction, if constraints are present). Essentially the same procedure, with suitable modifications to ensure covariance and globality, is actually available for field theory, no matter how many derivatives of fields enter the Lagrangian 5. Following an idea which can be traced back to Einstein 6, one can define conserved quantities along the flow of a vector field ~ in space-time by means of an appropriate vector-density EX(L,~), which shall be called here the energy-density flow. This will satisfy "weak conservation laws" (1) 0x EX(L,~) = 0, where "weak" means "on shell", i.e., along solutions of field equations. Moreover, the energy-density flow EX(L,~) can in turn be identified with the divergence of a skew-symmetric tensor-density UXp(L,~), usually called a superpotential, by a "weak equation" of the following type: (2) EX(L,~) = 0pUXp(L,~). For obvious reasons, for any given L there is no