 
Summary: physics/99060042Jun1999
Mechanics.  On the analytic expression that must be given to
the gravitational tensor in Einstein's theory
Note by the Fellow
T. LeviCivita
translation and foreword by
S. Antoci
and A. Loinger
Foreword. While most textbooks of general relativity and research articles discuss at length the relative merits
of the pseudo tensors proposed by Einstein and by other authors for representing the energy of the gravitational field,
Levi Civita's definition of a true gravitational energy tensor has succumbed to Einstein's authority and is nearly
forgotten. It seems however worthy of a careful reexamination, due to its unquestionable logical soundness and to
the unique manner of propagation for gravitational energy that it entails.
In the present Note, after having recalled, for the reader's convenience, the leading idea and
the mathematical framework of general relativity, I show how some identities (involving the deriva
tives of the Riemann symbols) discovered by Bianchi offer a sure criterion for introducing the so
called gravitational tensor. From the analytic standpoint one has to do with a double symmetric
system Aik (i, k = 0, 1, 2, 3), whose ten elements completely define the gravitational contribution
to the local mechanical behaviour. In fact they determine the stresses as well as the energy flow
and the energy density (of gravitational origin). The mechanical meaning of the system requires
