A New Gravitational Energy-Momentum Pseudotensor in Tetrad Form

Abstract

It is shown that (1) the new gravitational energy-momentum pseudotensor t,'(x) derived from the Hayashi-Shirafuji gravitational Lagrangian density in the extended Weitzenb5ck spacetime is the unique theory which can give the gravitational energy') within any spatial region a definite value independent of the spatial coordinate system and satisfy certain other physical postulates, and (2) toO(x) takes nonnegative values at any point in spacetime where goo>O**' for some well-known gravitational fields. The gravitational energy is not localizable at a point in spacetime because of the principle of equivalence. However, a true gravitational field in any non-zero region of space with torsion cannot be eliminated or created by a mere transforma­ tion of the spacetime coordinates and, of course, energy is not produced or annihilated by a mere change of the names of spatial points. Furthermore, from the quantum theoretical viewpoint that the gravitational field is an assemblage of gravitons with nonnegative energy, the gravitational energy within any nonzero region of space must be nonnegative, and hence, as has been proved in the positive energy theorem/) for a closed system (see § 3 for definition) having locally nonnegative matter energy density, the total energy of matter plus gravitational field should be positive. Thus, in addition to a conservation law for the energy and momentum of matter plus gravitational field, the gravitational energy-momentum pseudo tensor ti h(X), which can only be a tensor under linear coordinate transformations but cannot be a tensor under general coordinate transformations, must satisfy the following properties: 0) the gravitational energy within any nonzero region of space at a certain time is independent of the spatial coordinate system, and (ii) toO(x) which is the continuous function of spacetime coordinates takes nonnegative value at any point in spacetime where goo> O. N ow, Einstein's gravitational energy-momentum pseudotensor Eti h(X) (see (2·31) for definition) derived from the Einstein-Hilbert gravitational Lagrangian density2) has been proved to be the only possible theory which is expressed by the metric tensor only and also satisfies certain physical requirements. 3 ) However, EtOO(X) does not transform as a scalar under purely spatial transformations, so that Et/(X) cannot satisfy the property stated in 0). Furthermore, if we calculate EtoO(X) for the Schwar­ zschild exterior spacetime in a harmonic 4 ) or isotropic coordinate system,5) it takes a *> The gravitational energy is not the negative gravitational interaction energy but the energy caused