Positive definite problem of energy density and radiative energy flux for pulse cylindrical gravitational wave

Abstract

By using the general expressions of energy-momentum pseudo-tensor of the cylindrical gravitational waves (GW) given by Rosen and Virbhadra in Cartesian coordinates, the concrete forms of energy density and radiative energy flux of the pulse cylindrical GW are obtained. Their physical properties, suitable range and asymptotic behaviour are considered. It is found that: For the region in which space radial coordinates to origin are greater than the pulse width of the pulse cylindrical GW, the energy density and radiative energy flux of the outward travelling pulse cylindrical GW propagating along a light-cone are positive definite. However, for the region in which the space radial coordinates are less than the pulse width, there is no guarantee for the positive definite property of the radiative energy flux of the outward travelling wave. Moreover, we show that the asymptotic behaviour of the energy and energy flux densities of the pulse cylindrical GW and that of the Riemann curvature tensor have good self-consistancy in space-like, time-like and null infinity regions.