Analytic solutions for static spherically symmetric distribution of liquid in its gravitational self-field

Tolman's differential equations for the two components of the metric tensor of a spherically symmetric distribution of liquid are reduced to equations for two functions in which the derivative of one of them is expressed in terms of the other, and not only the components of the metric tensor but also the physical characteristics of the continuous medium are expressed in terms of these functions. Arbitrary choice of the second function generates different self-consistent solutions. By means of the simplest choices of this function, two single-parameter solutions are found — one for a gas and the other for a liquid.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 95, No. 1, pp. 135–145, April, 1993.     

Generation of modes on the three-dimensional sphere

We use the method of Lie generation of tensor fields, which works for fields of different tensor structures, to construct the complete system of scalar, vector, symmetric tensor, and spinor fields on the three-dimensional sphere. We construct the Pauli operator explicitly. We demonstrate the role of spin in forming the mode series. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 150, No. 3, pp. 417–440, March, 2007.     

Energy flows of a gravitational wave in a liquid sphere

We consider the energy transfer problem in a static space-time background created by a liquid spherical body. In the accompanying reference frame, we derive the Lagrangian of gravity and the liquid that is of the fourth order in perturbations. We impose gauge conditions and integrate over the angular coordinates at the level of the action, which makes the problem two-dimensional. We derive the density and flow of the perturbation energy. Different gauge choices are considered. The energy conservation law is ensured by the static property of the metric and by the vanishing of the Lagrange variations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 2, pp. 342–351, August, 1999.     

Energy flows of gravitational waves

The problem of the energy transfer produced by weak gravitational waves on a static background is solved by using the Killing vectors without introducing the momentum-energy pseudotensor. The Lagrangian, which is of the second order in metric excitations and which determines the flow and density of the energy of gravitational waves on a static spherically symmetrical space-time background, is derived. Integration over angular variables in the action integral thus reduces the problem to a two-dimensional effective problem. Energy flows on the Schwarzschild metric background are calculated. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 1, pp. 168–176, July, 1999.