The Gravitational Energy-Momentum Density of Radially Accelerated Observers in Schwarzschild Spacetime

A hybrid machinery that is useful for calculations in teleparallel theories when the spacetime is spherically symmetric is developed. Using this machinery, the gravitational energy-momentum tensor density of the Schwarzschild spacetime is evaluated in a frame adapted to observers that accelerate in the radial direction. The energy density, the total energy, and the gravitational energy-momentum flux are obtained. The regularization procedure and the limit where gravity is absent is discussed. It turns out that the regularized energy and energy-momentum flux are consistent in the whole spacetime. The continuity equation for the gravitational energy-momentum also holds for any point outside the black hole. Finally, the static and freely falling cases are discussed. It is found that a static observer measures a negative gravitational energy density, while a freely falling one measures a vanishing density.