Energy and momentum in general relativity

Abstract

In general relativity, conservation of energy and momentum is expressed by an equation of the form∂ℐμν/∂xν= 0, whereℐμν≡√−gTμν represents the total energy, momentum, and stress. This equation arises from the divergence formula\(\mathop{{\int\!\!\!\!\!\int\!\!\!\!\!\int}\mkern-31.2mu \bigodot} {}\) ∐μνdV v = ∬∬(∐μν/∂x v )d4d. Here we show that this formula fails to account properly for the system of basis vectors eμ(x). We obtain the (invariant) divergence formula\(\mathop{{\int\!\!\!\!\!\int\!\!\!\!\!\int}\mkern-31.2mu \bigodot} {}\)eμ ℐμνdV v = ∬∬eμ(∂ℐμν/∂x v + Γ νλ μ ℐνλ)d4d. Conservation of energy and momentum is therefore expressed by the covariant equation (∂ℐμν/∂x v ) + Γ νλ μ ℐνλ = 0. We go on to calculate the variation of the action under uniform displacements in space-time. This calculation yields the covariant equation of conservation, as well as the fully symmetric energy tensorℐμν. Finally, we discuss the transfer of energy and momentum, within the context of Einstein's theory of gravitation.