Mathematics
Scientific paper
May 1989
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1989gregr..21..533d&link_type=abstract
General Relativity and Gravitation, Volume 21, Issue 5, pp.533-544
Mathematics
7
Scientific paper
In general relativity, conservation of energy and momentum is expressed by an equation of the form∂? μν/∂xν= 0, where ? μν≡√-gTμν represents the total energy, momentum, and stress. This equation arises from the divergence formulamathop{{int int int}mkern-31.2mu bigodot} {} ∐μν dV v = ? ? (∐μν/∂x v )d 4 d. Here we show that this formula fails to account properly for the system of basis vectors eμ(x). We obtain the (invariant) divergence formulamathop{{int int int}mkern-31.2mu bigodot} {} e μ ?μν dV v = ? ? e μ(∂?μν/∂x v + Γ{/νλ μ} ?νλ)d 4 d. Conservation of energy and momentum is therefore expressed by the covariant equation (∂?μν/∂x v ) + Γ{/νλ μ} ?νλ = 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 tensor? μν. Finally, we discuss the transfer of energy and momentum, within the context of Einstein's theory of gravitation.
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