Generalization of Einstein’s gravitational field equations

Abstract

The Riemann tensor is the cornerstone of general relativity, but as is well known it does not appear explicitly in Einstein's equation of gravitation. This suggests that the latter may not be the most general equation. We propose here for the first time, following a rigorous mathematical treatment based on the variational principle, that there exists a generalized 4-index gravitational field equation containing the Riemann curvature tensor linearly, and thus the Weyl tensor as well. We show that this equation, written in n dimensions, contains the energy-momentum tensor for matter and that of the gravitational field itself. This new 4-index equation remains completely within the framework of general relativity and emerges as a natural generalization of the familiar 2-index Einstein equation. Due to the presence of the Weyl tensor, we show that this equation contains much more information, which fully justifies the use of a fourth-order theory.

4-Index theory of gravity and its relation with the violation of the energy–momentum conservation law

“…The integral of g ik δR ik will be vanished [1], and the coefficient of the δ(g ik R) will be zero whenever m = −dn, the primary and simple case studied in [1]. In this manner, combining this result with Eq.…”

“…Finally, in similarity with the definition of Ricci tensor (R jl = g ik R ijkl ), and just the same as Ref [1], we assume that there is a 4-index generalized energy-momentum tensor T ijkl satisfying the T jl = g ik T ijkl condition, in which T jl is the ordinary 2-index energy-momentum tensor representing all sources filling the background and obtainable by applying the action principle to the matter Lagrangian, i.e. [1] 2δI…”

“…Recently, a 4-index generalization of general relativity has been introduced to relate the Riemann tensor to the energy-momentum sources filling the spacetime [1]. Therefore, it is a theory may directly help us in modelling the evolution of geometry and thus its curvature.…”

“…and is zero for conformal flat spacetimes. Therefore, the view of Ref [1] claims that the gravitational field has not any effects in conformal flat spacetimes which are indeed curved. If we accept the Einstein idea that spacetime is curved by energy sources, then this property of Weyl tensor will establishes an inconsistency with his idea.…”

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Recently, a 4-index generalization of the Einstein theory is proposed by Moulin [1]. Using this method, we find the most general 2-index field equations derivable from the Einstein-Hilbert action.

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