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.
Highlights
The Riemann curvature tensor of general relativity Ri jkl can be split into the Weyl conformal tensor Ci jkl, and parts which involve only the Ricci tensor R jl and the curvature scalar R
We want that the new equation of general relativity necessarily contains the Riemann tensor linearly, and be a fourth-order tensor equation with the same index symmetries as Ri jkl
With Eq (40), simplified by (45), we find a simple connection between the energy-momentum tensor for the free gravitational field Ti(jFkl), and the Weyl tensor Ci jkl : aCi jkl = χ Ti(jFkl)
Summary
The Riemann curvature tensor of general relativity Ri jkl can be split into the Weyl conformal tensor Ci jkl , and parts which involve only the Ricci tensor R jl and the curvature scalar R. Because of the properties of the Weyl tensor, its contraction vanishes, gikCi jkl = 0, and the information it contains (namely the information as regards the gravitational field in vacuum) is not present in the famous Einstein equation. The aim of this paper is to find a generalized gravitational field equation explicitly containing the Riemann curvature tensor linearly. For this purpose, we have implemented a rigorous mathematical treatment with a classical variational principle using a generalized Lagrangian containing Ri jkl , R jl and R.
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