Rank-4 Structure of the Einstein Field Equations

Background:A simple approach to quantise gravitation is done.Material and methods:Simple rules of tensor algebra are used.Results:We represent every tensor \(X\) by a size \(S(X)\), the sum of the absolute values of its components, and a normalised tensor\(p(Xμν)=Xμν/S(X)\) whose components are signed shares in \([-1,+1]\) with \(∑|pμν|=1\). Division is defined with\(0/0:=1\), \(1/0:=∞\) and \(x/0=x\,∞\), and zero keeps its multiplicity in sums, so that the zero tensor has \(S=N·0\) and the uniformdistribution \(p=1/N\) in every component. Then \(Xμν=S(X)\,p(Xμν)\) holds for every tensor without exception, and the Einstein fieldequations (EFE) with cosmological constant are expressed in this form without restriction, including vacuum and flat space. All tensors of the equationsfollow from \(p(Rμν)\), \(S(Rμν)\) and \(Λ\). A decomposition of the Ricci tensor into several fields yields weights that add upto at least one, with equality exactly when no component cancels. The formulation is mathematically equivalent to the EFE.The field equations are also written in rank-4 form, \(Gμνρσ+Λ3\,gμνρσ=Eμνρσ\), where the source tensor \(Eμνρσ\) is the sum of the rank-4 versions of ordinary matter and of the electromagnetic field; the Riemann tensor splits into the Weyl tensor, the source tensor and a pure-trace term, and the complement of the Weyl tensor in the Riemann tensor is called the anti-Weyl tensor.This form is used to quantise the field equations as a whole, with gravity, matter and the electromagnetic field treated together.To lowest order in \(κ\), the commutators of the part of the curvature generated by the sources (the Ricci part) follow from the matter commutators without a separate gravitational postulate, and a classical consistency proposition with the Dirac bracket is proven. The freely propagating Weyl tensor (gravitational waves) requires an independent postulate, given for linear perturbations in transverse-traceless variables. The energy of the wave is written as a part of \(Eμνρσ\) (Isaacson tensor) with an anti-tensor as its complement, and an explicit commutator is derived for linear waves.Conclusion:Unfortunately, an anomaly-free closure of the constraint algebra, the inner product on the physical states, the operator ordering of composite operators, a non-perturbative definition and the problem of time remain open while the rank-4 form does not improve renormalisability.The normalised representation treats vacuum and flat space without special cases,gives every source a dimensionless, bounded shape with an entropy, and writes a tensor in the form of a normalised amplitude vector, which might be a first step from thegeometric language of general relativity towards the language of probability.