Einstein equations and Fierz-Pauli equations with self-interaction in quantum gravity

Abstract

The Einstein equations can be written as Fierz-Pauli equations with self-interaction,$W\gamma _{ik} = - G_{ik} + \tfrac{1}{2}g_{ik} g^{mn} G_{mn} - k(T_{ik} - \tfrac{1}{2}g_{ik} g^{mn} T_{mn} )$ together with the covariant Hilbert-gauge condition,$(\gamma _i^h - \tfrac{1}{2}\delta _i^k g^{mn} \gamma _{mn} )_{;k} = 0$ where W denotes the covariant wave operator and G _ik the Einstein tensor of the metric g _ik collecting all nonlinear terms of Einstein's equations. As is known, there do not, however, exist plane-wave solutions γ _ik(z)with g ^ik Z,_i Z,_k=0of these equations such that what is essential to the introduction of gravitons is not satisfied in general relativity. This nonexistence corresponds with the uncertainty relation,Δp(Δg*)²(Δx)³≥h hG/ c ³ concerning the total nonlinear gravitational field g ^*ik=g ^k+γ ^k.