From Gravitons to Gravity: Myths and Reality

Abstract

There is a general belief, reinforced by statements in standard textbooks, that: (i) one can obtain the full non-linear Einstein's theory of gravity by coupling a massless, spin-2 field \(h_{ab}\) self-consistently to the total energy momentum tensor, including its own; (ii) this procedure is unique and leads to Einstein-Hilbert action and (iii) it only uses standard concepts in Lorentz invariant field theory and does not involve any geometrical assumptions. After providing several reasons why such beliefs are suspect -- and critically re-examining several previous attempts -- we provide a detailed analysis aimed at clarifying the situation. First, we prove that it is \textit{impossible} to obtain the Einstein-Hilbert (EH) action, starting from the standard action for gravitons in linear theory and iterating repeatedly. Second, we use the Taylor series expansion of the action for Einstein's theory, to identify the tensor \(\mathcal{S}^{ab}\), to which the graviton field \(h_{ab}\) couples to the lowest order. We show that the second rank tensor \(\mathcal{S}^{ab}\) is {\it not} the conventional energy momentum tensor \(T^{ab}\) of the graviton and provide an explanation for this feature. Third, we construct the full nonlinear Einstein's theory with the source being spin-0 field, spin-1 field or relativistic particles by explicitly coupling the spin-2 field to this second rank tensor \(\mathcal{S}^{ab}\) order by order and summing up the infinite series. Finally, we construct the theory obtained by self consistently coupling \(h_{ab}\) to the conventional energy momentum tensor \(T^{ab}\) order by order and show that this does {\it not} lead to Einstein's theory. (condensed).