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 nonlinear Einstein 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 (EH) action; and (iii) it uses only 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 impossible to obtain the EH action, starting from the standard action for gravitons in linear theory and iterating repeatedly. This result follows from the fact that EH action has a part (viz. the surface term arising from second derivatives of the metric tensor) which is nonanalytic in the coupling constant, when expanded in terms of the graviton field. Thus, at best, one can only hope to obtain the remaining, quadratic, part of the EH Lagrangian (viz. the Γ² Lagrangian) if no additional assumptions are made. Second, we use the Taylor series expansion of the action for Einstein's theory, to identify the tensor S^ab, to which the graviton field h_ab couples to the lowest order (through a term of the form S^abh_ab in the Lagrangian). We show that the second rank tensor S^ab is 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 theory with the source being a spin 0 field, a spin 1 field or relativistic particles by explicitly coupling the spin 2 field to this second rank tensor 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 not lead to Einstein's theory. The implications are discussed.