From Gravitons to Gravity: Myths and Reality

Abstract

There is a general b elief, reinforced by statements in standard textb o oks, 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 pro cedure is unique and leads to Einstein–Hilb ert (EH) action; and (iii) it uses only standard concepts in Lorentz-invariant field theory and do es not involve any geomet-rical assumptions. After providing several reasons why such b eliefs are susp ect — 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 rep eatedly. 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 con-stant, when expanded in terms of the graviton field. Thus, at b est, one can only hop e to obtain the remaining, quadratic, part of the EH Lagrangian (viz. the Γ 2 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 ab h 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 con-struct the full nonlinear Einstein theory with the source b eing 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 do es not lead to Einstein’s theory. The implications are discussed.