juanrga
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atyy said:The reference only claims to show it for quantum GR in very strong gravity (eg. near spacetime singularities), for which it is agreed that GR may not be spin 2.
You are completely confused.
One can read already from page 2 of the reference cited (bold font from mine):
There are many articles and books dealing with GR but only a few papers discuss FTG. Perhaps it is a consequence of wide-spread opinion that FTG is equivalent to GR and hence we need not spend time to study field gravity approach. [...] Indeed in papers of Thirring(1961) and Deser (1970) there were claims that field theory approach is identical with the geometrical one and there are no gravitational effects which could provide grounds to distinguish between them.
[...]
However, as it will be shown here, reality turns out to be much more complex and interesting. Actually GR and FTG are two alternative theories with different bases and different observational predictions. Of course, for the weak gravitational fields, which are available for experiments now, both theories give the same values of the classical
relativistic effects, but they profoundly different in the case of strong gravity, which will be obtainable in near future.
First, FTG and GR agree on weak gravity, are different for intermediate gravity and «profoundly different in the case of strong gravity»
Second, he writes about GR. Nowhere he writes about «quantum GR» as you pretend {*}.
Indeed, the section 3 of the paper is titled «Classical theory of tensor field». The interesting part for this thread is the subsection 3.3 «Thirring and Deser about identity of GR and FTG», where is shown why the previous claims by both about GR being equivalent or derivable from a classical field theory are incorrect.
The «Quantum theory of tensor field» is presented in section 4 {**}, but I repeat, the proof that GR (a classical theory) is not equivalent to the classical field theory of gravity is given before in section 3.
The 'derivations' presented here by Deser, Schieve, and other people are incorrect: GR is not equivalent to a field theory of gravity (FTG).
{*} Moreover, does not exist a consistent and accepted «quantum GR».
{**} There exists a classical FTG and a quantum FTG, somewhat as there exists a classical electrodynamics and a quantum electrodynamics.