Einstein׳s Equations for Spin 2 Mass 0 from Noether׳s Converse Hilbertian Assertion

Abstract

An overlap between the general relativist and particle physicist views of Einstein gravity is uncovered. Noether's 1918 paper developed Hilbert's and Klein's reflections on the conservation laws. Energy-momentum is just a term proportional to the field equations and a "curl" term with identically zero divergence. Noether proved a converse "Hilbertian assertion": such "improper" conservation laws imply a generally covariant action. Later and independently, particle physicists derived the nonlinear Einstein equations assuming the absence of negative-energy degrees of freedom ("ghosts") for stability, along with universal coupling: all energy-momentum including gravity's serves as a source for gravity. Those assumptions (all but) imply (for 0 graviton mass) that the energy-momentum is only a term proportional to the field equations and a symmetric curl, which implies the coalescence of the flat background geometry and the gravitational potential into an effective curved geometry. The flat metric, though useful in Rosenfeld's stress-energy definition, disappears from the field equations. Thus the particle physics derivation uses a reinvented Noetherian converse Hilbertian assertion in Rosenfeld-tinged form.The Rosenfeld stress-energy is identically the canonical stress-energy plus a Belinfante curl and terms proportional to the field equations, so the flat metric is only a convenient mathematical trick without ontological commitment. Neither generalized relativity of motion, nor the identity of gravity and inertia, nor substantive general covariance is assumed. The more compelling criterion of lacking ghosts yields substantive general covariance as an output. Hence the particle physics derivation, though logically impressive, is neither as novel nor as ontologically laden as it has seemed.

The dynamical approach to spin-2 gravity

“…For instance, Pitts (2016) raises doubts about claims to the effect that the ontology of the spin-2 approach is obvious, and that the derivation shows that GR simply becomes another special-relativistic field theory. Pitts argues that it is not clear whether the unobservability (e.g., Thirring (1961)) of the flat metric, from which one starts in the spin-2 approach, accounts for it 'not being real' (which Pitts takes to be tacitly assumed), and, hence, for it 'ceasing to exist' (Pitts, 2016). However, in the dynamical perspective, the flat background is a non-entity from the start-it does not 'cease to exist'.…”

“…The features of the classical and the quantum version of the spin-2 approach as well as their relations to Einstein's (non-linear) theory of general relativity have been worked out by a number of physicists, e.g., Markus Fierz and Wolfgang Pauli (1939), Nathan Rosen (1940a;1940b), Achille Papapetrou (1948), Suraj Gupta (1952b;1952a;1957), Robert Kraichnan (1955;1956), Walter Thirring (1959;1961), Viktor Ogievetsky and Igor Polubarinov (1965), Walter Wyss (1965), J. Fang and Christian Frønsdal (1979), Richard Feynman (1995), Steven Weinberg (1964a;1964b;1972), Stanley Deser (1970;2010), Robert Wald (1986), and Brian Pitts and William Schieve (2001c;2007)-for a concise review see Preskill and Thorne (1995), for an extensive list of references see also Pitts (2016). In particular, it is suggested by Weinberg and others that the spin-2 approach to quantum gravity has a particularly significant feature: it is taken to provide an explanation for why the strong equivalence principle (SEP)-that is, the fact that locally all laws reduce to the laws of special relativity (SR)-holds (see Salimkhani (2018) for a sketch of the argument for a philosophical audience).…”

Friedman and Some of His Critics on the Foundations of General Relativity

“…These theories suggest two main challenges to Friedman's analysis. The first and most trenchant is implicit in the particle physics approaches to gravitation theory and in the work of Pitts (2016aPitts ( , 2016bPitts ( , 2018. This is the view that the equivalence principle is eliminable, and therefore unnecessary for the development of Einsteinian gravitation.…”

Kant, Schlick and Friedman on Space, Time and Gravity in Light of Three Lessons from Particle Physics

“…The massless spin 2 derivations of Einstein's equations from flat spacetime would eventually (Kraichnan, 1955;Gupta, 1954;Feynman et al, 1995;Weinberg, 1964a;Weinberg, 1964b;Deser, 1970;Pitts and Schieve, 2001) show why it might not be unreasonable to favor universal forces even given Einstein's equations because it isn't implausible that gravity would act in just that way without any peculiar premises. (Such derivations turn out to be built around what one can recognize as the converse of Noether's Hilbertian assertion (Pitts, 2016a).) Better yet, the already extant Neumann-Seeliger-Einstein 1890s/1917 modification of Newtonian gravity in principle showed the way to taking massive spin-2 gravity (not a hypothetical force) to be an almost-universal force (Ogievetsky and Polubarinov, 1965;Freund et al, 1969), in the sense of acting like a Poincaré-Reichenbach universal force if one is unable to perform experiments sensitive to long-range gravitational effects.…”