Einstein's Equations for Spin 2 Mass 0 from Noether's Converse Hilbertian Assertion
aa r X i v : . [ phy s i c s . h i s t - ph ] O c t Einstein’s Equations for Spin 2 Mass 0 from Noether’s ConverseHilbertian Assertion
November 9, 2016
J. Brian PittsFaculty of Philosophy, University of [email protected] in
Studies in History and Philosophy of Modern Physics
Abstract
An overlap between the general relativist and particle physicist views of Einstein gravity isuncovered. Noether’s 1918 paper developed Hilbert’s and Klein’s reflections on the conservationlaws. Energy-momentum is just a term proportional to the field equations and a “curl” term withidentically 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 as-suming the absence of negative-energy degrees of freedom (“ghosts”) for stability, along withuniversal 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 aterm proportional to the field equations and a symmetric curl, which implies the coalescence ofthe 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 fieldequations. Thus the particle physics derivation uses a reinvented Noetherian converse Hilbertianassertion in Rosenfeld-tinged form.The Rosenfeld stress-energy is identically the canonical stress-energy plus a Belinfante curland terms proportional to the field equations, so the flat metric is only a convenient mathematicaltrick without ontological commitment. Neither generalized relativity of motion, nor the identityof gravity and inertia, nor substantive general covariance is assumed. The more compellingcriterion of lacking ghosts yields substantive general covariance as an output. Hence the particlephysics derivation, though logically impressive, is neither as novel nor as ontologically laden asit has seemed. keywords: conservation laws; General Relativity; Noether’s theorems; energy-momentum tensor;particle physics; Belinfante-Rosenfeld equivalence
It is often held that there is a great gulf fixed between the views of gravitation held by generalrelativists and those held by particle physicists. Carlo Rovelli has commented on the resultanteffects on quantum gravity research programs. 1he divide is particularly macroscopic between the covariant line of research on theone hand and the canonical/sum over histories on the other. This divide has re-mained through over 70 years of research in quantum gravity. The separation cannot bestronger.. . . Partially, the divide reflects the different understanding of the world thatthe particle physics community on the one hand and the relativity community on theother hand, have. The two communities have made repeated and sincere efforts to talkto each other and understanding each other. But the divide remains, and, with thedivide, the feeling, on both sides, that the other side is incapable of appreciating some-thing basic and essential. . . . Both sides expect that the point of the other will turn out,at the end of the day, to be not very relevant.. . . Hopefully, the recent successes of bothlines will force the two sides, finally, to face the problems that the other side considersprioritary. . . (Rovelli, 2002).The following anecdote gives some background about the history of (non-)interaction betweenthe general relativity and particle physics communities:The advent of supergravity [footnote suppressed] made relativists and particle physicistsmeet. For many this was quite a new experience since very different languages wereused in the two communities. Only Stanley Deser was part of both camps. The particlephysicists had been brought up to consider perturbation series while relativists usuallyignored such issues. They knew all about geometry instead, a subject particle physicistsknew very little about. (Brink, 2006, p. 40)Unfortunately the pocket of convergence between the two communities in physics, though by nowexpanded well beyond Deser (and perhaps allowing for hyperbole), is still rather small.If one can show that the gulf is not as large as it seemed, might it be easier to imagine quan-tum gravity programs that also split the difference? This paper will not aim to say much aboutquantum gravity, but it will show that a certain part of the difference between general relativists’and particle physicists’ views is illusory, because a key part of particle physicists’ 1939-73 “spin 2”derivation of Einstein’s equations from flat space-time is basically Noether’s 1918 converse Hilber-tian assertion. Hilbert and later Klein had found that the conserved energy-momentum in GeneralRelativity consists of a term proportional to the Einstein tensor and hence having a value of 0 usingthe Euler-Lagrange equations (vanishing on-shell, one says) and a term with automatically van-ishing divergence (a “curl”). They worked with the most straightforwardly derivable expressionsfor gravitational energy-momentum: pseudo-tensors and what one now calls the Noether oper-ator (Hilbert, 2007; Klein, 1917; Klein, 1918; Pais, 1982; Olver, 1993; Rowe, 1999; Rowe, 2002;Brading and Brown, 2003). Noether also proved a converse to the “Hilbertian assertion,” alongwith converses to the two theorems associated with her name (Noether, 1918).Noether formulated the Hilbertian assertion and its converse as follows:If [action] I admits of the displacement group, then the energy relationships becomeimproper if and only if I is invariant with respect to an infinite group containingthe displacement group as subgroup. [footnote suppressed] (Noether, 1918, emphasisadded)The “only if” clause, the converse to which I call attention, offers the possibility of arriving atgeneral covariance from improper conservation laws. Such conservation laws consist only of terms2anishing by the field equations and terms with identically vanishing divergence (“curls”)—justthe kinds of things that separate the metrical and canonical stress-energy tensors (Belinfante-Rosenfeld equivalence), and just the kinds of things that one is often said to be free to modify totaste. Hence one could modify the stress-energy for vacuum General Relativity to be identically0 by that reasoning if one wished. One need not agree that there is anything improper about‘improper’ conservation laws (Pitts, 2010) to use the familiar term.During the 1920s-50s, the luster of General Relativity faded among physicists (Eisenstaedt, 1986;Eisenstaedt, 1989; van Dongen, 2010; Schutz, 2012), even if the popular image of Einstein was undi-minished. General Relativity was mathematical, hardly empirical, and tied to protracted fruitlessspeculation about unified field theories. Relativistic quantum mechanics and (at times) field theorywere flourishing, by contrast. In such a context it made sense, if one was not ignoring GeneralRelativity, to try to subsume it into the flourishing framework of particle physics. Apart fromquestions of quantization, this project succeeded. In 1939 Fierz and Pauli noticed that the lin-earized vacuum Einstein equations were just the equations for a particle/field of spin 2 and mass0 (Pauli and Fierz, 1939; Fierz and Pauli, 1939; Fierz, 1940). Even Nathan Rosen, long-time col-laborator with Einstein, appeared to be defecting in 1939: he proposed to reduce the conceptualdistance between General Relativity and other field theories by introducing a flat background ge-ometry and wondered aloud whether the nonlinearities of Einstein’s equations could be derived(Rosen, 1940). Einstein did not like such ideas when Rosen proposed them (Einstein, 1939). Nei-ther did he like it when his assistant Robert Kraichnan was in the process of executing such aderivation (Feynman et al., 1995, p. xiv). (Bryce DeWitt, who transcended the GR vs. par-ticle physics divide to an unusual degree, did like Kraichnan’s ideas (Feynman et al., 1995, xiv)(Seligman, 1949).)One might see Einstein as predicting failure for such projects, if success would be arriving at the1915 field equations without having to know them already: “. . . it would be practically impossiblefor anybody to hit on the gravitational equations” without the “exceedingly strong restrictionson the theoretical possibilities” imposed by “the principle of general relativity” (Einstein, 1954).Is the principle of general relativity really needed? As will appear below, (substantive) generalcovariance does play a key role, albeit as a lemma rather than a premise, but generalized rel-ativity of motion does not play any role . Such a derivation of Einstein’s equations was suc-cessfully carried out in the 1950s-70s at the classical level. What this derivation implies aboutspace-time and gravity is less clear, however. Feynman has some remarks that could be con-strued as conventionalist (Feynman et al., 1995, pp. 112, 113), shrinking the gulf between theinitial flat geometry and the final effective curved geometry because the most convenient geom-etry can shift more easily than can the true geometry. A complete story ought to take into ac-count a notion of causality suitable for quantum gravity (Butterfield and Isham, 2001, sect. 3.3.2)(Pitts and Schieve, 2002; Pitts and Schieve, 2004), but such issues will not be considered here.There is no hint that substantive general covariance is fed in and therefore easily recovered as has I thank Dennis Lehmkuhl of the Einstein Papers Project at Caltech for bringing this correspondence to myattention. The concept of general covariance has become problematic especially in the last decade (Pitts, 2006; Giulini, 2007;Pooley, 2010; Belot, 2011). For present purposes I am ignoring that problem. Anderson thought that this criterionwas equivalent to his preferred criterion of the absence of geometric objects that are the same (up to gauge equivalence)in all models (Anderson, 1967, pp. 88, 89), but it isn’t. formal general covariance is (inessentially) assumed, while sub-stantive general covariance is concluded, a quite different notion (Bergmann, 1957; Anderson, 1967;Stachel, 1993).The task of this paper is to show that this derivation of Einstein’s equations, though quitecompelling, is partly less novel and is not ontologically laden with flat space-time geometry than ithas seemed. Uncovering areas of overlap between general relativist and particle physics views mightlead to further rapprochement. A previous paper invoked the particle physics tradition as a foilfor Einstein’s 1913-15 physical strategy, which used somewhat similar ingredients including a keyrole for conservation laws and an analogy to Maxwell’s electromagnetism (Pitts, 2016a). There thethrust was how much light particle physics sheds on the processes of discovery and justification forEinstein’s equations. Here the direction of benefit is partly reversed: particle physicists could havearrived at Einstein’s equations much earlier if they had made use of Noether’s converse Hilbertianassertion. The derivation is also seen to be more ecumenical than one would have expected. Both ofthese works are thus early efforts at relating particle physics and the history of General Relativity.
While Noether’s work did not quickly get the explicit recognition that it deserved beyond initialendorsement by Klein (Kosmann-Schwarzbach, 2011), by now it is deservedly a standard topic inthe philosophy of physics (Brading and Castellani, 2003). Besides Noether’s first theorem derivingconserved currents from rigid symmetries (finitely many parameters) of the Lagrangian, and hersecond theorem deriving identities among the Euler-Lagrange equations, there are additional resultsof interest in Noether’s paper. These include, among other things (Petrov and Lompay, 2013),many often neglected converse results, a proof of Hilbert’s claim about the ‘improper’ form ofgeneral relativistic energy-momentum conservation (the Hilbertian assertion), and, crucially forpresent purposes, a proof of the converse Hilbertian assertion: improper conservation laws imply a(substantively) generally covariant action.Noether’s converses results seem to attract little attention even in works devoted to Noether’stheorems. There is an older literature that paid some attention to converses (Fletcher, 1960;Dass, 1966; Boyer, 1967; Palmieri and Vitale, 1970; Candotti et al., 1970; Candotti et al., 1972b;Candotti et al., 1972a; Rosen, 1972; Carinena et al., 1989; Ferrario and Passerini, 1990). The factthat such works are most readily found before Noether’s paper became widely available again(in terms of physical copies and language) due to a published English translation in 1971(Noether, 1918) is likely no coincidence. Such works might well be based on second-hand re-ports (Hill, 1951) that did not capture the full content of Noether’s paper (Olver, 1993), includingNoether’s own emphasis on the converses. Accord to Peter Olver, after 1922the next significant reference to Noether’s paper is in a review article by the physicistHill, [(Hill, 1951)], in which the special case of Noether’s theorem discussed in thischapter was presented, with implications that this was all Noether had actually provedon the subject. Unfortunately, the next twenty years saw a succession of innumerablepapers either re-deriving the basic Noether theorem 4.29 or purporting to generalize it,while in reality only reproving Noether’s original result or special cases thereof. The4athematical physics literature to this day abounds with such papers, and it would besenseless to list them here. (Olver, 1993, p. 282)Among modern and philosophical works, there are apparently few that emphasize converses. Oneis Katherine Brading’s dissertation (Brading, 2001, pp. 70-74), which emphasizes Noether’s proof ofthe converse of her first theorem and uses it to undermine other authors’ privileging of symmetriesover conservation laws. Likewise Harvey Brown and Peter Holland doubt that symmetries explainconservation laws, emphasize the converse, and include counterexamples to ideas that one mightloosely have associated with Noether’s first theorem (Brown and Holland, 2004).If others have neglected Noether’s converses, at least she thought them of special importance:her own intent in writing her article had been “to state in a rigorous fash-ion the significance of the principle and, above all, to state the converse . . . ”(Kosmann-Schwarzbach, 2011, p. 52).So she wrote in a referee report on a paper that covered similar ground to her own paper’s but didnot prove converses.The converse Hilbertian assertion is perhaps the most neglected of all; it is difficult to recall anyattention being paid to it at all. Perhaps it has seemed to be mathematical act of supererogationthat would not benefit the working physicist. What reason, after all, could one have for believingin improper conservation laws without already believing in Einstein’s equations? Even Kosmann-Schwarzbach’s book’s discussion of the Hilbertian assertion is not very interested in the converseHilbertian assertion (pp. 63, 64). Emphasis is placed rather on the rigor added to what Hilbert hadconjectured and the improper nature of the conservation laws as disanalogous to those followingfrom rigid symmetries along the lines of Noether’s first theorem (with antecedents in Lagrange andothers (Kastrup, 1987; Pitts, 2016a)).Can one use the converse Hilbertian assertion to derive Einstein’s equations? To my knowl-edge deriving Einstein’s equations via improper conservation laws has never been attempted out-side the particle physics tradition. The logical equivalence of the gravitational field equationsand the conservation laws has been noted (Anderson, 1967; Pitts, 2010), but that is still notenough. Such a derivation, to be sensible, would require some independent reason to believe in im-proper conservation laws. Such independent reasons are not plentiful. In the particle physicstradition such a derivation was achieved 60 years ago (Kraichnan, 1955), but how it workedcould use some clarification (Pitts and Schieve, 2001), especially to motivate the linear gaugefreedom by avoiding ghosts (Pauli and Fierz, 1939; Fierz and Pauli, 1939; Nachtmann et al., 1968;Nachtmann et al., 1969; van Nieuwenhuizen, 1973). Even with that clarification the connection tothe Noether was not made. Derivation(s)
Particle physicists have shown Einstein’s equations are what one naturally arrives for a localinteracting massless spin-2 field, with good explanations for the detailed nonlinearity, generalcovariance, etc . from nuts-and-bolts principles of (at least) Poincar´e-covariant field theory.(Poincar´e symmetry does not exclude a larger symmetry, as is especially clear from a Kleinian5ubtractive as opposed to Riemannian additive picture of geometry (Norton, 1999). ) In 1939 itbecame possible to situate Einstein’s theory vis-a-vis the full range of relativistic wave equationsand Lorentz group representations: Pauli and Fierz recognized the equation for a masslessspin 2 field as the source-free linearized Einstein equations (Fierz, 1939; Pauli and Fierz, 1939;Fierz and Pauli, 1939). That same year Rosen wondered about deriving General Relativity’s non-linearities from an initially special relativistic starting point (Rosen, 1940). Work by Kraichnan,Gupta, Feynman, Weinberg, Deser et al. eventually filled in the gaps, showing that, on painof instability, Einstein’s theory is basically the only option (eliminative induction, philosopherswould say), with contributions by many authors, not all with the same intentions (Weyl, 1944;Papapetrou, 1948; Gupta, 1954; Kraichnan, 1955; Thirring, 1961; Halpern, 1963b; Halpern, 1963a;Feynman et al., 1995; Wyss, 1965; Ogievetsky and Polubarinov, 1965; Weinberg, 1965;Nachtmann et al., 1968; Deser, 1970; van Nieuwenhuizen, 1973; Boulware and Deser, 1975;Pitts and Schieve, 2001; Boulanger and Esole, 2002). One could also consider massive spin 2 grav-ity if it works (Pauli and Fierz, 1939; Fierz and Pauli, 1939; Ogievetsky and Polubarinov, 1965;Freund et al., 1969), an issue apparently settled negatively in 1970-72 but reopened recentlyand now very active (Hinterbichler, 2012; de Rham, 2014). To recall a punchy expression byPeter van Nieuwenhuizen, “general relativity follows from special relativity by excluding ghosts”(van Nieuwenhuizen, 1973). Even apart from the empirical fact of light bending (which vanNieuwenhuizen mentions) needed to refute scalar theories, the claim is slightly exaggerated(Pitts and Schieve, 2001; Maheshwari, 1972; de Rham et al., 2011; Hassan and Rosen, 2011), butthe point remains that it is difficult to avoid negative energy instability without strong resemblanceto Einstein’s equations. Having negative-energy and positive-energy degrees of freedom interactseems likely to imply instability. And yet due to relativity’s − + ++ geometry, negative-energydegrees of freedom tend to crop up regularly if one is not careful (Fierz and Pauli, 1939;Wentzel, 1949; van Nieuwenhuizen, 1973). For a vector potential, if the spatial components havepositive energy, then the temporal component will have negative energy unless one engineers itaway, as occurs in Maxwell’s theory. Ghosts are almost always considered fatal (except in certaincontexts where they are introduced as a technical tool). What is to keep nothing from turningspontaneously into something and anti-something? What is possible soon becomes necessary inquantum mechanics; even energy conservation, supposed to exclude perpetual motion machinesof the first kind, fails to stop the catastrophe. Negative energy degrees of freedom were tacitlyassumed not to exist in 19th century formulations of energy conservation, it would seem. Lagrangeconsidered whether positive energy was required for stability; he showed that bad things couldhappen if the potential were indefinite (Lagrange, 1811). It did not occur to him to entertainnegative kinetic energy, however, something hardly conceivable apart from relativity and thede-materialization of matter into fields in the 20th century. Lagrange showed that positive energywas stable with a separable potential (Lagrange, 1811); the separability requirement was removedby Dirichlet (Dirichlet, 1846; Morrison, 1998). According to Boulanger and Esole,[i]t is well appreciated that general relativity is the unique way to consistently deform The Kleinian picture assumes initially that coordinates are quantitatively meaningful for lengths, volumes, etc. and then proceeds to strip them of meanings by larger symmetry groups. The Riemannian picture assumes thatcoordinates are quantitatively meaningless and then adds structures to define additional concepts such as volume,angle, length, etc. R L for a free massless spin-2 field under the assumption oflocality, Poincar´e invariance, preservation of the number of gauge symmetries and thenumber of derivatives in L [references suppressed] (Boulanger and Esole, 2002).Derivations based on the canonical energy-momentum tensor or some relative thereof(Gupta, 1954) perhaps do not savor as strongly of flat space-time as do derivations with a flatmetric tensor . This impression might be a mistake due to failure to notice the extra gauge groupthat makes the flat metric less significant than the tensor notion suggests (Grishchuk et al., 1984).The canonical tensor’s advantage (if such it is) of not savoring as strongly of flat space-time is tosome degree offset by the gruelingly explicit character of the derivation. This explicitness not onlymakes it harder to arrive at Einstein’s theory (the massless case), but also makes it difficult togeneralize the results to include a graviton rest mass(es). When efforts have been made to add amass term using a canonical-based Belinfante energy-momentum, the result has been consideredunique (Freund and Nambu, 1968; Freund et al., 1969). By contrast the Rosenfeld metric stress-energy approach used by Kraichnan has been generalized to show tremendous flexibility. Instead ofa unique result for the tensor and scalar cases, the tensor case has grown to two one-parameter fam-ilies (Pitts and Schieve, 2007), to four one-parameter families (Pitts, 2011b), to an arbitrary massterm with practically any algebraic self-interaction (Pitts, 2016b). The scalar case likewise has beengeneralized from one case to a one-parameter family. This one-parameter family is analogous to atensorial 2-parameter family because the covariance-contravariance parameter (a continuum withperhaps one hole (Ogievetsky and Polubarinov, 1965)) and the density weight parameter cover thesame ground; a 1 × The split of the stress-energy tensor into a curl and a piece vanishing on-shellto derive Einstein’s equations is a key step (Pitts and Schieve, 2001). This fact is what allows oneto recognize in effect Noether’s converse Hilbertian assertion. A connection to Noether’s converseHilbertian assertion has had to wait, however, to my knowledge.Given this link to Noether’s theorem and the Belinfante-Rosenfeld relation between canonicaland metrical stress-energy, one could envisage a parallel derivation of Einstein’s theory withoutthe flat metric tensor, albeit much less convenient. It is not coincidental that universal couplingderivations for massive scalar gravity using the canonical tensor have led only to a single theory(Freund and Nambu, 1968), because one needs to use the freedom to add a curl to the canonicaltensor to accommodate second derivatives except in one case (Pitts, 2011a). The ideal might be to7se the Belinfante-Rosenfeld equivalence the identity to permit using the Rosenfeld definition forcalculations and the Belinfante tensor (or it plus terms vanishing on-shell) for conceptual purposes.
One can take the leisurely pace of the linkage between Noether’s proof of the converse Hilbertianassertion and the spin 2 derivations as an indicator of the depth of the tragic split between generalrelativists and particle physicists, which this current paper aims to reduce somewhat. Hilbert wasinsightful in taking improper conservation laws as characteristic of General Relativity, a commentthat might be taken to suggest a derivation. Noether’s proof of the converse Hilbertian assertioncould be taken providing the core of a derivation of Einstein’s equations. There is, however, anobvious problem, of a sort discussed by Aristotle in the
Posterior Analytics .The remainder of
Posterior Analytics
I is largely concerned with two tasks: spellingout the nature of demonstration and demonstrative science and answering an impor-tant challenge to its very possibility. Aristotle first tells us that a demonstration is adeduction in which the premises are:1. true2. primary ( prota )3. immediate ( amesa , “without a middle”)4. better known or more familiar ( gnˆorimˆotera ) than the conclusion5. prior to the conclusion6. causes ( aitia ) of the conclusion. . . The fourth condition shows that the knower of a demonstration must be in somebetter epistemic condition towards [the premises]. . . . (Smith, 2015, emphasis in theoriginal)This fragment of Aristotle’s theory of demonstration has an insight that one would presumablywish to retain: if the premises are initially less plausible than the conclusion, then the argument isnot very good.In 1918 General Relativity was probably not known, but it was certainly seriously entertained.Improper conservation laws were entertained only as a consequence of General Relativity. Hencethere was little prospect for regarding improper conservation laws as better known than or priorto Einstein’s equations. Particle physics changed that situation, partly by supplying a taxon-omy in which one could fit General Relativity (massless spin 0) and easily conceive a rival the-ory (massive spin 2, at least prima facie ), thus making General Relativity less well known thanit must have seemed after the 1919 bending of light success. Shouldn’t one be open to rivaltheories that made the same prediction? Evidently some people were (Brush, 1989). Particlephysics also systematically implemented positive energy (avoiding ghosts) as a criterion of the-ory construction and theory choice (Pauli and Fierz, 1939; van Nieuwenhuizen, 1973) and showed8ow a massless spin 2 field satisfying the linearized Einstein equations is both a very naturalpath and one of very few paths to avoid ghosts with a symmetric rank 2 tensor potential. In-deed other plausible paths (unimodular/traceless massless spin 2/GR (Unruh, 1989), GR plussomething like a scalar field (Pitts and Schieve, 2001; ´Alvarez et al., 2006), and massive grav-ity (Maheshwari, 1972; de Rham et al., 2011)) are quite close to Einstein’s equations. In mas-sive spin 2 gravity, Einstein’s equations had at least one a priori and empirically plausible rival(Ogievetsky and Polubarinov, 1965; Freund et al., 1969). Unless stability can be achieved in someother way, the positive energy requirement seems non-negotiable and, in turn, makes the lineariza-tion of Einstein’s equations difficult to avoid, apart perhaps from a graviton mass term. Theprinciple of universal coupling, which is another key part of the particle physics derivations, in factwas a part of Einstein’s 1913-15 physical strategy as expressed in the Entwurf with Grossmann:These equations satisfy a requirement that, in our opinion, must be imposed on a relativ-ity theory of gravitation; that is to say, they show that the tensor θ µν of the gravitationalfield acts as a field generator in the same way as the tensor Θ µν of the material processes.An exceptional position of gravitational energy in comparison with all other kinds ofenergies would lead to untenable consequences. (Einstein and Grossmann, 1996)While ghost avoidance leaves nonlinearities quite unspecified as long as they aren’t too strong,universal coupling provides a close link between the linearized and exact nonlinear Einstein equa-tions. Improper conservation laws follow (Pitts and Schieve, 2001): the stress-energy tensor is apiece vanishing on-shell and a curl. Thus it became possible to regard improper conservation lawsas better known than Einstein’s equations. Almost as van Nieuwenhuizen said, “general relativityfollows from special relativity by excluding ghosts” (van Nieuwenhuizen, 1973). Aristotle’s prior-ity clause can be satisfied at least in the counterfactual history of how physics presumably wouldhave progressed without Einstein (Feynman et al., 1995; Ohanian, 2008). It isn’t clear why scienceshould be forever held captive to historical accidents, so that is good enough. Hence particle physicsmakes it plausible to argue for Einstein’s equations using Noether’s converse Hilbertian assumption,positive energy, etc. What about the principle(s) of equivalence? Sometimes universal coupling is associated withthe strong equivalence principle (Feynman et al., 1995, p. xiv). But universal coupling also leadsto massive gravities (Freund et al., 1969; Pitts and Schieve, 2007; Pitts, 2011b; Pitts, 2016b), forwhich there is a clear distinction between gravity and inertia. The identity of gravity and inertiais another strong meaning sometimes associated with the principle of equivalence (Friedman, 2001,pp. 37-39, 81). Avoiding multiply ambiguous terms like the principle of equivalence, one cansafely say that the identity of gravity and inertia is not assumed in the particle physics derivation(massless or massive), and that such identity is clearly false at the end of the derivation in themassive case.
It is not terribly obvious what the ontology suggested by the spin-2 derivations of Einstein’s equa-tions is. One often enough reads that the derivation shows that Einstein’s theory is renderedjust another field theory in Minkowski space-time, within special relativity, or similar expres-sions. Such conclusions are especially tempting if one uses a flat metric tensor (in the sense of9aving a nontrivial transformation rule under general coordinate transformations), not simply amatrix diag ( − , , , . Eventually it is concluded that the flat background metric is “unobserv-able,” which usually is supposed to mean or at least to imply that it doesn’t really exist, perhaps(Thirring, 1961). But how does coalescence of the flat geometry and the gravitational potentialmake the flat geometry cease to exist? One key issue pertains to whether one thinks in terms ofRiemannian additive or Kleinian subtractive pictures of geometry (Norton, 1999). General rela-tivists, historians and philosophers since the late 1970s have tended to default to a Riemannianadditive picture, according to which Special Relativity is about an enormously impressive crys-talline object, Minkowski Space-Time (or even Spacetime), which controls everything. Particlephysicists include toward a Kleinian subtractive picture, so that relativity is rather about havingthe Poincar´e covariance group, which is certainly compatible with having an even larger covariancegroup. Among philosophers, calling Minkowski space-time a “glorious nonentity” is reminiscentof Klein’s subtractive strategy (Brown and Pooley, 2006). Both Riemannian additive and Kleiniansubtractive strategies are sometimes useful and illuminating.While some of the mathematics of the spin-2 derivations of Einstein’s equations has a specialrelativistic feel—consider the title of Kraichnan’s classic paper “Special-Relativistic Derivation ofGenerally Covariant Gravitation Theory”—an additional gauge group emerges. The emergenceof an additional gauge group deprives of physical meaning the precise quantitative relationshipbetween the effective curved metric and the/a flat background metric, making the flat space-time(s) elusive (Grishchuk et al., 1984; Norton, 1994; Pinto-Neto and Trajtenberg, 2000), a pointmade early in little-known work by William Band (Band, 1942b; Band, 1942a) and conceded butstill insufficiently attended by Nathan Rosen in his application to gravitational energy localization(Rosen, 1963).
Which flat metric underlies the effective curved geometry? None in particular, onemight say. It is plausible that such a neo-traditionalist ontology is confusing to many, is (or wouldhave been) attractive to some such as Lotze (Lotze, 1879, pp. 248, 249) (see also (Torretti, 1978, pp.288, 299, 408)), perhaps
Poincar´e (Poincar´e, 1902), and Logunov (Logunov, 1998), and unattractiveto others. But the fact remains that the derivation of Einstein’s equations is quite compelling, muchbetter than the competition involving Principles. If one aspires to take the flat background seriously,then causality, ironically, requires reintroducing gauge freedom (Pitts and Schieve, 2007) (partlyakin to privileging the Stueckelberg form of massive electromagnetism with gauge compensationfields (Ruegg and Ruiz-Altaba, 2004) over the primordial non-gauge form, and then restricting thegauge freedom with inequalities). Hence some of the innovative features of General Relativity areobligatory anyway.One could reduce the ontological confusion by getting rid of the flat metric tensor (or should onesay, tensors, in light of the extra gauge group) in favor of the numerical matrix diag ( − , , , , such as one finds in some of this work already (Gupta, 1954; Ogievetsky and Polubarinov, 1965).Much of the value of using a flat metric tensor(s) pertains to defining the stress-energytensor. The Rosenfeld stress-energy tensor is defined using a flat background metric, butone momentarily relaxes flatness to take a variational derivative, and then restores flatnessagain (Rosenfeld, 1940; Kraichnan, 1955; Anderson, 1967; Deser, 1970; Gotay and Marsden, 1992;Pitts and Schieve, 2001), getting much of the benefit of Hilbert’s definition without requiring theexistence of the gravitational field. This definition is enormously convenient, but can cause con-fusion, both mathematical and conceptual. That the Rosenfeld stress-energy tensor is just a trick10as been urged in the context of Deser’s spin-2 derivation:. . . T µν is the stress-tensor of the linear action of equation (4). It is very simply com-puted in the usual (Rosenfeld) way as the variational derivative of I L with respect toan auxiliary contravariant metric density ψ µν , upon writing I L in ‘generally covari-ant form’, I L ( η → ψ ) , with respect to this metric. Note that this does not presupposeany geometrical notions, being merely a mathematical shortcut in finding the symmetricstress-tensor of I L . We could also obtain it by the (equivalent) (Belinfante) prescriptionof introducing local Lorentz transformations. (Deser, 1970)By the Belinfante-Rosenfeld equivalence theorem, one can replace the Rosenfeld stress-energytensor with the canonical tensor + certain terms with identically vanishing divergence + certainterms proportional to some field equations. Actually calculating the amended canonical tensormight be unpleasant, but one could use the Belinfante-Rosenfeld equivalence theorem in the otherdirection so that the Rosenfeld stress-energy tensor is just a calculating trick . All conceptual work isdone by the Belinfante modified canonical tensor (or it plus terms vanishing on-shell), but the cal-culations are done using Rosenfeld’s trick. Thus one evades an objection (Padmanabhan, 2008)that has been made to spin-2 derivations of Einstein’s equations; others have been addressed(Pitts and Schieve, 2007). One could avoid the additional gauge group, the topological limita-tions implied in introducing a flat metric (Ashtekar and Geroch, 1974), and most ontological con-notations of a flat geometry(s) by using the matrix diag ( − , , ,
1) instead of a flat backgroundmetric. This signature matrix is also intimately involved in nonlinear realizations of the ‘group’(using the term loosely) of general coordinate transformations (Ogievetsky and Polubarinov, 1965;Pitts, 2012), which permit spinors in coordinates without a tetrad. While one cannot take a vari-ational derivative with respect to this numerical matrix, one does not need to do so, using theBelinfante-Rosenfeld equivalence as described above. Conceptual matters are handled using themodified canonical tensor, while calculations can be handled with Rosenfeld’s trick. To implementthis project, one can insert the Belinfante-Rosenfeld equivalence theorem into some suitable spin-2derivation that uses the Rosenfeld tensor, preferably a version sufficiently flexible to display thepreviously unrecognized full generality of spin-2 derivations in cases where the graviton mass isnot 0 (Pitts, 2016b). Thus one overcomes a traditional weakness in spin-2 derivations using thecanonical tensor, namely, arriving at only one theory and even that with much gritty detailedcalculation (Freund and Nambu, 1968; Freund et al., 1969); such generality would otherwise likelyappear fiendishly difficult and/or unnatural. The canonical tensor and modifications thereof havethe additional virtues (compared to the Rosenfeld metric stress-energy tensor) of being directlyrelated to the translation symmetries that induces conservation and of being nontrivial (at leastoff-shell) even in topological (metric-free) field theories (Burgess, 2002).
It appears, then, that the particle physics derivation of Einstein’s equations should be quite attrac-tive to everyone. It involves principles that one could hardly avoid on pain of explosive instability.It avoids principles (including generalized relativity of motion or the identity of gravity and inertia)that could easily be false, though there is no harm if one finds them plausible. On the other hand11he derivation is surprisingly innocent metaphysically, involving no ontological commitment to aflat space-time metric tensor. Perhaps it should be more generally embraced, even by general rela-tivists, rather than viewed as the special property of particle physicists. Given that a core idea inthe particle physics derivation is Noether’s converse Hilbertian assertion, general relativists alreadydo have priority on part of the derivation. This surprising convergence might set an example foradditional fruitful work overcoming the general relativist vs. particle physicist divide.
The way that the split of the conserved stress-energy complex into a piece proportional to the grav-itational field equations and a piece with automatically vanishing divergence plays a role in the spin2 derivation of Einstein’s equations is worth recalling. This derivation (Pitts and Schieve, 2001) isbased largely on that of Kraichnan (Kraichnan, 1955) (Feynman et al., 1995, pp. xiii, xiv) but witha number of improvements. The basic variables in this approach are the gravitational potential γ µν and the flat metric η µν , along with (bosonic) matter fields u that can be any kind of geometricobject fields, though indices are suppressed. Gravity is assumed to have some free field equationsderived from a (presumably quadratic) action S f [ γ µν , η µν ].In an effort to avoid negative-energy degrees of freedom, one can require that S f change only bya boundary term under the infinitesimal gauge transformation γ µν → γ µν + ∂ µ ξ ν + ∂ ν ξ µ , (1) ξ ν being an arbitrary covector field. ∂ µ is the flat covariant derivative built from the flat metric η µν . (This condition is a bit stronger than necessary if the action does not imply higher derivatives inthe field equations, but can be too weak if there are higher derivatives. It is a good place to start,however.) In the special case that the Lagrangian density is a linear combination of terms quadraticin first derivatives of the γ µν , and free of algebraic and higher-derivative dependence on γ µν , therequirement of gauge invariance uniquely fixes coefficients of the terms in the free field actionup to a boundary term, giving linearized vacuum general relativity (Ohanian and Ruffini, 1994;Hakim, 1999). For any S f invariant in this sense under (1), the free field equation is identicallydivergenceless: ∂ µ δS f δγ µν = 0 . (2)This is the free field generalized Bianchi identity.As Einstein said in 1913, if a source is introduced, it is reasonable to have all stress-energy, forboth gravity and matter fields u , serve as a source in the same way (Einstein and Grossmann, 1996).Using the Rosenfeld stress-energy tensor from varying the (unknown) full action S with respect to η µν (with γ µν and u constant), one seeks the field equations from the (unknown) full action S forgravity: δSδγ µν = δS f δγ µν − λ δSδη µν , (3)where it turns out eventually that λ = −√ πG . This stress-energy tensor includes gravitationalstress-energy; at this stage one doesn’t know that an extra gauge group emerges and that the12ravitational energy-momentum has the peculiar properties that it has in General Relativity. (Ina non-Rosenfeld form of such a derivation, one would not yet know that the gravitational energy-momentum is only a pseudo-tensor rather than a tensor. One could derive the linear analog of thatfact if one takes the free field Lagrangian density to be quadratic in first derivatives of γ µν , however(Fierz, 1939).)One is free to make a change of variables in S from γ µν and η µν to g µν and η µν , where g µν = η µν − λγ µν . (4)No assumption is made that g µν has chronogeometric significance; it emerges later as a result that g µν is the only quantity that could be an observable metric. Equating coefficients of the variationsgives δSδη µν | γ = δSδη µν | g + δSδg µν (5)and δSδγ µν = − λ δSδg µν . (6)Putting these two results together gives λ δSδη µν | γ = λ δSδη µν | g − δSδγ µν . (7)Equation (7) splits the Rosenfeld stress-energy tensor into one piece that vanishes when gravityis on-shell and one piece that does not. That is one of the two types of terms that the converseHilbertian assumptions allows. Using this result in (3) gives λ δSδη µν | g = δS f δγ µν , (8)which says that the free field Euler-Lagrange derivative must equal (up to a constant factor) thatpart of the total stress tensor that does not vanish when the gravitational field equations hold.Using the linearized Bianchi identity (2), one derives ∂ µ δSδη µν | g = 0 , (9)which says that the part of the stress tensor not proportional to the gravitational field equations hasidentically vanishing divergence (on either index), i.e. , is a (symmetric) “curl” (Anderson, 1967).This is the other type of term allowed by the converse Hilbertian assertion. And there is nothingleft of the stress-energy tensor: those two terms, a piece proportional to the gravitational fieldequations and a piece with identically vanishing divergence, are the whole thing. The quantity δSδη µν | g , being symmetrical and having identically vanishing divergence on either index, is of theform δSδη µν | g = 12 ∂ ρ ∂ σ ( M [ µρ ][ σν ] + M [ νρ ][ σµ ] ) + b √− ηη µν (10)13Wald, 1984) (pp. 89, 429) (Kraichnan, 1955; Pitts and Schieve, 2001), where M µρσν is a tensordensity of weight 1 and b is a constant. This result follows from the converse of Poincar´e’s lemmain Minkowski spacetime. One can gather all dependence on η µν (with g µν independent) into oneterm, writing S = S [ g µν , ✟✟ η µν , u ] + S [ g µν , η µν , u ] . (11)If S = 12 Z d xR µνρσ ( η ) M µνρσ ( η µν , g µν , u ) + Z d xα µ , µ +2 b Z d x √− η, (12)then δS δη µν | g has just the desired form, while S does not affect the Euler-Lagrange equations(Kraichnan, 1955; Pitts and Schieve, 2001). ( R d xα µ , µ is any boundary term that one likes.) Thusthe Euler-Lagrange equations arise entirely from S = S [ g µν , ✟✟ η µν , u ]: the flat metric and the gravi-tational potential have merged, so the flat metric alone is unobservable and the only candidate fora metric is g µν .Thus the stress-energy tensor turns out to be just a term proportional to the Euler-Lagrangianequations and a term with identically vanishing divergence, and then one arrives at an actionwith Euler-Lagrange equations involving only a curved metric and matter fields, not a separateflat metric. Thus besides the (here trivial) formal general covariance, one has an additionalgauge freedom to alter the flat metric tensor while leaving the curved metric and matter fieldsalone, or, alternately, to alter the curved metric and matter fields by what looks like a coor-dinate transformation while leaving the flat metric alone (Grishchuk et al., 1984; Norton, 1994;Pinto-Neto and Trajtenberg, 2000; Pitts and Schieve, 2001). If one fixes the coordinates to beCartesian, then one has η µν = diag ( − , , ,
1) and the additional gauge freedom looks like a coor-dinate transformation in single-metric General Relativity (at least for transformations connectedto the identity). One thus de-Rosenfeldizes the result and arrives at what one usually considersa substantively generally covariant action S [ g µν , u ], as in the converse Hilbertian assertion. Thusthe spin 2 derivation is, apart from the well-motivated and crucial elimination of ghosts, largelythe converse Hilbertian assertion all over again with a glossy Rosenfeldized form using η µν and asymmetric curl term. One can show that λ = −√ πG by, e.g. , requiring proper normalization ofthe (Γ − Γ-like) bimetric General Relativity Lagrangian density of Rosen (Rosen, 1940).Universal coupling turns out to deform the free field gauge invariance into a nonlinear gaugeinvariance. (If one took a theory that had the linear gauge freedom but no nonlinear gauge freedom,it would have to violate universal coupling in order to escape the derivation above. A paper byWald is also relevant (Wald, 1986).) Chan and Frønsdal provide a helpful summary.The apparently miraculous success of the original Gupta program has been convincinglyexplained by the analysis of Thirring [reference to (Thirring, 1961; Thirring, 1959)] andothers. Namely, the structure of the full, nonlinear and non-Abelian gauge algebra ofgeneral relativity stands revealed upon completion of the first stage of Gupta’s program.The geometric interpretation is immediate, and the full nonlinear action follows from it.The essence of Gupta’s method is to notice that the invariance of the free Lagrangianis equivalent to a degeneracy of the free wave operators (linearized Bianchi identities)and that this degeneracy leads to strong constraints on the form of interactions. This in14urn implies the existence of a deformed invariance group of the perturbed (interacting)Lagrangian. (Chan and Frønsdal, 1996)That preservation of gauge symmetry is an important resource for avoiding ghosts at the nonlinearlevel. It is possible to have nonlinear ghosts without linear ghosts (Boulware and Deser, 1972), afact much discussed, and in some cases circumvented, in massive gravity (de Rham et al., 2011;Hassan and Rosen, 2012). Another way to have nonlinear ghosts without linear ghosts would beto take, say, General Relativity, expand it perturbatively (which gives an infinite series exceptin the case of two fractionally weighted choices of fields (DeWitt, 1967)), and then truncate theseries at any finite order. For all but the first few choices, such a theory will satisfy our empiricalevidence for the weak principle of equivalence, but the theory will be bimetric (Blanchet, 1992).Thus it will have no gauge freedom and so will have six field degrees of freedom, not two asin GR. One of them will be a ghost due to the indefinite kinetic term of GR (Wald, 1984) andany approximation thereof. Whereas a single-metric imagination renders one unable to conceiveof such theories, particle physics enables one to conceive and refute them. A few authors havesuggested that one can live with ghosts in certain cases (Hawking and Hertog, 2002). Perhapsso. But that is a difficult road, especially under quantization where the threat of spontaneousproduction of arbitrarily many positive-and negative-energy quanta arises, so one needs to providea detailed story for why such a theory is stable. In short, mere empirical evidence is too weak torule out infinitely many theories, almost empirically equivalent to GR, that likely are unstable dueto nonlinear ghosts. The no-ghost condition does important work to motivate Einstein’s equationsat both linear and distinctively again at nonlinear orders, work that mere empirical evidence cannever do. Again the superiority of the spin 2 derivation over principles of equivalence and the likeappears. Norton has noted that Einstein’s argumentation is often leaky and that one is well advisedto seek eliminative inductions (Norton, 1995). The spin 2 derivation provides them.This spin 2 derivation admits some generalizations, such as covariant or contravariant metricsof (nearly) arbitrary density weight (Kraichnan, 1955; Kraichnan, 1956; Pitts and Schieve, 2007),an orthonormal (co)tetrad (of almost any density weight), thus facilitating spinors (Deser, 1980;Pitts, 2011b), and nonlinear field redefinitions (Pitts, 2016b). While these derivations don’t yieldanything new for massless spin-2 gravity, they yield an enormous variety for massive spin 2 gravityif one requires only the kinetic term (the part with derivatives) to have gauge symmetry to try toavoid ghosts. Whether the theories are viable requires additional investigation beyond these criteriabecause they might have ghosts, tachyons, a bad massless limit, or some other pathology. Massivetheories, if viable, violate both the principle of generalized relativity and the identification of gravitywith inertia. One can also weaken the universal coupling condition to cover only the traceless partand leave an extra scalar (density) degree of freedom in the theory (Pitts and Schieve, 2001), thusarriving at scalar-tensor theories with the cosmological constant as a constant of integration.One could also consider global topological issues. Flat metric tensors do make some demands ontopology (Ashtekar and Geroch, 1974). But these demands are no stronger than spinors require,because “every flat space-time has spinor structure.” (Geroch, 1970)15
Acknowledgements
This work was supported by John Templeton Foundation grant
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