The relativity of inertia and reality of nothing
aa r X i v : . [ phy s i c s . h i s t - ph ] J a n The relativity of inertia and reality of nothing
Alexander Afriat and Ermenegildo CacceseOctober 30, 2018
Abstract
We first see that the inertia of Newtonian mechanics is absolute and troublesome.General relativity can be viewed as Einstein’s attempt to remedy, by making iner-tia relative, to matter—perhaps imperfectly though, as at least a couple of freedomdegrees separate inertia from matter in his theory. We consider ways the relationist(for whom it is of course unwelcome) can try to overcome such undetermination,dismissing it as physically meaningless, especially by insisting on the right trans-formation properties.
The indifference of mechanical phenomena and the classical laws governing them toabsolute position, to translation has long been known. This ‘relativity’ extends to thefirst derivative, velocity, but not to the second, acceleration, which—together with itsopposite, inertia —has a troubling absoluteness, dealt with in § § § T ab —rather than by U µν = T µν + t µν , which includes the gravitational energy-momentum t µν whose transformation properties make it too subjective and insubstantialto count. The role of distant matter is looked at in § § § § For motion is inertial when acceleration vanishes. Inertial motion can also be understood, in moreAristotelian terms, as “natural” rather than “violent” (accelerated) motion. Weyl (2000, p. 138) has a cor-responding dualism between inertia and force : “gravity, in the dualism between inertia and force, belongsto inertia, not to force.
In the phenomena of gravitation therefore the inertial- or , as I prefer to say, theguidance-field [ . . . ].” Cf . Weyl (1924, p. 198): “Dualism between guidance and force.” (The translationsfrom German and from French are ours.) Indices from the beginning of the Roman alphabet are abstract indices specifying valence, contraction(the once-contravariant and twice-covariant trilinear mapping A abc = B abdc C d : V ∗ × V × V → R , forinstance, turns one covector and two vectors into a number) etc. , whereas Greek letters are used for space-time coordinate indices running from to , and i , j and l for ‘spatial’ coordinate indices from to .Sometimes we write V for a four-vector V a , α for a one-form α a , g for the metric g ab , and h α, V i forthe scalar product α a V a = h α a , V a i . The abstract index of the covector dx µ = dx µa , whose valence isobvious, will usually be omitted. § +
16, which isconsidered in § § § t µν : while an observer in free fall sees nothingat all, an acceleration would produce energy out of nowhere, out of a mere transfor-mation to another ‘point of view’ or rather state of motion. To take advantage of thisfragility of gravitational waves, the relationist wanting motion (inertia in particular) tobe ‘entirely relative’ to matter will be mathematically intransigent and attribute phys-ical significance only to notions with the right transformation behaviour—and noneto those that can be transformed away—thus allowing him to dispute the reality of theunwelcome freedoms separating matter and inertia, which he can dismiss as mere opin-ion, as meaningless decoration. If general covariance has to hold , matter would seemto determine inertia rather strongly . . .In the early years of general relativity, Hilbert, Levi-Civita, Schr¨odinger and othersattributed physical meaning only to objects, like tensors, with the right transformationbehaviour. Einstein was at first less severe, extending reality to notions with a moreradical dependence on the observer’s state of motion. With a mathematical argument( § § § For instance we hesitate to help the relationist with the distant masses (which of course constrain inertia) Absolute inertia
Newton distinguished between an “absolute” space he also called “true and mathemat-ical,” and the “relative, apparent and vulgar” space in which distances and velocitiesare physically but imperfectly measured down here (rather than exactly divined by theDivinity). Absolute position and motion were not referred to anything. Leibniz iden-tified unnecessary determinations, excess structure in Newton’s ‘absolute’ kinematicswith celebrated arguments resting on the principium identitatis indiscernibilium : as atranslation of everything, or an exchange of east and west, produces no observable ef-fect, the situations before and after must be the same, for no difference is discerned. Butthere were superfluities with respect to Newton’s own dynamics too, founded as it wason the proportionality of force and acceleration. With his gran navilio , Galileo (1632,Second day) had already noted the indifference of various effetti to inertial transforma-tions; the invariance of Newton’s laws would more concisely express the indifferenceof all the effetti they governed. Modern notation, however anachronistic, can help sharpen interpretation. Thederivatives ˙ x = d x /dt and ¨ x = d ˙ x /dt are quotients of differences; already the po-sition difference ∆ x = x ( t + ε ) − x ( t ) = x ( t + ε ) + u − [ x ( t ) + u ]is indifferent to the addition of a constant u (which is the same for x both at t and at t + ε ). The velocity ˙ x = lim ε → ∆ x ε is therefore unaffected by the three-parameter group S of translations x x + u actingon the three-dimensional space E . The difference ∆ ˙ x = ˙ x ( t + ε ) − ˙ x ( t ) = ˙ x ( t + ε ) + v − [ ˙ x ( t ) + v ]of velocities is likewise indifferent to the addition of a constant velocity v (which is thesame for ˙ x both at t and at t + ε ). The acceleration ¨ x = lim ε → ∆ ˙ x ε is therefore invariant under the six-parameter group S × V which includes, alongsidethe translations, the group V of the inertial transformations x x + v t, ˙ x ˙ x + v acting on the space-time E = E × R . whose influence in the initial-value formulation has been so abundantly considered by Wheeler and others. For a recent treatment see Ryckman (2003, pp. 76-80). Cf . Dieks (2006, p. 178). Newton (1833),
Corollarium V (to the laws) On this distinction and its significance in relativity see Dieks (2006), where the effetti are called “factualstates of affairs.” ∆ x = x ( t + ε ) − x ( t ) = x a ( t + a + ε ) − x a ( t + a ) = ∆ x a is also invariant under the group T of time translations . The translation t t + a ∈ R can be seen as a relabelling of instants which makes x , or rather x a , assign to t + a the value x assigned to t : x a ( t + a ) = x ( t ). The difference ∆ ˙ x = ∆ ˙ x a has the sameinvariance—as do the quotients ∆ x /ε , ∆ ˙ x /ε , and the limits ˙ x , ¨ x .Newton’s second law is ‘covariant’ with respect to the group R = SO( S ) of rota-tions R : E → E , which turn the “straight line along which the force is applied” withthe “change of motion,” in the sense that the two rotations F RF, ¨ x R ¨ x , takentogether, maintain the proportionality of force and acceleration expressed by the law: [ F ∼ ¨ x ] ⇔ [ RF ∼ R ¨ x ] . We can say the second law is indifferent to the action of theten-parameter Galilei group G = ( S × V ) ⋊ ( T × R ) with composition ( u , v , a, R ) ⋊ ( u ′ , v ′ , a ′ , R ′ ) = ( u + R u ′ + a ′ v , v + R v ′ , a + a ′ , RR ′ ) , ⋊ being the semidirect product. So the absolute features of Newtonian mechanics—acceleration, force, inertia, the laws—emerge as invariants of the Galilei group, whosetransformations change the relative ones: position, velocity and so forth. A largergroup admitting acceleration would undermine the laws, requiring generalisation withother forces.Cartan (1923) undertook such a generalisation, with a larger group, new laws andother forces. The general covariance of his Newtonian formalism (with a flat connec-tion) may seem to make inertia and acceleration relative, but in fact the meaningfulacceleration in his theory is not d x i /dt , which can be called relative (to the coor-dinates), but the absolute (1) A i = d x i dt + X j,l =1 Γ ijl dx j dt dx l dt ( i = 1 , , and the time t is absolute). Relative acceleration comes and goes as co-ordinates change, whereas absolute acceleration is generally covariant and transformsas a tensor: if it vanishes in one system it always will. The two accelerations co-incide with respect to inertial coordinates, which make the connection components “Mutationem motus proportionalem esse vi motrici impressæ, et fieri secundum lineam rectam qua visilla imprimitur.” For Newton’s forces are superpositions of fundamental forces F = f ( | x − x | , | ˙ x − ˙ x | , | ¨ x − ¨ x | , . . . ) , covariant under G , exchanged by pairs of points. See L´evy-Leblond (1971, pp. 224-9). See also Friedman (1983, § III), Penrose (2005, § In Baker (2005) there appears to be a confusion of the two accelerations as they arise—in much thesame way—in general relativity. The acceleration d x µ /dτ = 0 Baker sees as evidence of the causalpowers possessed by an ostensibly empty space-time with Λ = 0 is merely relative ; even with Λ = 0 freebodies describe geodesics, which are wordlines whose absolute acceleration vanishes. The sensitivity ofprojective or affine structure to the cosmological constant Λ would seem to be more meaningful, and canserve to indicate similar causal powers. The abstract index representing the valence of the ‘partial derivation’ vector ∂ i = ∂ ai = ∂/∂x i tangentto the i th coordinate line will be omitted. ijl = h dx i , ∇ ∂ j ∂ l i vanish. The absolute acceleration of inertial motion vanishes how-ever it is represented—the connection being there to cancel the acceleration of non-inertial coordinates.So far, then, we have two formal criteria of inertial motion: • ¨ x = in Newton’s theory • A i = 0 in Cartan’s.But Newton’s criterion doesn’t really get us anywhere, as the vanishing accelerationhas to be referred to an inertial frame in the first place; to Cartan’s we are about toreturn.Einstein (1916, p. 770; 1988, p. 40; 1990, p. 28) and others have appealed to the simplicity of laws to tell inertia apart from acceleration: inertial systems admit thesimplest laws. Condition ¨ x = , for instance, is simpler than ¨ y + a = , with aterm a to compensate the acceleration of system y . But we have just seen that Car-tan’s theory takes account of possibile acceleration ab initio , thus preempting subse-quent complication—for accelerated coordinates do not appear to affect the syntacticalform of (1), which is complicated to begin with by the connection term. One couldargue that the law simplifies when that term disappears, when the coefficients Γ ijl allvanish; but then we’re back to the Newtonian condition ¨ x = . And just as that condi-tion requires an inertial system in the first place, Cartan’s condition A i = 0 requires aconnection, which is pretty much equivalent: it can be seen as a convention stipulatinghow the three-dimensional simultaneity surfaces are ‘stitched’ together by a congru-ence of (mathematically) arbitrary curves defined as geodesics. The connection wouldthen be determined, a posteriori as it were, by the requirement that its coefficientsvanish for those inertial curves. Once one congruence is chosen the connection, thusdetermined, provides all other congruences that are inertial with respect to the first. Sothe initial geodesics, by stitching together the simultaneity spaces, first provide a no-tion of rest and velocity, then a connection, representing inertia and acceleration. TheNewtonian condition ¨ x = presupposes the very class of inertial systems given bythe congruence and connection in Cartan’s theory. So we seem to be going around incircles: motion is inertial if it is inertial with respect to inertial motion .We should not be too surprised that purely formal criteria are of little use on theirown for the identification of something as physical as inertia. But are more physical,empirical criteria not available? Suppose we view Newton’s first law, his ‘principleof inertia,’ as a special case of the second law F = m ¨ x with vanishing force (andhence acceleration). So far we have been concentrating on the more mathematicalright-hand side, on vanishing acceleration; but there is also the more physical left-handside F = : can inertial systems not be characterised as free and far from everythingelse? Even if certain bodies may be isolated enough to be almost entirely uninfluencedby others, the matter remains problematic. For one thing we have no direct accessto such roughly free bodies, everything around us gets pulled and accelerated. And Cf . Dieks (2006, p. 186). See Earman (1989, §§ Einstein (1916, p. 772; 1988, p. 40; 1990, p. 59) which is what we were after in the first place.Just as the absence of force has been appealed to for the identification of inertia,its presence can be noted in an attempt to characterise acceleration; various passages in the scholium on absolute space and time show that Newton, for instance, proposedto tell apart inertia and acceleration through causes , effects , forces . In the two ex-periments described at the end of the scholium , involving the bucket and the rotatingglobes, there is an interplay of local causes and effects: the rotation of the water causesit to rise on the outside; the forces applied to opposite sides of the globes cause thetension in the string joining them to vary. But this doesn’t get us very far either; ourproblem remains, as we see using the distinction drawn above between absolute ac-celeration A i and relative acceleration d x i /dt , which surprisingly corresponds to adistinction Newton himself is groping for in the following passage from the scholium :The causes by which true and relative motions are distinguished, one fromthe other, are the forces impressed upon bodies to generate motion. Truemotion is neither generated nor altered, but by some force impressed uponthe body moved; but relative motion may be generated or altered withoutany force impressed upon the body. For it is sufficient only to impresssome force on other bodies with which the former is compared, that bytheir giving way, that relation may be changed, in which the relative restor motion of this other body did consist. Again, true motion suffers al-ways some change from any force impressed upon the moving body; butrelative motion does not necessarily undergo any change by such forces.For if the same forces are likewise impressed on those other bodies, withwhich the comparison is made, that the relative position may be preserved,then that condition will be preserved in which the relative motion consists.And therefore any relative motion may be changed when the true motionremains unaltered, and the relative may be preserved when the true sufferssome change. Thus, true motion by no means consists in such relations. An anonymous referee has pointed out that inertial systems can be large and rigid in flat space-times,but not with curvature; where present, tidal effects prevent inertial motion from being rigid, and even ruleout large inertial frames; but see § “Distinguuntur autem quies et motus absoluti et relativi ab invicem per proprietates suas et causas eteffectus”; “Causæ, quibus motus veri et relativi distinguuntur ab invicem, sunt vires in corpora impressæad motum generandum”; “Effectus, quibus motus absoluti et relativi distinguuntur ab invicem, sunt viresrecedendi ab axe motus circularis”; “Motus autem veros ex eorum causis, effectibus, et apparentibus differ-entiis colligere, et contra ex motibus seu veris seu apparentibus eorum causas et effectus, docebitur fusius insequentibus.” Cf . Rynasiewicz (1995). “Causæ, quibus motus veri et relativi distinguuntur ab invicem, sunt vires in corpora impressæ ad mo-tum generandum. Motus verus nec generatur nec mutatur nisi per vires in ipsum corpus motum impressas:at motus relativus generari et mutari potest absq; viribus impressis in hoc corpus. Sufficit enim ut impri-mantur in alia solum corpora ad quæ fit relatio, ut ijs cedentibus mutetur relatio illa in qua hujus quies velmotus relativus consistit. Rursus motus verus a viribus in corpus motum impressis semper mutatur, at motusrelativus ab his viribus non mutatur necessario. Nam si eædem vires in alia etiam corpora, ad quæ fit relatio,sic imprimantur ut situs relativus conservetur, conservabitur relatio in qua motus relativus consistit. Mutariigitur potest motus omnis relativus ubi verus conservatur, et conservari ubi verus mutatur; et propterea motusverus in ejusmodi relationibus minime consistit.” motus of a body β requires a force on β , but to produce relative motus the force can act on the reference body γ instead; and relative motus can even be can-celled if force is applied to both β and γ . The translators, Motte and Cajori, render motus as “motion” throughout, but the passage only makes sense (today) if we use acceleration , for most occurrences at any rate: Newton first speaks explicitly of thegeneration or alteration of motion, to establish that ‘acceleration’ is at issue; havingsettled that he abbreviates and just writes motus —while continuing to mean acceler-ation. And he distinguishes between a true acceleration and a relative accelerationwhich can be consistently interpreted, however anachronistically, as A i and d x i /dt .Of course Newton knows neither about connections nor affine structure, nor even ma-trices; but he is clearly groping for something neither he nor we can really pin downusing the mathematical resources then available. It may not be pointless to think of a‘Cauchy convergence’ of sorts towards something which at the time is unidentified andalien, and only much later gets discovered and identified as the goal towards which theintentions, the gropings were tending.When Newton states, in the second law, that the mutationem motus is proportionalto force, he could mean either the true acceleration or the relative acceleration; indeed itis in the spirit of the passage just quoted to distinguish correspondingly—pursuing ouranachronism—between a true force F i = mA i and a relative force f i = m d x i /dt .This last equation represents one condition for two unknowns, of which one can befixed or measured to yield the other. But the relative force f i is the wrong one. The‘default values’ for both force and acceleration, the ones Newton is really interestedin, the ones he means when he doesn’t specify, the ones that work in his laws, are the‘true’ ones: true force and true acceleration. And even if F i = mA i also looks like onecondition for two unknowns, the true acceleration A i in fact conceals two unknowns,the relative acceleration d x i /dt and the difference A i − d x i /dt representing theabsolute acceleration of the coordinate system. Nothing doing then, we’re still goingaround in circles: the inertia of Newtonian mechanics remains absolute, and cannoteven be ‘made relative’ to force.But what’s wrong with absolute inertia? In fact it can also be seen as ‘relative,’ butto something—mathematical structure or the sensorium Dei or absolute space—thatisn’t really there, that’s too tenuous, invisible, mathematical, ætherial, unmeasurableor theological to count as a cause, as a physically effective circumstance, for most em-piricists at any rate. The three unknowns of F i = mA i are a problem because inNewtonian mechanics affine structure, which determines A i − d x i /dt , is unobserv-able. By relating it to matter Einstein would give inertia a solid, tangible, empiricallysatisfactory foundation. General relativity can be seen as a response to various things. It suits our purposes toview it as a reaction to two ‘absolute’ features of Newtonian mechanics, of Newtonian All sorts of questions can be raised about the direct measurability of the true force. Cf . Einstein (1916, pp. 771-2; 1917b, p. 49; 1990, p. 57), Cassirer (1921, pp. 31, 38, 39), Rovelli (2007, § § motus —in his ex-position of the thought experiment at the beginning of “Die Grundlage der allgemeinenRelativit¨atstheorie” (1916, p. 771). There he brings together elements of Newton’s twoexperiments—rotating fluid, two rotating bodies: Two fluid bodies of the same size andkind, S and S , spin with respect to one another around the axis joining them whilethey float freely in space, far from everything else and at a considerable, unchangingdistance from each other. Whereas S is a sphere S is ellipsoidal. Einstein’s analy-sis of the difference betrays positivist zeal and impatience with metaphysics. Newton,who could be metaphysically indulgent to a point of mysticism, might—untroubled bythe absence of a manifest local cause—have been happy to view the deformation of S as the effect of an absolute rotation it would thus serve to reveal. Einstein’s epis-temological severity makes him more demanding; he wants the observable cause ofthe differing shapes; seeing no local cause, within the system, he feels obliged to lookelsewhere and finds an external one in distant masses which rotate with respect to S .ii. Einstein (1990) also objects to “the postulation,” in Newtonian mechanics, “of athing (the space-time continuum) which acts without being acted upon.” Newtonianspace-time structure—inertial structure in particular—has a lopsided, unreciprocatedrelationship with matter, which despite being guided by it does nothing in return.General relativity responds to absolute inertia by relating inertia to matter , whichhas a more obvious physical presence than mathematical background structure or the sensorium Dei . In “Prinzipielles zur allgemeinen Relativit¨atstheorie” (1918a) Einsteingoes so far as to claim that inertia in his theory is entirely determined by matter,which he uses T ab to represent:Since mass and energy are the same according to special relativity, andenergy is formally described by the symmetric tensor ( T µν ), the G -field isdetermined by the energy tensor of matter. Einstein (1916, p. 771); cf . footnote 22 above. Einstein wants visible effects to have visible causes; cf .Poincar´e (1908, pp. 64-94), who sees “chance” when “large” effects have “small” causes—which can evenbe too small to be observable; and Russell (1961, p. 162): “a very small force might produce a very largeeffect. [ . . . ] An act of volition may lead one atom to this choice rather than that, which may upset some verydelicate balance and so produce a large-scale result, such as saying one thing rather than another.” P. 58: “Erstens n¨amlich widerstrebt es dem wissenschaftlichen Verstande, ein Ding zu setzen (n¨amlichdas zeitr¨aumliche Kontinuum), das zwar wirkt, auf welches aber nicht gewirkt werden kann.” Cf . Weyl (1931,p. 51): “Space accordingly acts on [things], the way one necessarily conceives the behaviour of an absoluteGod on the world: the world subject to his action, he spared however of any reaction.” In fact he speaks of the “ G -field” (1918a, p. 241), “the state of space described by the fundamentaltensor,” by which inertia is represented: “Inertia and weight are essentially the same. From this, and fromthe results of the special theory of relativity, it follows necessarily that the symmetrical ‘fundamental tensor’( g µν ) determines the metrical properties of space, the inertial behaviour of bodies in it, as well as gravita-tional effects.” Ibid . p. 241: “
Mach’s principle:
The G -field is completely determined by the masses of bodies.” SeeHoefer (1995) on “Einstein’s formulations of Mach’s principle.” Ibid . 241-2: “Da Masse und Energie nach den Ergebnissen der speziellen Relativit¨atstheorie das Gleiche
8e explains in a footnote (p. 241) that this
Machsches Prinzip is a generalisation ofMach’s requirement (1988, § So we seem to be wondering about what Einstein calls
Mach’s principle , whichprovides a convenient label, and is something along the lines of matter determinesinertia . We have seen what a nuisance absolute inertia can be; to remedy Einsteinmade it relative, to matter; we accordingly consider the extent and character of his‘relativisation,’ of the determination of inertia by matter . To begin with, what is matter? Einstein (1918a), we have seen, used T ab to characteriseit, but maybe one should be more permissive and countenance less substantial stuff aswell. Einstein proposed(2) t µν = 12 δ µν g στ Γ λσρ Γ ρτλ − g στ Γ µσρ Γ ρτν for the representation of gravitational mass-energy; matter without mass, or mass awayfrom matter, are hard to imagine; so perhaps we can speak of gravitational matter -mass-energy. How about U µν = T µν + t µν then, rather than just T ab ? Several drawbacks cometo mind. The right-hand side of (2) shows how such ‘matter’ would be related to thenotoriously untensorial connection components. In free fall, when they vanish, thepseudotensor t µν does too, which means that gravitational matter-mass-energy wouldbe a matter of opinion , its presence depending on the state of motion of the observer.The distribution of matter-mass-energy in apparently empty space-time would accord-ingly depend on the choice of coordinates. To be extremely liberal one could even fillthe whole universe, however empty or flat, on grounds that matter-mass-energy is po-tentially present everywhere, as an appropriate acceleration could produce it anywhere.A superabundance of matter would help constrain inertia and hence make ‘Mach’sprinciple’—indeed any relationist claim or principle—easier to satisfy, perhaps to apoint of vacuity. The relationist would also be brought uncomfortably close to his ‘ab-solutist’ opponent, who believes there is more to inertia than one may think, that it sind, und die Energie formal durch den symmetrische Energietensor ( T µν ) beschrieben wird, so besagt dies,daß das G -Feld durch den Energietensor der Materie bedingt und bestimmt sei.” Barbour & Pfister (1995) is full of excellent accounts; see also Earman (1989, pp. 105-8),Mamone Capria (2005) and Rovelli (2007, § Cf . Russell (1927, p. 82): “We do not regard energy as a “thing,” because it is not connected with thequalitative continuity of common-sense objects: it may appear as light or heat or sound or what not. But nowthat energy and mass have turned out to be identical, our refusal to regard energy as a “thing” should inclineus to the view that what possesses mass need not be a “thing.”” Its convenient form is assumed with respect to coordinates satisfying √− g , where g is the deter-minant of the metric. Issues related to the domain of
Wegtransformierbarkeit are considered in § Wegtransformierbarkeit or ‘away-transformability’ is a useful notion for which there seems to be no English word. Cf . Earman & Norton (1987, p. 519). Webegan with Newton, Leibniz and Galileo, have been guided by a continuity connectingtheir preoccupations with Einstein’s, and accordingly adopt a notion of matter that dif-fers as little as possible (within general relativity) from theirs: hence T ab , rather thanthe ill-behaved U µν = T µν + t µν . This paper is much more about general relativity than about Mach himself; it is cer-tainly not about Mach’s own formulations of his principles. The vagueness and ambi-guities of Mach (1988, § ρ ; and ii. ‘field-theoretical holism.’i. Einstein’s equation G ab ( x ) = T ab ( x ) seems to express a circumscribed (direct)relationship between inertia and matter at (or around) point x . The matter-energy-momentum tensor T ab ( x ) = ρ ( x ) V a V b ,for instance, describing a dust with density ρ and four-velocity V a , would (directly)constrain inertia at x , not at other points far away. But much as in electromagnetism,the ‘continuity’ of ρ is deceptive. Once the scale is large enough to give a semblanceof continuity to the density ρ , almost all the celestial bodies contributing to the deter-mination of ρ ( x ) will be very far, on any familiar scale, from x . Einstein (1917a) sees ρ as an average, and speaks of ‘spreading’:The metrical structure of this continuum must therefore, as the distributionof matter is not uniform, necessarily be most complicated. But if we areonly interested in the structure in the large, we ought to represent matteras evenly spread over enormous spaces, so that its density of distributionwill be a function that varies very slowly. Needless to say, all the matter involved in the determination of ρ ( x ) will be very closeto x on the largest scales; but matter far from x even on those scales has a role too, a This is no peculiarity of general relativity, as an anonymous referee has pointed out: even in oldertheories the local energy density can disappear and reappear under coordinate changes. P. 135: “Die metrische Struktur dieses Kontinuums muß daher wegen der Ungleichm¨aßigkeit derVerteilung der Materie notwendig eine ¨außerst verwickelte sein. Wenn es uns aber nur auf die Struktur imgroßen ankommt, d¨urfen wir uns die Materie als ¨uber ungeheure R¨aume gleichm¨aßig ausgebreitet vorstellen,so daß deren Verteilungsdichte eine ungeheuer langsam ver¨anderliche Funktion wird.” D , with de-numerable elements D , D , . . . . Of course the value ϕ r = ϕ ( D r ) of a (scalar) field ϕ at D r will be completely unconstrained by the values ϕ s at other points D s if norestrictions are imposed. On its own the ‘boundary condition’ ϕ s → as s → ∞ —oreven the stronger condition ϕ s vanishes for s > —will not constrain ϕ at all. Butthe further requirement that, say, | ϕ r − ϕ s | <
12 min {| ϕ r | , | ϕ s |} for adjacent points ( i.e . | r − s | = 1 ) gives, by heavily constraining either value oncethe other is fixed, the crudest idea of how boundary conditions act.Of course the manifolds involved in general relativity are continuous, with smoothfields on them, which leads to subtler, less trivial constraint: such fields can undulate,propagate perturbations, drag and so forth; the constrained relationship between neigh-bouring values can ripple across the universe at the speed of light. The value R ( x )of a field R at point x can be indirectly constrained through restrictions imposed byanother field T on the values R ( x ′ ) at points x ′ far away; or directly, by the physi-cist, who may require for instance that R itself vanish somewhere—here one speaks of‘boundary conditions.’ If the universe foliates into spatially non-compact simultaneitysurfaces, such boundary conditions have to be imposed, typically asymptotic flatness.But this, wrote Einstein (1917a), is at odds with the relativity of inertia: “inertia wouldbe influenced but not determined by matter” —since the full determination requiresthe ‘additional,’ physically ‘extraneous’ stipulation of boundary conditions. So he didaway with boundary conditions by doing away with the boundary: he proposed a uni-verse foliating into spatially compact simultaneity surfaces (without boundary), whichlend themselves to ‘global’ Machian interpretations by favouring the determination ofinertia by matter. Even if the determination is partly field-theoretical, holistic, global, non-local, wewill concentrate on the ‘punctual’ determination, on the arithmetic and comparison offreedom degrees at a point . Words like “determination,” “over/underdetermination” or“freedom” are often referred to a single point—by Einstein and others—even in field-theoretical contexts (where more holistic influences are also at work), and seem neitherillegitimate, meaningless nor inappropriate when applied so locally. It is worth mentioning that Einstein’s own position on the matter of punctual ratherthan field-theoretical, non-local determination is confusing. In “Kosmologische Betra-chtungen zur allgemeinen Relativit¨atstheorie” (1917a), which is all about fieldtheoretical P. 135: “Somit w¨urde die Tr¨agheit durch die (im Endlichen vorhandene) Materie zwar beeinflußt abernicht bedingt .” Both kinds of foliation have received ample attention in the literature; see Wheeler (1959),Choquet-Bruhat (1962), ´O Murchadha & York (1974), Isenberg & Wheeler (1979), Choquet-Bruhat & York(1980), Isenberg (1981), York (1982), Ciufolini & Wheeler (1995, § Lusanna & Pauri (2006a, pp. 719-20 for instance) consider such non-locality in the Hamiltonian formu-lation. Specification of circumstances at a point is not enough, as an anonymous referee has pointed out, for prediction in a field theory, where much more (Cauchy data on a Cauchy surface) would have to be indicated. And he often counts degrees freedom at a point, saying that one object there over- orunder-determines another.
Inertial motion is free and not force d by alien influences to deviate from its naturalcourse. The characterisation is general, its terms take on specific meaning in particu-lar theories: in general relativity, inertial motion is subject only to gravity and not toelectromagnetic or other forces; we accordingly identify inertia with the structures thatguide the free fall of small bodies (perhaps the hands of clocks too) by determiningthe (possibly parametrised) geodesics they describe. We have seen that Einstein identifies inertia with the metric g , which in generalrelativity—where ∇ g vanishes (along with torsion)—corresponds to the affine struc-ture given by the Levi-Civita connection ∇ = Π , with twenty degrees of freedom.It gives the parametrised geodesics σ : ( a , b ) → M ; s σ ( s ) through ∇ ˙ σ ˙ σ = , and represents the ‘inertia’ of the parameter, hence of the hands of clocks,along with that of matter. ( M is the differential manifold representing the universe.)But time and clocks may be less the point here than plain free fall. Weyl identifiedinertia with the weaker projective structure Π , which gives the ‘generalised geodesics’ σ : ( a, b ) → M ; s σ ( s ), through ∇ ˙ σ ˙ σ = λ ˙ σ . Projective structure just representsfree fall, in other words the inertia of bodies alone, not of bodies and the hands of ac-companying clocks. One can say it is purely ‘material,’ rather than ‘materio-temporal.’In the class Π = { Π α : α ∈ Λ ( M ) } of connections projectively equivalentto ∇ , a particular connection Π α is singled out by a one-form α , which fixes theparametrisations s of all the generalised geodesics σ . So projective structure hastwenty-four degrees of freedom, four—namely α , . . . , α —more than affine struc-ture; α µ = h α, ∂ µ i . We can write h dx µ , Π α ∂ ν ∂ κ i = Γ µνκ + δ µν α κ + δ µκ α ν ,where the Γ µνκ are the components of the Levi-Civita connection. The most meaningful P. 135: “Der metrische Charakter (Kr¨ummung) des vierdimensionalen raumzeitlichen Kontinuums wirdnach der allgemeinen Relativit¨atstheorie in jedem Punkte durch die daselbst befindliche Materie und derenZustand bestimmt.” We only know that test bodies follow geodesics, as an anonymous reviewer has emphasised. Bodieslarge enough to influence projective structure may be guided by it in a different way: “Since we do notknow how to solve Einstein’s equations with matter, we do not know whether ‘dynamical masses’ followgeodesics.” Cf . Dorato (2007). See footnote 1, and Weyl (1921); or Malament (2006, p. 233) for a more modern treatment. Or alternatively the unparametrised geodesics, in other words just the image I ( σ ) = I ( σ ) ⊂ M . λ = − h α, ˙ σ i = − α µ dσ µ ds = − (cid:18) dsds (cid:19) d s ds of the parameter s along the generalised geodesic σ determined by Π α .In fact not all of the added freedom in projective structure is empirically available:as ‘second clock effects’ are never seen, α really should be exact. We have to makea choice, and will take affine structure to represent inertia; but if (duly restricted) pro-jective structure is preferred, the arithmetic can be adjusted accordingly. Before moving on to the underdetermination of inertia by matter we should consider theextent to which curvature interferes with low-dimensional (zero- or one-dimensional)idealisations that have a role here. We have associated inertia with the geodesicsof a connection; and a geodesic is a (parametrised) one-dimensional manifold, aworld line that (if timelike) can be described by an ideally small—essentially zero-dimensional—object with negligible mass and spatial extension. Masses can be largeenough to produce observable distortions of space-time—or small enough to distortonly unmeasurably: whatever the threshold of instrumental sensitivity, masses fallingbelow the threshold can always be found. And even if the relationships between theworldlines making up the worldtube of an extended object may not be uninteresting—their geodesic deviation will not always vanish—there will always be geodesics whoseseparation is small enough to bring geodesic deviation under the threshold of measur-ability.Synge (1964, pp. ix-x) wasnever [ . . . ] able to understand th[e] principle [of equivalence]. [ . . . ] Doesit mean that the effects of a gravitational field are indistinguishable fromthe effects of an observer’s acceleration? If so, it is false. In Einstein’stheory, either there is a gravitational field or there is none, according asthe Riemann tensor does not or does vanish. This is an absolute property;it has nothing to do with any observer’s world-line.It is doubtless right to distinguish between curvature and flatness; but also betweenmathematical distinguishability and experimental distinguishability.[ . . . ] The Principle of Equivalence performed the essential office of mid-wife at the birth of general relativity, but, as Einstein remarked, the in-fant would never have got beyond its long-clothes had it not been for See Afriat (2009) and Ehlers, Pirani & Schild (1972). We thank an anonymous referee for reminding usabout second clock effects. The four additional degrees of freedom would be subject to the differential restriction dα = 0 ; thetwo-form dα has six independent quantities. Cf . Lusanna (2007, p. 79): “a global vision of the equivalence principle implies that only global non-inertial frames exist in general relativity [ . . . ].” In other words: since low-dimensional frames are too smallto make sense, they have to be global; global frames are too large to be inertial; hence only non-inertialframes can be countenanced in general relativity . Of course an elevator that’s smallenough for one level of instrumental sensitivity may not be for another. The strategy isfamiliar from analysis: for any tolerance ε > one can always find a δ that gives riseto effective indistinguishability by falling under the tolerance. Mathematical physics isfull of linear approximations; one often takes the first term in a Taylor expansion andignores the others.Tidal effects already get ‘idealised away’ in the sixth corollary (to the laws), whereNewton points out that a system of bodies will be indifferent to a common “ac-celerative force.” He presumably means a ‘universal’ force subjecting all of them tothe same acceleration, and clearly has gravity in mind—which he doesn’t mention ex-plicitly, however, as it would produce tidal effects at odds with the claimed invariance.He idealises the difficulty away by specifying conditions that would (strictly speak-ing) be incompatible if the accelerations were indeed gravitational: they have to be“equal” —which would put the bodies at the same distance from the source—and inthe same direction —which would align them along the same ray. Together the twoconditions would confine the bodies to the same spot. Here too, then, there is a sensein which gravity can only be transformed away at a point. The absence of curvaturenonetheless makes inertia easier to represent in Newtonian mechanics, where it can be‘global’ (rather than low-dimensional), since geodesic deviation vanishes everywhere;but as we are wondering to what extent the ‘relative’ inertia of general relativity repre-sents a satisfactory response to the absolute inertia of Newtonian mechanics, we haveto represent inertia in general relativity too. Affine structure seems to capture it well—even if real objects are extended and distort space-time.Then there is the Wegtransformierbarkeit of gravitational energy. Though punctual(zero-dimensional)
Wegtransformierbarkeit has the merit of being logically clean— some objects satisfy it, others don’t —it may perhaps be too easily satisfied to be mean-ingful. Larger domains tend to make it harder; they complicate the logic and mathemat-ics of
Wegtransformierbarkeit by introducing differential constraints tying the fates ofcertain points to those of others. Curvature might appear to prevent broader
Wegtrans-formierbarkeit quite generally, but non-vanishing connection components do not keep t µν from vanishing: Schr¨odinger (1918) proposed coordinates that make t µν vanish ev-erywhere in an entirely curved universe; so one should not even think of a ‘bump in the Cf . Lusanna (2007, p. 80): “Special relativity can be recovered only locally by a freely falling observerin a neighborhood where tidal effects are negligible,” and p. 91: “[the equivalence principle] suggested [ . . . ]the impossibility to distinguish a uniform gravitational field from the effects of a constant acceleration bymeans of local experiments in sufficiently small regions where the effects of tidal forces are negligible.” “corpora moveantur quomodocunque inter se” “pergent omnia eodem modo moveri inter se, ac si viribus illis non essent incitata,” “corpora omniaæqualiter (quoad velocitatem) movebunt per legem II. ideoque nunquam mutabunt positiones et motus eoruminter se.” “a viribus acceleratricibus æqualibus” “æqualibus,” “æqualiter” “secundum lineas parallelas” § t µν from vanishing everywhere. Butsince useful general statements (like a satisfactory classification of cases) about how t µν is affected by coordinate transformations over an arbitrary region seem hard to make,one is tempted to stick to a single point—where the logic of Wegtransformierbarkeit is simplified by depending on the object in question alone. Though many quasi-localcharacterisations of matter-energy have been proposed, they all appear to have theirshortcomings; Szabados (2004, p. 9) writes:However, contrary to the high expectations of the eighties, finding an ap-propriate quasi-local notion of energy-momentum has proven to be sur-prisingly difficult. Nowadays, the state of the art is typically postmodern:Although there are several promising and useful suggestions, we have notonly no ultimate, generally accepted expression for the energy-momentumand especially for the angular momentum, but there is no consensus in therelativity community even on general questions (for example, what shouldwe mean e.g. by energy-momentum: only a general expression containingarbitrary functions, or rather a definite one free of any ambiguities, even ofadditive constants), or on the list of the criteria of reasonableness of suchexpressions. The various suggestions are based on different philosophies,approaches and give different results in the same situation. Apparently,the ideas and successes of one construction have only very little influenceon other constructions.The impressive efforts devoted to such constructions are no doubt due to a sense thatthe legitimacy energy and its conservation rightly have in the rest of physics must beextended to general relativity, however badly they get complicated or even compro-mised by curvature and path-dependence. Without attempting a serious evaluation ofthe fruits such efforts have yielded we will confine ourselves to punctual
Wegtrans-formierbarkeit , which is mathematically more straightforward and tractable, and logi-cally much cleaner than broader kinds.The physical significance of tensors is, incidentally, not unrelated to these matters—a tensor being an object that cannot be transformed away ; but at a point . A field that’s wegtransformierbar at a point may not be more broadly.
We can now try to characterise and quantify the underdetermination, at a point, ofinertia by matter. The relationship between affine structure and curvature is given by B µνκλ = 2 Γ µν [ λ,κ ] + Γ τνλ Γ µτκ − Γ τνκ Γ µτλ . The curvature tensor B abcd has ninety-six ( × ) independent quantities, eighty if theconnection is symmetric, only twenty if it is metric, in which case B abcd becomes theRiemann tensor R abcd . Einstein’s equation expresses the equality of the matter tensor15 ab and Einstein tensor G ab = R ab − Rg ab ,where the Ricci scalar R is the contraction g ab R ab of the Ricci tensor R ab = R cacb .Many Riemann tensors therefore correspond to the same Ricci tensor, to the sameEinstein tensor. By removing the ten freedom degrees of a symmetric index pair, thecontraction R ab = R cacb leaves the ten independent quantities of the Ricci tensor; thelost freedoms end up in the Weyl tensor C abcd = R abcd − g a [ c R d ] b + g b [ c R d ] a + 13 Rg a [ c g d ] b . To the disappointment of the relationist, local matter would therefore seem to underde-termine inertia by ten degrees of freedom—some of which may prove less meaningfulthan others, however, as we shall soon see. But whatever the meaning of the laxitybetween inertia and matter, their relationship already looks more balanced than before,for now there is inter action: besides guiding matter, inertial ( i.e . affine) structure is alsoconstrained by it. Of course this impression of apparent balance or justice, however en-couraging, does not settle the issue—the guidance after all leaves no freedom, perhapsthe constraint shouldn’t either. In fact we still have every reason to wonder about theway matter constrains inertia in general relativity.Before we see how inertia is constrained by the simplest configuration of matter—its complete absence—in the linear approximation, let us consider a point raised byEhlers and others: matter-energy would appear to make no sense without the metric.How can matter-energy underdetermine a more fundamental object that it requires andpresupposes?To begin with, no metric is needed to make sense of one conceptually importantmatter-energy tensor, namely T ab = 0 . The next-simplest matter-energy tensor is T ab = ρV a V b (‘dust’), with matrix representation ρ . To rule out tachyonic dust one may seem to need the metric, to impose g ab V a V b < ;but since conformally equivalent metrics e λ g ab all agree, in the sense that [ g ab V a V b < ⇔ [ e λ g ab V a V b < for every λ , conformal structure is enough. The next-simplest matter-energy tensor is T ab = ρV a V b + p ( g ab + V a V b ) , Ehlers (1995, p. 467): “So far, any description of the properties and states of matter involves a metric asan indispensible ingredient. Consequently, quite apart from mathematical technicalities the idea that “matterdetermines the metric” cannot even be meaningfully formulated. Besides matter variables, a metric [ . . . ]seems to be needed as an independent, primitive concept in physics [ . . . ].” We thank an anonymous refereefor having brought this up. ρ p p
00 0 0 p . The number p typically gets identified with pressure , which does involve the metric,being defined as force per unit area , or distance squared. The metric is also needed toraise and lower indices: to turn V a into V a or g ab into g ab , or even (by converting T ab into T ab ) to speak of ρ or p as eigenvalues, or of V a as an eigenvector. Electromag-netism in general relativity also requires the metric, which appears in the second termof the energy-momentum tensor T ab = 14 π (cid:18) F ac F cb − g ab F de F de (cid:19) ,and is also needed to relate F ab or F ab to F ab . But even if we have decided to representmatter with T ab however it is constituted, the ‘materiality’ of pure electromagnetismis suspect and open to question; it can be viewed as lower-grade stuff than dust, forinstance. And it must be remembered that we are interested in the relationship betweenmatter and inertia : admittedly inertia is closely related to the metric in standard generalrelativity (by ∇ g = 0 ); but that relationship, which can be seen as contingent, has beenrelaxed by Einstein (1925) and others.Generally, then, the reliance of matter on the metric seems to depend on the kind of matter; in particular on how rich, structured and complicated it is. The simplestmatter—absent matter—can do without the metric; the more frills it acquires, the moreit will need the metric. We shall continue to explore the underdetermination of inertiaby matter, which will be altogether absent in § To understand how gauge choices eliminate eight degrees of freedom let us nowturn to gravitational waves in the linear approximation. Through Einstein’s equation, then, matter determines the rough curvature given bythe Ricci tensor. The absence of matter, for instance, makes that curvature vanish The connection and metric were first varied independently by Einstein (1925), but he, misled byPauli, wrongly attributed the method to Palatini (1919)—who had in fact varied the metric connection; seeFerraris et al. (1982). A world made of dust or nothing may seem a trifle arid. In principle it could be enriched by the sixfreedom degrees of the symmetric tensor T ij , whose eigenvalues p , p and p would, if different, indicate acurious spatial anisotropy; to avoid which T ij is taken to be degenerate, with eigenvalue p = p = p = p ,so that only a single quantity gets typically added to the four of dust. Less arid, but barely . . . For a recent and readable account see Kennefick (2007). Einstein (1917a, p. 132), it is worth mentioning, wrote that without matter there is no inertia at all: “In aconsistent theory of relativity there can be no inertia with respect to ‘space , ’ but only an inertia of the masses with respect to one another .” h µν = g µν − η µν would (being symmetrical) first appearto maintain the ten freedoms of the Weyl tensor. It is customary to write γ µν = h µν − η µν h , where h is the trace h µµ . A choice of coordinates satisfying the fourcontinuity conditions ∂ ν γ µν = 0 allows us to set γ µ = 0 , which does away with thefour ‘temporal’ freedoms. There remains a symmetric ‘purely spatial’ matrix γ γ γ γ γ γ γ γ γ with six degrees of freedom. We can also make h vanish, which brings us back to h µν = γ µν and eliminates another freedom, leaving five. To follow the fates of theseremaining freedoms we can consider the plane harmonic(3) h µν = Re { A µν e i h k,x i } obeying (cid:3) h µν = 0 . If the wave equation were (cid:3) c h µν = ( ∂ − c ∇ ) h µν = 0 in-stead, with arbitrary c , the wave (co)vector k would have four independent components k µ = h k, ∂ µ i : • the direction k : k : k , in other words k / | k | (two) • the length | k | = p k + k + k (one) • the frequency ω = k = h k, ∂ i = c | k | (one).Since c = 1 is a natural constant, the condition (cid:3) h µν = 0 reduces them to three, byidentifying | k | and ω , which makes the squared length k a k a = k k − | k | vanish.And even these three degrees of freedom disappear into the coordinates if the waveis made to propagate along the third spatial axis, which can be recalibrated to matchthe wavelength, leaving two ( − ) freedoms, of polarisation. The three orthogonalityrelations X j =1 A ij k j = X j =1 A ( ∂ i , ∂ j ) h dx ja , k a i = 0 ( i = 1 , , ) follow from ∂ ν γ µν = 0 and situate the polarisation tensor A with com-ponents A ij in the plane k ⊥ ⊂ k ⊥ orthogonal to the three-vector k ∈ k ⊥ . Oncethe coordinates are realigned and recalibrated so that h k , ∂ i = 1 and h k , ∂ i , h k , ∂ i both vanish, the three components A ( ∂ , ∂ j ) also vanish, leaving a traceless symmetricmatrix h h h − h
00 0 0 0 with two independent components, h = − h and h = h .The above gauge choices therefore eliminate eight degrees of freedom:18 the four ‘temporal’ coordinates γ µ eliminated by the conditions ∂ ν γ µν = 0 • the freedom eliminated by h = 0 • the three freedoms of k eliminated by realignment and recalibration.One may wonder about the use of an only ‘linearly’ covariant approximation in apaper that so insistently associates physical legitimacy with general covariance. Thelinear approximation has been adopted as the simplest way of illustrating how twodegrees of freedom remain after gauge choices eliminate eight; but the same count( − ) can be shown, though much less easily, to hold in general. Very briefly:The ten vacuum field equations G µν = 0 are not independent, being constrained by the four contracted Bianchi identities ∇ a G a = · · · = ∇ a G a = 0 ; another fourdegrees freedom are lost to constraints on the initial data, leaving two. For detailswe refer the reader to Lusanna (2007, pp. 95-6), Lusanna & Pauri (2006a, pp. 696,699, 706-7) and Lusanna & Pauri (2006b, pp. 193-4); but should point out that their(related) agenda makes them favour a different, ‘canonical’ (or ‘double’) arithmetic( · · [ − − ]) of freedom degrees provided by the ADM Hamiltonianformalism, which they use to distinguish between four—two configurational and twocanonically conjugate—“ontic” (or “tidal” or “gravitational”) quantities and the re-maining “epistemic” (or “inertial” or “gauge”) degrees of freedom. The ontic-tidal-gravitational quantities—the
Dirac observables —are not numerically invariant underthe group G of gauge transformations; Lusanna & Pauri seem to view a gauge choice Γ ∈ G as determining a specific realisation (or ‘coordinatisation’?) ‘ Ω = Γ ( ˜Ω ) ’of a single “abstract” four-dimensional symplectic space ˜Ω . The ontic state can per-haps be understood as an invariant point ω ∈ ˜Ω , which acquires the four components { q ( ω ) , . . . , p ( ω ) } ∈ Ω with respect to the coordinates q , q , p , p characterisinga particular Ω . At any rate, Lusanna & Pauri use the four ontic-tidal-gravitationalobservables to • individuate space-time points • ‘dis-solve’ the hole argument See Brading & Ryckman (2008) and Ryckman (2008) on Hilbert’s struggle, with similar constraints, toreconcile causality and general covariance. In the general nonlinear case the two remaining freedoms can be harder to recognise as polarisations ofgravitational waves; Lusanna & Pauri speak of the “two autonomous degrees of freedom of the gravitationalfield.” But having based our arithmetic on the linear approximation we will continue to speak of polarisation. Lusanna & Pauri also take the four eigenvalues of the Weyl tensor, and gravitational ‘observables’characterised in various ways by Bergmann & Komar, to express ‘genuine gravity’ as opposed to mere‘inertial appearances.’ But Lusanna (2007, p. 101): “Conjecture: there should exist privileged Shanmugadhasan canonicalbases of phase space, in which the DO (the tidal effects) are also
Bergmann observables , namely coordinate-independent (scalar) tidal effects.” See Lusanna & Pauri (2006a, pp. 706-7); and also Lusanna (2007, p. 101): “The reduced phase spaceof this model of general relativity is the space of abstract DO (pure tidal effects without inertial effects),which can be thought as four fields on an abstract space-time ˜ M = equivalence class of all the admissiblenon-inertial frames M containing the associated inertial effects .” They point out that the diffeomorphism at issue is constrained by the fixed Cauchy data to be purely‘epistemic’ and not ‘ontic’; the covariance is not general . argue that change is possible in canonical gravity, for the ‘ontic’ quantities can evolve. Since so much hangs on their four observables, Lusanna & Pauri emphasise—withdetailed metrological considerations—that they really are observable , and go into pos-sible schemes for their measurement. In § The relationist will take the eightdegrees freedom eliminated by the above gauge choices to be meaningless, to lessenthe underdetermination of inertia—and because as a relationist he would anyway. Wewill too, and concentrate on the status of the double freedom of polarisation. Matter still underdetermines inertia, then, by two degrees freedom, which obstructthe satisfaction of ‘Mach’s principle,’ as we are calling it. But are they really there?Or do they share the fate of the eight freedoms eliminated by gauge choices, which wehave dismissed as physically meaningless? The relationist may prefer to discard themtoo as an empty mathematical fiction without physical consequence; but we know theirphysical meaning is bound up with that of gravitational waves, whose polarisation theyrepresent.We should emphasise that the formalism of general relativity (especially in its La-grangian and Hamiltonian versions) distinguishes clearly between the eight degrees offreedom eliminated by gauge choices and the remaining two representing polarisation.We are not claiming that all ten ( = 8 + 2 ) are theoretically, mathematically on an equalfooting, for they are not; we are merely wondering about the physical meaning of the The Hamiltonian acting on the reduced phase space is not constant in asymptotically flat space-times,where consistency requires the addition of a (De Witt surface) term generating a genuine ‘ontic’ evolution;see Lusanna (2007, p. 97), for instance. Cf . Belot & Earman (2001, §§ X H = ( dH ) is after all generated by the differential dH of theHamiltonian, which vanishes if the Hamiltonian is constant—for instance if it vanishes identically. We thankan anonymous referee for added precision on this matter. See Earman & Norton (1987), Butterfield (1987, 1989), Norton (1988), Earman (1989, § § §
2) for instance. Cf . Rovelli (2007, § Cf . Earman (2006) p. 444: “In what could be termed the classical phase of the debate, the focus wason coordinate systems and the issue of whether equations of motion/field equations transform in a generallycovariant manner under an arbitrary coordinate transformation. But from the perspective of the new groundthe substantive requirement of general covariance is not about the status of coordinate systems or covarianceproperties of equations under coordinate transformation; indeed, from the new perspective, such matterscannot hold any real interest for physics since the content of space-time theories [ . . . ] can be characterisedin a manner that does not use or mention coordinate systems. Rather, the substantive requirement of generalcovariance lies in the demand that diffeomorphism invariance is a gauge symmetry of the theory at issue.”A distinction between physically meaningful and mere gauge is at the heart of the new perspective. Cf .Lusanna (2007, p. 104): “the true physical degrees of freedom [ . . . ] are the gauge invariant quantities, the Dirac observables (DO).”
To deal with the polarisation obstructing a full determination of inertia the relationistcan insist on the right transformation behaviour, which gravitational waves do not seemto have, in various senses. He will argue that as the generation and energy, perhaps eventhe detection of gravitational waves can be transformed away, they and the underdeter-mination of inertia by matter are about as fictitious as the eight freedoms that have justdisappeared into the coordinates.If gravitational waves had mass-energy their reality could be hard to contest. Wehave seen that general relativity does allow the attribution of mass-energy to the gravi-tational field, to gravitational waves, through the pseudotensor t µν ; but also that t µν hasthe wrong transformation behaviour.Is the physical meaning of t µν really compromised by its troubling susceptibility to disappear, and reappear under acceleration? A similar question arose in § t µν oncemore as mere opinion. But we have no reason to be fair, and are merely exploring cer-tain logical possibilities. Perhaps ‘matter’ was something stronger, and required more;maybe a quantity that comes and goes with the accelerations of the observer can be realdespite being immaterial; so we shall treat the physical meaning of t µν —as opposed toits suitability for the representation of matter—as a further issue.General relativity has been at the centre of a tradition, conspicuously associatedwith Hilbert (1924, pp. 261 (Teil I), 276-8 (Teil II)), Levi-Civita (1917, p. 382),Schr¨odinger (1918, pp. 6-7; 1926, p. 492), Cassirer (1921), Einstein (1990, pp. 5, 13)himself eventually, Langevin (1922, pp. 31, 54), Meyerson (1925, § § VII) and Weyl (2000, § Roots can be sought as far back as Democritus, who is said to haveclaimed that “sweet, bitter, hot, cold, colour” are mere opinion, “only atoms andvoid”—concerning which there ought in principle to be better agreement—“are real”; Cf . Dorato (2000): “Furthermore, the gravitational field has momentum energy, therefore mass (via theequivalence between mass and energy) and having mass is a typical feature of substances.” This issue is logically straightforward at a single point, where it only depends on the object in question(here t µν ); the logic of broader Wegtransformierbarkeit is much messier, depending on the nature of theregion, the presence of cosmic rods etc .; see § § See also Brading & Ryckman (2008) and Ryckman (2008). Covariance and invariance are rightly conflated in much of the literature, and here too. Whether it isa number or
Gestalt or syntax or the appearance of a law that remains unchanged is less the point than thegenerality— complete or linear or Lorentz , for instance—of the transformations at issue.
21r more recently in Felix Klein’s ‘Erlangen programme’ (1872), which based geomet-rical relevance on invariance under the groups he used to classify geometries. BertrandRussell, in his version of neutral monism, identified objects with the class of theirappearances from different points of view—not really an association of invarianceand reality, but an attempt to transcend the misleading peculiarities of individual per-spectives nonetheless. Hilbert explicitly required invariance in “Die Grundlagen derPhysik,” denying physical significance to objects with the wrong transformation prop-erties. Levi-Civita, Schr¨odinger (1918) and Bauer (1918, p. 165), who saw the relationof physical meaning to appropriate transformation properties as a central feature of rel-ativity theory, likewise questioned the significance of the energy-momentum pseu-dotensor. Schr¨odinger noted that appropriate coordinates make t µν vanish identicallyin a curved space-time (containing only one body); Bauer that certain ‘accelerated’coordinates would give energy-momentum to flat regions.Einstein first seemed happy to extend physical meaning to objects with the wrongtransformation properties. In January 1918 he upheld the reality of t µν in a paper ongravitational waves:[Levi-Civita] (and with him other colleagues) is opposed to the emphasisof equation [ ∂ ν ( T νσ + t νσ ) = 0 ] and against the aforementioned interpreta-tion, because the t νσ do not make up a tensor . Admittedly they do not; butI cannot see why physical meaning should only be ascribed to quantitieswith the transformation properties of tensor components. In February (1918c) Einstein responded to Schr¨odinger’s objection, arguing that withmore than one body the stresses t ij transmitting gravitational interactions would notvanish: Take two bodies M and M kept apart by a rigid rod R aligned along ∂ . M is enclosed in a two-surface ∂ Θ which leaves out M and hence cuts R (orthogo-nally one can add, for simplicity). Integrating over the three-dimensional region Θ , theconservation law ∂ ν U νµ = 0 yields ddx Z Θ U µ d x = Z ∂ Θ 3 X i =1 U iµ d Σ i : any change in the total energy R U µ d x enclosed in Θ would be due to a flow, repre-sented on the right-hand side, through the boundary ∂ Θ (where U µν is again T µν + t µν ,and d x stands for dx ∧ dx ∧ dx ; we have replaced Einstein’s cosines with a nota-tion similar to the one used, for instance, in Misner et al . (1973)). Since the situation is Accounts can be found in Russell (1921, 1927, 1956). But see also Russell (1991, p. 14), which wasfirst published in 1912. Cf . Cassirer (1921, p. 36). See Cattani & De Maria (1993). Einstein (1918b, p. 167): “[Levi-Civita] (und mit ihm auch andere Fachgenossen) ist gegen eine Beto-nung der Gleichung [ ∂ ν ( T νσ + t νσ ) = 0 ] und gegen die obige Interpretation, weil die t νσ keinen T e n s o rbilden. Letzteres ist zuzugeben; aber ich sehe nicht ein, warum nur solchen Gr¨oßen eine physikalischeBedeutung zugeschrieben werden soll, welche die Transformationseigenschaften von Tensorkomponentenhaben.” µ = 0 , , , .Einstein takes µ = 1 and uses Z ∂ Θ 3 X i =1 U i d Σ i = 0 . He is very concise, and leaves out much more than he writes, but we are presumablyto consider the intersection R ∩ ∂ Θ of rod and enclosing surface, where it seems ∂ isorthogonal to ∂ and ∂ , which means the off-diagonal components T and T vanish,unlike the component T along R . Since − Z ∂ Θ 3 X i =1 t i d Σ i must be something like T times the sectional area of R , the three gravitational stresses t i cannot all vanish identically. The argument is swift, contrived and full of gaps,but the conclusion that gravitational stresses between two (or more) bodies cannot be‘transformed away’ seems valid.Then in May we again find Einstein lamenting thatColleagues are opposed to this formulation [of conservation] because ( U νσ )and ( t νσ ) are not tensors, while they expect all physically significant quan-tities to be expressed by scalars or tensor components. In the same paper he defends his controversial energy conservation law, which weshall soon come to. Conservation is bound to cause trouble in general relativity. The idea usually is thateven if the conserved quantity—say a ‘fluid’ with density ρ —doesn’t stay put, evenif it moves and gets transformed, an appropriate total over space nonetheless persiststhrough time; a spatial integral remains constant:(4) ddt Z ρ d x = 0 . So a clean separation into space (across which the integral is taken) and time (in thecourse of which the integral remains unchanged) seems to be presupposed when onespeaks of conservation. In relativity the separation suggests a Minkowskian orthogo-nality(5) ∂ ⊥ span { ∂ , ∂ , ∂ } Einstein (1918d, p. 447): “Diese Formulierung st¨oßt bei den Fachgenossen deshalb auf Widerstand, weil( U νσ ) und ( t νσ ) keine Tensoren sind, w¨ahrend sie erwarten, daß alle f¨ur die Physik bedeutsamen Gr¨oßen sichals Skalare und Tensorkomponenten auffassen lassen m¨ussen.” See Hoefer (2000) on the difficulties of energy conservation. which already restricts the class of admissible transforma-tions and hence the generality of any covariance. However restricted, the class willbe far from empty; and what if the various possible integrals it admits give differentresults? Or if some are conserved and others aren’t?An integral law like (4) can typically be reformulated as a ‘local’ divergence law ∂ρ∂t + ∇ · j = 0 ,which in four dimensions reads ∂ µ J µ = 0 , where j stands for the current density ρ v ,the three-vector v represents the three-velocity of the fluid, J is the density ρ and J i equals h dx i , j i . But the integral law is primary ; the divergence law derived from it only really expresses conservation to the extent that it is fully equivalent to the morefundamental integral law. As Einstein puts it:From the physical point of view this equation [ ∂ T νσ /∂x ν + g µνσ T µν = 0 ]cannot be considered completely equivalent to the conservation laws ofmomentum and energy, since it does not correspond to integral equationswhich can be interpreted as conservation laws of momentum and energy. In flat space-time, with inertial coordinates, the divergence law ∂ µ T µν = 0 can beunambiguously integrated to express a legitimate conservation law. But the ordinarydivergence ∂ µ T µν only vanishes in free fall (where it coincides with ∇ a T aν ), and oth-erwise registers the gain or loss seen by an accelerated observer. If such variations areto be viewed as exchanges with the environment and not as definitive acquisitions orlosses, account of them can be taken with t µν , which makes ∂ µ ( T µν + t µν ) vanish by com-pensating the difference. The generally covariant condition ∂ µ ( T µν + t µν ) = 0 , whichis equivalent to ∇ a T aν = 0 and ∂ µ T µν + ∂ ν g ab T ab = 0 , can also be unambiguouslyintegrated in flat space-time to express a legitimate conservation law. But integrationis less straightforward in curved space-time, where it involves a distant comparison ofdirection which cannot be both generally covariant and integrable.Nothing prevents us from comparing the values of a genuine scalar at distant points.But we know the density of mass-energy transforms according to( ρ, ) ρ p − | v | ( , v ),where v is the three-velocity of the observer. So the invariant quantity is not the mass-energy density, but (leaving aside the stresses that only make matters worse) the mass-energy-momentum density, which is manifestly directional . And how are distant direc-tions to be compared? Comparison of components is not invariant: directions or rathercomponent ratios equal with respect to one coordinate system may differ in another.Comparison by parallel transport will depend not on the coordinate system, but on thepath followed. Cf . Einstein (1918d, p. 450). Einstein (1918d, p. 449): “Vom physikalischen Standpunkt aus kann diese Gleichung nicht als vollw-ertiges ¨Aquivalent f¨ur die Erhaltungss¨atze des Impulses und der Energie angesehen werden, weil ihr nichtIntegralgleichungen entsprechen, die als Erhaltungss¨atze des Impulses und der Energie gedeutet werdenk¨onnen.” Cf . Brading & Ryckman (2008, p. 136). .10 Einstein’s defence of energy conservation Einstein tries to get around the problem in “Der Energiesatz in der allgemeinen Rel-ativit¨atstheorie” (1918d). Knowing that conservation is unproblematic in flat space-time, where parallel transport is integrable, he makes the universe look as Minkowskianas possible by keeping all the mass-energy spoiling the flatness neatly circumscribed(which is already questionable, for matter may be infinite).Einstein attributes an energy-momentum J to the universe, which he legitimates byimposing a kind of ‘general’ (but in fact restricted) invariance on each component J µ ,defined as the spatial integral J µ = Z U µ d x of the combined energy-momentum U µ = T µ + t µ of matter and field (where U µν = U µν √− g etc ., and the stresses seem to be neglected). To impose it he separates time andspace through (5), and requires the fields T µν and t µν to vanish outside a bounded region B . Einstein is prudently vague about B , which is first a subset of a simultaneity slice Σ t , and then gets “infinitely extended in the time direction,” to produce the world tube B ∂ described by B along the integral curves of the “time direction” ∂ . The supports T and t of T µν and t µν are contained in B ∂ by definition; but T may be much smaller than t and hence B ∂ : we have no reason to assume that T does not contain bodies that radiategravitational waves—of which t µν would have to take account—along the lightconesdelimiting the causal future of T t = T ∩ Σ t . Gravitational waves could therefore, byobliging B ∂ to be much larger than T , spoil the picture of an essentially Minkowskianuniverse barely perturbed by the ‘little clump’ of matter-energy it contains.The generality of any invariance or covariance is already limited by (5); Einsteinrestricts it further by demanding Minkowskian coordinates g µν = η µν (and henceflatness) outside B ∂ . He then uses the temporal constancy dJ µ /dx = 0 of eachcomponent J µ , which follows from ∂ µ U µν = 0 , to prove that J µ has the same value ( J µ ) = ( J µ ) on both three-dimensional simultaneity slices x = t and x = t of coordinate system K ; and value ( J ′ µ ) = ( J ′ µ ) at x ′ = t ′ and x ′ = t ′ in anothersystem K ′ . A third system K ′′ coinciding with K around the slice x = t and with K ′ around x ′ = t ′ allows the comparison of K and K ′ across time. The invarianceof each component J µ follows from ( J µ ) = ( J ′ µ ) . Having established that, Einsteinviews the world as a ‘body’ immersed in an otherwise flat space-time, whose energy-momentum J µ is covariant under the transformation laws—Lorentz transformations—considered appropriate for that (largely flat) environment. Unusal mixture of trans-formation properties: four components, each one ‘somewhat’ invariant, which togethermake up a four-vector whose Lorentz covariance would be of questionable appropri-ateness even if the universe were completely flat. Einstein (1918d, p. 450) Flatness cannot reasonably be demanded of the rest of the universe, as can be seen by giving T ab thespherical support it has in the Schwarzschild solution, where curvature diminishes radially without evervanishing. For a recent treatment see Lachi`eze-Rey (2001). Despite Kretschmann (1917), who pointed out that even an entirely flat universe can be consideredsubject to general (and not just Lorentz) covariance. Cf . Rovelli (2007, § the community,which became and largely remains more tolerant of objects (including laws and calcu-lations) with dubious transformation properties.In §§ he may have been glad to do away with coordinates, if possible—butlike Cassirer he thought it wasn’t: “[ . . . ] cannot do without the coordinate system[ . . . ].” If he had known that one can write, say, ∇ V instead of(6) ∂ µ V ν + Γ νµκ V κ ,Einstein would simply have attributed ‘full’ reality to ∇ V (without bothering with con-fusing compromises). But he saw the complicated compensation of expressions like (6)instead, in which various transformations balance each other to produce a less obviousinvariance: “Only certain, generally rather complicated expressions, made up of fieldcomponents and coordinates, correspond to coordinate-independent measurable ( i.e. real) quantities.” He felt that “the gravitational field [ Γ µνκ ] at a point is neither realnor merely fictitious” : not entirely real since it has “part of the arbitrariness” of See Cattani & De Maria (1993), Hoefer (2000). Einstein (1918e), middle of second column Cassirer (1921, p. 37) Einstein (1918e, p. 699): “Die wissenschaftliche Entwicklung aber hat diese Vermutung nicht best¨atigt.Sie kann das Koordinatensystem nicht entbehren, muß also in den Koordinaten Gr¨oßen verwenden, die sichnicht als Ergebnisse von definierbaren Messungen auffassen lassen.” Bertrand Russell (1927, p. 71) was perhaps the first to see the possibility of a formulation we wouldnow call ‘intrinsic’ or ‘geometrical’: “Reverting now to the method of tensors and its possible eventualsimplification, it seems probable that we have an example of a general tendency to over-emphasise numbers,which has existed in mathematics ever since the time of Pythagoras, though it was temporarily less prominentin later Greek geometry as exemplified in Euclid. [ . . . ] Owing to the fact that arithmetic is easy, Greekmethods in geometry have been in the background since Descartes, and co-ordinates have come to seemindispensable. But mathematical logic has shown that number is logically irrelevant in many problemswhere it formerly seemed essential [ . . . ]. A new technique, which seems difficult because it is unfamiliar, isrequired when numbers are not used; but there is a compensating gain in logical purity. It should be possibleto apply a similar process of purification to physics. The method of tensors first assigns co-ordinates, andthen shows how to obtain results which, though expressed in terms of co-ordinates, do not really depend uponthem. There must be a less indirect technique possible, in which we use no more apparatus than is logicallynecessary, and have a language which will only express such facts as are now expressed in the language oftensors, not such as depend on the choice of co-ordinates. I do not say that such a method, if discovered,would be preferable in practice, but I do say that it would give a better expression of the essential relations,and greatly facilitate the task of the philosopher.” Einstein (1918e, p. 699-700): “Nur gewissen, im allgemeinen ziemlich komplizierten Ausdr¨ucken, dieaus Feldkomponenten und Koordinaten gebildet werden, entsprechen vom Koordinatensystem unabh¨angigmeßbare (d. h. reale) Gr¨oßen.” A similar idea is expressed in Hilbert (1924, p. 278, D r i t t e n s. . . . ); cf . Brading & Ryckman (2008, p. 136): “Interestingly, Hilbert here cites the example of energy in generalwhere the (‘pseudo-tensor density’) expression for the energy-momentum-stress of the gravitational field isnot generally invariant but nonetheless, if defined properly, occurs in the statement of a conservation law thatholds in every frame, i.e., is generally covariant.” Einstein (1918e, p. 700): “Man kann deshalb weder sagen, das Gravitationsfeld an einer Stelle sei etwas”Reales“, noch es sei etwas ”bloß Fiktives“.” Ibid . p. 699: “Nach der allgemeinen Relativit¨atstheorie sind die vier Koordinaten des raum-zeitlichenKontinuums sogar ganz willk¨urlich w¨ahlbare, jeder selbst¨andigen physikalischen Bedeutung ermangelnde at a point , only tothe gravitational field in conjunction with other data.” In May 1921 Einstein seems to have gone a good deal farther, approaching, perhapseven exceeding the positions of his former opponents:With the help of speech, different people can compare their experiences toa certain extent. It turns out that some—but not all—of the sensory expe-riences of different people will coincide. To such sensory experiences ofdifferent people which, by coinciding, are superpersonal in a certain sense,there corresponds a reality. The natural sciences, and in particular the mostelementary one, physics, deal with that reality, and hence indirectly withthe totality of such experiences. To such relatively constant experiencecomplexes corresponds the concept of the physical body, in particular thatof the rigid body. Admittedly he only speaks of the “sensory experiences of different people” and notexplicitly of the transformations that convert sensations between them, nor of generalcovariance for that matter. Not explicitly, but almost: he eventually mentions physics; experiences in physics can be called measurements , and they tend to produce numbers;theory provides the transformations converting the numbers found by one person intothose found by another. For measurements yielding a single number, the interpersonal‘coincidence’ at issue can be interpreted as numerical equality: only genuine scalars —the same for everyone—would belong to the ‘superpersonal reality.’ With measure-ments producing complexes of numbers the notion of ‘coincidence’ upon which realityrests is less straightforward: since numerical equality, for each component of the com-plex, would be much too strong, it will have to be a more holistic kind of correspon-dence, to do with the way the components change together. Vanishing is an importantcriterion: a complex whose components are wegtransformierbar cannot be physicallyreal—one whose components all vanish cannot ‘coincide’ with one whose componentsdon’t. Of course the characteristic class of transformations is not the same in everytheory; in general relativity it is the most general class (of transformations satisfying
Parameter. Ein Teil jener Willk¨ur haftet aber auch denjenigen Gr¨oßen (Feldkomponenten) an, mit derenHilfe wir die physikalische Realit¨at beschreiben.” Ibid . p. 700: “dem Gravitationsfeld an einer Stelle entspricht also noch nichts ”physikalisch Reales“,wohl aber diesem Gravitationsfelde in Verbindung mit anderen Daten.” Einstein (1990, p. 5): “Verschiedene Menschen k¨onnen mit Hilfe der Sprache ihre Erlebnisse bis zueinem gewissen Grade miteinander vergleichen. Dabei zeigt sich, daß gewisse sinnliche Erlebnisse ver-schiedener Menschen einander entsprechen, w¨ahrend bei anderen ein solches Entsprechen nicht festgestelltwerden kann. Jenen sinnlichen Erlebnissen verschiedener Individuen, welche einander entsprechen unddemnach in gewissem Sinne ¨uberpers¨onlich sind, wird eine Realit¨at gedanklich zugeordnet. Von ihr, da-her mittelbar von der Gesamtheit jener Erlebnisse, handeln die Naturwissenschaften, speziell auch derenelementarste, die Physik. Relativ konstanten Erlebnis-komplexen solcher Art entspricht der Begriff desphysikalischen K¨orpers, speziell auch des festen K¨orpers.” only generally covariant notionsrepresent reality in general relativity .Eight pages on Einstein speaks of geometry in a similar spirit:In Euclidean geometry it is manifest that only (and all) quantities that canbe expressed as invariants (with respect to linear orthogonal coordinates)have objective meaning (which does not depend on the particular choiceof the Cartesian system). It is for this reason that the theory of invariants,which deals with the structural laws of invariants, is significant for analyticgeometry. Here “objective meaning” is explicitly attributed to invariance under the characteristicclass of transformations.In a letter to Paul Painlev´e dated 7 December 1921 Einstein will be even moreexplicit, claiming that coordinates and quantities depending on them not only have nophysical meaning, but do not even represent measurement results:When one replaces r with any function of r in the ds of the static spher-ically symmetric solution, one does not obtain a new solution, for thequantity r in itself has no physical meaning, meaning possessed only bythe quantity ds itself or rather by the network of all ds ’s in the four-dimensional manifold. One always has to bear in mind that coordinatesin themselves have no physical meaning, which means that they do notrepresent measurement results; only the results obtained by the elimina-tion of coordinates can claim objective meaning. The tension with the passages quoted in footnotes 74 and 75 above is not without itssignificance for the relationist, who at this point can really question the legitimacy of amathematical tolerance whose champion would develop an intransigence surprisinglyreminiscent of the severity expressed by his previous opponents.One can wonder what made Einstein change his mind, after Levi-Civita, Schr¨odingerand others had failed to persuade him. At the end of the foreword, dated 9 August “Offenbar haben in der euklidischen Geometrie nur solche (und alle solche) Gr¨oßen eine objektive (vonder besonderen Wahl des kartesischen Systems unabh¨angige) Bedeutung, welche sich durch eine Invariante(bez¨uglich linearer orthogonaler Koordinaten) ausdr¨ucken lassen. Hierauf beruht es, daß die Invarianten-theorie, welche sich mit den Strukturgesetzen der Invariante besch¨aftigt, f¨ur die analytische Geometrie vonBedeutung ist.” Einstein (1921): “Wenn man in der zentral-symmetrischen statischen L¨osung f¨ur ds statt r irgendeine Funktion von r einf¨ugt, so erh¨alt man keine neue L¨o[su]ng, da die Gr¨osse r an sich keinerlei phy-sikalische Bedeutung hat, sondern nur die Gr¨osse ds selbst, oder besser gesagt das Netz aller ds in dervierdimensionalen Mannigfaltigkeit. Es muss stets im Auge behalten werden, dass die Koordinaten an sichkeine physikalische Bedeutung besitzen, das heisst, dass sie keine Messresultate darstellen, nur Ergebnisse,die durch Elimination der Koordinaten erlangt sind, k¨onnen objektive Bedeutung beanspruchen. Die metri-sche Interpretation der Gr¨osse ds ist ferner keine ”pur imagination“, sondern der innerste Kern der ganzenTheorie. Die Sache verh¨alt sich n¨amlich wie folgt: Gem¨ass der speziellen Relativit¨ats-Theorie sind die Ko-ordinaten x, y, z, t mittelst relativ zum Koordinaten-System ruhenden Uhren unmittelbar messbar, also hatauch die Invariante ds , definiert durch die Gleichung ds = dt − dx − dy − dz die Bedeutung einesMessergebnisses.” Zur Einstein’schen Relativit¨atstheorie (1921) we discover that Ein-stein had read the manuscript and made comments. There he would have found thefirst thorough justification of the mathematical severity his opponents had expresseda few years before. We know how much the philosophical writings of Hume, Machand Poincar´e had influenced Einstein, and can conjecture that even here he was fi-nally persuaded by a philosopher after the best mathematical physicists of the day hadfailed.Be that as it may, it was too late to repent: the damage had been done, the (new)cause was already lost, and indeed the lenience Einstein promoted in 1918 continues tothis day. General covariance is often disregarded or violated in general relativity: ifa calculation works in one coordinate system, too bad if it doesn’t in another; if energyconservation is upset by peculiar coordinates, never mind. Before going on we can briefly consider what Einstein would have found in Cassirer’smanuscript.Cassirer welcomed general relativity as confirming, even consolidating a philo-sophical and scientific tendency he had already described in
Substanzbegriff und Funk-tionsbegriff (1910); a tendency that replaced the obvious things and substances fillingthe world of common sense, with abstract theoretical entities, relations and structures.Even the cruder objects of the na¨ıve previous ontology derived their reality from ‘in-variances’ of sorts, but only apparent ones—mistakenly perceived by the roughnessof our unassisted senses—which would be replaced by the more abstract and accurateinvariants of modern theory.Cassirer calls unity “the true goal of science.” It appears to have much to do witheconomy, of findinga minimum of assumptions, which are necessary and sufficient to providean unambiguous representation of experiences and their systematic con-text. To preserve, deepen and consolidate this unity, which seemed threat-ened by the tension between the principle of the constancy of the velocityof light, and the mechanical principle of relativity, the theory of relativ-ity abandoned the uniqueness of measurement results for space and timequantities in different systems. Introducing differences where there were none before would seem rather to undermineor disrupt unity than to produce it . . . See Howard (2005). Cf . Norton (1993). Cassirer (1921, p. 28): “[die Einheit] ist das wahre Ziel der Wissenschaft. Von dieser Einheit aber hatder Physiker nicht zu fragen, o b sie ist, sondern lediglich w i e sie ist – d. h. welches das Minimum derVoraussetzungen ist, die notwendig und hinreichend sind, eine eindeutige Darstellung der Gesamtheit derErfahrungen und ihres systematischen Zusammenhangs zu liefern [ . . . ].” Ibid . p. 28: “Um diese Einheit, die durch den Widerstreit des Prinzips der Konstanz der Licht-geschwindigkeit und des Relativit¨atsprinzips der Mechanik gef¨ahrdet schien, aufrecht zu erhalten und um sietiefer und fester zu begr¨unden, hat die Relativit¨atstheorie auf die Einerleiheit der Maßwerte f¨ur die Raum-und Zeitgr¨oßen in den verschiedenen Systemen verzichtet.” condition , through which the newinvariants of the theory are first found and established.
The foremost invariance is what we would typically call general covariance—whichCassirer considers “the fundamental principle of general relativity”:
Above all there is the general form itself of the laws of nature, in which wemust henceforth recognise the true invariant and as such the true logicalbasis of nature.
Again, Cassirer sees Einstein’s theory as a fundamental step in the transition betweena common sense world made of (apparently invariant) ‘things,’ to a more abstract andtheoretical world of generally invariant mathematical objects, laws and relations.
Only relations that hold for all observers are genuinely objective, they alone can beobjectively real “natural laws.”We should only apply the term “natural laws,” and attribute objective re-ality, to relationships whose form does not depend on the peculiarity ofour empirical measurement, on the special choice of the four variables x , x , x , x which express the space and time parameters. Cassirer even associates truth with general covariance:The space and time measurements in each individual system remain rela-tive: but the truth and generality of physical knowledge, which is nonethe-less attainable, lies in the reciprocal correspondence of all these measure-ments, which transform according to specific rules.
Truth is not captured by a single perspective:
Ibid . p. 29: “Aber alle diese Relativierungen stehen so wenig im Widerspruch zum Gedanken der Kon-stanz und der Einheit der Natur, daß sie vielmehr im Namen eben dieser Einheit gefordert und durchgef¨uhrtwerden. Die Variation der Raum- und Zeitmaße bildet die notwendige B e d i n g u n g, verm¨oge deren dieneuen Invarianten der Theorie sich erst finden und begr¨unden lassen.”
Ibid . p. 39: “den Grundsatz der allgemeinen Relativit¨atstheorie, daß die allgemeinen Naturgesetze beiganz beliebigen Transformationen der Raum-Zeit-Variablen ihre Form nicht ¨andern [ . . . ].”
Ibid . p. 29: “Vor allem aber ist es die allgemeine F o r m der Naturgesetze selbst, in der wir nunmehrdas eigentlich Invariante und somit das eigentliche logische Grundger¨ust der Natur ¨uberhaupt zu erkennenhaben.” ibid . pp. 34-5
Ibid . p. 35: “Wahrhaft objektiv k¨onnen nur diejenigen Beziehungen und diejenigen besonderen Gr¨oßen-werte heißen, die dieser kritischen Pr¨ufung standhalten – d. h. die sich nicht nur f¨ur e i n System, sondernf¨ur alle Systeme bew¨ahren.”
Ibid . p. 39: “Wir d¨urfen eben nur diejenigen Beziehungen Naturgesetze n e n n e n, d. h. ihnen ob-jektive Allgemeinheit zusprechen, deren Gestalt von der Besonderheit unserer empirischen Messung, vonder speziellen Wahl der vier Ver¨anderlichen x x x x , die den Raum- und Zeitparameter ausdr¨ucken,unabh¨angig ist.” Ibid . p. 36: “Die Raum- und Zeitmaße in jedem einzelnen System bleiben relativ: aber die Wahrheit undAllgemeinheit, die der physikalischen Erkenntnis nichtsdestoweniger erreichbar ist, besteht darin, daß allediese Maße sich wechselseitig entsprechen und einander nach bestimmten Regeln zugeordnet sind.”
Nor is it fully captured by an incomplete collection of perspectives; nothing short of all of them will give the whole truth:This will not be reached and ensured with respect to observations andmeasurements with respect to a single system, nor even with respect toarbitrarily many systems, but only through the reciprocal correspondencesbetween results obtained in all possible systems.
The point being that anything less than general covariance isn’t good enough: U µν , t µν and Γ µνκ are ‘linearly’ covariant, in the sense that they behave like tensors with respectto linear transformations; butMeasurement in one system, or even in an unlimited plurality of ‘priv-ileged’ systems of some sort, would yield only peculiarities in the end,rather than the real ‘synthetic unity’ of the object. And “overcoming the anthropomorphism of the natural sensory world view is,” forCassirer, “the true task of physical knowledge,” whose accomplishment is advancedby general covariance.
Earman (2006, pp. 457-8) is “leery of an attempt to use anappeal to intuitions about what is physically meaningful to establish, independently ofthe details of particular theories, a general thesis about what can count as a generalphysical quantity”; we have seen that Cassirer was less leery, and so—as Earman issuggesting—was Einstein . . .
One hesitates—with or without Cassirer—to attach objective reality or even importanceto things overly shaped by the peculiarities, point of view, state of motion or tastes ofthe subject or observer. Allowing him no participation would be somewhat drastic,leaving at most the meagrest ‘truly objective’ residue; but too much could make theobject rather ‘unobjective,’ and belong more to the observer than to the common reality.Appropriate transformation properties allow a moderate and regulated participation. Ibid . p. 50: “Denn nicht, das jedem wahr sei, was ihm erscheint, will die [ . . . ] Relativit¨atstheorielehren, sondern umgekehrt warnt sie davon, Erscheinungen, die nur von einem einzelnen bestimmten Systemaus gelten, schon f¨ur Wahrheit im Sinne der Wissenschaft, d. h. f¨ur einen Ausdruck der umfassenden undendg¨ultigen Gesetzlichkeit der Erfahrung zu nehmen.”
Ibid . p. 50: “Dieser wird weder durch die Beobachtungen und Messungen eines Einzelsystems, nochselbst durch diejenigen beliebig vieler solcher Systeme, sondern nur durch die wechselseitige Zuordnung derErgebnisse a l l e r m¨oglichen Systeme erreicht und gew¨ahrleistet.”
Ibid . p. 37: “Die Messung in e i n e m System, oder selbst in einer unbeschr¨ankten Vielheit irgendwel-cher ”berechtigter“ Systeme, w¨urde schließlich immer nur Einzelheiten, nicht aber die echte ”synthetischeEinheit“ des Gegenstandes ergeben.”
Ibid . p. 37: “Der Anthropomorphismus des nat¨urlichen sinnlichen Weltbildes, dessen ¨Uberwindungdie eigentliche Aufgabe der physikalischen Erkenntnis ist, wird hier abermals um einen Schritt weiterzur¨uckgedr¨angt.”
31s there an easy way of characterising how much participation would be too much?Of determining the ‘appropriateness’ of transformation properties? Again: vanish-ing, annihilation seems an important criterion, as to which the relationist can demandagreement for physical significance; he will deny the reality of a quantity that can betransformed away, that disappears for some observers but not others.But perhaps there is more at issue than just opinion or perspective. Much as one canwonder whether the different witnesses in Rashomon are lying , rather than expressingreasonable differences in perspective; whether their versions are incompatible , not justcoloured by stance and prejudice—here the relationist may even complain about some-thing as strong as inconsistency , while his opponent sees no more than rival points ofview.Of an object that’s at rest in one system but not in another one can say that it’smoving & isn’t , which sounds contradictory. Consistency can of course be restored withlonger statements specifying perspective, but the tension between the short statementsis not without significance—if the number were a scalar even they would agree. Similarconsiderations apply, mutatis mutandis , to covariance; one would then speak of formor syntax being the same, rather than of numerical equality.Consistency and reality are not unrelated. Consistency is certainly bound up withmathematical existence, for which it has long been considered necessary—perhapseven sufficient.
And in mathematical physics, how can the physical significanceof a mathematical structure not be compromised by its inconsistency? If inconsistencyprevents part of a formalism from ‘existing,’ how can it represent reality? The re-lationist will argue that an object, like t µν , whose existence is complicated—perhapseven compromised—by an ‘inconsistency’ of sorts (it’s there, and it isn’t), cannot bephysically meaningful. LEX I. [ . . . ]
Majora autem planetarum et cometarum corpora motus suos et pro-gressivos et circulares in spatiis minus resistentibus factos conservant diutius.
We can now turn from the reality of gravitational waves to their very generation, aboutwhich the relationist can also wonder.Belief in gravitational radiation rests largely on the binary star PSR 1913 + Observer Ξ with four-velocity V attributes speed w = p | g ( w , w ) | to body β with four-velocity W ,where the (spacelike) three-velocity w is the projection P V ⊥ W = X i =1 h dx i , W i ∂ i = W − g ( V, W ) V onto the three-dimensional simultaneity subspace V ⊥ = span { ∂ , ∂ , ∂ } orthogonal to V ; and the projec-tor P V ⊥ = h dx i , · i ∂ i is the identity minus the projector P V = g ( V, · ) V onto the ray determined by V .Another observer Ξ ′ moving at V ′ sees speed w ′ = p | g ( w ′ , w ′ ) | = k P V ′⊥ W k (all of this around thesame event). Here we’re supposing that one of the speeds vanishes. See Poincar´e (1902, p. 59).
32t any rate). If the kinetic energy is not to disappear without trace, it has to be con-verted, presumably into radiation. Since its disappearance is only ruled out by theconservation law, however, the very generation of gravitational waves must be subjectto the perplexities surrounding conservation.
If the conservation law is suspiciousenough to make us wonder whether the lost energy is really radiated into the gravi-tational field, why take the polarisation of that radiation—which stands in the way ofthe full determination of inertia—seriously? As we were wondering in § generally covariant and onlywork in certain coordinate systems.Even the ‘spiral’ behaviour, associated so intimately with the loss of kinetic energy,is wegtransformierbar . At every point along the worldlines σ and σ of the pulsarsone can always choose ( cf . Joshua x, 13: “the sun stood still, and the moon stayed”)a basis e rµ whose timelike vector e r coincides with the four-velocity ˙ σ r ( r = 1 , ).Since nothing prevents the bases from being holonomic we can view them as naturalbases e rµ = ∂ rµ of a coordinate system, with respect to which ˙ σ r will have components( , , , )—the three naughts being the components of the vanishing three-velocity v .The coordinate system can be chosen so as to leave the pulsars at, say, the constantpositions ( t, , , ) and ( t, , , ). If the pulsars don’t move, if they have no ‘kinesis,’how can they lose a kinetic energy (which is after all a quadratic function of the three-velocity v ) they never had in the first place? It may be felt that the pulsars have a genuine angular momentum, with the righttransformation properties; that they really are going around . But angular momentumis about as coordinate-dependent as quantities get—its transformation properties couldhardly be worse. The range of substitutions on which general relativity was built allowsus to choose a coordinate system that eliminates the rotation by turning with the pul-sars. If one feels instinctively that the rotation is real and legitimate, that it transcendscoordinates, one’s instincts are surreptitiously appealing—comparing the motion—toa background that general relativity was conceived to do away with (but since seemsto have found its way back). We are not really saying that such a backdrop is necessar-ily wrong or absent or unphysical or absurd, only that it should not be appealed to ingeneral relativity, which was invented to get rid of it; the point we are making is moretheoretical, conceptual and mathematical than physical. Certain coordinate systemsmay seem artificial, pathological, even perverse; but general relativity is precisely thetheory of such perversions, or rather of a generality encompassing so much that manysurprising substitutions are admitted along with more mundane ones. Some transfor-mations may savour of dishonest trickery; it might seem we are unscrupulously takingadvantage of the full range of possibilities offered by general relativity, of substitu-tions lying on the fringe of legitimacy, out on the dark edges of a class too enormousto take seriously in its entirety. We can only repeat that the point of general relativ-ity is precisely its full generality . Abstract talk of diffeomorphisms may make certain Cf . Hoefer (2000), Baker (2005). The pulsars are a bit large for low-dimensional idealisation (see § Cf . Weyl (1924, p. 198). geodesic deviation to express the relationship between neighbouring worldlines; but the pulsars are much too far apart for the construction ofa well-behaved , tensorial acceleration of one pulsar with respect to the other. To question the reality or generation of gravitational waves, the relationist woulddemand general covariance—one of the central principles of general relativity— as amatter of principle , whereas his opponent will fall back on the more tolerant day-to-day pragmatism of the practising, calculating, approximating physicist, who viewsthe theory more as an instrumental collection of recipes, perturbation methods, tricksand expedients, by which even the most sacred principles can be circumvented, thanas a handful of fundamental and inviolable axioms from which all is to be deduced.General covariance may have been indispensable at first (it seems a whole crowd ofmidwives was assembled for so demanding a birth), but surely general relativity hasnow outgrown it . . .
The absolutist will be doubly satisfied by the discovery of gravitational waves, whichwould not only reinforce his belief in the underdetermination of inertia, but even allowabsolute motion, as we shall now see.We began with Newton’s efforts to sort out absolute and relative motus , first took(certain occurrences of) motus to mean acceleration , and accordingly considered ab-solute acceleration; but are now in a position to countenance absolute motion moreliterally. The four ontic-tidal-gravitational observables of Lusanna & Pauri may evengive us absolute position : an observer capable of measuring them would infer his ab-solute position from the ontic-tidal-gravitational peculiarities of the spot—and even anequally absolute motion from the variation of those peculiarities. But their measure-ment is anything but trivial, as one gathers from § Gedankenexperiment that’s as simple as it is impossi-ble: Let us say that relative motion is motion referred to some thing —where by ‘thing’we mean a material object that has mass whatever the state of motion of the observer(materiality, again, is not an opinion ). Otherwise motion will be absolute . Supposean empty flat universe is perturbed by (3). Changes in the frequency ω measured bya roving observer would indicate absolute motion, and allow a reconstruction, through ω = k a V a , of the observer’s absolute velocity V a . Affine structure allows the (unambiguous) comparison of neighbouring , not distant, directions.
34s this undulating space-time absolute, substantival,
Newtonian? It is absoluteto the extent that according to the criterion adopted it admits absolute motion. Butits absoluteness precludes its substantival reification, which would make the motionrelative to some thing and hence not absolute. Newton, though no doubt approving onthe whole, would disown it, for “Spatium absolutum [ . . . ] semper manet similare etimmobile,” and our undulating space-time is neither ‘similar to itself’ ( R abcd oscillates,though R bd vanishes identically) nor immobile.We may remember that Newton spoke of revealing absolute motus through itscauses and effects, through forces. Absolute motion is precisely what our thoughtexperiment would reveal, and through forces, just as Newton wanted: the forces, forinstance, registered by a (most sensitive) dynamometer linking the masses whose vary-ing tidal oscillations give rise to the described Doppler effect.The absolutist will claim, then, that gravitational waves are so real they wiggle thedetector, and in so doing reveal absolute motion. But wiggling, the relationist willobject, is not generally covariant: it can be transformed away. Let us continue to sup-pose, for simplicity, that the masses (two are enough) making up the detector are inthe middle of nowhere, and not on the surface of the earth—whose gravitational fieldis not the point here. In what sense do they wiggle? As with the binary star, we canfind coordinate systems that leave them where they are, say at ( t, , , ) and ( t, , , ).Both masses describe geodesics; how can things wiggle if they neither accelerate nor move? The absolutist will reply that each mass, despite moving inertially, accel-erates absolutely with respect to the other, for the tensorial, generally covariant ex-pression d ξ a /dτ = R a c ξ c representing geodesic deviation cannot be transformedaway (where ξ a is the separation, with components ξ µ = h dx µa , ξ a i , and τ is the propertime of the mass to which the acceleration of the other is referred). This puts the re-lationist in something of a corner, mathematically —from which he could only emerge experimentally by pointing out that the acceleration in question, however tensorial andcovariant, has yet to be measured . The reader may feel, perhaps uneasily, that these explorations have been. . . exactlythat; that they lack the factious zeal that so often animates the literature, giving itcolour and heat and sentiment. But the enthusiast remains free to take sides, withoutbeing discouraged by our hesitating ambivalence.Having viewed general relativity as a reply to the absolute inertial structure of New-tonian mechanics—which acts on matter despite being unobservable, and does not evenreact to it—we have wondered about the extent to which the inertia of general relativityis determined by matter and thus overcomes the absoluteness it was responding to.
Newton never seems to use words resembling ‘substance’ in reference to his absolute space, whereasthe literature about it is full of them. Cf . Lusanna (2007, p. 80): “all realistic observers are accelerated,” for unaccelerated observers wouldhave to be too small to be realistic; but see § Again, the very fact that matter constrains inertia at all makes their relationship more balanced thanbefore, as an anonymous referee has pointed out; but a full assessment of how good a response (to theabsolute features of Newtonian mechanics) general relativity proved should nonetheless consider the details
35e have chosen to concentrate on punctual determination, paying little attention tothe holistic, field-theoretical constraint contributed by distant circumstances and stip-ulations. And at a point the matter tensor T ab underdetermines inertia by ten degreesfreedom, eight of which can be eliminated by suitable gauge choices. The remainingtwo represent the polarisation of gravitational waves, whose reality the relationist cancontest by insisting on general covariance; for the generation and energy-momentumof gravitational waves can, in appropriate senses, be transformed away. Their (longawaited) detection, which may at first seem just as wegtransformierbar , would in factbe generally covariant.So gravitational waves have an awkward status in general relativity: though not asmathematically sturdy as one might want them to be, they aren’t so flimsy the relationistcan do away with them without qualms. If gravitational waves could be legitimatelydismissed as a fiction, the determination of inertia by matter would be rather complete;and general relativity could be viewed as a satisfactory response to the absolute featuresof Newtonian mechanics that bothered Einstein.Belot & Earman (2001, p. 227) write that “It is no longer possible to cash outthe disagreement in terms of the nature of absolute motion (absolute acceleration willbe defined in terms of the four-dimensional geometrical structure that substantivalistsand relationalists agree about).” Relationists and absolutists—as we call them—maywell agree that absolute motion, or rather inertia, is represented by affine structure;but disagree about the nature of its determination by matter: only a relationist wouldcontest the physical significance of the mathematical underdetermination at issue here.Questioning the reality of gravitational waves is neither orthodox nor usual; buttheir bad transformation behaviour, which does not seem entirely meaningless, is worthdwelling on. While we await convincing, unambiguous experimental evidence, our be-lief in gravitational waves will (or perhaps should) be bound up with our feelings aboutgeneral covariance, about general intersubjective agreement.We thank Silvio Bergia, Roberto Danese, Dennis Dieks, Mauro Dorato, John Earman,Vincenzo Fano, Paolo Freguglia, Pierluigi Graziani, Catia Grimani, Niccol`o Guicciar-dini, Marc Lachi`eze-Rey, Liana Lomiento, Luca Lusanna, Giovanni Macchia, AntonioMasiello, John Norton, Marco Panza, Carlo Rovelli, Tom Ryckman, George Sparlingand Nino Zangh`ı for many fruitful discussions; and anonymous referees for helpfulsuggestions and comments.
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