1 / 1'/1'1" ( ; , Jl1 Tg111111111 ------w l•: su :\' <: . .'IA I.MON------space d efi ned 011 a 1'011r-di111< · 11 sio11al 111 a 11il'old : whal l1 as ph ys i"al signifi cance is the q11 o ti ent ' '" ''-'' ' of llw constra int hyp <" rsml:u·1· wilhin this function space over th e mappings assodated with tlH' 1'1111 g:111 g1· gronp of the theory. Even on a three-dimensional Cauchy hype rsnrfocc ii ap The Curvature of Physical Space pears ri sky to think primari ly of a diagonalized confi guration space (i.e., ol' a sharply defin ed th ree-dimensional metric g,,. ,, on a th ree-dimensional spacelike hypersurface). Although the constraints restrict to so me ex tenl the range of the canoni call y conjugate vari ables, their uncertai nty is un bounded, suffici ently so that the assumed sharpness of the 3-metric does 11' one were se1iously to entertain, even in a highl y p rogrammati c fas h not p ropagate at all . ion , the thesis "there is nothing in the world except empty curved space. Perhaps it is irrelevant whether we think of well-defi ned world-points Matter, charge, electromagnetism , and othe r fi elds are onl y manifesta with a fuzzy light cone, or conversely, of a sharp light cone, wi th consid li ons of the bending of space," ' it would seem highly germane to examine erabl e uncertainty as to whi ch world-points lie on it. Mos t likely, both of 1h e nature of this curvature, which is to serve as "a kind of magic building 2 these viewpoints are too naive. Suppose we attempt a physical measure mate1ial out of whi ch everything in the physical world is made." Such an ment, by means of instiuments that had bette r not intiude too ciudely on <'Xaminati on has been carried out in depth by Adolf Gri.inbaum in "Gen- the physical situati on, lest their large masses and stresses (if they are to 1·ral Relativity, Geometrodynami cs, and Ontology," a chapter that ap contain any ri gid components) modify th e gravitational fi eld fa r beyond pears for the first ti me in the new edition of his Philosophical Problems of 3 the minimal effects requi red by the uncertainty relati ons. In elaborating Space and Time. The present discussion is intended p rimarily as an ad 4 what such an instru me nt measures we must discuss in detail not only dendum to that chapter -although, I should hasten to add, not necessar which components of the fi elds are to be observed, but also in which ily one that he would endorse. space-time region these observations are to take place. Pe rhaps it is just as well if I. conclude my introductory remarks on this 1. M e tiical Amorphousn ess uncertain note, with all the technical and nontechnical connotati ons of The question I shall be addressing can be phrased, "Does physical "uncertain" you can imagi ne. It is this uncertai nty that makes the whole space possess intri nsic curvature?" This way of putting it is liable to fi eld of quantum gravitation attractive to me. serious misunderstanding on account of th e te rm "in hinsic," fo r it would be natu ral to call the Gaussian curvature of a surface "intrinsic" because, as Gauss showed in hi s theorema egregium , it can be defin ed on the bas is of the metric of the surface itself, without reference to any kind of embed ding space. The mean curvature, in contras t, is not inhinsic in this sense. It is enti rely uncontroversial to state that, in this sense, any Riemannian space-not just a two-dimensional smface-possesses an intrinsic curva ture (possibly identically zero) which is given by the type (0, 4) covariant
NOTE: The author wishes t.o express his gra ti tu de to th e National Science Foun dation for support of research on scientific expl anation and related matters, and to his fri end and colleague, Dr. Hanno Run d, Head of th e Departm ent of Mathemati cs , University of Arizona, for extremely helpful information an d advice. Dr. Ru nd is, of course, not responsi ble fo r any e rrors that may occur herein . His thanks also go to David Lovelock and Hanno Run d fo r making available a copy, prior to publi cation, of th e manu script of their book, Tensors, Differential Forms , and Variational Principles (New York: John Wi ley & Sons, 1975). 280 281 W1 ·sl1·11 C. S11 / 1111111 Tiii•: l :l lHVA 'l'll lli': Ill' l'll YS ICA I. S l'AC I•: curvature tensor R;mhk· For present purposes I shall , however, d1·lil wr Crii11h 11 11111l 11os11 rg1wd li>r lh c ex lrinsica lity of the metric, and in conse ately avoid use of the term "intrinsic" in this sense, and shall use th e te rm quence, lh e eurvalllrt', 011 lh e hasis of what he has called "the metrical int.ernal curvature to characterize those types of curvature that can be amorphousness of space." Appealing to an argument similar to one ad defined in terms of the metric tensor g.,- or more gene rally, those types vanced by Ri emann (which he calls RMH for Ri emann's me trical of curvature that do not depend upon an embedding space. This use of the hypoth esis), 10 he maintains that the congruence or incongruence of two word "internal" is nothing more than a terminological stipulation made for intervals in a continuous homogeneous manifold cannot be an intrinsic purposes of this particul ar discussion. I hope it will p rove conveni ent and property of those intervals. Since I plan to use a similar argument below, not misleading or confusing.• it will be well to state it explici tly. I shall not present the argument in the In line with the foregoing stipulation, whe n I now ask whethe r physical same manner as Griinbaum, but I intend to offer a schematic restatement space has intrinsic curvature, I am asking a question that is similar and of his argument, not an alternative argument. Again, he might not regard closely related to one discussed by.Grii nbaum in vari ous of his works, it as a mere reformulation. namely, "Does physical space possess an intrinsic metric?"• In the con Although Ri emann was obviously unaware of Cantor's theory of the tex t of .Riemannian geometry, this questi on is motivated by the fact that cardinality of the linear continuum; he did seem to recognize that any the salient geometrical properties of a space--such as its Eucl idean or closed linear interval is isomorphic to any other closed linear interval. In non-Euclidean character, or the type of internal curvature it possesses the jargon of set theory, any simply ordered, dense, denumerable set are determined entirely by congruence relations to which its metric gives containing its end points is of the same order type as any other simply rise. The converse is, of course, not true. The internal curvature--e.g., ordered , dense, denumerable set with end points. Even though the linear the Gaussian curvature for a two-dimensional surface--does not dete r continuum is no longer considered denumerable, Riemann's basic notion mine a metric uniquely (even up to a constant factor k). This fact does is not invalidated , for it is easily proved that any closed, continuous, linear nothing to unde rmine Gt ii nbaum's claim about the nonintrinsicality of interval is of the same order type as any other closed , continuous, linear the Ri emannian curvature. Although a given curvature tensor R;mhk of interval. Indeed , Cantor's proof that any two line segments ("regardless of type (0, 4) does not dete rmine a unique metric tensor g.;, it does deter length") have equal cardinali ty proceeds by establishing an order mine a unique class of metric tensors. Given two metric tensors from two preserving one-to-one correspondence between the points of the two such distinct classes, they do determine distinct curvature tensors. intervals. H ence, if one regards a linear continuum merely as a point set In his well-known "theory of equivalent descrip tions," Reichenbach that is ordered by a simple ordering relation, it foll ows that any closed maintains that, by employing diffe rent definitions of congruence, one and interval of any linear continuu:n is isomorphic to any other closed interval the same physical space can be described equivale ntly by the use of of any linear continuum. alternative geometries. 7 D ifferent choices of coordinating definitions of In dealing with "Zeno's metrical paradox of extension," Griinbaum congruence lead to different metrics, and these different metrics are such takes pains to show how it is possible without contradiction to regard a as to lead to different curvatures (or geometries). Reichenbach is not linear continuum of finite nonzero length as an aggregate of points whose making merely an epistemological clai m when he points to the existence unit sets have "length" or measure zero . 11 This is done, essentiall y, by of equivalent descriptions. H e is arguing instead that since the d escrip assigning coordinate numbers to the points on the line, and identifying tions are genuinely equivalent, they describe the same aspects of the the length (measure) of a nondegenerate closed interval with the absolute same reality. 8 Given that the resulting descriptions are equivalent value of the difference of the coordinates of the end points. 12 Given th e including the fact that they are eithe r both true or both fal se--a geometry arbitra rin e~ s of the coordinati zation of the line, it is evidently possible to involved in one but not the other of these descriptions cannot, in itself, ass ign coordinates to that line so as to yield an y desired positive value as represent an intrinsic characteri stic of the space. This leads Reichenbach the length of any segment of that line. Since the coordinate number to his thesis of the relativity of geometry. 9 assigned to a point on the line obviously does not represent any intii nsic 282 283 '1'111 •: C :llllVAT\lll l·: I II' l'llYSICAI. Sl'ACI O: W1 ·sll't/ C. S1tl1111111 s:1111t · as ll1al d1 · .~lµ. 1wd lo .'i llcl\<\' till· iso111orpliism of' nonoverlap ping seg- property of the point, we may say that the length or a segrne11t or elos .. d 111t•11ls in ll1t· 11 1w- cli1111·11 sio11al line. This isomorphis1n among coordinate linear inteival is an extrinsic property. And since the line can also he 11 eighhorhoods or '"I""' dimension wi ll play a major role in the sub coordinatized in such a way that any two nonove rlapping segments n· sequent discussion . ceive the same length, we may add that equality or inequality of length is If physical space is , in fact, metrically amorphous, then obviously it an extrinsic relation between them. 13 can, with equal legitimacy, be endowed with metrics that differ from each We could say that two line segments are nonisomorphic only if we other nonhivially, even to the extent of giving rise to different cuivatures. co uld invoke some further property or relabon of the segments or of their In that case, neither the metric nor the cuivature based thereon can be constituent points such that no property-and-relation-preseiving one-to held to represent intrinsic properties of the space. It is tempting to argue one correspondence between their members exists when the new prop for the converse proposition: if a given manifold can, with equal legiti erty or relation is taken into account. This is Griinbaum's reason for macy, be metrized by means of two different metric tensors, ghJ and ghJ, insisting emphatically that continuity by itself is not sufficient to guaran which are associated, respectively, with two distinct cuivature tensors, tee metrical amorphousness. Homogeneity is required in order to exclude Rimhk and RJmhk • then the curvature tensor Rimhk does not reflect an intrin possible further properties or relations that would destroy the isomor sic characteristic of that manifold. Appealing as this principle is , it must phism of any closed inteival with any other nonoverlapping closed inter be treated with caution, as is shown by Griinbaum's discussion of the val.1' Although it may be hard to certify that, as points on the line, the ele logical relations between alternative metrizability and metrical amor ments have no prope 1ty or relation that would render one segment noniso phousness. 16 However, for spaces composed of homogeneous elements morphic to another nonoverlapping segment, the obvious arbitrariness of the most likely case for physical spac~van Fraassen has offered a plausi the coordinatization lends strong prima facie plausibility to the supposition ble account of alternative metrizability and metrical amorphousness that that no such properties or relations ex ist. The absence of any reasonable equates the two concepts. 17 The success of his program hinges upon our suggestions as to what properties or relations might render two nonove r ability to recognize the difference between trivial variants of the same lapping segments intrinsically equal or unequal in length lends stronger metric and pairs of metrics that differ significantly from each other. I have presumptive evidence to the claim that, as geometrical inteivals on a line, offered a different account of alternative metrizability which is also de any two nonoverlapping segments are isomorphic to each other, and that signed to exclude all cases in which the alternative metrizabili ty rests this isomorphism holds with respect to all of the intrinsic spatial prop solely upon the existence of trivial variants of a single mehic. 18 erties and relations among the elements. 15 Even if we refuse to admit that, with suitable explicitly stated caveats, The same considerations apply whether we are dealing with segments alternative metrizability of a space entails metrical amorphousness, it still of a single one-dimensional eontinuum, or with all sorts of finite closed seems reasonable to construe alternative metrizability as usually or fre inteivals on one-dimensional cuives (which do not intersect themselves) quently symptomatic of metrical amorphousness. Since I am not attempt in a continuous manifold of higher dimension. Given the fact that any ing to repeat in full detail th e arguments for the metrical amorphousness segment of any curve is isomorphic to any other nonoverlapping segment of physical space, and the consequent extrinsicality of the Ri emannian of any cuive, we have extremely wide latitude in the choice of a congru curvature given by the type (0, 4) curvature tensor, I shall simply accept ence relation and a metric. This exhibits a facet of the metric amorphous the conclusion that alternative metrizability is, in this case, indicative of ness of physical space. The full force of this metric amorphousness is the relevant sort of amorphousness. I am thus inclined to agree with revealed by the fact that a Riemannian space is coordinatized by regions, Griinbaum et al. that such extrinsic curvature does not seem to constitute and that any coordinate region of an n-dimensional Riemannian space is a fundamental property of empty space that qualifies it as a "magic build isomorphic to any other coordinate region of any n-dimensional Ri eman ing material" from which everything else in the physical world is to be nian space. The argument to support this latter claim is essentially the 285 284 \\l,.s/1 ·11 <:. S11/111011 '1'111 •: !;l l ll \l/\'1'1 1111 ·: Ill' l'llYSI C/\ I. Sl'/\Ci': constructed. The main purpost• or this disc11 ss io11 is to show that ("Oll sid did wi ll1 n·sp1 •1·I In 1111 · 111dri1 ', llial tl1i ."i l ypt• ol' c 11rval11n· is also no11in e rations of the same type furnish equall y strong grounds for claiming that lri11 sie, li1r ii ca11 vary 11011lri vially ;11 11011 g c q11all y legitimate d escriptions anothe r type of curvature is likewise ex hinsic. of one and th e " """ 111 a11ili1ld . This amounts to an argument, similar to that hased on alte rnative me trizability, which might be said to rest upon 2. The Mixed Curvature T e n sor alte rnative connectability. Clark Glymour has quite properly pointed out that the re is a type (1, 3) 3 . Affine Amorphousne ss mixed curvature tensor Ki.imk that can be defined on a differentiable man ifold e ndowed with an affine connection, even if it does not possess a I shall now atte mpt to make a case for the view that the curvature me tric. 19 The existence of such a curvature tensor is not controversial. associated with the type (1, 3) mixed te nsor is extrinsic-in other words, This shows that there is a type of curvature that does not depe nd upon a that differentiable manifolds are amorphous, not only me trically, but also metric; conseque ntly, it does not follow immediately from the thesis of with respect to their affi ne connections. Let an n-dimensional differenti the metrical amorphousness of space that space lacks intrinsic curvature of able manifold X,, be given. This manifold can, by definition, be covered this type. At the same time, to show that curvature represented by the by a finite number of overlapping coordinate neighborhoods; in each of type (1, 3) tensor may exist independently of a metric does not show that these neighborhoods, every point can be assigned coordinates b y means this type of curvature is indeed an intrinsic property of space. Glymour's of a biunique continuous mapping of the points of the neighborhood onto consideration shows simply that the intrinsicality of this type of cu rvature n-tuples of real numbe rs. These n-tuples constitute an open subset of the is still an open question. 11-dimensional space of real numbers R,, . Since R,, is obviously isomorphic G lymour's argument does nothing to vindicate the original geomet to the Euclidean n-space E,, , the coordinatization of any coordinate rodynamic program of constructing everything in the physical world out neighborhood of our diffe re ntiable manifold X,, establishes an ismor of curved empty space; at best, it provides a temporary reprieve. For the phism between that coordinate neighborhood and an open n-dimensional question now becomes: is the curvature represented b y the mixed te nsor region of E,,. This is, of course, a local isomorphism between a region of a genuinely intrinsic property of space, or is it extrinsic in precisely the X,, and a region of E,,; it is not possible in general to extend this isomor same sense as the metric is ex trinsic to the Ri e mannian manifold? The phism to the entire manifold X,,, for it is not possible in general to cover question can be rephrased: is space as amorphous with respect to the the entire manifold with any single system of coordinates. curvature tensor furnished by the affine connection as it is with resp ect to The question I am raising is, however, a local q uestion. 1 am attempting the me tric tensor? This is the crucial question, but neither Glymour nor to clarify the relationship be tween the curvature associated with the type Gri.inbaum has addressed it. 20 (0, 4) covariant curvature te nsor R !mhk and that associated with the type (1, The answer to this question could be furnished, I believe, by means of 3) mixed curvature tensor K,/hk· The two types of tensors are defined at the following consideration. We asked above whether a Rie mannian man each point of their respective manifolds. The question of the metiical ifold, endowed with a particular metric, could with equal legitimacy be amorphousness of space is a local matter; it does not involve the global described by means of a different metric-one that leads to a diffe rent topological characteristics of the space. In raising the question of the curvature and a different geome try. In a completely parallel fashion, we intrinsicality of the curvature represented by the mixed curvature tensor, can now ask whe the r a diffe re ntiable manifold, endowed with a particular I am putting as ide the global considerations in precisely the same fas hion. affi ne connection yielding a particular mixed curvature tensor Ki.imk> In dealing with the nature of the affine connection and the curvature could with equal legitimacy be endowed with a diffe rent affine connection based thereon, it will the refore be sufficie nt to restrict attention to one that would yield a differe nt curvature tensor R himk· If so, we can provide a coordinate neighborhood. pair of equivale nt descriptions of the sam e manifold embodying different To state this point explicitly is to give the whole show away. As already curvatures. W e could then argue, along the same lines as Reichenbach re marked , the coordinate neighborhood of X,, is isomorphic to a region of 286 287 W1·s/1·11 ( :. S11/111u11 '1'111': C'l l ll\' /\'1'11 111 ·: Euclidean 11-space. If I am correct in saying that tl1<· f1111da11u ·11 ta l hasis liir 11u·nsio11 11 . T l111 s, II' 1111v t·oonli11alt· 1H'iµ;lahorl1ood of s11d1 a manif(>ld ca n claiming that Ri emannian space is metrically amorphous is th e isomor "" l'ndowl'd with 1111 alii "" con1H.:ction which gives rise to a nonvanishing phism of any finite closed interval to any other, then it would seem pla11si cnrvature, then any other coordinate neighborhood of equal dimensional ble to maintain that the isomorphism of any coordinate region ofX,. to some ity can be endowed with the same connection and the same nonvanishing region of Euclidean n-space has a fundam ental bearing upon th e question curvature. Given the obvious fact that some spaces are so endowed with a of whether the differentiable manifold is amorphous with respect to th e metric or affine connection that they are Hat, 21 we may thus conclude that curvature based upon th e affine connection. It shows that any differenti absence of nonvanishing curvature (i.e., the presence of zero curvature) of able manifold that can be endowed with any affine connection whatever the sort associated with the type (1, 3) , mixed curvature tensor is not an may be endowed locally with a connection whose components are identi intrinsic local property of a differentiable manifold. In other words, a cally zero in some given coordinate system. Obviously, th e curvature coordinate neighborhood of a differentiable manifold is neither intrinsi tensor based upon this connection will also have components that vanish cally Hat nor intrinsically nonHat. identically, and this property holds in all admissible coordinate systems. The affine connection is not a tensor; under special circumstances it 4. Parallelism may therefore vanish with respect to some sets of coordinates but not with I do not wish to rest the argument there, however, for I believe it can respect to others. For instance, while it vanishes identicall y for a Euclid be made more compelling by considering the nature and function of the ean space with Cartesian coordinates, it does not vanish for the same affine connection. Let us, th erefore, look at the grounds for introducing space referred to curvilinear coordinates. But the related curvature is such connections. In order to deal with certain kinds of physical and tensorial, and if it vanishes in one coordinate system it will vanish in all. geometrical problems, we introduce tensors of various types, including This means that any coordinate region of any differentiable manifold may vectors as special cases. At each point of our differentiable manifold X,. we legitimately be provided with a set of coordinates and an affine connection construct a series of vector spaces-type (r, s) tensors at a given point such that th e type (1, 3) mix ed curvature tensor vanishes. It is easy to see constituting the membe rs of an n" + ' -dimensional tangent vector space. an important analogy here between the two types of curvature. Given The vectors or tensors that are elements of these tangent vector spaces even a non-Euclidean space, such as the surface of a sphere or th e surface are not elements of the differentiable manifold Xn ; they are members of of a torus, it is possible, on account of metrical amorphousness, to remet abstract vector spaces associated with the points p of X,.. 22 Such algebraic rize an arbitrary region (provided it is not too large) in such a way that the operations as addition, multiplication, and contraction are performed region becomes Euclidean and its Ri emannian curvature vanishes on the elements of the vector spaces associated with one and the same throughout that region. In a completely analogous way, any coordinate point p of x• . At this stage of the analysis, the vectors that are elements region of a differentiable manifold can be coordinatized and endowed with of these various vector spaces have no physical or metrical significance; an affine connection such that its mixed curvature tensor vanishes they are simply elements of an abstract mathematical structure. throughout that region. This shows, I believe, that the presence of non There are many circumstances in which we must deal with relation vanishing curvature of the type indicated by the type (1, 3) mix ed curva ships among vectors or tensors located at different points of X,,. For this ture tensor cannot be an intrinsic local property of a coordinate neighbor purpose, we introduce vector or tensor fi elds. For the present discussion hood of a differentiable manifold. it will be sufficient to confine attention to contravariant vectors, i. e., type One further point should be added. Because of the transitivity of the (1, O) tensors. A contravariant vector fi eld is simply a collection of isomorphism relation, the foregoing argument shows that any coordinate uniquely specified contravariant vectors, each of which is associated with neighborhood of a differentiable manifold Xn is isomorphic to any other a distinct point of a region of Xn . The vector fi eld thus consists of members coordinate neighborhood of any differentiable manifold of the same di- of the various tangen t spaces of contravariant vectors associated with the 288 289 W1 ·slt:lf <: . Sa/mo 11 w lH·n· I' ,1,. is llw 1tlli111 · 1·1111111 ·1· tin11 . and 1 l'OITt·s p u 1Hl s lo tl 1t · poi11t / J. different points of the region ofX. over whi ch the vector fi eld is defin ed , 1 1 11 Tloi s t•q11ali1111 s .. rv1 ·s t11d1 ·fi1H' tl1t · diff1·rt ·11li al ti.\ , t'llahli11 g 11s to write (2) one member being selected from each such tangent vector space. Of particular importance is the so-called parallel vector field. i11 diff1·n ·11lial li1n11 : We may, indeed, defin e the concept of parallel displaceme nt. Tia· _tl X; + r ,,!,. X,, dx'· = 0 along C(t) (3) general idea is this. We say loosely th at a vector can be moved about in a dt dt Euclidean space, and as long as it retains th e same length and direction it For arbitrary points p and If , the speci fi cation of a vector at t/ as parall el to 23 is still the same vector. In th e very special case of Euclidean space and a given vector at p may not be unique, but may depend upon th e choice of Cartesian coordinates, this condition is equivale nt to saying that its com th e curve C(t ). For a point l/ in the neighborhood of p , however, a unique ponents are always the same, regardless of its point of applicati on. A vector Xi + dX; corresponding to the vector X; at p is given by the better way to say th e same thing is to say th at we can deli ne a parallel condition vector fi eld consisting of one vector at each point of th e space, (1 nd that (4) relative to the Cartes ian coordinates, each of these vectors has preci sely /.- th e same components. If this same Euclidean space is recoordinatized Thus f 1 provides a unique local mapping of the tangent space T,, (p) onto with curvilinear coordinates, the same parallel vector fi eld is defin ed in a T,, (q) . diffe rent way. Given a contravariant vector Xi at a point p with coordinates . Since the function of the affine connections is to establish isomor- 1~·, a vector X; + dX; at a neighboring point l/ with coordinates 1~· + d1~· phisms between th e members of vector spaces located at neighboring results from parallel displacement or is a membe r of th e same parallel points of our differentiable manifold, it is natu ral to ask what restrictions vector fi eld as X; if its components satisfy th e equations are to be imposed. Given any two vector spaces of equal dimension, there is obviously a vas t array of possible isomorphisms to choose from . In a (1) Euclidean space (whe re th e metri c is given) we want all of th e membe rs of where {hid is a Christoffel symbol of the second kind. Vectors related in a parallel vector fi eld to have th e sam e magnitude and d irection. In the this way are said to have the same magnitude and direction. It is of crucial absence of a metri c, however, no sense can be attached to th e question: importance to be cl ear about what is goin g on here. Given a particular are the members of the two spaces that are correlated by a given isomor vector at p , whi ch is a membe r of a tangent space at p, we associate with it phism really parallel (equal) to each other? As entities th at are correlated a unique me mber of the corresponding tangent space at q. Indeed , by with one anothe r by th e isomorphism of (4), th ey are parallel by defini means of th e Ch1i stoffel symbol we establish an isomorphism between th e tion, regardless of the particular isomorphism involved. membe rs of the tangent space at p and th ose of th e tangent space at the As equation (4) shows, the parallel vector fi eld is defined in terms of neighboiing point q. It is in this sense th at we specify vectors at q which relati ons between the respective components of the vectors at the two uniquely correspond with vectors at p. neighborhing points p and q. Given the continuity conditions imposed in An analogous procedure can be carried out, not only in Ri emannian characterizing a diffe rentiable manifold, it is natural to impose continuity spaces in general, but also in differentiable manifolds that do not possess a requirements upon the components of th e vector unde r parall el di s me tric, provided they are endowed with an affin e connection. To handle placement. This is, of course, built into th e definition of the affine co n this more general situation, one defin es parallel displacement of a vector necti on. In addition, th e connecti on must have certain properti es with from point p to point l/ along a curve C (defin ed by a paramete r t) by th e respect to coordinate transformations. This is obvious from the following condition consideration. If a particular vector X at a point p is chosen, it will have ce rtain compone nts X; relati ve to a system of coordinates .\J . If a coordi 1 Xj(t) = X coi - Ir ,,i .. X" - dxk dt, (2) I nate transformation to a system ;-.1; is performed, the same poi nt p will dt C 'o have a diffe rent set of coordinates , and the sam e vector X wi ll have 290 291 '1' 111 1: 1:1111\' il'l'tl lll-'. This new tensor field iti• is a different metric for our coordinate neighbor 2 1 2 2 2 ds = (d:;: ) + (dx ) ; g, •• = a~ (16) hood. We may now ask whether the two metrics gi,k and ghk are equally This constitutes a drastic change of metric, in the sense that the distance legitimate metrizations of our coordinate neighborhood. This is a difficult between two points p and q will in general be changed, and intervals that 298 299 \\11's/1 ·y ( .' . .'i11/1111J11 '1'111 •: 1 :1111\ ' 1\'l'lllll< IW l ' llYSIC:1\I . .~l'AC:I ·: c <111)..';no<'11I were congruent under th e old rnetrie will in )..'; Cll<'ral nol IH· :\. Adolf (; r1111! 11 111111 , J'/d/n.,·11 1ilii 1·11/ /'1 ·ohft·1w; 11 /' S111 w1 · fllul 'J'i11w , 2nd ed. (Dordredit/ under the new metric. Consider any two pairs of points (p,, q,) and (Ji ,, 11 ,) lluslon: H1·itl1·I . l!J7: 1), ~ · Imp . :.!:!. 111 this dtaptt'I' G riinbaum care fully docume nts Wheeler's r1·n·11f r1 •111111 t" ialio11 nf" li is t>arl i1·r ]Htr< • gcoine trndynamic thesis. that represent equal inte rvals in the old me tric, but unequal interval s in ·I. In pa rtil'ular, tl1t· pn·s1·11t disu1ssio n is a supplement to secti on 3b--an ex tended foot- the new metric. If there is no intrinsic property of these two inte1vals thal 1101!- lo p. 788, it mi1.d1t In: said . determines that they are either intrinsically congruent or th at th ey are .I) . Griinhaum, Philosophical Problems, p. 501, carefull y distingui shes between "intrinsic" (< ;t· rnun: i1iner) in the sense applicable to Gaussian curvature, and "intrinsic" or "implicit" intrinsically incongruent, then there is no valid basis for maintaining that (<;(· rrnc. m : schon enthalten) in the sense in which Ri e mann denied it wi t~ . res. p e~ t..t_o the one (at most) of th ese metrizations is legitimate--that at leas t one of th e m nH"lric of continuous space. lt is in this latter sense that I am using the term mtnns1c m the present discussion. For th e former concept, [ am using the term "internal. " must be ill egitimate. If it can be argued validly that there are no intrinsic 6. See chi efl y Griinbaum, Philosophical Problems, chap. 16, "Space, Time and Falsifiabil properties of this space that show that the congruences delivered by one ;1v." which also appeared in Philosophy of Science 37 (1970): pp. 469--588. . 7. Hans Reichenbach , Philosophy of Space and Time (New York: Dover Publica tions, of th ese metrics are correct while those produced by the other are incor 1958). This point had been argued with great cogency by Poincare n:iuch earli er. rect, then that argument for the legitimacy of this kind of remetrization is 8. Reichenbach does, it is true, adopt a verifiability criterion of equi valence, but he construes this as a criterion of what can be meaningfully said about physical reality. Thus th e an argument for the metiical amorphousness of the space. claim that ph ysical space actually has one of these geometries and not the other is without As I said at the outse t, I do not clai m to have provided necessary and any poss ible j ustification, and would constitute a totally unwarranted cl aim about reality (not merely about our knowledge) ...... ,, sufficient conditions for the applicability of such concepts as alternative 9. Some authors, including Gri.inbaurn, prefer th e term convent10nal1ty to relati vity metrizability and metric amorphousness in all generality. 37 I am pre in such contexts. Reichenbach may have eschewed the fo rmer term to avoid any suggestion pared, however, to offer the following sufficient conditions: that he was adopting a thoroughgoing conventionalism of the sort be found in Poincare. The term "relativity" emphas izes the fact that, in a given phys ical space, th e ge?metry is relative Condition 1. A coordinate neighborhood of Xn is alternatively metrizable to a choice of congruence standard-i.e. , given a coordinating definition of congruence, the geometry of the space is a matter of empirical fact. In the presence of suitable coordinating if it is unconditionally alternatively metrizable. definitions, there is nothing conventional about the geometry. Condition 2. A coordinate neighborhood of Xn is metrically amorphous if 10. Griinbaum, Philosophical Problems, pp. 495-99, 527-32, has offered a detail ed state ment and elaboration of RMH. it is unconditionally alternatively metrizable. 11. Gri.inba um , Philosophical Problerns, chap. 6, especiall y pp. 170-72. Condition 3. A coordinate neighborhood of X,, has no intrinsic metric if it 12. By a closed interval [a, h] on a line I shall mean the set co ~ sis t i n g of poin.ts a. and~ and is unconditionally alternatively metrizable. 38 all points lying between them. The in terval is degenerate only 1f a = b. In this d.1.scusswn .~ shall use th e term "interval" to refer onl y to nondegenerate interva ls. The term segment will be used as a synonym for "nondegenerate closed interval. " Unconditional alternative metrizability may be a fairly strong condition; 13. The continuity of the line, which is an int1insic structural feature, is re8ected in th e nevertheless, it seems to exhibit a type of alternative metrizability from fact that th e coordinatization of the line is a continuous one-one relation between th e points which metrical amorphousness can reasonably be inferred. It is strong of the line and the continuum of real numbers. We specify that the intervals be nonoverlap ping, for th e inclusion of one interval entirely wi thin another is an intrinsic relation between enough to rule out those cases of alte rnative metrizability in which all of them, and this relationship should be refl ected in the metri c. . the alternative metrics can be considered, in any sense, trivial variants of 14. This point is brought out clearly in Grlinbaum's comparison of th e geometncal con· ti nu um of points with the arithmetiC'al continuum of real numbers; Philosophical Problems , one another. Moreover, I am convinced that the arguments of Poincare, pp. 512-14. 526-31. Reichenbach, Griinbaum, et al. are sufficient to qualify such continuous 15. Griinbaum, Philosophical Problems, pp. 498-501, 529-31 explicitly acknowledges that RMH is only inductively confirmed, not a demonstrated truth. The strength of support for physical manifolds as physical space, physical time, and physical space RMH would seem at least adequate to shift the burden of the argument to its opponents. time for unconditional alternative metrizability. Conditions 1-3 thus 16. Griinbaum, Philosophical Problems, pp. 547-56. . .. seem to apply to the physical manifolds of interest. 17. Bas C. van Fraassen, "On Massey's Explication ofGriinbaum's Conception of Metnc Philosophy of Science 36, no. 4, December 1969. .. 18. [n the Appendix "Alternative Metrizability of Physical Space, I have defined uncon ~ Notes ditional alternative metrizability as a type of alternative metrizability that ex cludes cases in which the alternative metrics are simply trivial variants of one another. Unconditional 1. J. A. Wheeler, Geometrodynamics (New York: Academic Press, 1962), p. 225. alternative metrizability does, I claim, entail metrical amorphousness. 2. J. A. Wheeler, "Curved Empty Space-Time as the Building Material of the Ph ysical 19. Clark Glymour, "Ph ys ics by Convention," Philosophy of Science 39, no. 3, September World," in E. Nagel, P. Suppes, and A. Tarski , eds. , Logic, Methodology and Philosophy of 1972. Science (Stanford. Calif.: Stanford University Press, 1962), p. 361. 20. Griinbaum's answer to Glymour's attempt to impugn Griinbaum's philosophical attack 300 301 ------A DO 1.F C II 0 N BA ll M------w,.s/1 ·y C . S11h111111 upon the geome trodynamic.: program l(1l l11 ws a11 t•utirely clifft·rcnl taek; sc •c · Grii1dm11111 , Philosophica l Problems, pp. 77:w!8. 21 . When I use the term " fl at", it is to be construed in the technica l sense nf th t> vanishing of th e type (1, 3) curvature tensor. 22. This is true of the metric tensor g h k as well as any others- a fact which has not, to 111 y Absolute and Relational TheMies of knowledge, been mentioned explicitly in discussions of intrinsicality of metrics. 23. Because of its linearity, Euclidean space may be identifi ed with a tangent space. 24. Note that this is preci sely the sort of insensitivity to coordinate changes on the part of Space and Space-Time the affine connection as I discussed with respect to the metric in the Appendix "Alte rnative· Metrizability of Ph ys ical Space." It is as indicati ve of "affine amorphousness" in this case as it was indicative of me trical amorphousness in the other case. 25. Remembe r that the ori gin is not within our coordinate neighborhood . 26. In 2-space the curvature tensor has onl y one independent component. 27. Grlinbaum has argued in detail that Einstein's rods-and-clocks method, Synge's I wished to show that space-time is not necessarily some chronometric method, and the geodesic method of Wey] et al. depend equally upon extrin sic standards fo r the determination of the space-time metric. Philosophical Problems, ch ap. thing to which one can ascribe a separate existence, 22, secti on 2. independently of the actual objects of phys ical reality. 28. Just as one definiti on of congruence mi ght he preferable to anothe r on grounds of Physical objects are not in space, but these objects are descriptive simplicity, so a1so might one affine connection be preferable to anoth er on th e spatially extended. In this way th e concept "empty same grounds. Lack of descriptive simplicity does not render either a me tric or an affine connecti on inadmi ss ible as factually in con ect. space" loses its meaning. 29. In di scussion, Howard Stein made reference to a particle "sniffing out" a path through space. If we construe the te rm "path" as autoparallel, then the questi on becomes: on the June 9, 1952 A. Einstein basis of what "olfactory" characteri sti cs of phys ical space itself can an autoparallel be de tected ? To "sniff out" a path requires discriminable odors, as any bird dog knows, but in a (Preface to the fifteenth edition of Relativity: the Special space of homogeneous elements, such diffe rences in odor cannot possibly be intrinsic charac te ri stics. and the General Theory . New York: Crown Publishers, 30. Whethe r topological structure could provide the requisite curvature is. of course, an 1961 , p . vi.) entirely differe nt question. 3 1. Especiall y: Gerald J. Massey, "Toward a Clarifi cati on of Griinbaum's Concept of l. Introduction Intrinsic Metric," Philosophy of Science 36, no. 4, December 1969; van F raassen, "On Massey's Explicati on": Adolf Grunbaum , "Space, Time and Falsifi ability, Introducti on and Has the issue of ontological autonomy between Newton's absolutism Part A," Philoso,,hy of Science 37, no. 4, December 1970, reprinted in Adolf Grunbaum, and Leibniz's relationalism become otiose, defun ct, and perhap s even Philosophical Problems, chap. 16. 32. Massey, "Toward a Clarifi cati on," p. 332. In the end Massey seems to despair of spurious in the contex t of present-day th eories of space-time structure? finding a reasonable explicati on of "alternative metrizability." See p. 345. Or is the dichotomy between absolutisti c and relational ontologies as 33. GrUnbaum, Space, Time, and Fa lsifiahility, part A, section 3. ori ginally understood by Newton and Leibniz illuminatingly germane to 34. Van Fraassen, "On Massey's Explica ti on." 35. Henri Poincare, Science and Hypothesis (New York: Dover Publications, 1952); Hans space-time as much as it was to pre-Einsteinian space and time? Two Reichenbach, The Ph ilosophy of Space and Time (New York: Dove r Publicati ons, 1958); Adolf Grunbaum, Philoso,,hical Problems of Space and Time, 1st ed . (New York: Alfred A. N OTE: This paper is based on a much shorter version deli vered at a conference o f rt~ e sa".1 e Knopf, l 963). title, held on June 3-5, 1974 in Andover, Mass., under the auspices of the Bos ton Uni vers1ty 36. It is, of course, strictly nonsense to identify a coordinate system as such (without Institute of Relativity Studies, directed by John Stachel. specifying a metric) as Cartesian, polar, etc. I am nevertheJess usin g these intuitive notions Apart from specific other acknowledgments m a~ e within t~ e tex t, I am ve r~ grateful to the simply to try to give some feeling for the definiti on of unconditional alternative metrizabil various Andover Conference participants whose ideas are discussed or ment10ned by name ity. within the tex t, especially Howard Stein . In addition, I wish to thank John Stachel, th e 37. Unlike va n Fraassen's discussion, the present one applies (negatively) to various organizer of the co nfe r~ n ce, for th e opportunity ~o present the fi rst draft of this ~ap e r there inhomogeneous spaces, such as the arithmetical spaces of real numbe rs . I am not sure and fo r valuable private discussions of some of its content. And I am greatl y indebted to whether it can be ex tended to apply to discrete spaces. All en Jani s fo r havi ng read both the fi rs t and fin al drafts: my treatment of the debate 38. In this contex t I am, of course, referring onl y to nontrivial intrinsic metrics; see between the relationalist and the "semi-absolutist" had the benefit of some. mc1s1ve .ques Griinba um. Space, Time, and Falsifiability part A, section 2b, for an explication of triviality. ti ons from him. Both John Winnie and Albe rto Coffa stimulated me to deal with some iss ues in section 4 (2) which I would otherwise have neglected. Jeffrey Wi nicour mad: me aw ~ r e of some pitfalls in the use of the term "graviton." I also wis h to thank the Nati onal Science Foundati on fo r the support of research. 302 303