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.