The Cosmological Woes of Newtonian Gravitation Theory John D. Norton I N NOVEMBER 1894, THE ASTRONOMER HUGO SEELIGER sent the journal Astronomische Nachrichten a short article pointing out that Newton's law of gravitation could not be applied without modification to an infinite universe with a roughly uniform matter distribution. The problem Seeliger described was exceedingly simple. As we shall see in Section 1 below, it could be developed with just a few moments thought. Thus it is no surprise that Seeliger was not the first to notice this problem. But he was the first to lay it out with sufficient vigor that the need for a solution of some sort could not be avoided. Over the decades to follow, the problem lurked quietly in the corners of gravitational and cosmological theory, with proposals for its resolution reflecting whatever were the then current trends in physical theory. The simplest solution-possibly that of Newton himself-was just to deny that Seeliger's argument was valid. This untenable 'no-solution-needed' solution was in the minority of those views expressed in the historical record. The majority felt that the problem revealed that one or other of the. commitments of Newtonian cosmology required modification. These commitments can be collected into three groups: Cosmological. Space is infinite, Euclidean and filled with a (near) uniform mass distribution. Gravitational. All bodies attract one another according to Newton's inverse square law of gravitation. Kinematical. Newton's three laws of motion. The earliest escapes were sought in minute adjustments ofNewton's inverse square law of gravitational attraction. Seeliger and Neumann proposed augmenting the H. Goenner; J. Renn, J. Riller; T. Sauer (eds.) 271 The Expanding Worlds of General Relativity (Einstein Studies, volume 7) pp. 271-322 @1999 The Center for Einstein Studies 272 John Norton law with a tiny correction term whose effect would only become apparent at cosmic distances. More diverse escapes were also sought. Kelvin proposed that ethereal matter may not gravitate, allowing at least this form of matter to be spread uniformly through space. August Foppl suggested that there may be negative gravitational masses. Charlier, Selety and others proposed that the distribution of matter in the universe may have a hierarchical structure that allowed a vanishing mean matter density yet without concentrating all matter in a central island. Soon virtually every supposition buried in the 'cosmological' or 'gravitational' groups had been weighed and its modification proposed. Lense even explored the possibility of an escape through an alternative geometry for space. The problem achieved its moment of greatest glory when Einstein (1917a) thrust it into his first attempts at a relativistic cosmology. Agreeing with Seeliger, Einstein saw the problem as revealing a need for adjustment of Newton's inviolable law of gravitation. He used the adjustment as a foil to motivate the introduction of a cosmological term in the gravitational field equations of general relativity. However he also used the paradox to pose a dilemma for Newtonian cosmology: either the universe was homogeneous and gravitationally paradoxical or its matter was concentrated in a physically untenable island universe. Selety soon showed, however, that this was a false dilemma. The work on hierarchical cosmologies had already shown an escape between the horns of the dilemma. This work in the 1920s marked the end of the first phase of the problem posed by Seeliger; my purpose in this paper is to review the course of this first phase. 1 In a sequel I will review the later phase initiated by the discovery of the expansion of the universe and the advent of dynamical cosmologies. There it is found that the most satisfying escape from the problem lies not in modification of either 'cosmological' or 'gravitational' assumptions of Newtonian cosmology. Rather it lay in a modification of its kinematical core. The resolution depends on a hitherto obscured sense of relativity of acceleration in Newtonian cosmology and finds its fullest expression in the connection between gravitation and space-time curvature in Newtonian space-time theory. 1. The non-convergence of gravitational force in Newtonian cosmology In order to fix our topic, it will be helpful to give a brief and simple derivation of the problem that exercised Seeliger. In a Newtonian universe, the gravitational force exerted on a test body of unit mass is the resultant of the forces exerted by all the masses of the universe, which we shall assume to be distributed uniformly in space with mass density p. This force is computed by an integration over all these masses. This integral fails to converge. It can take on any value according to how we approach the limit of integration over all space. To see this lack of convergence, picture the uniform matter density p as distributed in concentric spherical shells of very small thickness !:!,.r all centered on 1 I gratefully acknowledge the assistance of the many before me who have mapped out various parts of the story in greater and lesser detail, especially Jaki (1979), Jammer (I 961: 127-29), North (1965: 16-23.180---185) Oppenheim (1920: 86-87). Sklar(l976) and Zenneck (1901: 51-53). The Cosmological Woes of Newtonian Gravitation Theory 273 A Figure 1. Non-convergence of force on a test mass in Newtonian cosmology. the unit test mass at O, as shown in Figure 1. Choose some arbitrary axis AA' and divide all the spherical shells into hemispherical shells by passing a plane B through O and perpendicular to AA'. Each hemispherical shell exerts a force on the test mass in direction A A' and its magnitude is independent of the radius r of the shell. To see the independence from r, consider how much matter in a shell at radius r is subtended by some small solid angle n at O. That amount of matter increases with r 2, but the gravitational force it exerts on the test mass decreases with l/r 2. So, overall, this force will be independent of r. This holds for each element in the shell, so we conclude that the net force exerted by the entire shell is a constant, independent of its radius r. The direct calculation reveals that the 274 John Norton value of the constant is G rr p t,r. 2 Thus the net force F on the test mass along an arbitrarily chosen axis AA' is given by an infinite series, each term representing the for:::e due to one hemispherical shell F = Grrp t,r - Grrp t,r + Grrp M - Grrp t,r + Grrp t,r - ... The series has alternating signs since shells on alternating sides exert a force in alternating direction. This alternating series is well known not to converge. Accord(ng to how one ~roups and reduces the tern:is in the series, the sum can have many different values. Each corresponds to a different way of approaching the limit of infinitely many masses in the associated integration. 2. Seeliger's formulations of the problem Seeliger's papers, especially his (1895a), contain the most detailed and general development of the problem. The price is that his exposition is the most cumbersome of all expositions; he alone resorts to infinite series expansions in Legendre polynomials and includes tidal forces in the analysis. Virtually all later commentators managed to reduce the exposition of the core difficulty to one or two lines of formulae. Seeliger initiated his discussion in his Seeliger 1895a (p. 129) by asking whether Newton's law of gravitation holds exactly for masses separated by "immeasurably great distances" [unermesslich grojJe Entfemung]. While observational astronomy gives the strongest reasons to believe the law within our planetary system, we have no similar foundation in experience for the law on the larger, cosmic scale. Nonetheless, he urged, the matter can be decided by applying the law to "simple and obvious examples" [einfachen und naheliegenden Beispielen] on the cosmic scale. It turns out that "thoroughly possible and conceivable assumptions lead to quite impossible or unthinkable conclusions" so that 2 The total mass in a ring with an angular position 0 and an angular thickness of dB in the shell is pr d0 2rr r sin 0 6.r. Therefore the total force exerted by the hemisphere at r is 1 2rcGpt>.r ,r/2 sin&cos0d&=Grcpt,.r. O 3 For example (Grcp 6.r - Grcp 6.r) + (Grcp 6.r - Gnp t,.r) + (Grcp 6.r - Grcp t,.r) + ... =0+0+0+. =0 and Grcp 6.r - (Grcp 6.r - Grcp 6.r) - (Grcp t:.r - Grcp 6.r) - ... = Grcp 6.r - 0 - 0 - ... = Grcp 6.r. The Cosmological Woes of Newtonian Gravitation Theory 275 Newton's law, applied to the immeasurably extended universe, leads to msupernble_ difficulties and irresolvable contradictions if one regards the matter d1stnbuted through the universe as infinitely great. 4 (Seeliger 1895a: 132) To arrive at these difficulties, Seeliger asked after the gravitational effect of the masses of the universe at a point O in space. To compute these effects, he laid out a spherical coordinate system (r, 0, y) centered on O. He represented the discontinuously distributed masses of the universe by an equivalent continuous distribution with density p. He found the gravitational effect of the masses between radial coordinate values r = Ro and r = R1 to be (la) !!1 1

0 a constant and n ::: 0. Different values for n lead to more or less rapid approaches to zero density as r becomes infinite. Table I shows how suitable selection of n can ensure convergence of virtually every quantity involved in the cosmology as the volume of space under consideration grows indefinitely. (See Appendix B for supporting calculations): Total Mass Mean Density Gravitational Potential Gravitational Force Tidal Force Zx n=0 n = I n=2 n = 3 n=4 Diverges for Converges for n~3 n> 3 K for Vanishes for n=0 n>0 Diverges for Converges for n~2 n>2 Diverges for Converges for n~I n>I Diverges for Converges for n=0 n > 0 Table 1. Convergence properties of quantities in hierarchic cosmology In particular, if we set a value of n for which 2 < n < 3, then we have a cosmology with • total matter of infinite mass • vanishing mean density • convergent gravitational potential, force and tidal force. Such a cosmology passes between the horns of Seeliger's dilemma in so far as it has infinite total mass but there is no need to forgo Newton's law since the relevant gravitational quantities are well defined. We see also that Kelvin's condition of vanishing mean density is not sufficiently restrictive, since this vanishing obtains when O < n < 2 for which not all the gravitational quantities are well defined. _50 More precisely, p(r) is the average matter density on the surface of a sphere of radius,. It is important not to confuse this density with the mean density of matter over the entire volume of the sphere. The latter mean density only cannot diminish faster that l/r 3-the case of single centrally located mass. 306 John Norton 9.4. How AN INFINITE WORLD MAY BE BUILT UP While it is easy to assume that cosmic matter dilutes with distance according to a rule such as (13), it is another matter to show that such dilution can be achieved within a matter distribution that is, in some significant sense, homogeneous. Modern work on this problem was initiated by an unlikely source. Fournier d' Albe 1907 is a somewhat inflated, semi-popular work intended to develop the notion that the world of the scale of our solar system is but one of an infinite hierarchy of worlds extending into the small and large. Proceeding into the small, the next level down is the atomic level; the atoms are the sun and the electrons the planets (pp. 84-85). Proceeding into the large, the units of the next level are stars clustered into galaxies. Fournier d' Albe 's essential purpose was to elaborate this grandiose vision through a seamless fusion of simple technical results and wild speculation. However, the hierarchic structure of his world just happened to allow him to escape two problems of an infinite cosmology. The first was Olbers' paradox of the darkness of the night sky (see below). The second was Kelvin's concern that stellar matter be not so densely distributed that stars falling into our system acquire too great a velocity. To escape both difficulties, Fournier d' Albe assumed his systems so distributed that the matter in a sphere of cosmic size increases only in direct proportion to the radius of the sphere (Part II, chap. II). This rate of dilution corresponds with the setting of n = 2 in ( 13). It escapes Olbcrs' paradox, but Fournier d' Albe apparently did not notice that it only just fails to ensure the convergent potentials needed to escape high stellar velocities. The latter requires any n > 2. The real value of Fournier d' Albe's work lay in the fact that Carl Charlicr read the work and, as he tells us in the opening paragraph of Charlier 1908, saw in the hierarchic proposal a way to escape the problems of an infinite cosmology. Charlier gave the escape a mathematically precise form in his (1908) "Wie ein unendliche Welt aufgebaut sein kann." The essential content of (1908) was repeated, corrected and extended in a different article, Charlicr 1922, but with the same title translated into English, " How an Infinite World May Be Built Up." His model proposed a hierarchy of larger galaxies and galaxy clusters, systems S1, S2 , ... such that N1 stars form a galaxy S1, N2 galaxies of type S1 form a second order galaxy S2, N3 galaxies of type S2 form a third order galaxy S3, etc. The systems S,, S2, ... arc presumed spherical with radii R1, R2, ... and have masses M,, M2, .... As galaxies of the (i - 1)th order are packed to form a galaxy of the ith order, empty space must be introduced in sufficient amount to ensure the dilution of the mean matter density. This is ensured by locating each galaxy of (i - 1)th order and radius R;_ 1 in a sphere of otherwise empty space of radius p; within a galaxy of ith order, as shown in Figure 5. Thus the mean density of matter will decrease as we proceed up the hierarchy if Ri-l / Pi < 1, for each i, since the ratio of mean density in systems of order (i - I) and i is (Ri- 1/ Pi )3. Charlier identified three problems facing an infinite cosmology. Each could be The Cosmological Woes of Newtonian Gravitation Theory 307 (i-1)th order galaxy ith order galaxy Figure 5. Charlier's hierarchic universe. (i+ 1)th order galaxy solved by requiring that the mean matter density dilute with increasing volumes of space at a suitable rate. Thus each problem could be restated as what he called a criterion that could be translated into a specific rate of dilution. The three criteria are: Olbers' Criterion. In 1826, Olbers had remarked that, if the universe were filled uniformly with stars, then the sky should glow as brightly as the face of the sun.51 To see how this arises, in a form relevant to Charlier's work, note that the intensity of light from distant stars diminishes with the inverse square of distance. But if stars are distributed uniformly in space, then the number of stars increases with the square of distance. Thus the total light incident from such stars on the earth is represented by a diverging integral. The intensity of starlight in the sky of such a universe would be infinite. 52 This intensity should be finite. Seeliger's Criterion. The gravitational force in a universe with a uniform matter distribution diverges, as Seeliger pointed out. This force should be finite. Velocity Criterion. As celestial objects fall into the cosmic gravitational field, the depth of the potential well determines how great these velocities are. In an infinite universe with a uniform matter distribution, this well is infinitely deep. Since Charlier knew only 51 The history of the paradox is a great deal richer than this simple remark suggests. See Jaki 1969. 52 If the intensity oflight due to a star at distance r is A/ r2 and the uniform density of stars is p. then the intensity of light due to stars at distance r > Ro is JR: (A/r 2)p4rr r2 dr = oo. The occulting of more distant stars by nearer stars reduces the intensity expected to that of the surface of the sun. 308 John Norton of small velocities observed among celestial objects, he required that this well not be so deep as to yield stellar velocities beyond the modest ranges then known. In effect this criterion calls for the convergence of the gravitational potential. We need not recapitulate Charlier's fairly complicated analysis since the essential results are already implicit in Table 1. Instead we can recover his results rapidly once we have connected the quantities of Charlier's model with the dilution rates of ( I3). To do this, when r = R;, we will approximate the density p (r) of (13) by the mean density 75; of the ith order galaxy. If the mass of a star is Mo, then the total mass of an ith order galaxy is M; = MoN 1N2 • • • N; and its mean density is MoN1N2·••N; P; = (4/3)7!' R; 3 It follows that the mean densities of (i + I )th and ith order galaxies are related by The Cosmological Woes of Newtonian Gravitation Theory 309 requiring that (13') with n = 1 set an upper limit for N;+i. Written after the form of Charlier 1922: 4, 6, the condition is Unfortunately Chartier did not see the equivalence of the two conditions in 1908. He gave the correct result for Seeliger's criterion ( 1908: 9). But he gave a result for the Olbers' criterion (1908: 8-9) which he later described as "inexact" ( 1922: 5) and then published a corrected analysis supplied to him by Seeliger in a letter of 28 April 1909. He also reported (1922: 5) that Franz Selety had informed him through a letter of the correct results, apparently at the time of publication of Charlier I922. Selety (1922: 299-300) recounts Selety's corrected analysis. The velocity criterion can be treated similarly. From Table I we see that it amounts to the stronger requirement that n > 2. Thus, within Charlier's model, the criterion amounts to setting (13') with n = I as an upper limit for N;+i: But we require that 75; dilute with distance as I/ rn. That is, that = P.'.._+I (_!!}_)n P; R;+1 Comparing the last two equations, it follows that the requirement that the mean density dilute as 1/ rn is equivalent to requiring that the number of ith order galaxies in an (i + I)th order galaxy be set b/3 R ; = R;+1 )J-n N;+1 ( (13') Without calculation, we can see that both Olbers' and Seeliger's criterion amount to the same constraint. For Seeliger's criterion amounts to requiring that matter dilute sufficiently rapidly so that the gravitational lines of force emitted by cosmic matter not build to infinite density. These lines of force dilute with the inverse square of distance. Light from distant stars also dilutes with the inverse square of distance. Therefore any star distribution that satisfies Seeliger's criterion will satisfy Olbers' criterion and vice versa. We know from Table 1 that gravitational force converges whenever we set n > 1. Thus the two criteria are satisfied by 53 It follows from ( 13') that Charlier's model admits a maximum n of 3. This limit arises since the density p(r) of (13), the density at the surface at radial distance, .. is being ap~roxi_mated by a mean density over the whole volume of the various orders of galaxies. Tots mean density dilutes at its fastest when the entire universe has just one star, which is the matter of every order galaxy. It 1s the case of n = 3. Charlier's (1922: 14-15) version of this result replaces the strict inequality> by an inequality~. apparently in error. Selety (1922: 300-301) again points out the error and in an addendum to proofs (1922: 322-324) criticizes Charlier's revised 1922 treatment. 9.5. THE E!NSTEIN-SELETY DEBATE The hierarchic cosmologies had already passed squarely between the horns of the dilemma Einstein had presented in his famous cosmology paper ( 1917a). Someone had to respond. So Franz Selety (1922) took on the task of dismantling Einstein's dichotomy and a great deal more. The paper is fairly long-winded and covers more material than can be reviewed here. However it does lay out quite clearly the options provided by Charlier's hierarchic model. As Selety summarized in his Introduction, this model allowed a Newtonian cosmology to satisfy all of 1. Spatial infinity of the universe, 2. Infinity of the total quantity of mass, 3. Complete filling of space with matter of overall finite local density. 4. Vanishing of the average density of the universe. 5. Non-existence of a singular center point or central region of the uni- verse.s4 (Selety 1922: 281) In addition, of course, he recalled how a suitable rate of dilution of matter with distance would escape Olbers' paradox and the divergence of gravitational force 54 "I. Riiumliche Unendlichkeit der Welt, 2. Unendlichkeit der Massengro6e, 3. Vollkommene Erftillung des Raumes mil Materie von tiberall endlicher lokaler Dichte. 4. Verschwinden der mittleren Dichte der Welt. 5. Nichtexistenz eines singuliiren Mittelpunktes oder Mittelgebietes der Welt." 310 John Norton noted by Seeliger. Finally Selety expended considerable effort to urge that the hierarchic universe provides a way of satisfying the Machian requirements that had influenced Einstein so profoundly in his work on general relativity. However Selety differed from the standard view of which divergences must be avoided in Newtonian cosmology. He recognized that diverging gravitational potentials could be avoided by allowing the matter density of (13) to dilute with distance faster than I/ r 2 (p. 287). However he was willing to entertain the divergence of the gravitational potential because he found the case of the 1/ r 2 dilution to have especially interesting properties. In particular, in this case, the mass M; of an ith order galaxy would be proportional to its radius R;. The (i - 1)th order galaxies within it would be in gravitational free fall and the order of magnitude of their velocities would be determined by the depth of the gravitational potential well of the ith order galaxy. The depth of that well was fixed by M; / R;. So if M; was proportional to R; that depth would be same at all levels. Thus Selety could conclude something very pretty about the velocity of any galaxy with respect to the next order galaxy that contains it: at all levels of the hierarchy, they have the same order of magnitude. Selety had arrived at a similarity between the structures of each order in the hierarchy. As one moved up the hierarchy the lengths, times and masses of processes would preserve their ratios. Selety was sufficiently impressed with this result to want to suggest a "relativity of magnitude" (Relativitiit der Crofte) in the hierarchic universe in which matter density diminishes as 1/ r 2 (p. 304).55 This relativity of magnitude allowed Selety to address a thorny problem of the hierarchic universe. Einstein had observed that a finite cluster of stars in empty space would scatter. Might not the same fate befall a hierarchic universe? Each galaxy is a region of greater density within a region of lesser density.56 Selety had to concede that it is of infinitely small probability that normal dynamical processes could bring about a hierarchic structure. But, once it was in existence and if it diluted its matter density as l / r 2, then this relativity of magnitude would ensure that it could not lose its hierarchic structure in any finite time. The reasoning was simple. Imagine some process through which a galaxy of some order is destroyed. That same process could also befall higher order galaxies. However the time required for the process to be completed would grow in direct proportion to the size of the galaxies. There is no upper limit to the orders of the galaxies and no upper limit to their size. Therefore, for any such process of destruction and any nominated finite time, there is some order in the hierarchy beyond which it has not had an effect by that time. 57 55 Selety did not give details of the derivation of this necessarily vague result. Presumably we must imagine that the velocity of an (i - I )th order galaxy is given as the ratio of some characteristic length and time. If the velocity is to remain roughly constant as we proceed up the hierarchy, these lengths and times must preserve their ratio. The masses of systems would also grow in direct proportion to the lengths and times since the total mass of a galaxy grows, by supposition, in proportion to its radius, as we move up the hierarchy. 56 Arrhenius (1909: 227) worried about exactly such instabilities, including also the possibility of eventual collapse of the galaxies under their own gravity. 57 I cannot resist observing the unhappy corollary. Since Newton's theory of gravitation is time The Cosmological Woes of Newtonian Gravitation Theory 311 The case of matter diluting as l/r 2 had further special importance. For it enabled Selety to mount a challenge to a major part of Einstein's 1917 argument. There, as we have seen in Section 5 above, Einstein (1917a: § 1) had argued against the possibility of the stars clustering in an island in an otherwise empty universe. If we assumed that the gravitational potential at spatial infinity is finite, then by statistical mechanical arguments, that island would evaporate. Each star would, over time, through random processes acquire sufficient energy to escape the island. Boltzmann's analysis of a kinetic gas in a gravitational field also yielded a relationship between the equilibrium densities p and Po of an isothermal gas in regions, where the potentials are

3. The mean density 75 for this shell is likewise 3K (R3-n - Ro3-n) _ M P = (4rr/3)(R3 - (3 - n) (R 3 - R0 3) = Ro 3) 3Klog(R/R0 ) 1 (R3 - Ro3) for n =f. 3, for n = 3. Hence 75 ---+ 0 just in case n > 0. The convergence of the potential 2. For the case of for n =f. 1, } Ro } Ro R Klog- Ro for n = 1. - Hence the force Fx converges as R ---+ oo just in case n > 1. For the case of the tidal force Zx we have K (R-n - Ro-") forn =/- 0, ! = = 1R p ,-1 dr 1R K,-n-1 dr Ro Ro n K log .!!._ Ro for n = 0. Hence the tidal force Zx converges as R ---* oo just in case n > 0. The Cosmological Woes of Newtonian Gravitation Theory 321 REFERENCES ARRHENIUS, Svante. (1909). "Die Unendlichkeit der Welt." Rivista di Scienza 5: 217-229. 'BACH, Rudolph' [Rudolf FORSTER]. (1918). "Die Anziehung eines unendlichen S1emsystems", Astronomische Nachrichten 206 (Nr. 4939): 165-172. BENlLEY, Richard. (1756). "Four Letters from Sir Issac Newton to Doctor Bentley: Containing Some Arguments in Proof of a Deity." 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