Weyl or Lemaitre - Who Gets Credit for Predicting the Hubble Law?

Background

Recently, Lemaitre has been lauded for having derived a theoretical prediction of the Hubble relation (linear relation between velocity and distance) and then analyzing the available data to compute the slope of that relation - the Hubble Constant - two years before Hubble. It is claimed that he failed to receive credit on account of the original paper being written in French, while the English translation that appeared four years later omitted the derivation of the Hubble constant (although the relation itself was fully presented.)

Is there ayone else besides Lemaitre who has also been overlooked? The answer may well be yes - Hermann Weyl. In an appendix to the fifth edition of his book Raum, Zeit, Materie (1923) Weyl presented an equation that, to first order, is the Hubble relation. He also made a crude estimate of the Hubble radius. In a parallel paper (1923, Phys. Zeitschr, 24, 230) he presented the full derivation of the Hubble relation, starting with an expanding model of the universe. In other words, Weyl made essentially the same calculations as Lemaitre, and he did so four years earlier. Shouldn't he receive credit?

Arguably the answer is yes, particularly since Lemaitre actually cited Weyl's paper in both the French and English versions of his paper and thus was aware of Weyl's derivation. Robertson (1933 Rev. Mod. Phys., 5, 62, p. 88) certainly gave him credit (Bibliography): "1923 H. Weyl ... Introduction of postulate concerning coherency of matter and deduction of linear velocity-distance relationship." Robertson also referenced Lemaitre's 1927 paper but made no mention of any contribution by him to the v-D relationship.

Regardless, it is interesting to compare the details of what Weyl and Lemaitre actually did.

Weyl

Weyl was analyzing a de Sitter universe, one that has no matter, just a cosmological constant (or, as we would say today, pure dark energy). Because dark energy has no preferred rest frame, a de Sitter universe by itself has too much flexibility to make a definitive prediction about what one would observe regarding redshifts v. distance, even if one inserts galaxies as massless test particles. Weyl made the somewhat bold conjecture that the observable universe should consist of a subset of all possible geodesics selected so that they emanate from a common point in the past. Such a universe is basically a big bang universe of test particles embedded in de Sitter space. Weyl indicated that he was driven to it by considerations of homogeneity of space and time and by considerations of causality. He then derived an equation relating redshift and distance which, to first order in distance, is simply the Hubble law. He then implied that this expansion is related to the observed redshifts of spiral galaxies and pointed explicitly to Eddington's table of velocities (1923, Math. Th. Rel.) as compiled by Slipher (p. 162).

In his book, Weyl made a crude calculation of the Hubble radius, which, for de Sitter space, is the same as the 4-d radius of curvature of that space. He used Lundmark's estimated distance to Andromeda (1 to 10 million light years), assigned a velocity of 1000 km/s (a typical value measured by Slipher), and combined them to get a radius of 10^9 light years. If we convert that to a Hubble constant, we get 978 km/s/Mpc. (Weyl next computed the ratio of the Hubble radius to the electron radius and thus entered the world of the Large Number Hypothesis.) Weyl's estimate is obviously of limited value - while the distance to M31 was quite reasonable, the velocity he used was an artificial assignment based on other galaxies. However, we have to remember the state of the field at this time. Distances to the spiral nebulae were still unknown in general. Hubble's Cepheid distance work was still in the future. Lundmark's distance was based on novae and Opik's method, both of which were considered speculative at the time.

Lemaitre

Now move forward four years to Lemaitre (1927, Annales de la Societe Scientifique de Bruxelles, A47, 49). Lemaitre's cosmological model was considerably more advanced than that of Weyl. Lemaitre constructed a model that incorporated both matter and a cosmological constant (i.e., dark energy); he also showed how to include radiation, although he did not have any data to constrain it. Lemaitre also rederived Friedmann's dynamical equations governing the expanding universe, implicitly showing that the end-point was a de Sitter model with the galaxies acting as test particles; this end state corresponded to the model constructed by Weyl. Following further in the footsteps of Weyl, he derived the Hubble law, this time for an evolving universe, and he also computed the Hubble constant from the available data. Finally, he pinned down all the free parameters of his model with observational data.

How does Lemaitre's work compare with Weyl's? On the theory side, Lemaitre took a big step forward by incorporating matter along with the cosmological constant. As far as the Hubble law, however, a linear relation between distance and velocity is a generic feature of any model of a homogeneous, expanding universe, so in that respect, Lemaitre added little to what Weyl had already shown. The Hubble constant itself now depended on two parameters of the theory (density of matter and lambda) instead of one, but because dark energy was dominant, Lemaitre gave an approximate equation for the Hubble constant that was identical to Weyl's.

On the observation side, Lemaitre also took a big step forward, but that is because the field itself had advanced and nearly caught up to the theory. In particular, Hubble's 1926 paper (ApJ., 64, 321), with apparent magnitudes and distance calibration, had been published. (Lemaitre also made use of Stromberg's 1925 paper, ApJ., 61, 353, which compiled a list of positions and velocities of galaxies, although nearly the same information was already available in Eddington's book. Lundmark's 1924, MNRAS, 84, 747 and 1925, MNRAS, 85, 865 papers deserve mention as well.) Hubble's paper, in particular, was crucial to Lemaitre. First, Hubble provided the distances to 7 local group galaxies; Weyl only had a distance to M31. Second, for more distant galaxies, in which Cepheids could not be detected, Hubble gave a tentative equation to convert apparent magnitude to distance (assuming a galaxy can be treated as a standard candle.) At best, Weyl could have used Andromeda alone as a standard candle (which is what Lundmark did, or at least, tried to do, in 1924.) Third, Hubble provided a convenient table where he assembled data for a large number of galaxies, including apparent magnitudes. (Lundmark was the first to rely on the works of Holetschek and Hopmann for magnitudes, which Hubble used as well, but Lundmark did not publish the actual magnitudes that he used.) Fourth, Hubble estimated the mean density of matter in the universe, and Lemaitre used it as one of two observational quantities needed to constrain the free parameters of his model (the other being the Hubble constant itself.) By combining the distances and magnitude data from Hubble's paper with the velocity and position data (the latter being needed for the vertex motion calculation) compiled by Stromberg, Lemaitre's task was greatly simplified.

Summary

Singling out Lemaitre's work on the Hubble law and Hubble constant for recognition really doesn't properly capture the scope of what he had accomplished, and arguably it assigns credit to the wrong person. The Hubble law was first derived by Weyl, and the Hubble constant calculation of Lemaitre, while superior to that of Weyl, depended almost entirely on the intervening work of ... Hubble. In fact, the desire to co-name the Hubble law after Lemaitre seems to stem from his one true contribution: he was the first to express the Hubble constant using the units "km/s/Mpc."

The real accomplishment of Lemaitre was that of taking disparate pieces of work - even if derived previously - and assembling them into a complete, self-consistent, evolving model of the universe that, for the first time, was matched to observations of both the density of matter and the rate of expansion. Further, unlike Weyl (1930, Phil. Mag., 9, 936), who considered his explanation of Hubble's "strange phenomenon" to be "speculative", Lemaitre was unequivocal in asserting that an expanding universe was the correct explanation.