## GUNNAR NORDSTRÖM (1881-1923)

## ON GRAVITATION AND RELATIVITY

Observatory

University of Helsinki

Presented at the XVIIth International Congress of History of Science

University of California, Berkeley, July 31 - August 8, 1985.Gunnar Nordström

Gunnar Nordström was certainly not the only physicist who made an attempt to formulate special relativistic theory of gravitation. He is distinguished by having been the first, and also the most successful at an attempt that was doomed to fail.

In 1916, when the only physical argument in favour of general relativity was the explanation of the perihelion advance of the planet Mercury, Nordström gave up his own, then still current theory in favour of Einstein's. His seemingly abrupt change of mind is a result of ten years of thinking in terms of Minkowskiian electrodynamics, and getting no further with it.

Gunnar Nordström was a contemporary of Albert Einstein. Both men took a circuitous path to university careers. During Einstein's early years at the Berne patent office, Nordström became an engineer. Never a very practically oriented man, he turned into studying physical chemistry. He arrived in Göttingen in April 1906 for about one year to study chemistry under Walther Nernst. In Göttingen, the young Nordström became a wholehearted believer in relativity in its Minkowskiian formulation. After having published only one paper in chemistry, Nordström's whole remaining published work was focused almost exclusively in issues of relativity, electrodynamics and gravitation.

Nordström's first major work was his dissertation on the energy equation for the electromagnetic field of moving bodies. It was presented in 1908 at the University of Helsinki, This work is an exposition of the Lorentz theory of electromagnetic phenomena as it was formulated by Max Abraham in his

Theory of Electricity. The Minkowskiian electrodynamics, along with the theories of Maxwell and Hertz and the theory of Cohn were discussed at a separate chapter, with no reference yet to Einstein.Nordström's role in Helsinki was that of an introducer of novel physical ideas. He established himself as a "Dozent" of theoretical physics at the university, at the same time teaching elementary physics to gym students. Between years 1916 and 1918, Nordström worked in Leiden, Holland. In 1918 he became a professor first of physics, then of mechanics at the Helsinki University of Technology. During his whole career, his Scientific colleagues were Central European, for no tradition of theoretical physics existed in Helsinki prior to Nordström. The Finnish intellectual climate of the 1910s is reflected in one academicians retort to a request for travel funds by Nordström: "One can study the fourth dimension at home, without any trips abroad." Meanwhile, Nordström had already tried to introduce a fifth dimension to the fabric of space-time in one of the very first attempts at a unified theory of electromagnetism and gravitation.

In 1909 Nordström published a lengthy semi-popular article on relativity for the Finnish Scientific audience: "Space and Time According to Einstein and Minkowski". There he at once recognizes the basic meaning of Einstein's formulation of the principle of relativity. He also states that Minkowski s "world postulate" is essentially identical with Einstein's principle of relativity, although there exists a formal difference. By now, Nordström had become fascinated by the Minkowskiian vision. He claimed that Einstein's version was designed for optical phenomena in ether, while Minkowski's version was generally valid for mechanics as well as for electromagnetic phenomena. He further commented that the Minkowskiian treatment for swiftly moving bodies differed from that by Lorentz. Nordström's argument seems to have been based in considerations of the principle of conservation of energy and basic concepts connected with it.

Having became an ardent supporter of Minkowskiian relativity, Nordström went on to argue in an article published in

Physikalische Zeitschriftin 1909 that the expression of electromagnetic ponderomotive force by Max Abraham was in contradiction with conservation of energy. Abraham has given a finishing touch to the structure of classical electrodynamics. He was also one of the very last eminent anti-relativists. Abraham was certainly a fair critic. He was able to point out inconsistencies in relativists' work and to straighten them out while advocating his own scepticismm. In his reply to Nordström, Abraham convinced his opponent by using strictly relativistic arguments. The critical issue was the time dependence of mass in motion and the consequent, interpretation of the energy equation from the point of view of relativistic thermodynamics. Abraham's paper contains one of the first satisfactory formulations of the energy-momentum tensor first conceptualized by Max Planck. Nordström closed the discussion in 1911 by adopting Abraham's version of the ponderomotive force.Abraham was soon to engage in another debate. In 1911, Einstein began to publish on gravitation after a four-year-long silence. His early papers on scalar gravitation with a variable velocity of light inspired Abraham both to attack relativity and to believe that even Einstein himself had now abandoned his own basic principles. Abraham designed a scalar theory of gravitation which was

neither relativistic nor consistent. Einstein abandoned scalar theories altogether and began his long struggle with tensor theory.Nordström's response to both was to suggest a scalar theory of gravitation in a rigid relativistic sense of gravitation in a rigid relativistic sense. Nordström required the constancy of the velocity of light in all frames of reference. Adopting Abraham's modification of the Poisson equation, Nordström arrived at a potential dependence of gravitational mass. This first version of the Nordström theory was sent to

Physikalische Zeitschriftin October 1912. The theory was notably imperfect in its definition of the source of the gravitational field, which had to be later defined as the trace ofthe energy-momentum tensor as pointed out by Max von Laue in hisRelativitätsprinzipof 1911. Nordström made this important modification in his theory in summer 1913 while staying in Zurich in collaboration with Einstein. In September 1913 Einstein gave a progress report on his work to the German Naturforscherversammlung, summarizing Nordström's theory at a great length alongside with his own tensor theory.The meaning of Nordström's scalar theory of gravitation is most clearly seen from its covariant formulation which was made in 1914 by Einstein and A. D. Fokker. This formulation was in effect a by-product of Einstein's own theory. It is in the footnote of this very paper where Einstein raises his former objection to the existence of generally covariant field equations of gravitation.

In terms of the formulation given by Einstein and Fokker, the Nordström theory can now be expressed in terms of a conformally flat line element:

with

The equations of motion can be derived from this line element with a variational principle

The field equations are found by equating the curvature scalar R with the trace of the energy-momentum tensor, multiplied by a constant:

This was the first time when the complex tensor machinery gave Einstein generally covariant equations. With this in mind he surely gained additional confidence for continuing his own chosen path which led to the final formulation of the field equations of the general theory of relativity in November 1915. The form of these equations is strikingly analogous with the scalar field equation:

Nordström did not at once see the advantage of this approach. He proceeded by trying to unify the known physical theories by treating the four-dimensional space-time as a projection of a five-dimensional continuum. in practice, this attempt to rewrite the field equations of electromagnetism and of gravitation in a mathematically analogous way by introducing a fifth dimension remained only a mathematical trick. Nordström concluded that the predictions of this new theory were either wrong or irrelevant. The theory is, however, interesting as an early scalar version of the ten years later Kaluza-Klein theory.

In 1917 Max von Laue published a comprehensive exposition of the Nordström theory. As late as over a year after the completion of general relativity, a scientist of Laue's standing could still consider the Nordström theory as a serious rival to the Einstein theory. Of course, the confirmation of the bending of light rays during the solar eclipse of 1919 changed all that. Nordström's theory predicted no deflection at all, because in his theory electromagnetism is not coupled to gravitational phenomena because the trace of the electromagnetic energy-momentum tensor vanishes.

Nordström himself had already abandoned his own theory in 1916, despite the fact that the question of bending of light still remained open, and despite the fact that Nordström had never been very worried about the other serious flaw in the predictions of scalar gravitation, namely the perihelion shift of the planet Mercury, which was of the wrong magnitude and even in the wrong direction.

A major factor in Nordström's conversion to the general theory of relativity was his growing consciousness of the importance of Hamilton's Principle for physics. After arriving in Leiden in 1916, Nordström entered a first-rate community of theoretical physicists. He soon began working with Einstein's new ideas on the Hamiltonian formulation of general relativity. Nordström first considered the case of mechanics of continua, studied by Gustav Herglotz for special relativistic cases, treating it first from the point of view of his own theory, then from that of Einstein's This work led Nordström to study the general problem of a finite matter source with an electric charge, which had already been considered by Hans Reissner and by Hermann Weyl for a point source. This significant contribution by Nordström can be found in modern textbooks of general relativity as the Reissner-Nordström metric for a nonrotating charge distribution.

After 1918, Gunnar Nordström disappeared from the international scientific arena. Instead of entering the competition for an assistant professorship in Berlin under Max Planck, he chose to return to Helsinki with his new Dutch wife, Cornelia van Leeuwen, a physics student of Lorentz's. Einstein and Max Born were helpful to the young couple when visas to Finland were urgently needed. Another physicist who provided help to Nordström was Niels Bohr who forwarded his mail during the war years by simply changing the envelopes.

Once in Helsinki, Nordström continued his work until his early death at an early age of 42 in 1923. Experimenting with radioactive substances had been one of his hobbies and very likely a cause of his illness. As a professor at Helsinki University of Technology, Nordström twice nominated Einstein for the Nobel prize in physics for his special and general theories of relativity. He also taught a course of general relativity to Finnish students, which inspired later scientists to continue the research begun by Nordström.

Nordström's scalar theory of gravitation was rediscovered in the 1960's when Robert Dicke, A. L. Harvey and others used scalar models in discussing basic concepts connected with theories of gravitation. Despite the fact that Nordström was wrong about gravitation, his theory still serves as a stepping stone to understanding Einstein.