John Wheeler was the inventor and promoter of some of physics’s most iconic terms and phrases, such as black hole, wormhole, or “it from bit”. He was a world-leading nuclear physicist who provided the first full theoretical analysis of nuclear fission together with Niels Bohr, and after World War II, he became the leader of the renaissance of general relativity.
To those who knew him, he was also famous for his notebooks. His students and collaborators frequently mention Wheeler’s notebooks in their recollections of him. Daniel Holz (Senior Thesis, 1992) remembers him “[h]unched over his notebook”. Edwin Taylor (who co-authored the book Spacetime Physics with Wheeler) remembered how “[o]ut would come the bound notebook” whenever they were working on a hard problem.
Charles Misner (Ph.D., 1957) recalled: “John did have this habit for I guess, all of his life of having bound notebooks. I looked at one at a bookstore the other day; it costs $90 to get one of these books — bound books of blank pages — but John had very nice bound books. They were always there. When he had a group of students in the office he would sit down and take notes as the discussion went on. He would also make notes to himself about the calculations he was doing, or the work he planned to do. What were the important questions in physics? and so forth.”
To give the reader an idea of the content of these notebooks, we will briefly look at the elaboration of one of Wheeler’s most famous concepts: the wormhole. In the mid-1950s, Wheeler was pursuing a program he called “daring conservatism”. It was based on the idea that, in general relativity, electrodynamics, and quantum theory, we already have of the necessary components to construct a final theory of physics, a theory of everything.
This was an explicit move away from the mainstream of high energy physics, where it was generally believed that a successful description the manifold new particles being discovered at accelerators (what would become known as the “particle zoo”) would require, at the very least, a new theory of nuclear forces. Wheeler, in contrast, wanted to explore whether the established theories of the electromagnetic and gravitational forces alone might not already provide the necessary building blocks to reconstruct, at least in principle, the newly discovered particles and the physics of the nucleus — from scratch, as it were.
With a quantum theory of gravity still unavailable, Wheeler began to try his hand by working only with gravitation and electromagnetism. The idea was to use the non-linearity of the Einstein-Maxwell equations to construct localized, particle-like solutions, i.e., to construct particles from pure field configurations. This idea harkened back to Einstein’s unified field theory, by which Wheeler was certainly inspired.
In contrast to Einstein, Wheeler did not seek to combine electrodynamics and gravitation into a new, unified mathematical entity; he was happy to work with Einstein-Maxwell theory in a “conservative” manner, leaving the theory as he found it. Another difference to Einstein’s program was that Wheeler ultimately anticipated the construction of a quantum theory. This made the construction of particles from fields more of a proof of principle, rather than an actual physical hypothesis; any discrepancy (especially quantitative ones) between the particles constructed in this way and the particles actually observed in experiment could be interpreted as arising from the neglect of quantum effects.
The first particle-like entity that Wheeler constructed in this manner was the geon (gravitational-electromagnetic entity), an (approximately) localized and stable solution to the Einstein-Maxwell equations, whose condensed energy could be interpreted as the mass of a particle; mass without mass, in Wheeler’s words. But these particles, solutions of the source-free field equations, did not carry any charge. In a long paper on geons, Wheeler also gave a brief sketch on how one might obtain charge without charge: by having electric field lines apparently emerge from a point in space (like a charge), when in fact they continued through a tunnel in a space-time that is not simply connected and re-emerged at some other, distant point in space, simulating a charge of opposite sign.
The idea of a tunnel connecting two distant points in space will be familiar to the contemporary reader. But while such wormholes are nowadays known primarily as physics-inspired portals for fast intergalactic travel in sci-fi movies and novels, they originated in Wheeler’s thinking as potential models of elementary charged particles. The idea of using spatiotemporal tunnels as particle models was not entirely new: Albert Einstein and Nathan Rosen had proposed a similar idea some 20 years earlier but their “bridges” had connected two different universes (rather than two distant points in the same one) and had not contained electromagnetic field lines.
In the geon paper, Wheeler had not fleshed out this model, nor had he given it a name. But one can track the further development of the idea through Wheeler’s notebooks. In Wheeler’s Relativity Notebook III, on page 63, we find an entry dated Saturday, 11 September (1954), shortly after completion of the geon paper. The entry is located on the “Ile de France”, the ship on which Wheeler was traveling back from the United States after a visit to Europe. The peace and quiet of the transatlantic travel clearly provided Wheeler with ample time to think. And what makes Wheeler’s notebooks such an intriguing historical resource is that he really thought with and through his notebooks, penning down his reflections in flowing (and perfectly legible) prose.
We can thus observe him reflecting again on his attempts to construct particles from classical general relativity and electrodynamics: “Recall picture of charge as connected with multiply-connected space-time.” We can see him aware of the difficulties involved in trying to do things at a purely classical level at first: “Could also fear that tunnels can only sensibly be treated on q. mech. basis.” And we see him brushing these worries away: “In spite of misgivings, make try”.
The page also contains the first drawing by Wheeler of a wormhole, which he was, however, still calling a “woodchuck hole” at the time. But Wheeler was not yet trying to construct a woodchuck hole as a solution of the Einstein-Maxwell equations; instead he was trying to integrate them into cosmology. This was characteristic of Wheeler’s approach, which very often tried to connect the macro- and the microphysical. In this case it meant trying to find a “relation between general curvature of space and number of woodchuck holes and `intensity’ of each hole”, i.e., between the size of the universe, the number of charged particles, and the elementary charge e.
Wheeler tried to find such a relation by drawing upon topological invariants (the number of woodchuck holes was connected to the Betti number) and the so-called large number relations first popularized by Arthur Eddington and Pauli Dirac, e.g., the empirical observation that the square root of the number of particles in the observable universe is of the same order of magnitude as the ratio between electrostatic and gravitational force in the hydrogen atom, namely about 1040.
This was a strange brew, and it is hardly surprising that Wheeler did not succeed. But while the ideas that Wheeler was trying out in his notebooks often seem crazy — crazier even than the ones he published — one sees here quite impressively Wheeler’s ability to connect apparently unrelated fields of knowledge. And while these connections often did not produce immediate results, they often proved to be very fruitful in the long run, as was the case for the connection between mathematical topology and field theory that Wheeler was pioneering here.
Wheeler was always pursuing many different, and often even contradictory, ideas in parallel. So one has to page a bit further for the woodchuck holes to resurface again. On page 113, we find another brief woodchuck hole calculation, which ends, however, just as inconclusively as the first one. A few pages on, we see a first mathematical breakthrough and it comes from a source, whose importance to Wheeler’s thinking is highlighted by the notebooks time and again: his students.
Wheeler was famous for his mentoring activity and he was also a master at making use of his students’ special abilities. Glued in between page 116 and 117, we find a typescript manuscript (dated 13 December 1954) by Wheeler’s PhD student Charles Misner, an expert in modern mathematics. It is a mathematical study of the simplest possible wormhole, a wormhole without a universe attached to it, that is — this wormhole closes in on itself and is really just a toroidal universe with electromagnetic field lines circling around it. Misner attempted to establish whether such a Wormhole could be a solution of the Einstein-Maxwell equations.
This calculation remained inconclusive (like many in Wheeler’s notebooks). But more inspirational than Misner’s concrete calculations on this toy wormhole were the new mathematical tools that he introduced to Wheeler. After another long lull, we find another insert on p. 209, apparently typed by Wheeler. It is dated 13 November 1955 and is entitled: “Questions incited by Charles Misner’s nice way (based on Cartan) of putting the mathematics of general relativity”, a reference to the formulation of general relativity in terms of differential forms. The list of questions also includes one on finding conserved fluxes through a tunnel, and this question also contains the first recorded use of the term “worm hole”, though still written as two words.
This renaming coincides with a great intensification of Wheeler’s work on wormholes in the notebooks, culminating in a joint paper with Misner. Soon we see Wheeler worrying about more detailed wormhole questions, such as travel through them; though as a problem, not as an opportunity. On 11 Februrary 1956 (page 246), he noted an issue his student Peter Putnam had alerted him to:
“[M]omentum flowing into one worm hole may come out another one an entirely different direction. Apparent violation of law of conservation of linear momentum, not to mention angular momentum. But field eq[uatio]ns that govern everything permit no violation. Therefore we know that there must be something like a recoil of the walls or generation of grav[itatio]n[a]l wave or similar effect that makes linear [and] ang[ular] mom[entum] come out OK. Would be most interesting to analyze this recoil effect.”
The problem of transport through wormholes would occupy Wheeler for several years to come. While he would conclude that this was not a problem, that not even light signals could travel through a wormhole, he ultimately abandoned the idea of modelling particles with gravitational and electromagnetic fields around 1970.
The concept of wormhole, now re-imagined as traversable portals, survived within science fiction, and in the 1980s the study of traversable wormholes (re-)entered the scientific discourse. The wormhole is thus a perfect example of how a new concept emerged in Wheeler’s thinking, was explored and tested from various angles in his notebooks, ultimately emerging (in spite of numerous inconclusive results) as an idea interesting and robust enough to henceforth lead a life of its own, even after Wheeler had lost interest in it.
I have here only provided a small glimpse at a short episode from Wheeler’s notebooks. The notebooks have served as the basis for a number of in-depth historical investigations, e.g., of how Wheeler came to work in general relativity in the first place, of how in the 1970s he came to question the concept of physical law, or even of Wheeler’s multifaceted reception of the philosophy of Leibniz. And physicists, too, should be able to profit from perusing Wheeler’s notebooks, from his eclectic yet never aimless method or from his crazy yet never one-dimensional ideas.
Thankfully, the Library of the American Philosophical Society in Philadelphia, where Wheeler deposited his notebooks, has begun digitizing the notebooks and making them freely available on their website. Now everyone can share the “wonderful experience” that used to be reserved for those few students who got to borrow one of Wheeler’s notebooks for a few days.
This article was originally published in Annalen der Physik’s ongoing “Then and now” series, which is dedicated to the history of physics. The article has been modified for this website version.
Access the full article here: Alexander Blum, From Wood Chuck Holes to Worm Holes—A Look into the Notebooks of John A. Wheeler, Annalen der Physik (2022). DOI: 10.1002/andp.202200244
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