Abstract
The renaissance of General Relativity witnessed considerable progress regarding both understanding and justifying Einstein’s equations. Both general relativists and historians of the subject tend to share a view, General Relativity exceptionalism, which emphasizes how General Relativity is novel and unlike the other fundamental interactions. But does some of the renaissance progress in understanding and justifying Einstein’s equations owe something to an alien source, namely, particle physics egalitarianism, which emphasizes how similar General Relativity is to the other fundamental interactions? If so, how should the historiography of gravitation and Einstein’s equations reflect that fact? Two areas of renaissance progress can be considered very briefly (gravitational waves and the question of energy and the justification of Einstein’s field equations in terms of particle physics) and one in detail: the longstanding confusion over the relation between Einstein’s cosmological constant Λ and a graviton mass, confusion introduced in 1917 with Λ.
Regarding gravitational waves, General Relativity exceptionalism discouraged trusting the linear approximation, in which the existence of gravitational radiation is evident, whereas particle physics egalitarianism encouraged such trust. While general relativists differed on this issue in the 1950s, the rare particle physicist seriously involved, Feynman, was among those who invented the sticky bead argument that largely resolved the controversy. Regarding the justification of Einstein’s equations in terms of particle physics, the spin 2 derivations of Einstein’s equations provided considerably support of the view that Einstein’s equations correctly describe gravitation.
The idea of a graviton mass has a nineteenth-century Newtonian pre-history in Neumann’s and Seeliger’s long-distance modification of gravity, which (especially for Neumann) altered Poisson’s equation to give a potential e −mr∕r for a point mass, improving convergence for homogeneous matter. Einstein reinvented the idea before introducing his faulty analogy with Λ. This confusion was first critiqued by Heckmann in the 1940s (without effect) and by Trautman, DeWitt, Treder, Rindler, and Freund et al. in the 1960s, and especially more recently by Schücking, but it has misled North, Jammer, Pais, Kerszberg, the Einstein Papers, and Kragh. The error is difficult to catch if one has an aversion to perturbative thinking, but difficult to make if one thinks along the lines of particle physics. Λ contributes predominantly a zeroth order term to the field equations (a constant source), whereas a graviton mass contributes a linear term. Nonperturbatively, massive spin 2 gravity is bimetric. The Λ-graviton mass confusion not only distorted the interpretation of Einstein’s theory, but also obscured a potentially serious particle physics-motivated rivalry (massless vs. massive spin 2). How could one entertain massive spin 2 gravity if Λ is thought already analogous to the Neumann-Seeliger scalar theory? Massive spin 2 gravity would encounter problems within particle physics in the early 1970s, which have been substantially resolved in recent years (though new problems have arisen). Other authors manage to avoid confusing Λ with the Neumann-Seeliger long-range decay but fail to recognize that the latter, with the physical meaning of a graviton mass, provides an interesting case of unconceived alternatives.
In sum, both the interpretation and the justification of Einstein’s equations owed some of their renaissance progress to particle physics egalitarianism. Historiography, like physics, is best served by overcoming the divide between the two views of gravitation.
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Notes
- 1.
The obvious exception was the Russian school of A. A. Logunov and collaborators starting in the late 1970s, which was largely ignored by others or occasionally subject to polemics (Zel’dovich and Grishchuk 1988), not without some justification. Logunov being the editor of the Russian original of Theoretical and Mathematical Physics and a Soviet and Russian Academician, he was able to maintain a noticeable research group with many publications.
- 2.
Mathematically with the graviton mass term one has a superposition of exponentially decaying and exponentially growing factors times \(\frac {1}{r},\) but one routinely discards the growing solution on grounds of physical reasonableness. With the zeroth order term in the field equations, by contrast, there are no solutions to spare and a (quadratically) growing solution cannot be discarded.
- 3.
“In diesem Zusammenhang sei folgendes bemerkt: Die von (21) verschiedene Gleichung Δ Φ + λ Φ = 4πGρ wird von EINSTEIN in der S. 2. Anm. 4 zitierten Arbeit zur Erläuterung der Einführung des Gliedes λg μν in seine Feldgleichungen herangezogen. Dieser bereits von C. NEUMANN gemachte Abänderungsvorschlag des NEWTONschen Gezetzes (vgl. S. 1 Anm. 1) ergibt sich aber nicht als Näherung aus den Feldgleichungen der Relativitätstheorie. Damit ist die Begründung, die HECKMANN und SIEDENTOPF [Z. Astrophys. 1, 67 (1930)] für ihre Gleichung (5, 18) gegeben haben, hinfällig.” (Heckmann 1942, 15, emphasis in the original).
- 4.
“Die Konstante λ kann nun aber nicht proportional k 2 gesetzt werden, wobei k −1 die Comptonwellenlänge des Gravitons wäre. Bilden wir nämlich das Äquivalent von (1.7) [the linearized wave equation, without the cosmological constant] zu den neuen Gleichungen (1.10) [Einstein’s equations with the cosmological constant], so erhalten wir
Damit aber λ im wesentlichen k 2 sein könnte, müßte anstelle von (1.11)
gelten” [references to papers by L. de Broglie and M.-A. Tonnelat].
- 5.
They were presumably not reckoning sufficiently with the ghost problem, though they did discuss it. But their appendix is evidently the first public appearance of a nonlinearly ghost-free, that is pure spin 2, massive gravity, singled out from among the OP 2-parameter family of theories. The nonlinear argument was published later (Maheshwari 1972) (submitted no later than early April 1971). This (Tyutin-Fradkin-)Boulware-Deser nonlinear ghost problem was pre-solved before it was proposed. But no one noticed and the problem had to be solved again in 2010. Thus the decades of darkness for massive gravity were quite contingent. Maheshwari was unaware of the van Dam-Veltman-Zakharov discontinuity, however.
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Acknowledgements
Many thanks are due to Alexander Blum for assistance with translations, editing, points of emphasis, and acquainting me with Bryce (Seligman) DeWitt’s dissertation; to Jürgen Renn, Roberto Lalli, and the rest of the Discussion Group on the Recent History of General Relativity at the Max-Planck-Institut für Wissenschaftsgeschichte for the opportunity to participate; to Michel Janssen and Dennis Lehmkuhl for discussion; to Karl-Heinz Schlote for help in finding Neumann’s works; and to Cormac O’Raifeartaigh for discussion and mentioning the Mamone Capria paper. This work was supported by the John Templeton Foundation grants #38761 and #60745 and the National Science Foundation (USA) STS grant #1734402; all views are my own.
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Pitts, J.B. (2020). Cosmological Constant Λ vs. Massive Gravitons: A Case Study in General Relativity Exceptionalism vs. Particle Physics Egalitarianism. In: Blum, A.S., Lalli, R., Renn, J. (eds) The Renaissance of General Relativity in Context. Einstein Studies, vol 16. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-50754-1_6
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DOI: https://doi.org/10.1007/978-3-030-50754-1_6
Publisher Name: Birkhäuser, Cham
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