Abstract
I reconsider Einstein’s 1912 “Prague-Theory” of static gravity based on a scalar field obeying a non-linear field equation. I point out that this equation follows from the self-consistent implementation of the principle that all forms of energy are the source of the gravitational field according to \(E=mc^2\). This makes it an interesting toy-model for the “flat-space approach” to General Relativity (GR), as pioneered by Kraichnan and later Feynman. Solutions modelling stars show features familiar from GR, e.g., Buchdahl-like inequalities. The relation to full GR is also discussed. This lends this toy theory also some pedagogical significance.
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- 1.
In his Prague papers Einstein gradually changed from the first to the second expression.
- 2.
Since here we will be more concerned with the mathematical form and not so much the actual derivation by Einstein, we will ignore the obvious objection that \(c\) has the wrong physical dimension, namely that of a velocity, whereas the proper gravitational potential should have the dimension of a velocity-squared.
- 3.
Einstein considers radiation enclosed in a container whose walls are “massless” (meaning vanishing rest-mass) but can support stresses, so as to be able to counteract radiation pressure. Einstein keeps repeating that equality of both mass types can only be proven if the gravitational field does not act on the stressed walls. That remark is hard to understand in view of the fact that unbalanced stresses add to inertia, as he well knew from his own earlier investigations [8]. However, as explained by Max Laue a year earlier [9], the gravitational action on the stressed walls is just cancelled by that on the stresses of the electromagnetic field, for both systems together form a “complete static system”, as Laue calls it. A year later, in the 1913 “Entwurf” paper with Marcel Grossmann [10], Einstein again used a similar Gedankenexperiment with a massless box containing radiation immersed in a gravitational field, by means of which he allegedly shows that any Poincaré invariant scalar theory of gravity must violate energy conservation. A modern reader must ask how this can possibly be, in view of Noether’s theorem applied to time-translation invariance. A detailed analysis [11] shows that this energy contains indeed the expected contribution from the tension of the walls, which may not be neglected.
- 4.
“Anderenfalls würde sich die Gesamtheit der in dem betrachteten Raume befindlichen Massen, die wir auf einem starren, masselosen Gerüste uns befestigt denken wollen, sich in Bewegung zu setzen streben” ([3], p. 452).
- 5.
Pioneered by Robert Kraichnan in his 1947 MIT Bachelor thesis “Quantum Theory of the Linear Gravitational Field”.
- 6.
Its Taylor expansion at \(x=0\) is \(1-6x/5+51x^2/35+\cdots \).
- 7.
This differs by a factor of 2 from (23) which we need and to which we return below.
References
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Acknowledgments
I sincerely thank the organisers and in particular Jiří Bičák for inviting me to the most stimulating and beautiful conference Relativity and Gravitation—100 years after Einstein in Prague.
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Giulini, D. (2014). Einstein’s “Prague Field Equation” of 1912: Another Perspective. In: Bičák, J., Ledvinka, T. (eds) Relativity and Gravitation. Springer Proceedings in Physics, vol 157. Springer, Cham. https://doi.org/10.1007/978-3-319-06761-2_10
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