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I've heard anecdotally that Bernhard Riemann was interested in applying his discovery/invention of Riemannian geometry to model gravitation about 50 years before Einstein managed to do so.

In this video, we hear Edward Witten say,

Riemann almost did it. Riemann had the conception that gravity was based on curved [...] space. He didn't conceive of curved spacetime, because special relativity hadn't happen yet, so time and space haven't been unified.

In this wikipedia article, we see written,

Similar to Newton, but mathematically in greater detail, Bernhard Riemann assumed in 1853 that the gravitational aether is an incompressible fluid and normal matter represents sinks in this aether. So if the aether is destroyed or absorbed proportionally to the masses within the bodies, a stream arises and carries all surrounding bodies into the direction of the central mass. Riemann speculated that the absorbed aether is transferred into another world or dimension.

These quotes sparked my interest in finding all the sources (preferably primary sources) where Riemann writes about gravity. This is for both where he considers curved space and the aether-based models.

Are there any sources to back up either one of the quotes?

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    $\begingroup$ These quotes look unjustified. I bet that Riemann never discussed gravity in his published work at least. $\endgroup$ – Alexandre Eremenko May 10 at 2:58
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Riemann discussed a "unified field theory", including light, electromagnetism and gravity, in the unpublished paper Neue Mathematische Principien der Naturphilosophie (New Mathematical Principles of Natural Philosophy, 1853, the title obviously alludes to Newton's), and in Gravitation und Licht (Gravity and Light), the last section of his Fragmente on Naturphilosophie (both published posthumously in 1876). For English translation see e.g. Collected Papers, translated by Baker, Christenson, and Orde, 2004. Useful summaries are given by Bottazzini-Tazzioli in Naturphilosophie and its role in Riemann's mathematics and Gray in Riemann on Geometry, Physics, and Philosophy.

Witten is largely off, what he says is more applicable to Clifford, Wikipedia is closer, although Riemann's ether is not exclusively "gravitational".

Most of Riemann's natural philosophy was motivated by early Herbart's interpretation of Kant (1802), including the idea that the worldspace (Weltraum) is filled with ether which flows through the atoms and then dematerializes (Herbart is also the only person, besides Gauss, credited by name in Riemann's famous geometry lecture). Some quotes:

"My main work concerns a new conception of the known laws of nature — an expression of them by means of other fundamental concepts — whereby the use of experimental data concerning the interaction between heat, light, magnetism, and electricity would make possible an investigation of their interrelationship. I was led to this primarily through the study of the works of Newton, Euler and, on the other side, Herbart. [undated, 1851?]

[...] Both classes of phenomena may be explained, if we suppose that the whole of infinite space is filled with a uniform substance, and each particle of substance acts only on its immediate neighbourhood. The mathematical law according to which this occurs can be considered as divided into
1) the resistance of a particle of substance to alteration in volume;
2) the resistance of a physical line element to alteration in length.
Gravitation and electrostatic attraction and repulsion are founded on the first part; propagation of light and heat, and electrodynamic or magnetic attraction and repulsion on the second.
" [1853]

Mathematically, according to Bottazzini-Tazzioli, Riemann’s model amounts to supposing that ether is an elastic, homogeneous, isotropic medium, a popular theory at the time, to which Cauchy, Lame and others contributed earlier. It was vaguely discussed already by Newton. Euler expressed even closer ideas in his Letters to German Princess (1760s), but they were largely forgotten. In 1858 Riemann wrote a paper deriving equations for a theory of electrodynamics, but withdrew it, probably due to a mistake in exchanging the order of integration. In his 1861 lectures he tried to relate it to the propagation of light. In Gravity and Light Riemann even writes "this substance can therefore be conceived as a physical space whose points move in geometrical space", but he never related gravity to curvature or curved space, and even wrote that "the basis for the metric relations must be sought outside it, in binding forces acting upon it".

The idea that dynamics of matter is determined by the curving of space itself was first mentioned by Sylvester in 1869, with credit to Clifford, who delivered his own lecture on the subject to the Cambridge Philosophical Society in 1870. Only a summary of it survives, published in 1876 On the Space-Theory of Matter, very vague and telegraphic, with explicit credit to Riemann for inspiration. He does not mention gravity specifically, only application to the "double refraction" (presumably, a reference to the then unexplained polarization of skylight, see commentary by Galindo and Cervantes-Cota). In the unfinished book The Common Sense of the Exact Sciences (1885), completed and posthumously published by Pearson, the latter wondered more pointedly "whether physicists might not find it simpler to assume that space is capable of a varying curvature, and of a resistance to that variation, than to suppose the existence of a subtle medium pervading an invariable homaloidal space". But even he does not mention gravity by name. Clifford did see himself as following in Riemann's footsteps, but, as Gray writes, there is no evidence that Riemann took that step.

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