Yes, the concept of a manifold was created by Riemann; however, that does not mean he thought the universe could be like that. The concept of manifold is a purely mathematical one, and I do not think Riemann made any explicit reference to physics or astronomy in this context. (I may be wrong, of course. I have not seen all of Riemann's works; not even a few of them.)
I agree with Charles. In my personal opinion Riemann did not received much credit for general theory of relativity (GTR) as Einstein did. However, if you read about the development of (GTR) you will find Hilbert was very close in obtaining the required equations (more or less, he actually did!), and he had been working really hard much before Einstein. But, Einstein after much help from Riemann's work was able to things put together in an "elegant" form. Einstein knew how to couple different concepts together. That was his true merit in my opinion.
Riemann was a mathematician that would resolve the Euclid's fifth postulate. Einstein used Riemannian geometry for general relativity, does it mean that Riemann has knew where his invention can be used?
Gauss! But even earlier than that, Kepler circa 1600 wondered why there are three dimensions to space, and concluded that it was a physical manifestation of the Holy Trinity. See http://arxiv.org/abs/gr-qc/9805018
I agree both with Charles and Vikash. Just want to add that Poincare had pretty much founded Special Theory of Relativity, before Einstein (https://en.wikipedia.org/wiki/Relativity_priority_dispute#Undisputed_and_well_known_facts), although not much credit was given to him for that. Not to mention Lorentz's contribution and Maxwell's, whose work had prepared the ground. In the end, Einstein did put it all together, but "had at least one of the above brilliant minds questioned the dimension of the universe?" is a different question.
Yes, trying to solve the problem of the parallel postulate has two thousand years of history.
Kaluza first introduced one more dimension for unifying General Relativity and electromagnetism. But there are additional terms and field. Einstein and Mayer then designed another five dimensional theory that is exact, though the space-time is no longer pseudo Riemannian but non holonomic.
In fact, Gunnar Nordström proposed his 5-dimensional solution years before Kaluza. It was in order to unify his scalar theory of gravitation with Maxwell’s theory of electromagnetism. It turned out that his gravitational theory got a very short lifetime when Einstein brought out the fully functional tensor theory of gravitation as the theory of general relativity.
Esa is correct: Nordstrom's paper appeared in 1914, Kaluza's was proposed in 1919 but held up for publication by Einstein until 1921. It's remarkable that all this happened 100 years ago, yet there is still no consensus about how, or even whether to incorporate extra dimensions into physics today.
Charles Francis makes an excellent point on this thread about the distinction between curvature and embedding. 4D spacetime curvature does not in itself imply the existence of more than 4 dimensions. However it may be attributing too much to Riemann to say that he saw a possible connection between curvature and gravity. In the available English translation of his habilitation lecture in 1854, "On the hypotheses which lie at the basis of geometry," the closest he comes to physics is to say that the curvature of 3D space on small scales might be suitable for "the domain of another science ... physic." I think we have to give Einstein the credit for linking geometry with gravity (though of course he could not have done it without Riemann's mathematical foundation).
Going back to the simpler question of who was the first to think about the dimensionality of space, I argued above that Kepler might get credit for being the first person with enough imagination to even see this as a question (as he was in so many other areas)---even if his answer seems prescientific today. But going back even farther, the ultimate historical source for thinking on extra dimensions is probably Aristotle, who in "De Caelo" (c. 350 BC) gave this "proof" for the non-existence of more than 3 dimensions: "The line has magnitude in one way, the plane in two ways, and the solid in three ways, and beyond these there is no other magnitude because the three are all."
Forty years before Einstein, W. K. Clifford wrote a seminal paper titled "On the space theory of matter", full of physical intuitions which point towards Einstein's General Relativity. It is reproduced in J. R. Newman: "Sigma, the World of Math. "
@ Raul Simon,
Could you please provide the two articles you have mentioned?
Vikash Pandey:
There is only one article; it is summarized in "Sigma: The world of mathematics", by James R. Newman (New York, Simon & Schuster, 1956), vol. 4. I only have a Spanish translation.
@ Raul Simon
I do not have access to the articles. Please send the articles. I will use google translate, and I am really keen to study this.
Actually, from the abstract of his paper, Clifford only refers to a theory of matter, similar to quantum foam. That is, small scale curvature while at large scale it is flat. It would be extraordinary that the idea of general relativity could be express way before the one of special relativity.
Clifford, Riemann, Hilbert, Einstein...did not understand Gravity.
“I am the first who Understood and Explained Gravitation with high speed gravitons v = 1.001762 × 10^17 m/s, with Negative Impulse, Negative Mass and Negative Energy” Adrian Ferent
https://www.researchgate.net/publication/287331589_Ferent_Gravitation_theory
@ Adrian, I just had a look at your article and I really have none words to say. If you think you are right, why don't you publish it in a reputed journal. You say things without giving any reference. It is like cooking without any ingredients.
Please see N.Lobachevsky' geometry.
He (born 1792 ) worked on that much earlier than Riemann.
Dear Natalia
Yes, you are right. But Bolyai and Lobachevskian propounded "Hyperbolic geometry" and Riemannian geometry is Elliptic geometry.
@ Vikash, my quantum gravity theory shows that the gravitons are too small to be detected by today’s technology.
Reputed journals do not publish a new theory, they publish only articles related to old theories like General Relativity
For gravity there have been vigorous debates about even the concept of graviton rest mass;
Goldhaber, A. S., & Nieto, M. M. (2010). Photon and graviton mass limits. Reviews of Modern Physics, 82(1), 939
A nonzero graviton mass would upset this remarkable agreement altering the predicted orbital decay, implying an upper limit on the graviton mass.
L. S. Finn and P. J. Sutton, “Bounding the mass of the graviton using binary pulsar observations,” Physical Review D, vol. 65, no. 4, Article ID 044022, 2002.
There is a link between the cosmological constant and the graviton mass.
http://www.hindawi.com/journals/isrn/2014/718251/
Graviton production through photon-quark scattering at the LHC
Phys. Rev. D 91, 035017 - 2015
http://journals.aps.org/prd/abstract/10.1103/PhysRevD.91.035017
In my theory, energy is "dense field" and mass is "Intensive energy".
https://www.researchgate.net/publication/268343642_A_new_Definition_of_Graviton?ev=prf_pub
Article A new Definition of Graviton
Well the first measurements that tried to test, whether Euclidian geometry did describe Nature, did involve Gauss, cf. http://mathpages.com/rr/s8-06/8-06.htm Cf. also http://www.springer.com/cda/content/document/cda_downloaddocument/9780387295541-c2.pdf?SGWID=0-0-45-301316-p86706747 for a more detailed account.
To the difference between Lobachevsky's geometry and Riemann's: the former is homogeneous, that is, you can transport a geometric figure of finite size without deforming it, from one place to the next. In other words, the curvature is constant, and, as turns out, negative. The other possibilities are constant positive curvature, corresponding to a sphere, and vanishing curvature, corresponding to a plane. Riemannian geometry, on the other hand, is a generalisation of the geometry of arbitrary curved surfaces, with (generally) non constant curvature, so that there are no congruences. It is thus very different from both Euclidean and non-Euclidean geometry.
Thank you for all your valid answers. I did PhD in Maths. If I want to do research on Quantum Gravity. What should I do?. Can I apply for another PhD?
another PhD? Nope. Just start reading articles and writing your own articles...thats it! The number of PhDs won't affect the quality of your research. Just be your own supervisor!
It was Rene Descartes (1596-1650), the French Mathematician, who invented the three dimensional Cartesian coordinate system, in which any given vector quantity can be represented in terms of three scalar numbers, representing three spatial dimensions. But others must have thought about the three spatial dimensions (length, breadth, height) much before this.
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