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I have a question about the definition of mass. In general relativity mass is closely related the curvature of spacetime, which indicates that mass might be interpreted as a geometric concept (maybe related with the connection defined on some fibre bundle structures above spacetime). I found only Gonzalez-Martin's related work on this topic in his book "physical geometry".

Gonzalez-Martin's work was quite old and two of his papers related to this topic are:

  • Physical Geometry: A Unified Theory of Gravitation, Electromagnetism and Other Interactions
  • A Geometric Definition of Mass

His book including these two papers can be found here: Physical geometry.

But I did not catch his idea. Can anybody help to explain his idea or provide other alternative work in this direction? BTW, I wonder if a geometric description of spacetime is valid, if the discrete space (~Planck length) somehow be related with the isoholonomic problem, which aims to find out the minimal loop to generate a given holonomy.

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closed as unclear what you're asking by CuriousOne, heather, ACuriousMind, Gert, Wolpertinger Aug 13 '16 at 15:34

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ You should provide us with the ideas he presented. We aren't going to read through an entire book just to see if we can answer this question. $\endgroup$ – Jim Aug 11 '16 at 16:05
  • $\begingroup$ Mass is defined by inertia trough Newton's second law. General relativity assumes that inertial mass and gravitating mass are the same. If that assumption is false, then there would have to be, at least, two different masses and it would not be clear, at this point, what the actual meaning of gravitating mass is, but it would definitely not be geometric, since the geometric aspects of general relativity are a direct consequence of the equivalence relation. Failing equivalence GR would become a true force theory with mass being an effective charge. $\endgroup$ – CuriousOne Aug 11 '16 at 18:54
  • $\begingroup$ Unfortunately those links are paywalled, or require an institutional subscription. Sadly this is true for a lot of papers before the arxiv took off. It's very hard to judge the content of the question without access to them (I personally find the idea attractive, but I was a classical GR person, so everything I think is wrong in suitable limits...). $\endgroup$ – tfb Aug 11 '16 at 19:10
  • $\begingroup$ @ CuriousOne The inertial mass defined by Newton's second law is essentially also defined by the space time structure, so it might also be 'geometric'. So it's natural to assume the equivalence principle. We can not say if the equivalence relation is the reason or the consequence. $\endgroup$ – XXDD Aug 12 '16 at 2:34
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    $\begingroup$ @X.Dong Mass is not geometry, mass and any other energy-like sources (momentum etc) determine the geometry, through Einstein. You can go to higher resolution, or energies, and maybe you sense the insides of elementary particles, and maybe they turn out to be strings or loops or or quantum gravity constructs. And then maybe those strings/etc maybe cause (however, not clear), spacetime, or if you wish geometry. But it is then clear that you need to have those strings/etc also account for the other forces, because you have energies that create spacetime and you've not accounted for them. $\endgroup$ – Bob Bee Aug 13 '16 at 5:06
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I admit that what I write here is not a an answer to the question but rather a comment on the ideas of treating mass (or matter) in geometric terms. Unfortunately, the comment feature has limited space so I am using this answer to voice these comments.

In the 1972 publication of Steven Weinberg's book "Gravitation and Cosmology -- Principles and Applications of the General Theory of Relativity", in the Preface Weinberg writes (April 1971) about his lament of the overemphasis (of authors) of the Geometrical approach to General Relativity, in particular he says

However, I believe that the geometrical approach has driven a wedge between General Relativity and the theory of elementary particles. As long as it could be hoped, as Einstein did hope, that matter would eventually be understood in geometrical terms, [Emphasis mine].

In addition, Weinberg writes

...it made sense to give Riemannian geometry a primary role in describing the theory of gravitation. But now the passage of time has taught us not to expect that the strong, weak, and electromagnetic interactions can be understood in geometrical terms and too great an emphasis on geometry can only obscure the deep connections between gravitation and the rest of physics.

So, Einstein apparently had this notion that matter (or, mass) could be described in geometrical terms yet it was never a fulfilled work in his lifetime. And, as far as I know, not fulfilled yet. Einstein's belief in the geometric approach in describing natural laws of physics is what occupied him for many years chasing down the wrong path.

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    $\begingroup$ If it is not an answer don't write it in the answer box. Write a comment if you want, not a history of why some people like Einstein wanted to make mass or all of it geometric. There's already a reference in the question to somebody who published and wrote on it extensively. At least read something and try to answer what was the idea in the author's writing. It is much more than making matter geometric. It's making electromagnetism geometric and other things. It is complex enough that I don't know if I believe or understand it. $\endgroup$ – Bob Bee Aug 13 '16 at 4:50
  • $\begingroup$ I can not fully catch the geometric meaning of the mass of Ganzalez-Martin. It seems that mass is defined as a function of connection, but what exactly is mass's geometric picture? $\endgroup$ – XXDD Aug 14 '16 at 9:45
  • $\begingroup$ @K7PEH Personally I would like not to regard gravity as a fundamental force, at least we have to admit it has a different structure with the other three forces. It seems that gravity as the geometry of spacetime is emergent, for example from quantum mechanics. $\endgroup$ – XXDD Aug 14 '16 at 10:02

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