Q&A: Geometry and Physics
Geometry and Physics
Question
Two questions:
Is it accurate to say that the accepted interpretation of general relativity tries to ground geometry in physics, in the sense that it sees geometry as a projection of the gravitational force?
Is it correct to say that this interpretation is not necessary, because one could, for example, argue that geometry and physics both rest on a shared principle (a third one), that is, that both of them are projections of it?
Answer
Both possibilities are incorrect. Geometry in itself is a mathematical field, and therefore it does not depend in any way on physics. The geometry of our world (which is a claim in physics) is connected to general relativity. It is not a projection of the gravitational force, but another way of presenting it.
Discussion on Answer
There is a misunderstanding here, so I’ll try to explain a bit more.
Geometry is a mathematical field that has nothing to do with reality. It describes properties of a space defined by axioms. Therefore in mathematics there are different geometries, each one describing a different space. There is no empirical dimension here at all, only an a priori logical one.
It turns out that one of these geometries describes our world. The question of which one describes our world is a question in physics, not mathematics. It is an empirical question. One has to observe and see whether our world is described by Euclidean geometry or not, three-dimensional or not, and so on. Relativity claims that the geometry in which our world should be described is not three-dimensional and not straight/Euclidean (our space is curved).
It’s like arithmetic: it is not connected to the world and cannot be empirically refuted, but we use it to do calculations at the grocery store or at the bank.
The splitting of the messages is because of a glitch on my phone.
Okay, it seems to me that you still haven’t addressed my question about the two different forms of presentation: gravitational force and geometry.
Is there some kind of equivalence between them, and can they be seen as two different expressions of a separate third (“objective”) element?
I’m not sure exactly what the question is. Gravitation can be presented in two equivalent forms: either as curvature of space-time or as a force in straight space (though with regard to light they are not equivalent, but I think they can be presented as equivalent for light too if you assign it mass, and then it also does not move in a straight line under a force).
There are a few things here that I don’t fully understand.
First, regarding your claim that geometry does not depend in any way on physics. I understand that this is your claim, but is that also what follows, in your view, from general relativity?
Second, you speak about the geometry of our world and emphasize that this is a claim in physics. A. What do you mean by “our world”? Does general relativity allow us to say something about the geometry of a world that is “not ours”? If the claim about the geometry of our world is “a claim in physics,” how does that fit with your claim that geometry does not depend on physics?
Third, suppose we adopt your terminology and don’t speak about a “projection” but about a “form of presentation.” The question I asked earlier still remains (is it possible that according to general relativity there is some third factor that “feeds” these two forms of viewing things?).
Fourth, even if we assume there are only two forms of viewing things (geometry and physics): as I understand it, relativity does not assume symmetry between them, but rather “guides” the scientist to derive the structure of space (expressed in geometric language) from the gravitational force. In this it is faithful to a methodological principle already in the background of special relativity (to derive entities or abstract states such as velocity or motion from actual observation). Do you agree?