Galaxy with no dark matter? (NGC1052-DF2)

[PeterDonis]

I propose that a reasonable definition of bends is something that does not follow a geodesic.

This is a reasonable definition of a path bending, yes; i.e., it's what it means to say that a circle, for example, is curved in Euclidean geometry, as opposed to a straight line. However, note that this definition of curvature is extrinsic--it depends on the curve being embedded in a higher dimensional space in a particular way.

However, when we say in GR that spacetime is curved, we are talking about intrinsic curvature--curvature that can be defined simply by the intrinsic features of the manifold, without making use of any embedding in any higher dimensional space. There is no such thing for a one-dimensional curve; the lowest dimension a manifold can have and have intrinsic curvature at all is 2. And in 2 or more dimensions, the definition of "curved" is "has a nonzero Riemann tensor"--or, to put it in more concrete terms, that parallel transporting a vector around a closed curve does not leave the vector unchanged.

Its shape, whatever it is, defines straight. Any two points in a spacetime are connected by a geodesic.

This is true, but it only means that we can always find a straight curve--straight in the sense of extrinsic curvature, i.e., no bending of the path, i.e., a geodesic--between any two points. It does not mean that there is no difference at all between, for example, a flat Euclidean plane and a 2-sphere like the surface of the Earth. There is; but that difference cannot be captured by just looking at individual geodesics. You have to look at how multiple geodesics "fit together", so to speak--for example, by looking at what happens to a vector when you parallel transport it around a closed curve composed of geodesic segments, which is what the Riemann tensor describes.

 

[Grinkle]

@PeterDonis Can you recommend a lay-person book or textbook on whatever math it is that is behind what you are describing?

I have in my (distant) past 4 semesters of calculus and 2 semesters (one undergrad and one grad) of engineering analysis, stated to give an idea of whether I am in any position to study this math.

 

[PeterDonis]

Can you recommend a lay-person book or textbook on whatever math it is that is behind what you are describing?

Carroll's online lecture notes on GR give a good introduction to the math of manifolds, tensors, and curvature in the first couple of chapters:

https://arxiv.org/abs/gr-qc/9712019

 

[Orodruin]

To clarify, based upon our understanding of quantum mechanics, everything in the universe is made out of fields, and fields can be quantized into particles.

I would not put it like this as it seems to put an equal sign between quantum fields and particles. Particles are a particular type of state of a quantum field, but the phenomenology would be so much more dull if they were the only type of state. Only considering particles you miss out on any non-perturbative effects as well as the coherent states (and thereby the classical limit).