With no intent to violate se rules, I wish to as specific questions related to this general question. For that reason, I will attempt to specify the kind of general answer I seek. It is most likely that the best answer to the current question will be a reference to a discussion of purely intrinsic properties of a differentiable manifold, or something similar.

This is Box 9.1 from Misner, Thorne and Wheeler's Gravitation.

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The discussion assumes a metric-free, geodesic-free spacetime. The authors never explain what properties this spacetime does posses. For example, what does it mean to multiply the displacement of $\mathcal{P}$ as $\lambda$ ranges from $0$ to $1/N$? With no concept of distance what does $\lambda$ even mean?

What properties are we to attribute to this spacetime? Must we assume that locally it approximates the spacetime of special relativity? Can we speak of open balls centered on an event? Can we speak of a neighborhood of an event becoming arbitrarily small?

The authors do speak of the possibility of a higher dimensional "flat" "embedding space", but call it extraneous.

  • $\begingroup$ it seems like this is just a heuristic idea to motivate the definition of a tangent vector (though I'm not a fan of such heuristics; I would much rather see the precise definition first and then look at the motivation). Also, I'm not really sure what your main concern is. Anyway, the concepts of differentiable manifolds, "curves" in a manifold (i.e just a smooth mapping of an interval into the manifold), and tangent space to a differentiable manifold are all standard topics, so once again, I'm not really sure what you're after. $\endgroup$
    – peek-a-boo
    Jun 19 at 5:35
  • $\begingroup$ I'm fairly convinced that more is intended than simply introducing tangent spaces, etc. In many places in the book, they show a great deal of enthusiasm for what can be accomplished without a metric. In Chapter 9 the notion of a tangent vector arrow in a tangent space becomes useful fiction. What remains is the ability to differentiate scalar functions with respect to a parameter. The books by Loring Tu and Frank Warner appear to share MTW's view. But they are formidable challenges. C. H. Edwards mumbles conspicuously in this area. Alfred Grey introduces arc-length almost immediately. $\endgroup$ Jun 19 at 6:20

You do not need distance. What is considered are curves on a manifold. The curve on a manifold ($M$) is map form real numbers into manifold, i.e. a map that takes real number and assigns a point in the manifold: $$P(\lambda):\mathbb{R}\rightarrow M.$$ $\lambda$ is simply parameter of a curve. The tangent vector is then considered to be $$\frac{dP}{d\lambda}=\lim_{N\rightarrow \infty}\frac{P(\frac{1}{N})-P(0)}{\frac{1}{N}}.$$

Problem is that you are subtracting two points and then dividing by number and it is not clear what exactly does it mean for general manifold. For Riemannian manifold, you can imagine this being embedded in a higher dimensional flat manifold, where the operation makes sense. I think this is also the origin of the name "tangent space", because in the limit, the vectors in this high dimensional flat space indeed become tangent to the (sub)manifold considered.

And if I remember correctly, such embedding always exists. But mathematically it is a little unsatisfactory definition, since it requires starting with higher dimensional space in which we are not interested, defining our tangent vectors and then throwing it away. The approach also requires the manifold to be Riemannian, but you can define vectors on any manifold whatsoever with no problem.

On the other hand, this approach is easier for our intuition, because then we can draw pictures like the one you posted. MTW goes for more intuitive explanation, but I think it would not be a terrible idea to supplement it with more mathematical approach to differential geometry.

  • $\begingroup$ My question is very hard to ask since there are a number of issues. One of them is: what is meant by curve? math.stackexchange.com/q/3431851/342834 Then what is meant by a parameterized curve? If we start talking about differentiable manifolds, we then encounter such things as the inverse and implicit mapping theorems. $\endgroup$ Jun 18 at 12:45
  • $\begingroup$ @StevenThomasHatton Have we talked about anything else then differentiable manifolds? Curve is a shorthand for parametrized curve. In differential geometry all curves are parametrized. Also, the definition I am familiar with all require curves to be smooth...I fail to see problem with inverse and implicit mapping theorems in definition of a curve, nor do I see how it is relevant to your original question. $\endgroup$
    – Umaxo
    Jun 18 at 13:07
  • $\begingroup$ The implicit function theorem is used to define a differentiable manifold as the solution set for a system of constraints. Each independent constraint removes one dimension (degree of freedom) form the number of dimensions in the embedding space. The developments I am familiar with use various norms, which are used to quantify displacements. For a regular parameterized open curve, there is, at a minimum a way to speak of relative arc-length. A point with greater parameter value has a greater arc-length from the point where the parameter is 0. It goes on and on. $\endgroup$ Jun 18 at 14:17
  • $\begingroup$ I accepted the answer, but the idea of an unembeded differentiable (non-flat) manifold still hurts my head. For example Milnor's Topology from a Differentiable Viewpoint requires a local parameterization of a smooth m-manifold embedded in $\mathbb{R}^k$ to be a mapping from an open subset of $\mathbb{R}^m$ into $\mathbb{R}^k,$ "so that the derivative is defined." In the case of physical spacetime, we are equipped with intrinsic local measuring devices called measuring rods and clocks. $\endgroup$ Jun 19 at 12:41
  • $\begingroup$ I now realize that part of the reason this was confusing to me is that most general to specialized developments of geometry begin with some notion of "straightedge". From the most general projective geometry; to affine geometry; to metric geometry. But MTW throw out everything. Previously when I tried to make sense of this, I assumed I had an infinitesimal meter-stick, and a wristwatch at my disposal. $\endgroup$ Jun 25 at 22:38

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