That’s what curvature means if you visualize a manifold as being embedded in a higher-dimensional ambient space, “bending” away from an idealized flatness. This is simply not a helpful way to think about it, IMO, and leads to a LOT of confusion/misinterpretation from novice physicists/lay people.

The extrinsic perspective isn't necessary. The Ricci scalar, for example, directly measures the distortion in the volume of small balls: en.wikipedia.org/wiki/…. Distortion/warping/bending/stretching/etc. aren't fundamentally extrinsic concepts (though they can sometimes be visualized that way).

Yes, intrinsic geometric properties can be described as “deformation” from flatness of certain geometric measurements, but the common popular explanation is that spacetime itself is warped/stretched/etc.. My point is that this explanation is highly misleading and often lead people to think of the oversimplified “trampoline” model often showed in popularized explanations.

Furthermore, the action-type sounds of words like bending/stretching/warping leads people to think of space as having curvature, which then evolves in time, which is a very not-relativistic way of looking at things. Instead, spacetime is a single object which doesn’t evolve, it just has curvature (at least classically — who the hell knows what’s happening at the quantum level).

Again, this is just a personal preference in how I teach. I think it’s important to be precise, else people start getting false ideas about trampolines and balls (and especially black holes).

+1 for the final quote!

@debo.stackoverflow ...and I understood most of it!

Now give critical review for this PBS documentary image! en.wikipedia.org/wiki/Big_Bang#/media/…

@cpburnz Thanks! I typed this on my iPad while getting out of bed this morning, so I wasn't the biggest on spell check.

I think I have to disagree with the dismissal of fabric as a useful picture of spacetime (or general riemannian manifolds) - the fact that they can be embedded in high enough dimensional space (of course preserving all the structure that we care about) means that it's still a useful picture to keep in mind... sure I can describe everything just by describing their algebraic properties - but pictures are good?

Indeed, any straightforward answer to this requires going beyond physics and into philosophy. If anyone is interested in the philosophy around "is spacetime real?", check out plato.stanford.edu/entries/spacetime-theories

@user76284 I find terms like "stretch" or "warp" provide an implication that space is supposed to have no curvature, and it is by an external action that they arrive in their final shape. They are terms which induce a curvature. Often this is a valid phrasing, so I can see why it is common, but when discussing very precise questions about whether spacetime is "real," they encourage unintended expectations about what reality should be. Terms like "curvature" do so to a much lesser degree.

@JoshuaLin How do you draw a 5 dimensional picture?

Lovely. Is that last quote also from Gannon's book?

@N.Steinle Nah, that last quote was just me rambling.

@BobKnighton I am a layperson, programmer by day, follower of everything science by night, and I have been avidly reading all physics papers and literature I can muster an understanding of for years. I always thought the fabric representation of space time was a nice image but couldn't possibly be right. This answer is the first time I understand well and truly the idea of a manifold, and WHY the fabric analogy is faulty. Thank you so much for this wonderful answer!

@jpmc26 with a little bit of difficulty