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So there's general relativity and Einstein's field equations that tell us "mass(or equivalently energy) warps space-time, and the warping tells mass how to move", but I'm still having trouble understanding how space (a thing I conceive to be 'empty' or nothingness) can be 'warped'.

After researching books, on-line articles, and this stack exchange itself I can't seem to find any descriptions or discussions that don't immediately resort to jumping into GR and its mathematics to provide a more physical intuitive understanding of what it means to for space to be warped.

Is it space, this empty thing, that's getting warped. Or rather is it some field (Higgs field?) that exists around matter that's getting warped?

Although this question appears similar to This Question , it's asking if it's empty space that's being warped by matter or rather a field in space that's being warped (the Higgs Field?). The other question and all 10 answers addressing it do not address fields vs. empty space - What's really getting warped?

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    $\begingroup$ possible duplicate of How exactly does curved space-time describe the force of gravity? $\endgroup$ – ACuriousMind Sep 21 '15 at 15:23
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    $\begingroup$ Have a look at my answers to Universe being flat etc and What is the universe 'expanding' into?. In those answers I've attempted to give an intuitive idea of intrinsic curvature. If you think that approach is helpful shout if you want me to expand on it. $\endgroup$ – John Rennie Sep 21 '15 at 16:00
  • $\begingroup$ @JohnRennie thanks for the links and illustrations of intrinsic curvature. I still have this hunch though that it's not the space but rather a field within the space. And this field coupled with the presence of mass. Fields I can imagine being warped, space not. Isn't this the Higgs field? $\endgroup$ – docscience Sep 23 '15 at 14:13
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1. Void vs Vacuum

The first thing that needs to be done is to distinguish between void and space (ie vacuum).

Space is not nothing, because you can move things in it; think of it as the medium in which particles can move.

For if space was exactly nothing; then where could you put a particle? There is no place you can put it.

2. Crumpling space

The second thing is to imagine how you can warp this space; this is difficult using the space we actually live in.

So let's imagine a page torn out of Ryders QFT is our space. This is easy enough to warp - you can roll it into a cylinder, or crumple it in some other way.

3. Geodesics or straight lines

But how to do physics on this surface? Well, let's just take Newton's first law: a particle without any forces acting on it moves in a straight line.

When the page is rolled out flat on the surface of a table, the path this particle takes is easy enough to imagine - it's just the straight line we can draw by eye.

But how about when it's crumpled? Well, to make things easy for us let's imagine that the page has been crumpled and glued into a sphere. So how can we draw a straight line on this sphere? We can't do the obvious thing and just 'drill' through the sphere in the obvious straight line - because the interior of the sphere is not space but void - so a particle can't go there.

And we can't do the second most obvious thing either, which is just draw the straight line by eye on the surface of this sphere, because between any two points it's not clear what is the straight line that connects them - because to our eye they all look curved.

We turn this around, by asking if there is some unique property that characterises the straight line on the surface; well: on the surface of the paper spread out flat on the surface of a table, our original set-up, we see straight-away that the straight line between two points is the shortest path between two points.

4. Newton's first law again

So, we use this property on the sphere; and now it's easy to see how a particle will follow Newton's first law on this 'curved' or 'warped' space; it's still moves in a straight line, but here we call them geodesics.

In fact, this sphere is 'warped' in a way that the crumpled page is not; because if you draw triangles on it - which we can now, since we know what straight lines look like - we discover that their interior angles always add up to more than 180 degrees.

5. And GR, briefly

And this is how GR works: given a mass distribution in a space, this tells what the geodesics are in this space - ie how it's curved; and then particles move along these geodesics.

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    $\begingroup$ The example of rolling or crumpling paper is misleading because these are examples of extrinsic curvature. The cylinder and crumpled piece of paper are intrinsically flat, very different from the intrinsically curved 2-sphere. $\endgroup$ – Evan Rule Sep 26 '15 at 21:59
  • $\begingroup$ @evan rule: sure; still the cylinder is flat but it's global topology is very different from the plane; the point was to help the OP visualise by analogy what warped space is like. $\endgroup$ – Mozibur Ullah Sep 26 '15 at 23:08
  • $\begingroup$ This is after all what he was struggling with. $\endgroup$ – Mozibur Ullah Sep 26 '15 at 23:11
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Describing ´space´ as ´empty´ does very little when you are juxtaposing that with a field.. Spacetime is not ´nothing,´ (though its emptiness can be used to describe it as empty at a localized point given[quantum fluctuations notwithstanding]no actual matter is within whatever frame of reference you are using)it is a field like the Higgs field you referenced. Einsteins field equations describe this field, but without math there exist only poor metaphors (¨spacetime is a trampoline¨) to describe its physical existence.

If you want an intuitive metaphor for something pretty unintuitive, imagine the aforementioned trampoline. You sit in the middle of the trampoline, and throw a ball towards the edge. The ball rolls toward you, because you have warped the trampoline-time around you, and gravity works in conjunction with that to bring the ball towards you.

However in actual spacetime this trampoline would be symmetrical under all observational transformations(any way you look at it) sort of like the person in google sketchup that always faces you unless you unlock the face. Terrible comparison, but it serves the purpose. The gravity that existed on the trampoline is a product of gravity being a product of spacetime, so you can see why gravity cannot be what holds things ¨down¨ to spacetime. It is easiest to describe it as simply following a path along spacetime, with the warp towards the more massive object being the path of least resistance.

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Physically what does warping (of space-time) mean?

Inhomogeneous space. See the Einstein digital papers where Einstein describes a gravitational field as a place where space is "neither homogeneous nor isotropic". And note that curved spacetime isn't curved time and curved space. See this Baez article for a bit about that: "Similarly, in general relativity gravity is not really a 'force', but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial". Also see Inhomogeneous vacuum: an alternative interpretation of curved spacetime.

So there's general relativity and Einstein's field equations that tell us "mass (or equivalently energy) warps space-time, and the warping tells mass how to move", but I'm still having trouble understanding how space (a thing I conceive to be 'empty' or nothingness) can be 'warped'.

It not so much warped as altered or "conditioned", this affect diminishing with distance in a 1/r² fashion. As a result the properties of space are not uniform, altering the coordinate speed of light so light doesn't go straight. Here's the more complete Einstein quote:

"According to this theory the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration. This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that "empty space" in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials g$_{μv}$), has, I think, finally disposed of the view that space is physically empty."

After researching books, on-line articles, and this stack exchange itself I can't seem to find any descriptions or discussions that don't immediately resort to jumping into GR and its mathematics to provide a more physical intuitive understanding of what it means to for space to be warped.

I recommend you read the original material by Einstein. Then you appreciate that space isn't curved. Unfortunately Wheeler confused space and spacetime, and as per the Baez article, the distinction is crucial.

Is it space, this empty thing, that's getting warped?

Einstein said space, he said it wasn't empty, and he said it gets conditioned rather than warped.

Or rather is it some field (Higgs field?) that exists around matter that's getting warped?

It's space that's getting conditioned. To get the gist of it, see the stress-energy-momentum tensor, and note the energy-pressure diagonal. That's pressure, not tension. The rubber-sheet analogy is back-to-front. When you step up to three dimensions, instead of a pull like the picture here, it's a push. Then zoom in to avoid getting distracted by the Earth's curvature, and the light beam curves downward because space is inhomogeneous:

enter image description here

Note that this depiction can be likened to Ricci curvature wherein a volume deviates from the Euclidean norm. When you plot the (vertical) deviation, your plot is curved, but the grid lines aren't. Imagine looking at the Riemann curvature depiction from the top, but with squashed length-contracted squares near the middle rather than stretched squares.

Then does that mean that a body in a 3-D world warps space into 4-D?

No. IMHO it's reasonable to say it warps the metrical qualities of the 4D spacetime continuum, but not space.

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