I ask your opinion? Does matter expand with space?[/quote]
What does that mean? We only see matter, we don't see space. It is the distance between matter that apparently increases.[/quote]


Well Does matter curve with curved space?

It is space that expands, and it carries matter along with it. In addition to being carried along in space as space expands, matter may also be moving with respect to the space that it in. But what is described as expansion of the Universe is that of space itself by virtue of the fact that it possesses energy over and above that possessed by the matter contained in that space. This energy is described by Einstein's cosmological constant.

For example, the Andromeda galaxy is being carried away from us by the universal expansion of space, but it also has its own local movement through space. In the case of the Andromeda galaxy, it is so close to us that the recession that the expansion of space imparts to it is so small that it is more than offset by its own local motion through space, which happers to be toward us instead of away from us. For much more distant objects, their local movement is overwhelmed by the expansion of space itself, so all of them display red shifts denoting recession.

In addition to the general constant expansion of space, it has recently been determined that space possesses what has come to be called "dark energy" that is causing the expansion of space itelf to accelerate.

As this acceleration causes the expansion rate to increase, it is expected to also affect matter on smaller and smaller scales. Currently, it affects large bodies such as galaxies uniformly, not affecting their contents differentially. But it believed that as time goes on, smaller structures will be affected differentially, so that planetary systems will be disrupted, planets being drawn away from their stars. Then planets will lose their satellites. Then planets will fall apart, then atoms will lose their electrons, then quarks comprising nuclei will separate, leaving only a "quark soup" like what existed back around 10 exp -43 seconds after the Big Bang. We won't be around to witness that.

I'll offer my point of view on your original question. I believe it will simplify the picture considerably. It was, "Is the expansion of space constant?"

The simple answer is, "No", but that answer won't mean much to you without some essential details.

For just a moment, picture the Universe as two- instead of the three-dimensional entity that we believe it to be. Picture it as occupying the surface of a balloon that is expanding. Galaxies are tiny dots distributed uniformly over its surface. The distance between any two dots is increasing as time goes on. Way back in the past, 13.7 billion years ago, the balloon was a mere dot, but over the course of time, it expanded to its present size. We have no idea how large it is, but we suspect that we can see only a tiny part of it, only the dots within a short distance of the dot representing our own galaxy. Light hasn't yet had time to reach us from the remaining dots.

The actual Universe isn't two-dimensional like the surface of the sphere, but three-dimensional. But it is still somewhat like the surface of a sphere, only the sphere is four-dimensional instead of three-dimensional, and our Universe is the "surface" of the four-dimensional HYPERSPHERE. Just as all points on the surface of the sphere are equidistant from the center of the sphere, all points in the three-dimensional space comprising the "surface" of the four-dimensional hypersphere are equidistant from the center of the hypersphere. Just as inhabitants of the two-dimensional surface of the sphere can visualize only of other points on that surface and cannot visualize the direction to the center of their sphere, we cannot visualize the direction in the fourth dimension to the center of the hypersphere on whose three-dimensional "surface" we live. But every point in our three-dimensional Universe is moving away from that point at the center of the hypersphere.

It used to be assumed that the hypersphere was expanding at a uniform rate. It was discovered within the last ten years that the rate of expansion has been increasing for at least the last seven billion years or so because of "dark energy" that space itself apparently possesses as an integral property of itself. By "the expansion of space", we mean the expansion of the three-dimensional "surface" of the hypersphere as the radius of hte hypersphere increases. Dark energy is believed to be a mysterious intrinsic property of space itself.

Does matter also curve with curved space?

I've never heard any put forth such a theory.

Well it is a general question ... are you claiming no?

What I am claiming is that I don't know of anyone who has put forth such a theory. GR isn't my area of expertise, but I think that the curvature tensor acts on space-time only.

The question is very straightforward. Does matter also curve with space-time?
I am asking for an opinion.

I personally have no opinion on that matter, no pun intended. Physics is a model of the universe, and is "correct" only to the extent that it makes useful predictions that agree with observation. If you can develop a theory in which matter curves with space-time, and find that it is useful in predicting and explaining observations beyond that of what current theories are capable, then your question becomes meaningful to me.

I am just asking a question. I think a fairly important one. Personally I have my belief in what the answer is and I wanted to see if that was what the mainstream theory also states.

First we might ask ourselves if spacetime "really" curves, or if that's isn't just a descriptive metaphor for the mathematics of the metric.
If it's just a story about the metric, then it's not a story about matter.

Grant Hutchison

I goofed in my answer to the original question. I said, "No" when I evidently meant to say "Yes". for I went on to argue for the "Yes" answer. I was evidently sleeping peacefully when I wrote that "No". To the extent that matter occupies spacetime, it follows the curvature of spacetime. I used the image of our Universe as laid out on the three-dimensional "surface" of a four-dimensional hypersphere. Objects on thaat "surface" follow its curvature. That curvature might be nearly negligible for a small object like a planet, but it becomes very significant for an entire galaxy.

I currently personally believe that there is another hidden dimension one that somehow is related to the contraction/curvature of space. Still working on the math though.

Well, good luck with that.
I believe the maths is already done and tested, in the form of general relativity. But whether the maths describes a real thing, curving in a way we can understand: I doubt if the maths can tell you.
To quote from one of Tim Thompson's excellent reading recommendations, Edward Harrison's Cosmology: The Science of the Universe:Grant Hutchison

tommac: I currently personally believe that there is another hidden dimension one that somehow is related to the contraction/curvature of space.

dcl: I agree. I suggest you look at the lead string in my thread, entitled "The Shape of the Universe", where I argue that the Universe is a four-dimensional hypersphere of which its three-dimensional "surface" is the Universe that we know. Its surface is curved, and its center is in a direction that we cannot point. I offered as what I believe to be a more plausible description of the shape of the Universe than the doughnut-shaped thing that Dr. Gay described in her podcast with the same title. The lecturer at a lecture on Big Bang cosmology that I attended told me that he was familiar with this model and regarded it as plausible.

I also agree about the plausibility.
There are times when I think that cosmologists are trying so hard to explain what they see, that they are forgetting that we don't see quite a bit.

It will have to wait until we can (technologically) make better observations but, I cannot help but think we are in for a few surprises.

Gentlemen, I hate to rain on this little higher dimensional parade, but I'm gonna have to. The whole higher point of differential geometry is *you don't need it*. However, it can be a very useful visualizing tool to help you understand.

For example, you can imagine a curved 2D space as a curved surface in a flat 3D space. The defining (and simplest) example is the surface of sphere (that turns out to be constant positive curvature) -- one imagines flat landers on the surface aware of only 2D dimensions. And one extends that up one dimension and imagines a closed 3D space of positive constant curvature that is the "surface volume" of a hypersphere.

One can then work out a mathematical framework about how to do things by using coordinates confined to the surface. The higher flat space is useful because it is Euclidean -- Pythagoras rules -- and you can work out how things work in the curved coordinates from your higher dimensional vantage point because you intuitively know how flat space works.

But then you get to the point. Once you've worked out the machinery for dealing with the curved coordinates that live in the curved space, you realize that you don't need the higher dimensional space at all. That was just a crutch so to speak. And then you realize something more fundamental: what so special about "flat" space in the first place? Once you have the general curved rules to guide you, you realize flat space is just a special case where things work out very simply, or maybe I should work out *as we expect*.

Our minds somehow declare that "real space" must be flat, must be Euclidean, and so if there's any "curvature" funny business going on, it must be because the real space is higher dimensional and we're somehow living on a curved hypersurface embedded therein. But, once you've "got it", you realize that's not the case. There's no reason space must be "flat", and no reason a curved space must be embedded in a higher flat space.

But that's just "space", and by that I mean something with a positive definite metric signature. IOW, the (invariant) notion of "distance" (the norm) is always positive and, in the most simple coordinates involved the Pythagorean sum of squares.

Well, space-time is not positive definite. Even flat space-time, Minkowski, is *non-Euclidean*. It's not positive definite. The invariant "distance"/norm there (s/ds) can be negative. The distance formula involves the differences of squares of the time-like and space-like parts.

So even in flat space-time, we've left our dear Euclid behind and moved on to some abstraction that is not like any "real space" our minds can visualize.

And then GR lets that get curved. But our minds persist in trying to paint pictures, filling in gaps of reality that don't have to be there.

-RIchard

Now, that said, you can play embedding space games in GR, and some space-times can be visualized as curved hypersurfaces in a higher dimensional, flat, Minkowski like space-time.

Indeed, dcl's hypersphere has some utility here. DeSitter space-time (empty space with a positive Lambda) can be visualized neatly as an expanding hypershphere in a 1,4 space-time. 1-4 means 1 time-like dimension and 4 space-like dimensions.

Imagine an observer in this 1-4 space-time at the "center". Let the radius of a hypersphere around him expand, and expand in a Rindler hyperbolic accelerating world line. The subspace of the surface of that hypersphere is the curved, 1-3 deSitter space-time.

[Because of the non-positive definite behavior here, hyperbolas behave like spheres. And you can see the space-like part of the hyperbolic structures can be nicely spherical. Anyway, this hyperbolic sub-hypersurface is one of constant space-time curvature. That is given by Lambda, and that relates directly to the proper acceleration of the radius. So you can think of Lambda as how fast the hypersphere is accelerating.]

All well and good and neat. But throw matter in the mix for a LCDM universe and it doesn't work so neatly anymore. We've just let the air out of our little hyperballoon.

And that's the problem. Many valid curved 1-3 space-times (and by valid I mean they are solutions of the EFE and so can represent some plausible space-time) cannot be embedded in a 1-4 flat space-time. In fact, some of them will require 2 or more time-like dimensions to do so.

I just get off the train at the thought of 2 or more time dimensions. Indeed, the opposite of deSitter space, one with constant *negative* curvature, requires an embedding space of 2 time and 3 space. A certain subsurface gives you a 1-3 sub-space-time that is anti-deSitter space-time (and that sucker is very weird, indeed).

And guess what. Even simple Schwarzschild cannot be embedded in a 1-4 higher space-time! Oddly, if you supress the tangential directions and just have a 1-1 space-time using r and t, *that* can be emdedded in a 1-2 flat space-time. And you can draw pictures of that. However, add the two additional space dimensions, and it no longer works.


-Richard

While embedding spaces are seductive at first blush, I've hope I've given you a taste of why they just don't work out like you would hope.

And finally, I've read that "GR is topologically incomplete". What that means exactly, I'm not sure, and we'd need the high priests to try to 'splain that (if they could). The way I understand it is that given space-times that are valid solutions of the EFE (such as our universe), there is some ambiquity in the "shape" in the sense that some topologist would use when he declared something topologically "complete".

IOW, all the information that GR provides is not enough to pin down the ultimate "shape of the universe".

Sorry for raining on the parade.

-Richard