Understanding Positive and Negative Intrinsic Curvature in General Relativity

[ktx49]

if gravity arises from normal accelerations due to the curvature of spacetime...what would the opposite of this "process" represent?

to clarify is it possible to describe the opposite of this curvature??

thanks

 

[WannabeNewton]

I'm not sure you clarified enough unfortunately. Gravity is a manifestation of the curvature of space-time itself, which is in turn dependent upon the dynamics of mass-energy-momentum distributions throughout space-time. What exactly do you mean by "opposite" of this curvature?

 

[ktx49]

We can both describe the curvature of spacetime through mathematical equations and also visualize it(conceptually) with the classic picture of a heavy ball placed in the center of taunt bedsheet etc

what I'm looking for is a description of the exact opposite "process"...what/how could we describe this?

at minimum, is there a term or description for the opposite of curving/bending of spacetime?

 

[WannabeNewton]

Humans cannot visualize space-time curvature (not a normal human anyways). That ball in a trampoline analogy is nowhere near a proper visualization of space-time curvature. I do, however, have a vague idea of what you are asking for. In those (albeit horrible) ball in a trampoline analogies, the curvature is shown as a right-side up bowl of sorts. Are you asking if we can have curvature resembling, at the least locally, an upside-down bowl?

 

[A.T.]

Are you asking if we can have curvature resembling, at the least locally, an upside-down bowl?

An upside-down bowl has the same geometry as the original. Turning it upside down changes nothing about the geodesics, and thus the effects of the curvature.

 

[Dale]

We is there a term or description for the opposite of curving/bending of spacetime?

Do you mean "flattening" or do you mean "curving" in the opposite direction?

 

[WannabeNewton]

An upside-down bowl has the same geometry as the original. Turning it upside down changes nothing about the geodesics, and thus the effects of the curvature.

I was simply asking the OP if that was what he/she was visualizing. I was not making any assertion of such sort. The OP is clearly visualizing spatial submanifolds embedded in higher dimensional euclidean spaces. Also, things which locally bend away from normal planes through a given point have positive normal curvature whereas things which locally bend towards normal planes through a given point have negative normal curvature. This is a trivial fact. It's analogous to the second derivative of $f(x) = x^2$ versus the second derivative of $f(x) = -x^{2}$. This is related to how the spatial submanifolds are embedded in higher dimensional euclidean spaces and their subsequent orientations.

 

[ktx49]

Do you mean "flattening" or do you mean "curving" in the opposite direction?

woot! someone is finally understanding what I'm getting at here...

I can easily conceptualize the flattening of space...if for example a massive object was instantly removed from existence I can imagine there's a description of spacetime "flattening" back out.

either way I'm definitely referring to the "curving" in the opposite "direction" in my posts...

thanks for helping to clarify

 

[A.T.]

We can both describe the curvature of spacetime through mathematical equations and also visualize it(conceptually) with the classic picture of a heavy ball placed in the center of taunt bedsheet etc

That is a very misleading picture. It shows at best space geometry not space-time geometry. And it has little to do with the observed gravitational attraction. Here is how that works locally:

https://www.youtube.com/watch?v=DdC0QN6f3G4

And here more globally:
http://www.adamtoons.de/physics/gravitation.swf

An inverse of this could be if the temporal distances were deceasing closer to the source. This would cause repulsion from the source and gravitational time acceleration, in such a model.

 

[ktx49]

maybe the term I should use here is "uncurling" or "unfurling" of spacetime?

 

[WannabeNewton]

either way I'm definitely referring to the "curving" in the opposite "direction" in my posts...

Perfect, so you were visualizing what I thought you were visualizing . What you are seeing is a geometrical shape embedded in a higher dimensional space and seeing how this shape bends in that higher dimensional space. Going back to the example of the graph of the function $f(x) = x^{2}$ versus the graph of the function $f(x) = -x^{2}$, we visualize the two as an upwards facing bowl and downwards facing bowl, respectively, as embedded in two dimensional euclidean space. Their extrinsic curvatures then differ by a sign because of the flip in orientation.

However, this extrinsic curvature is only defined when given such an embedding. Space-time curvature is an intrinsic curvature which you cannot visualize in the above sense; it is something you can measure locally using various geometrical tools e.g. the parallel propagator. It is not a curvature gotten by looking at how space-time bends in some higher dimensional space-time. As such, it is not physically meaningful to ask if it is oriented in one direction or in the opposite direction, as far as general relativity is concerned.

 

[ktx49]

AT I understand the visualization of gravity as the bending of spacetime like a ball on a trampoline is an incomplete picture at best...but its certainly good enough to jump start our discussion here.

I think your last post shows you do indeed understand what I'm attempting to get at here...

 

[Dale]

either way I'm definitely referring to the "curving" in the opposite "direction" in my posts...

I am glad I asked.

In this case, AT's post 5 applies. Basically, there are two kinds of curvature, "intrinsic" and "extrinsic". Here is an explanation: http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/non_Euclid_variable/index.html#Intrinsic

The idea of extrinsic curvature requires a surface to be embedded in some higher dimensional flat space. However, we, in our 4 dimensions, do not have any access to these higher dimensions. So all of the notions of curvature in general relativity are intrinsic curvature. For intrinsic curvature, there is no distinction between curvature in different directions in any hypothetical embedding space.

So the short answer is that there is no "opposite process" as you intend.

 

[ktx49]

It is not a curvature gotten by looking at how space-time bends in some higher dimensional space-time. As such, it is not physically meaningful to ask if it is oriented in one direction or in the opposite direction, as far as general relativity is concerned.

ok cool so I think you've answered my question(s) but to make sure...

you're saying a "flat" part of spacetime curving would be the exact same thing as it "uncurving" & that a "direction" to this curvature arises purely from our brains attempting to visualize spacetime in a higher dimensional manifold(ie. ball on trampoline)

 

[A.T.]

I'm definitely referring to the "curving" in the opposite "direction" in my posts...

If you mean that the bulge is mirrored and goes the other way, that changes nothing, as I said. The distances within the surface don't change.

If you mean negating the intrinsic curvature at every point, I don't how this would look like, and if it is even possible to do it for a bulge in a plane. Locally it means this:

http://www.britannica.com/EBchecked/media/90070/Intrinsic-curvature-of-a-surface

But a bulge in a plane has areas positive curvature (in the middle) and negative curvature (at the transition tot he plane). I don't see how you can invert it at every point.

 

[WannabeNewton]

ok cool so I think you've answered my question(s) but to make sure...

you're saying a "flat" part of spacetime curving would be the exact same thing as it "uncurving" & that a "direction" to this curvature arises purely from our brains attempting to visualize spacetime in a higher dimensional manifold(ie. ball on trampoline)

I'm not saying flat and curved are the same. I'm saying that the "direction" (i.e. orientation in some higher dimensional space) won't matter as far as space-time curvature goes because the space-time curvature is intrinsic to the space-time, unaffected by how it is embedded and oriented in some higher dimensional space. This is in the spirit of one of the most celebrated theorems in the differential geometry of curves and surfaces-Gauss's Theorema Egregium: http://en.wikipedia.org/wiki/Theorema_Egregium

This is why it is important to have a distinction between intrinsic curvature and extrinsic curvature. And yes what you naturally try to picture is the extrinsic curvature.

 

[ktx49]

I'm not saying flat and curved are the same. I'm saying that the "direction" (i.e. orientation in some higher dimensional space) won't matter as far as space-time curvature goes because the space-time curvature is intrinsic to the space-time, unaffected by how it is embedded and oriented in some higher dimensional space.

I understand that flat and curve are not the same...i think you missed the word curving in my sentence here...

you're saying a "flat" part of spacetime curving would be the exact same thing as it "uncurving"

anyways, your responses still apply...although to make sure, I'm still asking whether there is any description of spacetime "uncurving"...ughh its the only word I can come up with to describe what I'm thinking of here.

maybe inverting the curvature would be better terminology or help to clarify my questions?

either way, thanks for responses and taking the time to help explain these things. great stuff

 

[WannabeNewton]

The notion of space-time curvature bending "upwards" or "downwards" in one way or another is not something we can meaningfully define physically because, as far as GR is concerned, space-time is all there is so we can't embed it in some natural higher dimensional space and see how it bends ("upwards", "downwards" etc.) in that space. Mathematically this can be done but it has no physical meaning in GR. We care only about the intrinsic curvature of the space-time manifold. This was in fact one of the motivations for Riemannian geometry as a whole, even before GR came along.

 

[ktx49]

ok so that link one of you posted about variable curvature is proving to be very useful...

since we have access to only the 4 dimensions we know of, we obviously treat spacetime curvature as intrinsic...which makes perfect sense to me.

so if we visualize ourselves like some 2D flatlanders drawn on a sheet of paper, the intrinsic curvature of spacetime would be analogous to rolling the flatlander's sheet of paper into any number of curved geometric shapes. is this correct?

i'll stop there for now, make sure we are on same page

 

[WannabeNewton]

I am still unsure of what it is you are asking exactly but it seems like you are asking: how do we actually measure/quantify intrinsic curvature? If so, see the following passage from the textbook "General Relativity"- M.Wald: http://postimg.org/image/58tua25td/

A creature living on a given Riemannian manifold could construct such an small closed loop in some region of the manifold, parallel transport the vector around the loop, and see how the vector changes when it comes back to its initial starting point. This will allow the creature to measure the Riemann curvature in that region (the Riemann curvature is a type of intrinsic curvature).

 

[Dale]

so if we visualize ourselves like some 2D flatlanders drawn on a sheet of paper, the intrinsic curvature of spacetime would be analogous to rolling the flatlander's sheet of paper into any number of curved geometric shapes. is this correct?

Actually, that is extrinsic curvature. Suppose that you take a flat sheet of paper (intrinsically flat), and you draw a triangle on that flat sheet of paper. You then measure the angles in the triangle and find that they sum to 180º.

Now, take that paper with the triangle and fold it into an oragami crane. Now, measure the angles in the triangle and find that they still sum to 180º. So even though the crane is folded (complicated extrinsic curvature) in 3D, the 2D paper is still intrinsically flat.

Now, take a globe and draw a triangle on it. Measure the angles in the triangle and you will find that they sum to more than 180º. A sphere has positive intrinsic curvature. You can cut the triangle off the globe and fold it in any way you like and that 2D measurement of the triangle geometry still shows that the globe is intrinsically curved. Furthermore, you cannot flatten the globe without stretching some part of it.

Does that help understand the difference between intrinsic and extrinsic curvature?

 

[ktx49]

Actually, that is extrinsic curvature. Suppose that you take a flat sheet of paper (intrinsically flat), and you draw a triangle on that flat sheet of paper. You then measure the angles in the triangle and find that they sum to 180º.

Now, take that paper with the triangle and fold it into an oragami crane. Now, measure the angles in the triangle and find that they still sum to 180º. So even though the crane is folded (complicated extrinsic curvature) in 3D, the 2D paper is still intrinsically flat.

Now, take a globe and draw a triangle on it. Measure the angles in the triangle and you will find that they sum to more than 180º. A sphere has positive intrinsic curvature. You can cut the triangle off the globe and fold it in any way you like and that 2D measurement of the triangle geometry still shows that the globe is intrinsically curved. Furthermore, you cannot flatten the globe without stretching some part of it.

Does that help understand the difference between intrinsic and extrinsic curvature?

sorry for delayed response.
yes I believe I have a good understanding of the difference between extrinsic and intrinsic curvature...however in the flat-lander example I think you have misunderstood what I was getting at.

to be sure...if we draw that triangle on the sheet of paper AFTER it has been rolled/curved/folded it will certainly have intrinsic curvature, along with the rest of the 2D universe of these flatlanders, correct?

how does something embedded within this plane/manifold make intrinsic changes to its own intrinsic curvature? maybe I'm looking for an analogy that doesn't exist.

as far as my original questions...lets forget all this talk about varying directions, orientations, or even "negative" curvatures(meaningless).

to put it simply, what is the term or description for the opposite process we know as gravity?

if spacetime bends, curves, or distorts in the presence of matter to produce what we call gravity, then what happens to a curved region of spacetime if/when the matter is removed?

thanks

 

[WannabeNewton]

No, the intrinsic curvature will not change if you for example roll up a flat piece of paper into a cylinder. You are again thinking of extrinsic curvature. If matter is moved away from a region then the space-time curvature in that region simply changes to reflect that; for example it can become "weaker" in that region as the matter is moved away. I'm not sure what kind of effect you're after with regards to "opposite" of gravity. In classical general relativity gravitation can be shown to contribute to an attractive effect on nearby particles much like Newtonian gravity is attractive.

 

[A.T.]

to put it simply, what is the term or description for the opposite process we know as gravity?

By "opposite to gravity" to do you mean repulsion or just lack of gravity?

what happens to a curved region of spacetime if/when the matter is removed?

It becomes flat spacetime.

 

[ktx49]

ok forgetting all the curvature etc for now and just focus on what exactly spacetime is doing when it changes from an initial state of being curved by the presence of matter/energy to a state of little to no matter("empty space")...initially it was "curved" by the matter and yet at some point it became a region of flat spacetime like AJ said.

and yes AJ I'm asking for the opposite process by which spacetime is curved to create what we call gravity...

What I'm asking is how we describe spacetime that is uncurving or moving from a state of curvature to a state of flatness. sorry for the terrible terminology but for something like electromagnetism we have clear terms like attraction and repulsion to describe the opposing forces but in the case of gravity we consider the force to be a consequence of the bending of spacetime itself so I'm not sure something like repulsion is adequate to describe what I'm looking for here.

either way I had an interested thought when one considers that matter is neither created or destroyed and is limited to moving around at the speed of light(as are gravitational waves).
then one can also imagine that spacetime has always been "curved" starting from the initial moments of the big bang and that the distribution of this curvature is the only thing changing and seemingly diffusing as matter coalesces(if spacetime itself is truly expanding).

 

[WannabeNewton]

Space-time asymptotically approaching flatness in some region due to localized matter being moved farther and farther away from said region will be described by Einstein's equations. Note that even in general relativity, gravity can be shown to be attractive.

 

[Dale]

sorry for delayed response.
yes I believe I have a good understanding of the difference between extrinsic and intrinsic curvature...however in the flat-lander example I think you have misunderstood what I was getting at.

to be sure...if we draw that triangle on the sheet of paper AFTER it has been rolled/curved/folded it will certainly have intrinsic curvature, along with the rest of the 2D universe of these flatlanders, correct?

No, the embedding in the higher dimensional (3D) space is completely irrelevant for intrinsic curvature in the lower dimensional (2D) space. It doesn't matter if the folding or unfolding is done before, after, or during the drawing (although it may be hard to draw on a folded piece of paper). There is nothing relevant about the higher dimensional embedding when determining intrinsic curvature.

how does something embedded within this plane/manifold make intrinsic changes to its own intrinsic curvature? maybe I'm looking for an analogy that doesn't exist.

That would depend on how the intrinsic curvature of the manifold is related to things that the flatlander can change. In the case of GR, you would change the intrinsic curvature of spacetime by moving stress-energy around.

to put it simply, what is the term or description for the opposite process we know as gravity?

What is the opposite process to electromagnetism?

if spacetime bends, curves, or distorts in the presence of matter to produce what we call gravity, then what happens to a curved region of spacetime if/when the matter is removed?

It flattens.

 

[pervect]

A slightly over-simplified way of looking at intrinsic curvature of a plane slice is this:

Draw a small triangle on the plane slice. (How do you draw a triangle? Pick three points, and draw or compute the curve of shortest distance between each pair of vertices).

Compute the angular excess (the sum of the angles of the triangles) minus 180 degrees, and divide it by the area of the triangle. This is the intrinsic curvature of the plane slice.

If you do this on a flat sheet of paper, you'll get zero. If you do this on the surface of a sphere, you'll get a positive number. This might not be obvious at first glance, a bit of reading might be in order. I'll give one example to demonstrate, that example is a triangle on the surface of a sphere that has three right angles. You draw this triangle by starting at the north pole, going down to the equator, making a 1/4 circuit around the equator, an going back up to the north pole.

(On a sphere, the curves of shortest distance between any two points are great circles. ALl the curves we draw are arcs of great circles, so it satisfies our requirements that about the triangle being constructed by the curves of shortest length connecting the three points).

All the angles of this (spherical) triangle are right angles, and 3*90 = 270, so this triangle has an angular excess.

The area of this triangle is 1/8 the surface area of the sphere.

Thus we can quantify the fact that the surface of a sphere is curved, while a plane is flat, with this notion of intrinsic curvature, which we can compute by actual measurements on the surface.

As far as I know, there aren't any hidden gotchas (as long as you use _small_ triangles), and I think the above is equivalent to the more formal defitions that involve "parallel transport", while avoiding this unfamiliar term which would probalby require at least a post (and more preferably a book) to explain. But I don't at this point have a rigorous proof that this simplified definition is completely equivalent to the more formal one.

I think I got the sign reight too (a sphere has a positive curvature).

Note that if you have a space of higher dimension than 2, you'll have a different intrinsic curvature for every different plane slice. It turns out there is a rank 4 tensor that will give you the curvature of all possible plane slices and that this rank 4 tensor defines the general notion of intrinsic curvature in a space of dimension higher than 2. This may be too advanced to get into more details at the moment, it ight be better to stick with the above notion of curvature of a plane slice, which is known as "sectional curvature".

 

[ktx49]

I truly appreciate all the replies as well as the abundant amount of patience displayed by those who've responded to this thread...sorry for my vagueness and absence of formal terminology.

I now see the difference between this:

and rolling/bending a 2D sheet of a paper into a similar shape...i think it's easiest to visualize this difference when one imagines trying to flatten them out.

anyways when I asked for the opposite description of GR gravity, I'm looking for what happens to spacetime when matter is moved away...you are quick to say it "flattens", but what does that even mean unless you are visualizing spacetime as being embedded in a higher dimension?
maybe I'm missing something, but flattening would inherently imply a direction to the curvature...

 

[WannabeNewton]

You don't need to embed anything in a higher dimension in order to talk about intrinsic curvature. That's what intrinsic curvature means: it is independent of any embedding because it is intrinsic to the manifold. It is intrinsic curvature that we speak of when we say that the aforementioned region of space-time asymptotically flattens out i.e. the curvature of that region asymptotically approaches that of flat space-time.