# Curvature of Spacetime?

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1. Jun 13, 2013

### 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

2. Jun 13, 2013

### 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?

3. Jun 13, 2013

### 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?

4. Jun 13, 2013

### 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?

5. Jun 13, 2013

### A.T.

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.

6. Jun 13, 2013

### Staff: Mentor

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

7. Jun 13, 2013

### WannabeNewton

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.

8. Jun 13, 2013

### ktx49

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

9. Jun 13, 2013

### A.T.

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:

And here more globally:

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.

10. Jun 13, 2013

### ktx49

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

11. Jun 13, 2013

### WannabeNewton

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.

12. Jun 13, 2013

### 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...

13. Jun 13, 2013

### Staff: Mentor

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.

14. Jun 13, 2013

### ktx49

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)

15. Jun 13, 2013

### A.T.

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 [Broken]

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.

Last edited by a moderator: May 6, 2017
16. Jun 13, 2013

### WannabeNewton

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.

17. Jun 13, 2013

### ktx49

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

18. Jun 13, 2013

### 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.

19. Jun 13, 2013

### 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

20. Jun 13, 2013

### 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/ [Broken]

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).

Last edited by a moderator: May 6, 2017
21. Jun 13, 2013

### Staff: Mentor

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?

22. Jul 8, 2013

### ktx49

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 doesnt 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

23. Jul 9, 2013

### 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.

24. Jul 9, 2013

### A.T.

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

It becomes flat spacetime.

25. Jul 11, 2013

### 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).

Last edited: Jul 11, 2013