# Conservation in space-time curvature

Pardon this possibly naive question.

I'm starting to poke around in the topic of General Relativity (as soon as I can pull myself back up out of the vortex of underlying mathematics that I've gotten sucked into) and started to wonder this: is there any sort of "conservation" law(s) associated with space-time curvature?

Perhaps I'm stuck trying to visualize the effects of mass (or acceleration), so let me explain my question a bit more. If the observable universe is expanding from every observer's viewpoint, one model that supports this consists of the observable universe on the surface of an expanding "sphere." In trying to visualize the curvature of the surface of that "sphere," I began to wonder if the "inward" bulging of the "sphere" might not somehow need to be compensated for by a corresponding "outward" bulge elsewhere?

Does this question make sense?

## 2 Answers

Your visualization is a good one for exploring the "no-center" concept of the universe - that is, if you only count the universe as the boundary of the hyper-sphere. Technically, though, it could be wrong.

As you'll find as you look at GR (if you haven't already) is that there are three types of curvature: positive, negative, and flat. A positively-curved surface is like the surface of a sphere. An example of a negatively-curved surface is a saddle. A surface with flat curvature is ordinary Euclidean space - like a perfectly flat tabletop.

So why would a physicist call your idea possibly incorrect? Well, while the "shape of the universe" has been debated for decades, there are some signs that it is flat (Wikipedia covers this pretty well in https://en.wikipedia.org/wiki/Shape_of_the_Universe - see the data for WMAP). Now, this data is not conclusive proof that the universe is flat - other curvatures are still possible - but it seems to swing in favor of a flat-universe. There's still one thing to answer, though - your main question about "conservation" of curvature. Well, on a flat universe, what is the curvature? There is none, and so - at least in our universe - the question is moot.

I hope this helps.

• @Daphne, while I really appreciate the "best answer" vote, I would caution you in the future to wait for other answers. I do find it odd that there hasn't been much activity on this question, or other answers or comments, but this is an exception to the rule. Again, I appreciate the "best answer" vote. Aug 10, 2014 at 21:15
• I would also say that while the curvature of constant time spatial sections in comoving coordinates appear to be flat, the universe appears to be flat, the spacetime curvature of the universe most definitely is NOT flat. Aug 10, 2014 at 21:20
• Whoops, should have been more specific regarding the distinction between space-time curvature and the curvature of - as you put it - "constant time special sections". Does it warrant an edit? Aug 10, 2014 at 21:22
• Wow. Quite the typo - I meant "spatial sections", not "special sections" - although I think the spell-checker changed that. Aug 10, 2014 at 21:55
• Thanks for the all of the comments as well as the tip on "best answer." This helps set a direction for what I want to research. Aug 10, 2014 at 22:46

Curvature dosen't have to balance out with positive balancing negative. A good way to look at it is to see what causes curvature. Firstly, mass, energy, momentum, stress, and pressure are sources of curvature, but they are not the only things that create curvature, curvature itself can create further and additional curvature. A gravitational wave can propagate or even spread in a vacuum of empty space devoid of all mass, energy, momentum, stress, and pressure.

The region outside a symmetric nonrotating static star is curved, even the parts far from any mass or energy or momentum or stress or pressure. The space remains curved because the existing curvature is exactly shaped so as to persist (or otherwise cause future curvature exactly like itself).

So curvature allows and sometimes requires more and/or future curvature, just as a travelling electromagnetic wave allows and/or even requires there be more electromagnetic waves elsewhere and/or later. The vacuum allows curvature far from gravitational sources just as it allows electromagnetic waves far from electromagnetic sources. What electromagnetic sources allow is for electromagnetic fields to behave differently (namely to gain or lose energy as well as move in different ways and gain and lose momentum and stress). Similarly what gravitational sources do is allow curvature to react differently to itself than it otherwise would.

Imagine a flat region of space shaped like a ball, then imagine a funnel type curved space where two regions of surface area are farther apart than they would be if flat (like a higher dimensional version of a funnel and on a funnel surface two circles of a particular circumference are farther away as measured along the funnel then if two similarly sized circles were in a flat sheet). On its own, spacetime doesn't allow itself to connect those two kinds of regions together, but that mismatch is exactly the kind or not-lining-up that putting some mass or energy right there on the boundary fixes. So without mass those two regions can't line up, with mass they can. Just like an electromagnetic field can have a kink if there is a charge there.

So your curvature likes to propagate a certain way, and if you want it to deviate from that, you need mass, energy, momentum, stress, and/or pressure. And you'd need the right kind to get it to match up, the kind you want might be available, and might not even exist, so not all kinds of curvature will be allowed. But the point of a source is that it changes the balance between nearby curvature and not that affects future curvature. So there is a kind of balance, and there are things that can warp that a balance. Those things that warp that natural vacuum balance are called gravitational sources.