# Is the curvature of space around mass independent of gravity?

Is the curvature of space caused by the local density of the energy in that area?Could gravity be a separate phenomenon only arising from the curvature of space? For instance if the density of energy in a particular area cause that area of space to ”curve” but the effect that we understand as gravity, (causing anything with mass to be attracted to each other) is only arising as a consequence of that space being curved. I guess it seems to me that things other than mass can cause the curvature of space (electromagnetic fields, an enormously high density of photons in a small area or at least I think so, but I'm not sure about the photons, and if a black hole rotating causes frame dragging (which I'm assuming means the surrounding physical metric of space is probably some mechanism, or thought experiment where you could ball up space tight enough to become a black hole even without any matter in it. I guess it's another question I Could ask.

As understood by Einstein's general theory of relativity completed in 1915-16, gravity is indeed a manifestation of (nothing else than) the curvature of space and I have some doubts about your implicit claim that you have made this discovery "independently" of Einstein. According to the precise equations of general relativity, the so-called Einstein's equations $$G_{\mu\nu} = \frac{8\pi G}{c^2} T_{\mu\nu},$$ what influences the curvature of spacetime is the stress-energy tensor that knows about the density of energy and momentum and the flux of energy and momentum. Terms like "flux of momentum" may sound obscure but they are described by well-defined mathematical formulae. In particular, "flux of momentum" is nothing else than the component of pressure. So pressure also influences the curvature of spacetime – and therefore the gravitational field and the behavior of objects in this field – according to general relativity.

On the other hand, it is irrelevant for the curvature and gravity whether the same stress energy tensor – the density of mass, energy, momentum, and components of pressure and stress – are achieved by the electromagnetic field, one material, or another material. However, it's still impossible to "create" curvature of space without any material (or energetic) carrier. The equations explicitly show that the Ricci tensor is zero if there's no energy/momentum density in the space. So one can't create a "black hole out of nothing".

Nevertheless, black holes may suck all the material and make the spacetime around Ricci-flat; the Ricci (or Einstein) tensor is equal to zero almost everywhere in the space. This Ricci-flatness is still importantly violated at the black hole singularity which is the reason why the black holes still carry a nonzero mass/energy.

The question is getting increasingly impenetrable as one continues to read it so what you exactly wanted to do with the frame-dragging effect remained unknown to me (and I guess that not only me). Frame-dragging is a particular new gravitational effect that occurs in the gravitational field induced by rotating bodies.

• I can barely discover how to spell many of the words you used, I certainly don't think I'm discovering anything new! LOL much to my regret I seem to be unable to learn the language of mathematics But I am very curious and I'm simply trying to visualize how things work, I'm sure that will leave me unable to comprehend some of the more counterintuitive concepts. But I enjoy trying to learn! Thank you very much for taking the time to trying to help me understand. Jan 25 '12 at 9:44
• @ToddBurkett The ability to visualize is not necessirely a measure for understanding.
– Revo
Jan 25 '12 at 10:49
• In a couple of places you talk about the curvature of space. Shouldn't that be the curvature of spacetime? Certainly, space is curved when spacetime is curved, but sometimes people get the impression that the spatial component of the curvature is much larger than it actually is. E.g., a planet's orbit is a closed space curve, but the spatial curvature of the orbit is much larger than the spacetime curvature of the planet's worldline. Aug 18 '18 at 5:01
• Yes, the spacetime, not just space, is curved, and the warping of time is actually more important for normal gravity than the curvature of the space itself. I used the phrase "curvature of space" because the OP mentioned it. There is still a curvature of space, too. Aug 28 '18 at 5:22

According to GR, in the vicinity of the Earth, for things that are not moving very fast, the gravity we experience is roughly 99.9999% due to the curvature of time and 0.0001% due to the curvature of space. Space near the Earth is extremely flat; the main effect of the Earth's mass is to cause a time dilation field, and the time dilation curves geodesics in a way that looks like a 1-over-R-squared "force". Thus, anyone who simplifies General Relativity down to "gravity is caused by curved space" is almost entirely wrong.

You don't need the full Einstein field equations to see this. If you're willing to assume a spherical non-rotating mass, you can just use the Schwarzschild metric and take the low speed and weak field limits. What you end up with is just flat Minkowski space-time (which has zero gravity) PLUS a time-only term that describes the time dilation. And yet that simplified metric still gives Newtonian gravity. That means that gravity, in the Newtonian limit, is ENTIRELY caused by the time dilation field. The phrase "gravitational time dilation" gets the causality precisely backwards; we should probably actually be talking about "time-dilational gravity". The reason that your butt is being pressed into your seat is that time is moving faster at your head than at your feet.

Caveats: This analysis is NOT valid near black holes (where you can't assume weak field) or for light (which violates the low speed assumption). But for people or planets, it is very very close to correct.

Curvature is curvature, and curvature is one of the very things that can cause curvature, what gravitational sources do is change how different kinds and regions of curvature interact with each other. They interact one way on their own and in a different way when sources are around.

For example, 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.

Having curved spacetime is something we observe. Having gravitational sources that can change the normal or usual way curvature evolves is something else entirely. We can make theories about how the sources evolve, and then the curvature is forced to co-evolve with it, and that's what gravity is about, about gravitational interactions (source and curvature together) changing how the curvature evolves changing the evolution that the curvature otherwise would have evolved a different way.