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Mass (Energy) creates space-time curvature and thus it forms the reason for gravity. Can it be vice-versa too? Like, mass created gravitational field, gravitational field created space-time curvature?

I would like to agree, because, in that sense, I can say electromagnetic field also produces space-time curvature! This can lead to a unification between them! (that's worth a question)

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4 Answers 4

I disagree with your premise that fields and curvature are different at all.

The gravitational field strength tensor is (or, can be seen as, but usually isn't) the Riemann curvature tensor of spacetime.

Likewise, the electromagnetic field strength tensor is the curvature tensor of a gauge principal bundle.

The fields and the curvatures are not distinct objects, and one cannot meaningfully talk about which of them is fundamental, and which of them is derived. By gauge theory, unification in the sense of viewing both gravity and all other forces coming from connections and their induced curvatures on bundles has already been achieved. The problem is that gravitational theories are non-renormalizable, and thus not as easily incoporated into the QFT framework as gauge theories, who presume a fixed (Minkowski) metric on spacetime.

However, as the electromagnetic field inherently contains energy/momentum, it is, by Einstein's famous formula, inherently equivalent to mass, so, indeed, the presence of electromagnetic fields will curve spacetime. There is a portion of the stress-energy tensor due to electromagnetic fields. However, this is not all the EM field does, as it acts on charged bodies as something which is not the curvature of spacetime, but the curvature of the gauge bundle.

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Although this is a decent answer, I do not think it captures exactly what the OP is asking about. I think the main point is whether the EM field contribution to spacetime curvature is "the same" as the contribution from mass (the question is somewhat ill-defined...). In your answer you address whether we can view EM and gravity in a similar way (i.e. as gauge theories). –  Danu Jul 21 '14 at 18:09

Spacetime curvature and gravity are not two distinct concepts, they are one and the same. You can say that one "translates into" (can be represented and looked at as) the other, they do not have a cause-and-effect type of relationship. Gravity simply is the curvature of spacetime.

To do a quick analogy - it's like someone saying either "Au revoir" or "Goodbye" to someone who speaks and understands both French and English - and you're asking if "Au revoir" means "Goodbye" OR vice-versa.

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Mass and energy (and momentum and stress and pressure) are not the only things that create curvature, curvature itself can create further and additional curvature. A gravitational wave can propagate or even spread.

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 or even requires there be more electromagnetic waves. 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 surface where two circles of a particular circumference are farther away as measured along the funnel then if they 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.

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Space-time is affected by energy forms, like gravity, electro-magnetic flux-but in yet unknown ways. Space-time shifts under influence of these fields, but you will agree that the mass and electro-magnetic field too shifts mass and electro-magnetic fields! Hence, we may notice immeasurable change in mass under the influence of the energy-for mass and space-time are related with mutual affectation.

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