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.
General Relativity – Is Space Curvature Independent of Gravity?
curvatureenergygeneral-relativitygravitymass
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If we consider the thought experiment where we take the classic 2 dimensional plane curved in a graphic representation of the curvature of space-time, copy it and arrange the copy so the lowest points of the gravity wells are aligned. These copies can be arranged any way you like, as long as the low point, or gravity wells are aligned, then the imagery still works. Then we can see what i think you are asking about. With just a single image, it appears space-time can be stretched and pulled 'downward.' But in 3 dimensions, the second image, or many others, are also possible, implying that space is being curved down and up in the same place.
And so, i assume, your question. My thought was, where did the space go?
The other things that i consider are distance and time. Using the earth for example, over time the earth moves around the sun. In the summer the earth is curving different space-time than it does in the winter. As the earth leaves a place in space, space returns to the shape it was before the earth was present, so it seems that space-time was 'compressed' and then returned to it's original shape.
I find it interesting that space can be expanded, inflated or curved but not compressed. The confusion comes from the comparison of space to a fluid, and fluids cannot generally be compressed. Keep in mind that my first example, the rubber tarp example, and comparing space to a fluid all give us a idea of what space is like and what it is doing, but none of them are perfect.
GR describes curvature as being caused by stress-energy.
This statement is slightly wrong and is the cause of your confusion here.
Technically, in GR the stress energy tensor is the source of curvature. That is not quite the same as being the cause.
An easy analogy is with Maxwell’s equations. In Maxwell’s equations charge and current density are the sources of the electromagnetic field. However, although charges are the source of the field there exist non trivial solutions to Maxwell’s equations that involve no sources. These are called vacuum solutions, and include plane waves. In other words Maxwell’s equations permit solutions where a wave simply exists and propagates forever without ever having any charges as a source.
Similarly with the Einstein field equations (EFE). The stress energy tensor is the source of curvature, but just as in Maxwell’s equations there exist non trivial vacuum solutions, including the Schwarzschild metric. In that solution there is no cause of the curvature any more than there is a cause of the plane wave in Maxwell’s equations. The curvature in the Schwarzschild metric is simply a way that vacuum is allowed to curve even without any sources.
Now, both in Maxwell’s equations and in the EFE the vacuum solutions are not particularly realistic. Charges exist as does stress energy. So the universe is not actually described by a vacuum solution in either case. So typically only a small portion of a vacuum solution is used to describe only a small portion of the universe starting at some matching boundary. A plane wave can match the vacuum region next to a sheet of current, and the Schwarzschild solution can match the vacuum region outside a collapsing star.
So realistically, the cause of the curvature would be stress-energy that is outside of the vacuum solution, in the part of the universe not described by the Schwarzschild metric. This would be in the causal past of the vacuum region including the vacuum inside the horizon. Since it is in the causal past it can be described both as the cause and the source of the curvature, with the understanding that it is strictly outside of the Schwarzschild metric which is a pure vacuum solution in which the curvature has no source.
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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.