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Is Space time curvature responsible for gravity or Gravity responsible for the curvature in space-time.

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    $\begingroup$ Gravity is a physical phenomenon. Spacetime curvature is merely a human description for how it behaves. $\endgroup$ – CuriousOne Apr 17 '16 at 2:45
  • $\begingroup$ oh can you explain the phenomenon of gravity just tell me why gravity exists $\endgroup$ – Faisal Iqbal Baba Apr 17 '16 at 2:53
  • $\begingroup$ Science finds descriptions for reality, not existential reasons. Having said that, some people believe gravity is one of the remnants of one fundamental force, from which every other force can be derived, others think that space, time and gravity are one emergent phenomenon that has something to do with either quantum physics or thermodynamics (probably both, actually). The problem with that is that we don't know what "the thing" is that all these are emerging from and nature hasn't given us any clues, yet. If you answer "We don't know, yet." to your own question, you make a true statement. $\endgroup$ – CuriousOne Apr 17 '16 at 2:59
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Per Einstein's theory, they both are one and the same thing. There is some, yet not known property of space, and mass, that causes this phenomenon. Even if the property becomes known today, the question will be why that property exists?

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As general relativity says, gravity is nothing but a geometry of space time. The bending of spacetime is caused by mass and energy distributions, which we see as gravity. Both accounts to the same thing. The spacetime is affected by mass as well as condensed energy. It could successfully explain why there is gravity and almost everything related to that. But if you ask the source of spacetime, and why it is affected by mass, it's still unknown.
In the case of special relativity, which consists of only inertial frames (no gravity), you can see that the spacetime is flat.

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If we were discussing electromagnetic forces, you might be asking whether electromagnetism causes the forces or vice versa. To make this less of a chicken-egg question, I'd assume "electromagnetism" here means the existence of nonzero values for electric charges and the electric and magnetic fields. The electromagnetic force on a particle of charge $q$ of velocity $v$ in an electric field $E$ and a magnetic field $B$ is $q\left(E+v\times B\right)$. It makes sense to say the force is caused by the aforementioned details of electromagnetism rather than vice versa. In the special case of a force due to a point charge $Q$ a distance $r$ from the charge $q$, the force has magniture $\frac{k_C qQ}{r^2}$, where $k_C :=\frac{1}{4\pi\varepsilon_0}$ is the Coulomb constant. This is because the charge $Q$ generates an electric field proportional to $k_C$.

The analogous constant for gravity is Newton's constant $G$. In general relativity, the equivalent of a "force" is the expression $G_{\mu\nu}+\Lambda g_{\mu\nu}$ (note that only the first $g$ should be capitalised). The Einstein field equation equates this expression to $\frac{8\pi G}{c^4}T_{\mu\nu}$, where $T_{\mu\nu}$ is a description of the space's matter content. This is analogous to a charge distribution that would generate electromagnetic fields. In short, our equivalent of a "force" is due to matter and proportional t the gravitational constant $G$. This is what people mean when they say matter tells space how to curve (that's what the "force" side really described) and space tells matter how to move.

If you'd like to learn more, you'll want to first learn about covariant derivatives and Christoffel symbols, followed by the Riemann tensor, Ricci tensor and Ricci scalar.

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