I was wondering a hypothetic situation where I imagine our sun vanishing at an instant. Now, from relativity, information cannot travel faster than light, so the fact that, not only we known that our sun is disappeared but also earth stop rotating because of sun's gravity after 8 minutes.
But, according to General Theory of Relativity, gravity is the distortion of spacetime due to massive object so, If sun disappear, would not the spacetime instantly and would affect the Earth's orbit?
General relativity isn't just the statement "mass curves spacetime." It has an equation that describes exactly how mass curves spacetime (which is frankly very hard to understand and one kind of needs to take a whole course to be able to use it): $$G_{\mu\nu}=\frac{8\pi G}{c^4}T_{\mu\nu}$$ The whole point of Einstein's venture to make general relativity was to unite special relativity with gravity. And one of the predictions of general relativity was that gravity, like electromagnetism, propagates at the speed of light. So for example when two black holes collide, the gravitational waves produced will reach an observer at a later time $d/c$, where $c$ is the speed of light and $d$ is the distance between the observer and the collision event.
Unfortunately you can't just take the statement "gravity is the distortion of spacetime due to a massive object" and try to come to conclusions about how spacetime changes in response to a situation you've come up with. You have to use Einstein's equation to see how the shape of spacetime responds to your situation. And Einstein's equation has a lot more going on than just mass curving spacetime - it has gravitational waves for example (moving distortions in spacetime in the absence of any mass).
But there's a difficulty with the idea of deleting the sun... just deleting mass/energy is actually not allowed in general relativity [ignoring the expert-level discussion about how spacetime expansion doesn't conserve energy]. In some sense, general relativity doesn't say anything about what happens when you delete the sun, because deleting the sun isn't compatible with general relativity. That's why I instead went with the collision of two black holes, where total energy is conserved - although a lot of energy is converted into gravitational waves (unlike if you were to just delete one black hole).
No. The principle of relativity applies to everything, not only light. So it will take the same 8 min until gravitational perturbation (aka "gravitational wave") will reach the Earth.
