So I get the analogy of the fabric of space-time bending as a sheet would bend with the weight of a ball on it, but that explains the "outside" how would the fabric warp inside the earth? Like the "line" passing through the center of the earth, would it be bent? What about some meters around it?
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1Fabric is not a great analogy. Four-dimensional spacetime is what is actually curved, and at every point it takes 20 numbers to fully describe the curvature in general. So, really, if you want to go beyond the fabric analogy, you have to start learning differential geometry. Fortunately, this requires little more than calculus. – G. Smith Sep 25 '19 at 23:48
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Thanks will take a look on that. – Caio Keto Sep 25 '19 at 23:48
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The main mathematical idea is that the Pythagorean Theorem gets generalized. A “curved space” is basically one where the Pythagorean Theorem doesn’t hold. The curvature measures how badly it fails. – G. Smith Sep 25 '19 at 23:50
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The analogy of a sheet bending with a ball on it is just popsci physics.
In mainstream physics, spacetime is the collection of events that happen at certain places at certain times, but spacetime itself does not have a fabric that would stretch like a sheet (at least not like in that example with the ball).
In GR we use the expression, spacetime curvature, meaning the geometry is non-eucledian.
To answer your question, yes, inside the earth, there is spacetime curvature just like outside it. The curvature depends on the stress-energy of the earth (not the mass contrary to popular belief), and the gravitational potential will determine the strength of the gravitational field.
When you are outside the Earth, the closer to the center of mass (closer to the surface of Earth), the curvature will be stronger, but this does not mean that in your question, inside the earth, the gravitational field and the curvature will be stronger then outside the earth.
It is non-intuitive, but inside the Earth, the curvature can actually be less then on the surface of the Earth. It is because if you are actually inside the Earth, the matter (its stress energy) from all directions will affect you, and the net curvature might even be less then on the surface of Earth.
Árpád Szendrei
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I kind of get that, but what confuses me is that the spacetime where there is mass is curved, but the mass in there doesn't "curve", or does it? If you have a material that is flat will it curve if the spacetime where it is is curved too right? – Caio Keto Sep 25 '19 at 23:59
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@CaioKeto the flat material will exist inside the curved spacetime, now looking at it, like that, from inside, it will look flat. But it is because it exist inside the curved spacetime. Looking at it from outside the gravitational field, you will see it is curved, and the flat material becomes curved too. The curvature is so small in the case of the Earth, that you do not see it like that. This is because the curvature inside the object is cancelled from all directions. The curvature is the highest near the surface inside. – Árpád Szendrei Sep 26 '19 at 00:05
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@now if you let light shine through a hole that you dig inside the object, near the surface, you will see that the light will travel a path that is bent, non-eucledian. If light's path is bent, that means that spacetime itself is curved, and you are correct, the material inside will exist inside the bent spacetime, meaning that if you look at the material from a 3D view outside the object, you will see it bent. – Árpád Szendrei Sep 26 '19 at 00:07
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@CaioKeto now, your question is, if you put a straight rod inside a neutron star, will it bend? Yes it will because of the curvature of spacetime. – Árpád Szendrei Sep 26 '19 at 00:11