In all the illustrations regarding space distortion, I find an extra dimension that depicts space distortion. Is there any mathematical equation that proves the creation of an extra dimension ?



Just because there's an extra dimension on a diagram doesn't mean that it's real. No such thing has been created. It's just an artifact of embedding a manifold with a non-Euclidean geometry into a Euclidean space.

One of the biggest stumbling blocks of intuition for people learning general relativity is that the physics only cares about the intrinsic geometry. For example, imagine an ordinary ball in three (Euclidean) dimensions, and take its surface. In jargon, the surface is the two-dimensional sphere $\mathrm{S}^2$, and it has geometric properties that can be described by reference to it alone, e.g., lengths of curves drawn on it, angles between intersecting curves, that if one starts at some point and goes in a single direction, eventually one would be back to the starting point, etc.

One should consider the two-sphere as a valid geometry by itself, and the fact that it can be pictured as the surface of a three-dimensional Euclidean object as purely incidental--it $\mathrm{S}^2$ can be embedded in the Euclidean space $\mathrm{E}^3$ in that simple manner, yes, but that's a fairly arbitrary choice. It can be embedded in other spaces as well, or not embedded in anything at all.

For Riemannian manifolds (the purely spatial geometries), there are some nice mathematical results regarding embedding into Euclidean spaces, but nevertheless, they're usually impractical and not relevant to general relativity. Even worse, general Lorentzian manifolds (spacetimes) have no comparably 'nice' results regarding embedding spacetime geometries into flat pseudo-Euclidean spaces $\mathrm{E}^{n,m}$, so it's doubly useless to worry about it, except perhaps in the cases where the spacetime is extraordinarily simple.

In the end, such extra dimension(s) are just artifacts of making a picture. They're not physically real, so they're not 'created' in any physical sense.


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