The Einstein gravitational field equations in the context of general relativity interpret gravity as equivalent to the curvature of three dimensional space, and precisely relate that curvature to the “material content” that is present in that space. In general relativity there is also the additional complexity of mixing of space and time in a space-time. The precise form of the gravitational field equations is:

Ricci – constant x metric = Stress-energy

The ‘Stress-energy’ on the right side of the field equations give precise form to the material content of the universe, and the left side contains the geometric information about the curvature of space. An intuitive picture for the general relativity approach can be understood by considering the analogy of a two dimensional surface sitting on top of a soft surface, such as foam. If we put a metal ball on the surface, then the flat surface will dip into the soft surface, producing a curved region near it with higher curvature closer to the ball.

From the S4 theory perspective, one notes that any three dimensional smooth submanifold of a four dimensional sphere automatically produces these field equations when one computes the Ricci curvature of the immersed manifold in terms of the Ricci curvature of the ambient S4. The ‘Stress-energy’ term is produced in terms of the extrinsic curvatures of the immersion via the second fundamental form of the submanifold. Here the Einstein equations can be interpreted as describing a ‘structure equation’ describing how the three dimensional universe sits inside a four dimensional one.

The consequences of this reinterpretation are quite far-reaching. Now, the Einstein equations decoupled from gravity applies at all scales throughout the three-dimensional universe. In particular, now they describe subatomic phenomena. Indeed, one can see from this reinterpretation why it makes sense to consider whether quarks are truly stable particles or merely describing principle curvatures of hadrons.