### General Relativity as extrinsic curvature and other lies my professors told me

This post is about how General Relativity (GR) is explained to the masses, specifically how one should picture curvature. Most popular science accounts use the 'rubber sheet' analogy.

Extrinsic curvature, curvature by embedding 2D in 3D. Extrinsic curvature or curvature by embedding

Consider the picture of the Earth rotating around the Sun. This is the classic picture most 'scientific american' type articles will throw your way to explain what curvature is.  It is the bending of a 2D surface in 3D. If you take a 2D rubber sheet and put some mass, it will deform and a particle will orbit around it.  This is a good picture in the sense that it is based on classic visual 3D intuition and reproduces the correct result for 2D sheets that deform.  But try generalizing it to 3D.

GR as extrinsic curvature by embedding 3D in 4D?

A finer problem with this image is that it seems to imply that you should abstractly extend this construction from 3D to 4D.  The curvature is extrinsic coming from the bending of 3D in a higher (4D?) space.  You lose the visual guidance. Humans simply cannot visualize in 4D (except Hawkins). The math can guide you though. Another point is rather ontological. If you need a 4D to create the curvature by embedding, considering only extrinsic curvature, isn't that proof that 4D of space exists?

Intrinsic 3D. Intrinsic curvature

We just looked at extrinsic curvature and by definition of 'ex' it needs an extra dimension.  But there is also 'intrinsic' curvature that lives purely in 3D.  Can we use it to construct gravity? The picture shows how flat space, the cartesian grid gets deformed by matter. Send a photon by the earth, and it will follow geodesics (the lines) and it will be going 'straight in that curved space'. This does not depend on a hypothetical 4th dimension of space.  Why isn't this picture, that of intrinsic curvature, more used amongst practitioners to explain gravity?

The problem with smooth 3D intrinsic curvature: the Kaluza-Klein example

There is something missing in the intrinsic construction.   GR is modeled by a Riemann geometry.  Smooth 3D deformations of space cannot give the proper mathematics.  We cannot create the proper GR curvature with the only assumption of smooth fields in 3D.  In order to obtain a proper unification, Kaluza and Klein in the 50's had to hypothesize a 4th dimension.  There, with smooth fields they could recreate GR and incorporate EM.

Non-smooth deformations, non commutative algebra

However non-smooth deformations of space in 3D arrive at the proper curvature.   The mathematics of non smooth deformations (non-holonomic theories) have been explored during the 80s in the solid state physics.  The non-commutative algebra that result is fancy mathematics.  The main result however is that the presence of certain defects leads to a proper Riemann curvature. The proper Riemann curvature arises in 3D if one considers singular deformations of space, as if brought about by defects of a certain particular shape.

The ontological reality of defects as curvature.

Since we observe gravity and we model it with curvature (GR). Here are some choices on how to assign an element of reality to this elastic metric of space (Einstein's and MTW words).  It can either A/ come from embedding in higher dimensions.  Curvature is the result of smooth deformations. B/ Assume defects in 3 dimensions and be intrinsic deformation with no further appeal to extra dimensions. Einstein was looking for answers within smooth fields (A) and the math for non-commutative algebras, arose much later within the standard model and solid state communities. The non-commutative algebra turns out to be relevant in GR and the standard model.

Where the defects come about is a story for another day.