The No-Curvature Interpretation of GR

There is an interesting analogy between the theories of quantum
mechanics and general relativity involving apparent non-linearity.
Both theories can be regarded as being based on a purely linear
foundation, represented by the Schrodinger wave equation for quantum
mechanics and the flat Minkowski spacetime metric for general
relativity.  For quantum mechanics we then traditionally imagine
something like a "collapse of the wavefunction" when a measurement
takes place, and this operation introduces non-linearity to the
theory.  Analagously in general relativity we imagine "curvature"
of the observational spacetime manifold in accord with the non-linear
Einstein field equations.

In the case of quantum mechanics there are alternative interpretations
that do not explicitly involve a collapse of the wave-function, and
that assume instead that the overall wavefunction always continues
to evolve in accord with the purely linear Schrodinger equation.
These are sometimes called "no-collapse" interpretations.  According
to these approaches, we are typically asked to imagine a "branching"
of the wavefunction into multiple alternate histories, leading to
the so-called "many worlds" or "many histories" interpretations of 
quantum mechanics.  Of course, it's necessary for these theories to 
somehow account for the APPARENT collapse and non-linearity in the 
course of events.  In other words, they must explain why we seem to
observe only a single world with a single history.  For this purpose,
the idea of "decoherence" is sometimes invoked.

Much less well-known is a somewhat analagous approach to general
relativity, which we might call the "no-curvature" interpretation.
This also involves "branching" into multiple layers which maintaining
purely linear local evolution of each layer.  To understand how this
approach yields the effects of curvature while always dealing in terms
of flat manifolds, it's useful to consider a simple two-dimensional
surface.  For any such surface, curved or not, we can partition the
surface into a network of triangles, each sufficiently small so that
the surface can be regarded as flat over an individual triangle.

The curvature of the surface arises from how these flat triangles
fit together, just like the triangles used in surveying hilly terrain.
As an example, if we find six equilateral triangles meeting at a
given vertex, then the surface is flat at that vertex, whereas if
we find only FIVE equilateral triangles meeting at a vertex, it is
curved, congruent to a vertex of an icosahedron.  Notice that we
referring here to intrinsic Gaussian curvature, and there is no
intrinsic curvature along the edge between two faces.  We can always
flatten out two faces meeting along a straight edge, showing that
the surface is metrically flat across such boundaries.  All the
curvature of the surface is concentrated at singularities at the 
verticies, as shown by the Gauss-Bonnet theorem for loops surrounding
various point on the surface.  Only loops circling verticies can
reveal any curvature.  (This is an interesting illustration of how
"curvature" can be regarded as a non-local attribute, since a path
can reveal curvature topologically even though the local curvature
is everywhere zero on the path.)

But suppose that, instead of attaching all five of those equilateral
triangles together along their edges, we branch into a new layer of
the surface, so we are continuing to circle the vertex, not returning
to the original face after the first circuit, but arriving at a face
coinciding with the first but on a different layer.  We can then
continue to circle the vertex and never implicate any metrical 
curvature of the actual multi-sheeted manifold.  For the specific
case of an icosahedral vertex we can actually close the manifold
around the vertex by joining the edges after six windings in the
icosahedral interpretation, which corresponds to five windings in
the flat interpretation.  (For a more detailed discussion, see the
note on The Tetratorus.)

The simplest illustration of a complete closed surface with two
alternative interpretations is the tetratorus, which can be described
as either a two-layered surface in the shape of a sphere (with 
positive intrinsic curvature), or a one-layered torroidal surface
(with no curvature).  Incidentally, for those familiar with complex
analytic functions, this corresponds exactly to the Riemann surface
of the two-valued function sqrt[p(z)] for a cubic polynomial p with 
distinct roots, which gives a two-sheeted sphere homeomorphic to a 
one-sheeted torus.

In general it is always possible to interpret any arbitrarily curved
surface, closed or open, with any topology, as an intrinsically flat
surface wrapped into suitably connected layers.  Of course, this
implies that each face of the interpreted surface actually represents
multiple layers, and the "curvature" emerges from identifying these
layers modulo the interpreted shape.

For purposes of expressing a physical theory of spacetime, we now
face a task similar to the task in quantum mechanics of explaining 
the physical and phenomenological significance of the multiple linear
branches, and accounting for the APPEARANCE of just a single branch.
One trivial approach would be to simply stipulate empirical
indistinguishability of the layers, which would make the entire
construction merely conventional.  In other words, it serves as an
un-defeatable example in support of Poincare's thesis that geometry
is conventional, and in particular the fact that we can always 
conceive of a curved surface as a flat one, provided we are willing
to conceptually decompose the surface into layers which may be
empirically indistinguishable.

Another, more interesting, approach would be take seriously the idea
of multiple sheets, regarding the spacetime manifold as something 
like the Riemann surface (in the sense of analytic function theory) 
of a multi-valued function whose zeros correspond to the vertices
(particles?) of the manifold.  It's worth noting that the quantum
wave functions of some particles have the property that they are
returned to their original state not by a rotation of 360 degrees
but by "going around twice", i.e., by a rotation of 720 degrees.
This might lead us to associate multiple sheets of spacetime around
a vertex with some aspect of the quantum wave function of a particle.