Interesting that it seems to be not possible to 'geometricize' mass-energy in only R^4. Heim apparently managed to geometricize all known types of particles, but that required R^6. Those two extra dimensions are not spatial or rolled up into a tiny ball like in Kaluza-Klein theory however, they are just associated with organisational properties... i.e they are 'informational', not spatial or temporal. That's a novel concept i think, that there is another type of dimension other than spatial and temporal.
I think Heim theory would be well worth reading. Unfortunately i can't because my math isn't good enough (he basically invented his own math because in his theory he never deals with infintismals) his books are written in German and i wouldn't want to study a 'far out' theory like that until i have a solid grounding in more established theories like GR. Here is a brief abstract for anyone interested about Heim's idea that matter might be purely made up of space-time itself. The 'metron' referred to is the fundamental, smallest area that can exist in nature according to Heim. Hence his theory never deals with infintismals.
"The essence of Heim’s theory is its complete geometrization of physics. By this is meant the fact that the
universe is pictured as consisting of innumerable small, locally confined geometric deformations of an otherwise
unpertubed 6-dimensional metronic lattice. The influence these deformations have on our 4-dimensional world,
or the effects of their projections into it, constitute the structures we interpret as gravitons and photons, as well as
charged and uncharged particles. The theory ultimately results in a formula from which the masses of all known
elementary particles and a few unknown ones may be derived. In addition, it provides a picture of cosmology
differing widely from the established one. Despite the insight gained into particle physics, the theory is not entirely equvalent to modern quantum theory. For this reason Heim has extended the theory to 12 dimensions. Only this extension allows full quantization, and as a consequence it becomes possible to unite relativity and quantum theory. Even 6 dimensions are not
sufficient to accomplish this."
Different particles are made by the deformation of different combinations of dimensions of the 'metron'. Here's a brief summary:
a) The first type is a lattice deformation involving only the 5th and the 6th coordinates. In the 4 remaining
dimensions the metronic lattice remains undisturbed. Physically, this may be interpreted as a structure
existing in the two transdimensions. Since our senses are not attuned to events in the two
transdimensions this may difficult to visualize. Although the deformation exists in dimensions 5 and 6 only, and does not project directly into our 3 dimensions, its effect may be occasionally be felt in the rest of the world. Under certain conditions it
may be extended into the four remaining dimensions in the form of quantized gravitational waves, so called
gravitons.
b) The second type of deformation again involves dimensions 5 and 6, and in addition time, the 4th
dimension. Again, this particle like structure does not project directly in our 3-dimensional world, but
is felt here only in the form of waves. Heim derives the property of these waves and shows that they
are identical to those of electromagnetic light waves or photons. It follows that case (b) describes a
particle like structure in the 4th, 5th and 6th dimensions, extending into the remaining 3 dimensions in
the form of photons.
c) The third possible deformation involves 5 dimensions, i.e. all coordinates except time. This 5-
dimensional structure projects into 3-dimensional space of our experience, i.e. it forms a condensation
here, and it is reasonable to assume that we are sensitive to such condensations. This is indeed the case,
and Heim shows that they give rise to uncharged particles with gravitational mass and inertia.
d) The final deformation involves all 6 coordinates. This again leads to condensations in the space of our
experience, giving rise to particles, but as in case (b), the inclusion of time leads to electric phenomena
as well. Heim can show that 6-dimensional lattice distortions lead to charged particles.
I guess this highlights what jgraber said, about Wheeler having found it difficult to explain the many different types of mass and energy using only R^4. If you add extra dimensions it might be possible though. However since GR exists in R^4 i guess any attempt to geometricize other properties of the world other than gravity would have to require extra dimensions... which is one thing that string theory, LQG, and Heim theory all have in common i think.
