Types of quantum field thories
This disappearance and regeneration of space in time and of time in space is motion;– a becoming, which, however, is itself just as much immediately the identically existing unity of both, or matter. (PdN§261)
The term geometrodynamics has been coined, or at least promoted, by John Wheeler as a description for the dynamics of gravity according to general relativity. Since the field of gravity is entirely encoded in the spacetime metric geometry, there is some justification for thinking of the dynamics of the gravitational field as being “the dynamics of geometry” itself. Hence the term.
More specifically the term geometrodynamics is associated with Wheeler’s speculation that all of physics might fundamentally be described by configurations of gravity coupled to other fields, notably the field of electromagnetism, but without any matter: one can see that certain spacetimes without any matter content but with certain nontrivial topology may locally effectively look as if they contained massive and possibly charged bodies.
For instance on a spacetime that is obtained from two copies of Minkowski space connected by a thin (as measured by the metric) throat – often called a wormhole – an electric field configuration whose field lines all converge to the throat’s mouth in one of the two Minkowski sheets, pass through the throat and then emerge concentrically in the other Minkowski sheet may have no divergence anywhere, hence according to Maxwell's equations have no charge sources anywhere, and still effectively look to an observer constrained to one of the two Minkowski sheets but relatively far away from the throat’s mouth as if they were the field lines of a positively or negatively charged point source located where the mouth of the throat is.
These kinds of ideas Wheeler liked to describe by phrases such as charge without charge and mass without mass. Later these basic ideas have continued a life notably in the context of attempts to describe gravity by a topological quantum field theory, for instance in approaches to describe gravity as a BF-theory.
A rather similar perspective arises in Kaluza-Klein compactification of systems of pure gravity to effective field theories. In the case of supergravity these reductions give rise to effective field theories which contain not just extra force field (such as the electromagnetic field in the original KK-mechanism) but also fermionic matter. For instance models such as the G2-MSSM consist entirely of pure 11d supergravity KK-compactified on G2-manifold fiber bundles and the resulting effective field theory contains all of the standard model of particle physics.
[[Eric]: Of the four “X without X”s above, the one that is not in Wheeler’s “Classical Physics from Geometry” is “Spin without Spin”. This is described in Section 3.4 of Matter from Space. It would be great to expand on that here.
Bruce Bartlett: Another example of this phenomenon seems to be the fact that Maxwell’s equations in matter are the same as Maxwell’s equations in curved space without matter! This is the basis of cloaking technology, see article by Leonhardt and Philbin. You can read this equivalence both ways. Either you can conclude that there is no such thing as curved space: it’s just a piece of dielectric material causing the light rays to bend which gives the illusion of curved space. Or you can conclude even more radically that there is no such thing as matter: what we think of as a block of wood is just a radically curved region of space (Maxwell’s equations can’t tell the difference). Or you can just think of it as a formal equivalence :-)