Dear Dr. Dreyer:
I have just read your very interesting essay; and it seems to me that, since your internal relativity recognizes a "Newtonian"-like background time, it would dispense with many problems of time, because time has a definite direction. Each space-time point would be in exactly one hypersurface of simultaneity and each world line would intersect each hypersurface exactly once (with certain reasonable assumptions).
As you may know, Lorentz was an ether theorist, as was Poincaré. This does not mean that they believed in a "luminiferous" medium, but they did believe in a rest frame, only with respect to which light travels with the same speed, in all directions. You did not say so, in your essay; but, from the description of your work, taking the Lorentz perspective and recognizing a background time, it seems that you are really following the Lorentzian ether theory. (I consider this a good thing; since I too am an ether theorist, in the tradition of Lorentz and Poincaré.)
Given the above, I thought that you might find the following of interest:
As you no doubt understand, better than I, quantum mechanics is often preferable in a finite universe, for reasons of boundary conditions. Well, it has long been known that each finite universe has a rest frame, which means that the Lorentz-Poincaré ether theory is the correct theory, for the flat subspaces of such universes. To my knowledge, the first understand this were Brans and Stewart; however, I believe that the most insightful work was done by Peters [see references below, for both]. I also discuss this, in my essay, which is at, http://fqxi.org/data/essay-contest-files/Sasaki._TDoT.pdf.
C.H. Brans and D. R. Stewart, Phys. Rev. D 8 (6), 1662-1666 (1973), Unaccelerated-Returning-Twin Paradox in Flat Space-Time.
P.C. Peters, Am. J. Phys. 51 (9), 791-795 (1983), Periodic boundary conditions in special
Relativity.
Also, in your essay, you state, "This raises the question whether there exists a Lorentz type version of general relativity that does without this split? Internal relativity [1] is our attempt at constructing such a theory." Given this, you might be interested in work, by a guy in Belgium named Jan (B.) Broekaert (webpage: http://www.vub.ac.be/CLEA/Broekaert/). From his web page:
The development of a scalar-vector gravitation model with Lorentz- Poincaré type interpretation in concordance with General Relativity Theory. This model requires gravitationally modified Lorentz transformations ("GMLT") and enables to give -at present- a Hamiltonian description of particles and photons till 1-PN of GRT. Like standard Lorentz transformations in Special Relativity, the GMLT endorse an underlying non-observable presentist space and time ontology (due to the Poincaré Principle of relativity of movement any preferred frame remains unobservable). While this physical analysis of Relativity Theory conceptually contrasts its usual geometrical analysis, Poincaréan geometric conventionalism should be able to bridge these with one common observable empiry, i.e. the experimentally confirmed one from SRT and GRT.
I am not familiar with this theory, but from his description, I gather that he is trying to reconstruct the predictions of GR, in his scalar-vector theory, so I thought that you might be interested.
Finally, I thought that you might be interested in work on synchronization, done by Reichenbach (Poincaré also did similar work, but Reichenbach's is more formal and organized). Work surrounding his "epsilon" definition of synchronization can be found in:
H. Reichenbach, Axiomatization of the Theory of Relativity, [pg. 35, 44-45] (Univ. of California Press, Berkeley, 1969), translated by M. Reichenbach, from Axiomatik der relativistischen Raum-Zeit-Lehre (Vieweg, Braunschweig, 1924); The Philosophy Of Space And Time [pg. 126-127, 125-127, 126-127] (Dover, New York, 1957), translated, with omissions, by M. Reichenbach and J. Freund, from Philosophie der Raum-Zeit-Lehre (de Gruyter, Berlin, 1928).
If you are really interested, I have photocopied these sections and can send them to you.
I hope that you have found this interesting.
Take care,
Ken.