Although I have philosophical objections to being considered "Against The Mainstream" as I considered some questions seen on this board I have found some problems that will fit this category. At the very least I hope to articulate the range of possibilities. Of particular interest to me are the skeptics.
Questions about expansion and observability have continually came up with varying levels of sophistication. Some deal with how the speed of light is affected by expansion, others simply by the observability. grav has done some fairly sophisticated thinking about the issues. The standard model says that space itself is expanding. I then set out to define it relativistically. The question I'll try to address is can this be incorporated into standard relativity in such a way that we can begin asking empirical questions? To do this I will try to relate the Hubble expansion to an expansion in the spacetime interval that varies with time in the same way GR relates a change in the spacetime interval that varies with position in the feild.
The Hubble expansion is an expansion in the spacetime interval.
Under GR we are familiar with expanding spacetime intervals in terms of a change in curvature or depth of feild. Here we will relate it to a change in time due to expansion. Since this is a preliminary model let's look at known GR effects as seen under Newtonian gravity as a litmus test. Specifically those that lead to the correct advance of the perihelion of mercury. This is mathematically done by applying the Lorentz transform to the instantaneous acceleration of g. As you move a meterstick away from the sun its length is Lorentz Transformed from the suns proper time to smaller units. The observer at the meterstick sees no change in the meterstick units because it represents that observers proper units. That observer will note that the sun will appear to gain mass though because of his smaller proper units. Locally these transformation of units are purely mathematical and don't represent real changes in parameters. However if we consider a mass (A) at the sun and in the proper units of (A) observes an equal mass (B) in outer space then as mass (B) approaches the sun mass (A) will notice that their mass is no longer equal. In fact both masses agree that mass (B) shrank in comparison to (A) even though both masses in their proper frames didn't change. It is this actual change in parameters that leads to the correct advance of the perihelion of mercury.
Now we will consider a uniform expansion in the spacetime interval over time. An observer outside our universe would watch our universe expand. Localy we would see no change because the spacetime interval defines our proper time. This is physically equivalent to the depth of the gravitational field increasing uniformly throughout the universe. If we consider a distant light signal from the past then the information is from a time when the spacetime interval was shorter. The gravitational redshift is z=(ωo-ωe)/ωe, where ωo is the wavelength defined by the spacetime interval of a distant observer and ωe is the wavelength in proper units at the source of emission. Modeling expansion this way ωo is defined by proper units in the present while ωe is defined by proper units in the past. This indicates that space is in fact expanding but as the distance between points expands so does our meterstick.
Taken at face value this leads to some strange conclusions.
1) The universe is expanding.
2) This Hubble expansion is observed via the redshift.
3) The proper interval between masses are not increasing as a result of this Hubble expansion.
I am not very familiar with the coordinate system used by cosmology. I must remedy this for certain comparisons. However given the addition of velocities as defined by SR this would make it appear as if the farther we look back in time the slower the Hubble expansion will appear or accelerated expansion. This could provide a test if the addition of velocities correctly predict the magnitude of the accelerated expansion. Although the comoving coordinates of cosmology are explicitly designed to wash out the relativistic addition of velocities it doesn't include the above described effects of expansion. I need more data on this.
It is usefull to determine how we might question the legitamacy of tying expansion of space to the spacetime interval in this way. If we assume the absolute magnitude of proper distance is increasing with expansion but maintain a relationship with the spacetime interval it leads to more empirical consequences. It would mean that fundamental physical constants are decreasing with time. In fact the Hubble constant would define the rate at which they decrease. The only remaining approach I can think of is that the spacetime interval has a value independent of particles that exist within it. This would mean that the Big Bang was an event that occurred in a spacetime that had preexisting properties. The Hubble expansion would then be an actual increase in the magnitude of proper distance. The notion that space is what is expanding then becomes untenable, it would merely be a seperation of the masses in it. None of these alternatives are very satisfactory upon inspection but I will entertain them for review. The question here is, "How can we increase the total spacetime without increasing the spacetime interval or proper units of an observer?". Any prior work on this issue would be appreciated.
-"Proper" is always intended here to indicate proper spacetime intervals as defined by a local observer not necessarily the proper distance as defined by Weinberg.