The Copernican Principle and the Hubble Law
Background: Our observable universe can be shown on a time / distance plot such as Fig. 1., where, our observable universe is a very, very thin line that intersects the time / distance axis at “Here” and ”Now” and progresses equally in time (into the past) (x-axis) and distance (y-axis). The line which currently extends over 13 billion light years into the past splits the figure into two areas. The upper-left area is not observable now: There has not been enough time for the light from these objects to reach us. The lower-right area is also not observable now: It was observable in the past, but the light has passed us now. Every observation we make of the distant universe comes from some point along this line. That is, every single observation of our observable universe is an observation from the past. As we observe galaxies that are removed from us in the past, we observe a redshift. This redshift increases the further in the past we look (with increasing distance). That is, the further in the past we look, the greater the apparent speed of recession of the object we observe. Where the speed of recession is significantly less than the speed of light, the speed of recession can be obtained from the equally well-known equation:
v (speed of recession) = H0 x D (distance), where H0 is the Hubble constant.
Note: ‘Moving towards’ or ‘away from’ or similar such phrase in the text are not meant to imply any absolute frame of reference for the relative observable movement but are simply convenient linguistic terminology.
The nearer an object is to our galaxy (or the younger the observation is) the less it is moving away from us to the point where, now, local gravitational influences dominate. Andromeda is, for example, actually moving towards us. Where however, this apparent speed of recession is a significant percentage of the speed of light, the equation in its simplest form is no longer valid and for the observations of very old objects considerable adjustment must be made. Nevertheless, the younger the observation and specifically where red shifts are comparatively small, the closer the result approximates to the equation in its simplest form. This equation has one very important property, that is, a plot of v against D intersects the x- and y-axes at its origin (i.e., v = 0 when D = 0) and we observe for local galaxies v ≈ 0 when D ≈ 0.
Argument: If we were to pick a random galaxy, that had at the time of observation a significant red shift, that is its speed of recession from us was, at the point of observation, significant, we can ask the question, what would this property of its observable universe look like to a hypothetical observer within this galaxy, now?
There are only two possible scenarios: That is, either its inherent motion would have an effect on the redshift of its “local” observable galaxies (in its very recent past) such that, irrespective of the value of an equivalent of any “Hubble-Like” parameter, or indeed the form of the hypothetically derived equation itself, the resulting speed of recession, v1, of the galaxies observable from its location would be such that v1 > 0 when D1 ≈ 0. Alternatively, the motion would not have an effect, and as is also the case for our observable universe, v1 ≈ 0 when D1 ≈ 0. This is all we need to consider – no other supposition of the properties of this hypothetical observable universe is required. The following discussion about equivalence or uniqueness of perspective refers only to this one property and this property alone.
Let us consider these two cases in more detail.
In the first case, this inherent motion of the random remote galaxy, now, within the universe is such that even comparatively close galaxies would be receding from it in such a manner as to still have an observable redshift such that v1 > 0 when D1 ≈ 0. To whatever extend this is the case, we must assume that an even more remote galaxy (that is currently presumed to be moving at an increased speed) must, now, have an even greater deviation and its equivalent speed of recession, v2, would deviate even further such that v2 > v1 when D2 ≈ 0. This creates a problem. That is, it makes our observable universe very special, since we would necessarily have to assume that only galaxies very close to our own would have a chance of their equivalently derived relationship between v and D, such that, irrespective of the size of any observable “Hubble-like” constant, v ≈ 0 when D ≈ 0. This in turn makes our place within the universe very special, indeed unique. This is the special case.
In the second case, that is the inherent motion of such a randomly chosen galaxy, now, would have no effect on this property of its observable universe. In this case, a hypothetical observer from within that random galaxy, would also derive the relationship between the relative motion of galaxies in its observable universe and would observe, irrespective of the nature of that relationship and the value of any “Hubble-like” parameter, that there also v1 ≈ 0 when D1 ≈ 0. This is the none-special case. That is, at the very least, this property of our observable universe is not dependent on our absolute position within the universe.
In the second case, an extension of the Copernican principle to the non-specialness of the location of our galaxy in the universe, any randomly chosen galaxy would necessarily have this same observable property of the relative motions of local galaxies, to reiterate, the motion is, to all intents and purposes, only governed by local gravitational influences, from which they are not receding with any great speed and may indeed be approaching. If this is true for any randomly chosen galaxy, now, it would necessarily be true for every galaxy, now. Further to this, we must still draw the same conclusion of non-uniqueness and equivalence of perspective for any randomly placed “hypothetical” galaxy anywhere in the universe, now. These hypothetical galaxies can be placed at every conceivable location within the universe, and the hypothetical local observation would still have to be the same, that is vany ≈ 0 when Dany ≈ 0.
Here as well, there are only two scenarios where this could be the case. The first is that no (hypothetical) observer anywhere in the universe could ever draw any conclusion about its inherent speed of recession from any other galaxy now, based on the relative motion of “local” galaxies. It would be an interesting conclusion indeed, that whilst we can easily draw conclusions about the past expansion of the universe based on the motions of distant galaxies, we could never draw any conclusion about the present based on the observation of local galaxies.
The alternative is simply that the hypothetical result vany ≈ 0 when Dany ≈ 0, is exactly the result we would expect, if there were to be little or no remaining expansion in the universe, now. That is, if we were to conclude that it is not possible to have the same “local observation” of “little or no local expansion” from anywhere, and thus everywhere within the universe, in a universe that is rapidly expanding.
The first “special case” is the only case that would directly support the argument that the universe is expanding, now. However, this necessitates that our view of this properties of the universe be unique. It creates an absolute frame of reference within which, we would effectively be at the only location within that frame of reference where v 0 when D = 0. In effect placing us within an absolute frame of reference where we would necessarily be located at (or at least very close to) the center.
The second “none-special” case, implies we can either never know anything about the current status of the universe, it could have already started contracting, it would simply be unknowable; or we would be able to conclude, that the universe is no longer significantly expanding.
Of all the possible alternatives, this last case, is the only case that directly correlates to the information we have about the relative movement of galaxies with respect to time. The universe was expanding: The further we look into the past the more rapid that expansion was. The closer we look to the present day, the lower the rate of expansion we observe.
Conclusion: The above arguments are based on one and only one, unprovable assumption: The lack of specialness of the location of any randomly chosen galaxy, including our own, within the universe, implies an observer within any randomly chosen galaxy, would draw the same conclusions, now, about the relative movements of galaxies the more recently in the past they are observed.
The special case, where this property of our observable universe is unique, is the only case that directly supports the assumption that the universe is expanding now.
The Copernican equivalence of perspective is entirely consistent with a view of the universe, where in the first instance we think of remote galaxies as being far away in time (the past). Here then, we may use the change of motion with respect to time (the only information directly available to us) to draw the conclusion that the universe is now expanding only minimally or indeed not at all.
If we choose not to do this, if we choose to ignore the time aspect of the data to hand, then this would be to accept that the current status of the expansion or contraction of the universe would be unknowable.