The process of assembling the collection of ideas that would go into my paper for the 2008 ASSA symposium vividly brought back to me how much I feared some of the concepts in cosmology that were placed before me as a student. I didn’t think I would ever master Relativity, and I was terrified of being exposed as an idiot. Being scared wasn’t the answer though; I should have accepted the inevitable. I did not master Relativity, and there are many more people in the world of astrophysics who today regard me as congenitally inept than those few who wriggle in their straight-jackets and drool uncontrollably as they blither away that someone should be listening to me. Poor souls.
Like the sly old fox that I am, I am going quote from my soon-to-be-completed collaboration with Sir Patrick Moore, The Static Universe. In it I address the problems inherent in trying to calculate cosmic velocities, and the critical role played by where we anchor our logic, and upon what. Please let me know if this is easy to understand, because if it isn’t, I must make changes now!
We need a frame of reference in order to be sane. There has to be concrete ground for our ideas. Those who tease the boundaries of abstract thought come back nervous, and admit that even at distant horizons, there are no absolutes. On the largest conceivable scale, things don’t relate sensibly to each other at all. Albert Einstein applied his clairvoyant genius to the problem, and gave us the equations to deal with it. My approach is somewhat different, and certainly far less imaginative. We do have an inertial frame on which we can calibrate the extent of our environment. In fact, it would appear that we have a great number of them, each unambiguously applicable to its own field of endeavour. The onus is on us to choose wisely.
The structural asymmetries and heterogeneity of our relatively stable environment provide us with a pegboard in which to fix our beacons. It is important to note very carefully that were it not so, if our cosmic environment were perfectly smooth as we are told it is, then it would be quite impossible to calibrate an axis there, or in fact know anything at all about the spatial distribution of matter. It would be a fog of all-encompassing opacity, but fortunately for us, the Universe that is subservient to the Cosmological Principle is very clearly not the one that astronomers, you and I, look at. There are uniquely arranged patterns on the sky that we have identified, named, and accurately determined the co-ordinate positions of. Thus we have a background map against which we can plot our plots, and derive our frames reference. Astronomers have done so with great success.
Any system in apparent equilibrium for the duration of our measurements will do, although of course, the longer things maintain their spatial arrangement, the better. The ecliptic disk of the Solar System gives us a handy horizontal line, and thus a basis on which to declare “north and south”, and from that, if we like, “up and down”, “clockwise and anticlockwise”. We can state that the galaxy Andromeda is inclined 13° from the horizontal only because we do have a horizontal axis upon which to reference our measurement. To measure further afield, we can take our frame up a gear. The mean plane of the Milky Way’s galactic disk can take over from the solar ecliptic, and we can spread the net of our quantitative understanding a great deal further. As we proceed radially outward, so does the universe appear ever more motionless by human standards. For our practical purposes, we can safely ignore relative motion of superclusters.
Now, in theory at least, objects at great remoteness are indeed moving around, and therefore cannot delineate an absolute frame. No argument. We don’t have an absolute anything, but for the purposes of empirical science, we can choose a frame that applies best to our field of investigation. Let me give you an example of how astronomers use these principles of mechanical relativity. Say we were in our research to analyse the redshifts of galaxies in the Messier Catalogue. The redshifts in most tables are given as radial velocities (RV) rather than z-numbers, although it ultimately makes no difference for they are derived one from the other. How do we express relative motion in this scenario? The Messier objects range in remoteness from ~2 million light years (Mly) to ~60 Mly. Take the interesting (I won’t say anomalous!) case of M31, the Andromeda galaxy.
Depending on the method used, distance estimates vary from about 2.2 Mly to 2.9 Mly, with the latter (higher) figure the one used to determine velocity on a redshift table. Andromeda is travelling towards us, and thus has a mean Doppler blueshift. The rate at which it moves relative to us depends very much on what exactly “us” means. If by “us” we mean our point of observation (Earth), then M31 is heading towards “us” at 300 kilometres every second. If however, our measurement seeks to determine when Andromeda and the Milky Way will collide, then “us” needs to be the Milky Way itself, not the tiny component buried deep within it we call the Solar System. Using the larger inertial frame of reference gives us a significantly different approach speed—only about 122 km/sec, less than half that obtained using the Solar System. The differential comes about because of the motion of the Solar frame within the encompassing frame of the galaxy, meaning that the Sun and the Earth are at the present time at a point on their orbit of the galactic nucleus where they travel towards M31 at a speed, relative to the nucleus of the Milky Way, of 178 km/sec. Although the two galaxies close at only 122 km/sec, the actual approach velocity of M31 to me when I look at it at night from Mother Earth is the sum of 122 and 178 km/sec, that is, 300 km/sec.
It won’t always be like that. In fact it changes constantly as the Sun goes around the Milky Way. In 100-odd million years’ time, we will have circled around to the other side of our home galaxy, and will then be moving away from M31 at a speed that more than cancels out the rate of convergence of the two galaxies. Of course, the arithmetic is easy because the two galaxies in our study spin on very nearly the same plane. Nevertheless, it is a practical, entirely realistic illustration from everyday astrophysics of mechanical relativity, the cascade of inertial frames, and what the choice of frame can do to our numbers.
So, we have measurements of relative motion that differ significantly depending on which inertial frame we choose. If we take the Sun as our point of reference, we get a redshift velocity of minus 300km/sec, and if we choose the Milky Way’s inertial frame, we find that it is only minus 122 km/sec. If you think about it, there’s no problem really, other than thinking about it too much! The physics involved is entirely classical Newtonian Mechanics, and we can do the sums with high school mathematics. Question: Where does Einstein’s Relativity come in? Answer: Very, very far away!