Relativity of Length Scales July 30, 2008
Posted by tarun in Uncategorized.trackback
A problem given in The Fundamentals of Physics by Resnick and Halliday asks the following question. If someone told you that overnight the size of the Universe, alongwith the size of everything contained inside it, has doubled, then how would you go about verifying it? If you have access only to length measuring devices then clearly this task is impossible, since all that is important here is the relative size of objects, and obviously the length of the standard metre too has doubled. So whoever told you that could not have known it by conducting any experiments involving only lengths. Fortunately there are other physical quantities in our world, such as mass, time, charges of various kinds; and physical constants such as speed of light, Planck’s constant, Newton’s gravitational constant and so on. Clearly, if only the lengths have changed and not the time then a measurement of speed of light should reveal if the assertion is true or not.
Our Universe is indeed expanding and the previous argument suggests that it can’t be that everything in our Universe is expanding, else how could we figure out that there is more room out there today than was tomorrow? In fact, not only our metre rods are not expanding, even more loosely bound objects such as galaxies and clusters of galaxies are not expanding, and have reached a semi-stable configuration where they do not participate in the general expansion of the Universe any more. The usual explanation of an expanding Universe by using an expanding balloon and dots drawn on it works only if the dots are infinitesimal. For instance, if I make a rather largish dot that occupies half the surface on the balloon then as the balloon expands it will still occupy half the available space.
Incidentally, it might seem that we require other forces of nature to produce a non-expanding metre rod; and indeed it is true that a metal rod is held together by electromagnetic forces, and the atoms inside it acquire a fixed physical length through the principles of quantum mechanics (Planck’s constant and so on), but this is not really true. As I mentioned earlier, a galaxy is held together only by gravitational forces, and it too stops expanding after it has become dense enough to oppose the pull of the rest of the matter in the Universe that is trying to pull it apart. So even if there were no other forces but only gravity, the expansion of the Universe can be still figured out by making a simple experiment of comparing lengths using galaxies as a metre rod. If this were not true we would be led to a Universe that expands but there would be no experimental method to discover it. And that would be a bit metaphysical!
I’m not really clear on why expansion happens only on the super-cluster scale. Is it only because of the electromagnetic forces that bind matter and gravitation that holds large masses together? Shouldn’t it be possible to then measure expansion by considering the repulsion between two like charges? And in some sense, would an expanding universe then mean that matter is being less bound as time passes?
When there are no other forces acting on a particle it will move on a geodesic. In an expanding universe geodesics are the trajectories that keep the particles glued to their comoving coordinate position while the space stretches, so they separate according to the Hubble’s law. With additional forces the particles no longer move on geodesics. For example, two particles released at the same height on Earth would move on straight lines heading towards the centre. If there is a repulsive or attractive force between them this wont happen. This is simple enough, but then you might wonder what happens when there are no other forces and only gravity. The expanding solution applies only when there is an absolutely homogenous and isotropic matter distribution. In reality there are perturbations in the smooth background. With only gravity on particles still move on geodesics, but the geodesics of this perturbed universe do not separate according to the average expansion law but can cluster. This clustering slows down when the particles form virialized structures, that is configurations where 2T + V =0, where T is the total kinetic energy and V is the total potential energy. Hope this helps?
Yes, thanks.
“would an expanding universe then mean that matter is being less bound as time passes?”
Always thought that was the basic problem (for life, not for the theory) with the “open universe” theory, and why people keep looking for dark matter. Wasn’t it?
The density of the Universe decreases due to expansion, however, bound structures shall remain bound in the standard theories. There are theories involving exotic dark energy models where the dark energy increases with time in such a manner that the repulsive force due to the dark energy eventually exceeds the binding energy of even atoms and subsequently nuclei to tear them apart. BTW, there is no problem with an open Universe other than the fact that our Universe is so close to the flat one that it requires fine tuning to get it right, so most people believe that it is indeed flat. The amount of matter that we can see falls far short to make it flat. There are other reasons to seek dark matter that I don’t have time to include here at the present time, perhaps I shall write a post on this subject later.
“There are other reasons to seek dark matter that I don’t have time to include here at the present time, perhaps I shall write a post on this subject later.”
In short, the relation between orbital velocities of stars around galactic centers and the distance of the star from the center doesn’t fit in with Newtonian calculations based on *observed* (visible) mass of the galaxy. But it fits with models in which there are invisible “dark” matter halos surrounding the galaxies. We’re not sure what dark matter is made of, but we know it doesnt interact with light (electromagnetic radiation) and we also know there’s a lot of it in the universe.