Earlier I was killing a little time today by reading about MOND. It’s an ad hoc attempt at a solution so the galaxy rotation problem. Essentially the outer portions of galaxies rotate a lot faster than they should given the their visible mass distribution. The obvious solution is that there’s mass which isn’t easily visible which accounts for the extra gravity. This approach is called dark matter, and there’s various searches underway to detect it.
The MOND proposal is an alternative out of left field. It assumes that in fact it it not true that F = ma for small enough values of a. Instead F = m(a2/a0) for some constant a0. This is more or less equivalent to saying that the force of gravity falls off as ~1/r at large distances instead of the usual ~1/r2. It’s more than a little far-fetched, and there are observational reasons involving gravitational lensing which make the hypothesis seem quite implausible. Never hurts to try though.
It does however give us the opportunity to think about changes in the behavior of fields at various distances. Take the electric field, for instance. The electric field of a point charge is just
For a continuous charge distribution, you can take the field as being produced by the charge density integrated over the charge region. Consider a finite line charge of length a. The above mentioned integration tells us that the field experienced if you start at the center of that line charge and move perpendicularly outward is
And so for very large a (or equivalently, very small r) this asymptotically approaches
That’s what we call the field of an infinite line charge. But here’s the thing: there’s no such thing as an infinite line charge. It’s a very good approximation when you’re close to a long line, but as you move farther and farther away that line charge starts to look more and more just like a finite little piece. As that happens the field stops falling off as 1/r and begins to fall off as 1/r2. We can see this mathematically by taking the variable a to be very small and checking the behavior in r. (If finding “large r” by taking “small a” bothers the mathematicians in the audience, recast the problem in terms of some dimensionless ρ = r/a and take ρ large)
This is a good example of a function changing regimes as some other parameter varies. As in this example, lots of these types of functions tend to die out faster as distance increases. This instance in particular is related to the general idea of multipole moments. Are there any examples of a physical function’s decay changing from faster to slower as distance increases, as hypothetically in MOND? Actually with a tiny bit of modification our charged wire is also an example of this. In real life, a line charge is not a continuous charge distribution but a collection of charged particles strung out at very tiny regular intervals. On a macroscopic level it looks continuous but on an atomic level it’s not even close.
In the immediate neighborhood of each charged particle (much closer to the particle than the particles are to each other) the field will be dominated by the closest particle. It will look essentially like the 1/r2 field of that particle. As you move away from that particle and the wire, the field from the closest particle ceases to be the main influence and you experience the field of the wire as a whole - which behaves as 1/r. The decay has entered a slower regime. Assuming the wire is not actually infinite, the field will eventually revert back to the inverse square behavior.
Though inspired by a theory that very probably isn’t true, this kind of thing is fun to think about. In the professional world, the mental effort behind even a theory that turns out to be wrong can end up leading to something better later on.
1 response so far ↓
1 Carl Brannen // Jun 15, 2008 at 4:12 pm
The funny thing about MOND is that it involves a square root function in a place where we don’t expect to see one.
In comparing Newtonian-like theories like MOND, where the equations of motion are written as a force F=ma, with a symmetry based theory like Einstein’s gravitation, the obvious thing to do is to rewrite the symmetry theory as a set of equations of motion. So I took the Schwarzschild black hole and converted its equations of motion into F=ma form.
In doing this, you get to make a choice of the coordinate system. I used Schwarzschild and Painleve. I wrote a cool simulation to verify the rather complicated equations of motion.
Painleve coordinates are special for people working in Hestenes’ version of Clifford algebra (i.e. gamma matrices), he calls it “geometric algebra” (GA). When you rewrite GR into GA you end up with a flat space gravity theory (that is equivalent to GR to all orders except it’s missing the topological crap solutions that have never been observed like worm holes). The non rotating black hole ends up with Painleve coordinates.
Like MOND, the Painleve equations of motion have square roots in them. I should add that when you rewrite GR in gamma matrix form it becomes very easy to do relativistic electron wave functions on curved space, and so the Cambridge geometry group (which invented the idea) has written a lot of papers on electron interaction with black holes.
They also generalized the theory to rotating charged black holes but to rewrite those as F=ma was beyond my (synthetic) calculus tools. The Painleve equations of motion are a sum of things that go as 1/r^{n/2}. The Schwarzschild equations are functions of things that go like 1/(r^n(r-2)^m), where “2″ is the event horizon radius for a unit mass and n and m are integers.
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