Sunday, March 22, 2009

Questions for Physicists, part one: The Lorentz Factor vs. Relativity

As I alluded to in my previous post, I used to be a physicist. I never got a PhD, or even a bachelor's degree, but as a freshman physics major intending to teach physics to high school students (and going to a public university with a modest scholarship and zero parental assistance), I needed money. Because I had a very high ACT score, I was offered a position in a research unit doing work for the university in theoretical physics. I even made a little discovery, in that I saw all the numbers we were putting into a computer and recognized the relationships between those numbers, so I was able to make a couple semi-educated guesses as to how the numbers could be juggled to yield interesting results, and one of those guesses panned out. So because I was doing physics, I feel justified in saying that I was a physicist.

I ended going in a different direction, but the point of all this is that I still like science, so from time to time I come up with some interesting questions that involve putting one esoteric bit of knowledge together with another esoteric bit of knowledge. One such question arose while writing about the Lorentz factor, and the other one that I'm going to discuss later arose while discussing the Bell experiment with some friends. To be sure, these questions probably have answers, but some cursory Google searches haven't yielded anything promising, so I'm going to be approaching some old contacts in the physics department to find out what's up.

OK, first question: does the Lorentz factor imply that absolute velocity exists? My last post explains the Lorentz factor (γ, the Greek letter gamma) in some depth, so I won't re-tread that old ground. What I will take a little time to explain is relativity: in particular, one of the upshots of relativity is that time and motion are relative, in that you can't speak of an object's velocity without respect to some observer (or "frame of reference"). On Earth, we speak of velocity relative to the Earth; in the Solar System, we speak of velocity relative to the Sun; in the Milky Way, we speak of velocity relative to the galactic core. But each of these scenarios suggests an obvious frame of reference, specifically, the body around which we are travelling. Let's take an example where a frame of reference does not so easily present itself to see what I'm talking about.

Suppose that a spacecraft is travelling from planet A in galaxy X to planet B in galaxy Y. Planets A & B, as well as galaxies X & Y, all provide us with equally valid frames of reference, and roughly equivalent ones to boot (for the sake of argument, if nothing else). In the darkness between the galaxies, our spacecraft begins accelerating towards c to such a point that γ starts to appreciably show itself in the forces experienced by the passengers relative to their actual acceleration with respect to A, B, X, and Y (let's just lump them all together as coordinate system Z). For the sake of argument, let's say that the spacecraft is travelling at 0.5c, half the speed of light. Here's my question: suppose the spacecraft then separates into two pieces, P and Q. Part P then begins to accelerate further, while Q stays at the same velocity - i.e. Q does not accelerate. After the spacecraft separates into parts P & Q, these two parts are travelling at exactly the same velocity with respect to each other, so with respect to Q, P is not moving and so has a velocity of zero. But with respect to Z, P is moving at 0.5c, which is substantially more than zero! Time to plug and chug! Forgive the MS Paint equations, but I don't feel like busting out the tablet right now. Let's just solve for γ to see how acceleration is affected:
So at .5c, γ=1.155, so F=1.155ma; we can juggle this to yield a=.866F/m. The point of the this is that, for a given force F exerted by P (casting off mass to change velocity), P will experience about 13% less acceleration with respect to Z than it will with respect to Q! Now, this is only initially, as P's mass will change, as will its velocity as it continues to accelerate - but for small masses lost and small accelerations, v won't change by much with respect to c and m won't change much at all, which means we should be able to see a much different a for the same F, depending on which frame of reference we use.

Now, this is weird, because one or the other frames of reference should be "chosen" in that P will experience some acceleration (a), and that acceleration will bear a relationship to the force (F) exerted, so we can calculate which frame of reference is being preferred - but relativity states that there is nothing special about any particular frame of reference. I'm probably fucking up somewhere, likely having to do with a Lorentz contraction that I'm not taking into account (because Lorentz basically assumed relativity to determine his γ, I suspect this may be relevant): P and Q will be experiencing Lorentz contractions which may cause the discrepancies to disappear under actual observation, in which case I am erroneously assuming some absolute frame of reference by not taking all the relevant relativistic effects into account. Still, that's just my best guess for where I may be wrong if I am, and I'm not quite sure about that.

OK, this is getting long, so I'm cutting it in two. In about three hours, I'll be starting a work week that has me going pretty much around the clock, so I probably won't be able to finish my bit on the happiness machine until next weekend. I haven't forgotten, though - that research supports the whole thrust of my argument, so I hope I get it soon!

2 comments:

Zach L said...

Ow. My head hurts ;_; I never had much of a mind for math. I love physics in concept (it is, after all, a description of the functioning of the universe, and you know how much of a pragmatist I am), but damn, I just can't wrap my brain around it.

D said...

Yeah, I'm having trouble putting this particular two together with that particular two at points like this as well. And on second thought (rather, after trying to deduce where I had gone wrong all day), I'm unsure how a Lorentz contraction (which only alters an object's perceived length) can fully account for the discrepancy described above... unless I'm forgetting to take some seriously relativistic effects into account... which I guess is what I was getting at in the first place...

Anyway, if that made your head hurt, wait until I give a crash course in quantum. It's gonna be fun on a bun!