Saturday, March 21, 2009

101 Interesting Things, part eleven: The Lorentz Factor

Like any young physicist, I spent my high-school years desperately attempting to break the very laws I was studying: perpetual motion machines, FTL drives, doomsday devices that swallowed universes, cold fusion reactors, the works. Because I failed to break any of the laws I learned, I imagine this would be like a DEA agent trying desperately to score some pot, but being totally unable to do so.  Somehow.  I don't know, maybe he was wearing his badge or something.  Every scientific principle I learned was either a catapult to becoming a Nobel laureate, or the roadblock I would inevitably hear about from my physics teachers later that day.

One of the earliest of these endeavors involved faster-than-light (FTL) travel.  You see, back in the Pleistocene, before the internet had infected everyone's house and I had to go halfway across town to an actual library to look something up (holy crap, do I feel old), I would take my designs to the physics department and ask the teachers why they wouldn't work.  I actually succeeded in stumping a few of them once or twice, but I would inevitably be shot down by some bit of trivia like Lenz's law (which is basically electromagnetic friction).  Right after we learned about vectors, we jumped into the application of actual forces (which was followed throughout the semester by figuring out those forces, roughly in order of ascending complexity and/or descending magnitude), where I learned the mighty F=ma.  For the uninitiated, or those who simply don't remember high school science (don't feel bad, I don't remember anything else), F is force, m is mass, and a is acceleration.

What this means is that if you know the mass of an object, as well as its current acceleration, you can calculate the force acting upon it.  Conversely, if you have some mass and want it to accelerate at a certain rate, then you can calculate the force that must be applied to do so.  Actually, as long as you know any two of those terms, you can calculate what the third one must be.  I reasoned that as long as we kept applying force, we could accelerate to any velocity we could choose.

So one day I asked my teacher why we couldn't go faster than the speed of light.  I was told that as we accelerated towards the speed of light, the forces involved in our acceleration would crush us.  Obviously, the more a mass accelerates, the greater the force acting upon it - but what if we accelerated really, really slowly?  It would take some time, but it should work, shouldn't it?  And, to jump ahead just a bit, subjective time "slows down" as one approaches the speed of light, so shouldn't the passengers experience the trip as a relatively quick one?

There were two things that I hadn't taken into account.  First, that modern spacecraft accelerate by casting off mass:  by throwing something in the opposite direction that you want to go, you move yourself in the direction you want to go, because for every action there is an equal and opposite reaction.  Spacecraft don't need fuel for a trip like cars do, because cars need fuel to travel a certain distance, but spacecraft only need the fuel required to accelerate to the desired travel speed; then they wait, and then maybe decelerate when they're about to arrive.  But the speed of light is really, really fast, and accelerating anything to such a speed would require a lot of mass to be cast off, which would in turn require a very massive craft which would in turn require a lot of fuel to take off in the first place.  I think you can see where this is going.

But that was merely a technical limitation, a problem for the engineers to work out.  The second thing I had not taken into account was the Lorentz factor, which actually makes accelerating to the speed of light impossible - for any living thing, anyway.  You see, as it turns out, F=ma isn't the whole story.  The whole story is F=γma, where γ (gamma) represents the Lorentz factor.  γ itself is defined as the inverse of the square root of the quantity one minus the square of the ratio of velocity to the speed of light.  Huh?  Yeah, I'm not very good with words, either.  Here it is mathematically (v is velocity, c is the speed of light):
OK, so there are a couple of important things with this new insight.  The above is γ, and since F=γma, we can do some juggling to yield γ=F/(ma).  Since F is precisely equal to m*a in most Earthly applications, it seems reasonable to assume that γ=1 most of the time.  Fortunately, this is true!  Looking at the above equation, v/c is really tiny when velocity isn't close to the speed of light, because c is huge and v is not huge.  This means that (v/c)2 is going to be even tinier, so that 1-(v/c)2 is going to be really close to 1 (but just a tiny bit more).  The square root of this will also be close to 1, and 1 divided by something really close to 1 is also really, really, really close to 1.  This means that γ doesn't make much of a difference in our day-to-day lives, because we don't go anywhere near the speed of light.

But what if we did?  If v was equal to c, then v/c would equal 1, and 1-12 is zero, and... OK, now we're dividing by zero, which isn't allowed.  OK, let's say that v is really close to c instead.  With v being close to c (but not equal to it), v/c is going to be close to one, but just a little bit less.  Squaring something close to one will make it only a little bit less than one than it is right now, and so 1-(v/c)2 is going to be a tiny, tiny number.  The square root of a tiny, tiny number (that is less than one but ever so slightly more than zero) is also a tiny, tiny number - it's only a bit bigger.  And one divided by a tiny, tiny number is a really, really big number.

Moral of the story:  as v approaches c, γ becomes huge.  And when γ is huge enough, even a tiny m & a will make for a huge F.  What this means for our travelers is that, as they cast off a little bit of mass at a time, for them to accelerate to the speed of light is going to require increasing amounts of force.  Furthermore, the opposite side of that coin is that the travelers will also experience this force, and tiny bits of acceleration will result in huge amounts of force, to the point where just the margins of error in even our most precisely controlled accelerations could be deadly.  So, no, we mortals will probably never get appreciably close to the speed of light.

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