Built on Facts

Sunday Function

If you want to kill some time, try to think of all the different definitions of the word “set”. You have chess sets, sets in tennis, sets of dishes, sets musicians play, sets as abstract mathematical entities, television sets. You can set a table, set a clock, set someone straight, set a price, set out on a journey. That’s just scratching the surface. The Oxford English Dictionary lists literally hundreds of different senses of the word. And yet if I set out (ha!) a particular instance of the word, you’ll process its meaning instantly.

You’d think mathematics could avoid the ambiguity of multiple meanings for the same symbol, but I regret to say it’s not so. There’s much less ambiguity to be sure, but there is some out there. Even the venerable π symbol has more than one common meaning. So you’ll not be surprised that when physicists talk about gamma as a function, it’s not necessarily the gamma function. The one we’ll talk about today is generally just called gamma even when treated as a function to avoid confusion. On paper it’s more clear because our gamma today is lowercase and the gamma of the gamma function is upper case in the Greek. Not that a lower-case gamma is definitive either. When in doubt today’s Sunday Function has its own proper name: the Lorentz factor.

Our gamma is a property of a moving object. It’s a dimensionless function of velocity, and it’s defined like this:

i-d4bb9fe09d6b78eb832985821d8a307f-1.png

It’s not a very complicated function, which is nice. Lorentz was the first person to do much with it, and when Einstein put together his theory of special relativity this function turned out to be a central feature permeating the mathematical structure of his theory. There’s only two constants in the equation: v is the relative velocity between two non-accelerating frames of reference (we call these inertial frames), and c is the speed of light. Relativity predicts that time and space aren’t the same for two observers in different inertial frames, but instead time and space are scaled by a factor of gamma via the Lorentz transformation. Notice two things. First, if v is very small than the fraction under the square root is very tiny. Thus the denominator is pretty much 1, and so gamma itself is pretty much one. Second, if v gets close to c, then the ratio in the denominator becomes close to 1, leaving the denominator as a whole close to 0. And this makes gamma huge. We can graph the rest of the behavior, with units in meters per second. Roughly speaking, gamma close to 1 means that the classical non-relativistic treatment is a very close approximation to reality. When gamma gets starts to get much higher than one, classical physics is a bad approximation and you need relativity:

i-7b7142f1efa09b47c3b8242f89bcb9f4-graph.png

Let’s do an example. It turns out that in relativity the kinetic energy of an object in special relativity is not the usual one-half mass times velocity squared. Instead it’s:

i-88516ca505d84f969b7ab15a965c3fb3-2.png

So the kinetic energy of an object isn’t just the usual gently sloping parabola as a function of velocity*, instead at velocity near the speed of light the kinetic energy will look pretty much like the graph above. Without going into the why of the Lorentz factor, this is why it’s not possible to go faster than the speed of light. The graph goes rapidly to infinity, and thus the energy you need to approach the speed of light goes rapidly to infinity as your speed gets close to c.

There are some subtleties to thinking about it this way. There’s no absolute reference frame, so it’s not as though your spaceship suddenly gets near the speed of light and refuses to accelerate further. Instead, time and space are simply differently scaled in your reference frame and the reference frame of the galaxy around you. You need more and more energy to go faster in the galaxy’s frame, but in your frame you keep accelerating fine while the galaxy around you is distorted due to the gamma-factor in the scaling of space and time.

It’s hard to explain precisely without doing the math. Which we in fact will do at some point in the future. For now, just be glad we had Lorentz and Einstein thought this stuff up so we wouldn’t have to do it on our own. On the other hand if we had thought of it ourselves it would have been a free trip to Sweden…

* Yes, I’m thinking the same thing you are, so it’s officially on the to-do list for Sunday Function: Taylor expansion of the Lorentz factor, showing equivalence to classical kinetic energy

Comments

  1. #1 Tyler DiPietro
    June 21, 2009

    Have you done the sigmoid/logistic function yet?

  2. #2 Crux Australis
    June 21, 2009

    “Relativity predicts that dime and space”

    I wish my ‘dimes’ were relative.

    [If I had more dimes I'd hire a proofreader! -Matt]

  3. #3 Paul Murray
    June 22, 2009

    The logistic function – is that (e^x)/(e^x + 1) ? I discovered that one while I was trying to make sense of the notion “to double a probability”.

  4. #4 csrster
    June 22, 2009

    I remember back in my day of teaching undergraduates to do SR that my only piece of consistent advice was “use gamma whenever you can”. It’s still good advice, I think. If you remember that the total energy of a particle is gamma*m*c² and the momentum is
    gamma*m*v then you won’t go far wrong in any basic dynamical calculation.

  5. #5 Tyler DiPietro
    June 22, 2009

    “The logistic function – is that (e^x)/(e^x + 1) ?”

    It’s actually 1/(1 + e^x). My current interest in it has to do with my current interest in non-linear regression models.

  6. #6 Tyler DiPietro
    June 22, 2009

    Whoopsie! I forgot the exponent is actually negative in that formulation, so your formulation is actually quite correct. Sorry about that.

  7. #7 Winter Toad
    June 22, 2009

    Ummmmm… what’s in Switzerland?

  8. #8 CCPhysicist
    June 22, 2009

    I’ll reinforce what csrster@4 said: Those two facts, along with the relationship that displays the invariant “length” of the energy-momentum 4-vector

    (mc^2)^2 = E^2 – (pc)^2

    or E^2 = m^2 + p^2 if you rewrite and use c=1 units

    is pretty much what you need to know for a typical 1-D problem because it leads naturally to the use of the invariant Mandelstam variable s to simplify problems involving collisions. The “length” of the (E,p) 4-vector is just the unchanging mass for one body or, for two bodies, the total energy available to create massive particles at rest in the center-of-momentum frame.

  9. #9 CCPhysicist
    June 22, 2009

    I thought about stealing your thunder about the expansion of gamma, but instead I will pose the following challenge to those who have never done the calculation:

    Use your calculator to find the kinetic energy K of a 5 kg mass traveling at 3 m/s using gamma and see if you get the same answer as you get from good old 1/2 mv^2.

    Repeat for v = 300,000 m/s or other speeds that are not too small and you will see that the two agree extremely well for these “low” velocities as long as this flaw in your calculator (which is that it does not use real numbers) does not come into play. You will start to see differences above 3,000,000 m/s.

    The proof of this equality at “low” velocities is the exercise in basic calculus that Matt promises for the future.

  10. #10 Matt Springer
    June 22, 2009

    #7: Woah, egregious brain glitch there. I have fixed it. Great physics wins you a free trip to Sweden, which was of course the homeland of Alfred Nobel.

  11. #11 rob
    June 22, 2009

    i figured it was a free trip to a patent office in Bern.

  12. #12 Hypocee
    July 22, 2009

    And, of course, the classic SF example is Poul Anderson’s novel Tau Zero, about a Bussard ramjet that gets stuck “on”. You’re quite free to “travel” at arbitrary speeds through classical physics from your point of view, it’s just that time cranks up around you. Eventually our heroes are having to dodge in realtime through a fizzing mass of stars, which are of course shifted up to gamma-ray frequencies. This is before they fly into the next universe. Yes.

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