Monday, April 28, 2008

Relativistic mass in introductory physics

There seems to be a slight issue with how some principles in physics are introduced nowadays.

If you've already had some exposure to Einstein's Special Relativity, you'll likely be familiar with the following equation (if not, stay tuned -- I'll give an overview in later posts):

Here we have m, the rest mass of the object, m[rel] - the relativistic mass of an object in motion, v - the velocity of the object, and c - the speed of light.

In a nutshell, the basic idea is usually introduced roughly like so: "the apparent or observed mass of an object or system that's moving will change from a stationary frame of reference as that object's velocity changes." The premise was that as the velocity of the object or system increased relative to the frame of reference, so did the relativistic mass. You can see, in looking at the equation, that as the velocity approaches the speed of light, the relativistic mass approaches infinity. (For those of you who haven't had the pleasure of delving into this area yet, wait for it -- time operates the same way! Yes, as that object travels faster and faster, time for it slows down, compared to the static frame of reference!)

All seems well and good as the ideas are explained one by one...but then a problem arises. You may have heard this yourself: "...and so we can see that it's impossible for an object to accelerate to the speed of light, since the closer the object gets to that speed, the greater its mass, and so more and more fuel would be required for it to continue to accelerate -- so the amount of fuel you'd need would approach infinity, too!"

On the surface, it seems reasonable. Take a moment to think about it, though -- if the object is carrying its "gas tank" with it, then the relativistic mass of the "gas" on board is approaching infinity, too. This isn't to say that it's gaining fuel as it goes -- just that the rate at which the apparent mass of the carried fuel is increasing matches the increase observed in the object itself! With that in mind, this argument for an object not being to accelerate so certainly appears to be false.

Don't go jumping about thinking that warp speed is around the corner, though -- there are a few other factors that need to be considered, too. While the "infinite fuel" argument in this fashion certainly seems to be false, there are other things at work: For an observer on that moving object, they may very well see a smooth and steady increase to the speed of light, maybe even beyond, too. If they had a means of constant acceleration, and the fuel to do so theoretically, given the object's rest mass, the change in the traditionally viewed relativistic mass of the system should have no bearing on whether it could achieve that velocity. "Oh good, we've achieved light speed, and just in time for tea."

For the stationary observer, though, they're not so lucky. If we take into account the time dilation side of things, the stationary frame of reference will never see that object achieve light speed. Why? Well, for the object in motion it's easy -- it's accelerating smoothly through light speed at a constant acceleration. For the stationary observer, though, the increase in velocity appears to be slowing down more and more as it approaches light speed, but due to time dilation rather than anything mass related.

Just to put some numbers to it: at 90% of the speed of light, what seems to take 1 second on the object takes roughly 2 seconds for the stationary observer. At 99%, it appears to take 7.01 seconds. At 99.99%, 70.7 seconds....and so on. (At 99.99999999999996% light speed, one second for the object equates to over 1 year for the stationary observer!) So that smooth, continuous acceleration from the perspective of the moving object appears to be an ever diminishing acceleration from the outside, getting ever slower as it approaches light speed!

If you could undertake such a journey, what would occur? Ever closer to lightspeed -- 1 second passes...oh, that's 10,000 years. Another -- oh, there went a million years. Who knows what you'd find upon your "return"? >>ZIP<< "Oh, lightspeed! Hooray!" -- at that moment, though, the traditional view will basically have it that an infinite amount of time will pass for that stationary where does that leave you..."Oh dear, where will I get crumpets from, and what happened to home?" Hrmm, perhaps that's not such a good idea ;)

More recently, there have been some modifications to the view on this -- taking a broader view for considering momentum and energy of a system...that's not relevant to our discussion at the moment, though. (What might happen to the object reaching "c" is beyond the scope of the article, too -- sorry ;)


  1. This treats Relativistic mass as if it was a real changing thing.
    Science does not treat mass that way, nor did Einstein.
    Better to think of the formula needing to factor both the mass and the speed by the Gamma Factor to give a result and not that any “Mass” is actually changing form any perspective.
    Only that different observers must use different Gamma Factors.
    Think of “relativistic mass” as an abstraction that can be used in the math.

  2. Thank you for your response!

    On first read, what you've said makes sense, and is a better fit for my current level of understanding of relativity.

    By your comments, it seems safe to assume that you also feel that the original example provided is not a good one, albeit for slightly different reasons, and that it shouldn't be used in today's school system.

    [Your view appears to refute it based on how things should actually be considered, whereas my attempt to refute it was based on a logical fallacy..."Well, if that's how things work, then _this_ should be the case, not what you've proposed"]