Physicist: The laws of the universe are relativistic. That is, Einstein was right and Newton, although accurately and intuitively describing the world around us, was wrong. When you try to translate the cleaner Einstein laws into a Newtonian form you find that the mass, M, has a sitting next to it. Normally it has almost no effect, but at very high speeds it becomes important. The Newtonian interpretation (where you don’t allow for messed up time and space, but you do allow for alchemy) of that is that the mass is “exaggerated” by a factor of .
is an excellent way to talk about how much relativity is having an impact on distance, time, and mass. For those of you who’d like to do some back-of-the-envelope relativity , where is the fraction of the speed of light that the object in question is traveling ().
The source of the may be obscure, but you’ll find that when you try to write the laws of the universe (which prefer relativity) in a form we’re used to it always seems to show up.
Matter does two big things: it has inertia, and it makes gravity. Every attempt to measure matter always comes down to one or both of these properties.
Inertial mass means that matter is difficult to push around. This is summed up by Newton’s second law of motion: F=MA.
(Historical side note: Newton himself originally wanted to call this the “second law of suck it Leibnitz“, but in committee he was talked down to the more demure “Newton’s second law”. At one time Newton was discovering as many as 9 new Leibnitz-insulting laws a week.)
Here’s a thumbnail sketch of an example of showing up:
It turns out that, in relativity, the true law is: f=Ma. (Notice the difference?)
Einstein realized that in order to rewrite Newton’s laws for relativity he needed to be careful about whose time he was talking about. As it happens, the correct choice is the on-board time of the moving mass (which is what a clock on the object would read). So here I’m using lower-case to talk about force and acceleration in terms of on-board time, and upper case to talk about them in terms of “world time”, which is the usual time we’re used to talking about.
The ratio of the “on-board time” () to the “world time” (T) is .
As a (non-obvious) result, . A (hand-waving) way to think about this is “if I push (F) that space ship for a while (T), they’ll think I was pushing (f) longer, because their clock () is ticking slower”.
At the same time, that acceleration (a) is in terms of on-board time, but if you want A (the acceleration to everyone else) you have to write it in terms of world time. Acceleration is distance per second per second, so:
So if you look at the forces involved in pushing an object (that’s moving so fast that is noticeable), you find that it’s harder to push than it “should” be (by a factor of ). The easiest way to interpret that is as an increase in mass.
You could just as easily say that “force mysteriously gets weaker” by looking at this last equation as: . But that wouldn’t be consistent with the interpretation of…
Gravitational mass, which creates gravity and pulls other matter toward it, is the second property of matter. But matter isn’t really what creates gravity, it’s energy.
For example, if you had a crazy big laser beam (all energy, no matter), that beam would have some gravity. Similarly, if you have a moving rock (one with kinetic energy) it will have more gravity than a perfectly stationary one. And the kinetic energy of an object with mass M is given by , which is very, very close to (the “rest mass energy” plus the “Newtonian energy“). Point is: there’s another .
These last statements are wrist-breakingly hand-wavy, but they do hold up. The Frankenstein created by combining gravity and relativity is called “general relativity”, and it’s nasty enough that cute tricks like “multiply by ” don’t come close to telling the whole story, or even enough to be useful.