## Relativistic mass [Fact or Fiction]

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### Relativistic mass [Fact or Fiction]

I thought I'd amplify the relativistic mass idea, as my words have been misquoted and mangled in the past. Relativistic mass is a pernicious misconception, ironically most prevalent among the most educated of the lay audience. Partly this is because of how the idea is taught (and how I learned it the first "real" time in college.) Only later, when a deeper grasp of relativity was necessary, did I come to realize how the pedagogical approach necessary to introduce relativity's odd behavior is "not quite right".

In some sense, the discussion is semantic. But the cartoon of increasing mass at high speeds is one of those analogies with limited strength and should be put aside by a serious student of relativity.

Basically the problem is the following. We teach to students the Newtonian concept of momentum, denoted p, which is simply mass (m) times velocity (v). Or, in the concise language:

$\Large p = m \times v$

[or momentum equals mass times velocity for the math-phobic.]

This mass is what you're used to...a measure of how much "stuff" is involved. It is an immutable quantity. However, Einstein showed that this Newtonian idea was wrong, although it works very well at lowish speeds (note lowish still means many thousands of miles per second). He found that the correct equation was

$\Large p = \gamma \times m \times v$

where ($\gamma$) holds all of the relativity stuff. And, at low speeds, (gamma) is equal to 1, so Newton's equation is just fine. So here is were the pedagogical approach comes in. Relativity is weird, gamma is weird. In order to make students a little more comfortable, we decide to not monkey with the equations and so we define a new quantity, relativistic mass. Relativistic mass is defined:

$\Large {\rm relativistic mass} = \gamma \times m$

And thus the momentum equation becomes

$\Large p = \rm{(relativistic mass)} \times v$

which is our old Newtonian buddy. So you tell the students that we've just defined a slightly different mass, their equation is what they're used to and everything is both hunky and dory. Relativistic mass does increase with velocity. This is where mid-level students of relativity stop.

The question is "Does relativistic mass have any real significance?" The answer is no. What is really increasing as the velocity increases is inertia. Einstein's equation is right and it is gamma that changes, not mass (the rest mass).

Partially, this is a math thing. There is a property of mathematics, called commutativity, that says when you multiply three things, you can multiply them in different orders and get the same thing.

For instance 5 x 2 x 3 = 30. But you can write that as ( 5 x 2 ) x 3 = 10 x 3 = 30. You can also write it as 5 x (2 x 3) = 5 x 6 = 30. The question becomes, do the things inside the parentheses have a physical significance? Sometimes yes and sometimes no.

The volume of a rectangular cube is length times width times height

$\Large V = l \times w \times h = (l \times w) \times h = l \times (w \times h)$

In this case, the parentheses denote a physical quantity. (l x w) or (w x h) denote the area of a face of the rectangular solid. So it makes sense to say

V = (area of one face) x h
or
V = (area of a different face) x l

In this case, putting something inside the parentheses really means something.

On the other hand, the volume of a sphere is $\frac{4}{3} \times \pi \times r^3$ (or four thirds times pi times r (cubed)).

Now if you can certainly mathematically put parentheses around parts of the equation

$\Large V = \frac{4}{3} \times \pi \times r \times r \times r = (\frac{4}{3} \times \pi \times r) \times r \times r$

But the stuff inside the parentheses doesn't really mean anything.

Similarly in

$\Large p = ( \gamma \times m ) \times v$

$( \gamma \times m )$...which is defined to be relativistic mass, doesn't have a physical significance.

In any event, it is INERTIA that increases as the velocity increases, not MASS.

Now you can't imagine how much this discussion has caused in the past. There are a lot of people out there who figure that if Einstein, laboring as a lowly patent clerk, could come up with relativity then they can work out something even better, sitting in their cubicle. The difference was that Einstein was a far-seeing, visionary genius. Oh, yeah...and his theory of special relativity has been experimentally proven...the ultimate arbiter of a physical theory. But when I tell people that mass doesn't increase without bound, although inertia does, an irritating few take the first half of the sentence and now I'm their best buddy, as they've somehow proven that Einstein was wrong. Now interstellar travel becomes more tenable, blah, blah, blah.... I've even been quoted (against my will and without my permission) in books on the topic, with the quote taken out of context and the clear intent of the original text completely subverted. Hence my long-time hesitation about getting into bulletin boards.

I wouldn't be shocked if it turns out that relativity isn't the whole story. But the piece of the story we understand is very solid, well tested, and all of the goofy time and space dilation bits of relativity have been observed. My colleagues and I use time dilation every day to study longer-lived particles...for instance b quarks.

So, in summation: Relativistic mass is not real. But it is a helpful pedagogical concept, which is why it is still used. But it's only useful as an analogy for someone encountering the whole "can't go faster than the speed of light" for the first time.

The reality is that the Newtonian equation $p = mv$ is a special case of the correct Einsteinian equation $p = \gamma m v$, when the velocity is low and therefore $\gamma = 1/\sqrt{1 - (v/c)^2} \simeq 1$

Lincoln
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