JohnDenker
commented
8 years ago This is labeled as a Ch13 issue, and we agree Ch13 needs attention ... but Ch14 is where changes are most urgently needed. Actually this issue has ramifications in many chapters from Ch3 onwards, as discussed in item #134.
Suggestions:
- The term «inertia» should be replaced by "mass" everywhere, as discussed in item #134.
- There are other inertia-related suggestions in item #134.
- The whole idea of velocity-dependent mass (or velocity-dependent inertia) needs to go away.
- See also item #107 for a catalog of issues related to the equivalence principle.
- The concept of "time dilation" should be dropped, as discussed in item #202.
- Ditto for «length contraction», as discussed in item #202.
@JohnDenker wrote:
Here is the smart way to simplify the equations of motion. It does not stick gamma-factors in places where they don't belong (such as in the ratio if inertia to gravitational mass).
Let me foreshadow the bottom line: Eventually we will write the four-vector equations:
where p is the four-momentum, m is the mass, u is the four-velocity, x is the four-vector position, and τ is the proper time.
These equations could not possibly be any simpler. Physics does not get any better than this.
Note that the mass (m) is a Lorentz-invariant scalar. It is the same thing as inertia. Inertial mass is equal to gravitational mass. It has to be, in accordance with the equivalence principle. This is verified by Eötvös experiments to verrry high accuracy.
Everything to this point is independent of whatever reference frame (if any!) you choose.
Remark: Consider the spatial part of p, i.e. the projection of p onto the spatial part of some chosen coordinate system. This is denoted p_xyz and is equal to the conventional three-dimensional momentum.
In contrast, the spatial part of u is not the conventional three-dimensional velocity! This is a trap for the unwary. It is one of the few tricky things about relativity.
That's fine as an ending point. Now let's discuss the starting point. We can write on Day One that
where p is the 3-momentum, v is the 3-dimensional "reduced velocity" in some chosen frame, t is the time projected onto the chosen frame, and 🐌 (the snail emoji) indicates that the equation is valid only in the low-speed limit.
Again, these equations could not possibly be any simpler. On the other hand, it would be good to attach a gloss or a footnote to equation [3]. It is the first word on the subject, not the last word. The generalization to arbitrary speed will be discussed in a later chapter.
NOTE: We discussed the value of giving consistent meaning to symbols. This is valuable, but not always possible. There are more concepts than there are available symbols.
In particular, in chapter 1 we use p to denote the 3-momentum, whereas in a later chapter we use p to denote the 4-momentum. This is unfortunate, but unavoidable. We minimize the damage by flagging all the 3D equations with a 🐌 symbol.
The best generalization of equation [3] is equation [1], which we restate here:
p = m u [5] aka [1]
It should be obvious that this reduces to [3] in the low-speed limit. Another generalization of equation [3] is:
p_xyz = γ m v [6]
which is consistent with the spatial part of equation [1]. It reduces to equation [3] in the low-speed limit. Equation [6] is more complicated than one might like ... but nobody cares, because it isn't very useful. Equation [1] is the useful, simple, and elegant equation. The general rule is, if your equations have gamma-factors in them, you are probably formulating the problem in a suboptimal way.
The same gamma-factor shows up in the relationship: u_xyz = γ v [7]
We call v the "reduced" velocity because it is smaller than u_xyz by a factor of gamma. Neither equation [6] nor equation [7] is worth remembering, because they can be instantly rederived if/when needed. Equation [7] is an obvious consequence of the definition of v as a t-derivative rather than a τ-derivative, and [6] is an obvious consequence of [7].
The book tries to simplify the concepts by simplifying equation [6]. In particular it absorbs a factor of γ into the definition of inertia, which is a disaster.
The smart way to simplify equation [6] is to write equation [1] instead!
If a gamma-factor is going to show up at all, it is associated with the velocity (as in equation [7]) ... not the mass. If you fold it into the mass, you violate the equivalence principle. In any case, the important point is that the fundamental equations don't need any gamma-factors at all.
Equation [1] is easier to learn, easier to teach, easier to use, and in every way better. It is simultaneously simpler and more sophisticated.
This is the modern (post-1908) approach. I can explain it to high-school students in less time than it takes to tell about it. (Explaining it to high-school teachers is more complicated.)