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Yes, it is not immediately obvious how mass ought to be undershood in the context of relativity. First attempts treated mass as a directed vector, with transverse and longitudinal components for an accelerated particle (now we would say that the mass is fixed, it is the momentum that changes non-linearly and just use F=dp/dt).

Mass should rightly be considered more inherent in SR that energy. Energy in relativity is coordinate dependent, but mass is not: all observers agree on a system’s mass, because it is invariant.

Re: the last paragraph, yes you can project the four vector onto time and space, and you get energy and momentum respectively. So energy is not the four vector itself but its projection onto the time axis (and mass is of course its magnitude).

(Relativistic) mass is totally frame dependent in SR. That's why SR normally talks about the rest or invariant mass, which is the mass in a special frame, the rest frame of the object.

You can't construct a reference frame such that a rotating object is at rest.

You can construct a reference frame such that a rotating object is rotating but not translating. You can also construct a non-inertial reference frame such that it's not even rotating.

Not quite, the laws of physics don't apply in a rotating reference frame. For example the speed of light stops being a constant.

So, if you wanted to use one you would need to redo all your math and effectively use a non rotating frame in the middle of all your calculations, or add fictitious forces and flexible constants.

> Not quite, the laws of physics don't apply in a rotating reference frame.

Yes they do and it's precisely the point of GR.

Consider a reference frame rotating at say 1 RPM. At 1 light year from the center a 'stationary' particle in a non rotation reference frame is now moving in a circle at 2 * π light years per minute. At 2 light years from the center it's moving 4 * π light years each minute and that continues the further from the center you get.

GR is fine with most reference frames as long as it's translation. Rotation is however not ok.

GR is fine with that - what will happen is that there'll be a sign change in the metric (the 'time' coordinate will become spacelike, whereas the 'angular' coordinate will become timelike), making it impossible to sit at a fixed value of the rotating frame's angular coordinate (which would correspond to faster-than-light motion in the non-rotating, inertial frame).

That's just a notation for doing exactly what I am describing.

GR deals with curved spacetime, but the only way to work in rotating spacetime is by changing how you calculate what's going on. Which means your calculations must be mapped and don't generalize.

EX: Try and do an actual calculation for say two electrons hitting each other at say 0.5c. In rotating, non rotating inertial, and non rotating non-inertial reference frames of your choice.

So, if you wanted to use one you would need to redo all your math and effectively use a non rotating frame in the middle of all your calculations, or add fictitious forces and flexible constants.

That's what the Christoffel symbols already do - it's part of the formalism, there's nothing to add!

This. The quantity that “truly” exists is the energy-momentum 4-vector P. A vector exists independently of any particular reference frame. Its components, on the other hand, depend on such a reference frame.

Seen from a given reference frame, the temporal component of P is the energy E, its spatial components are the momentum 3-vector p, and its magnitude is the rest mass |P| = sqrt(E^2 - p^2) = m. The minus sign rather than plus sign is due to the Lorentzian signature. It is very helpful to visualize these quantities geometrically as 4-vectors in 4-dimensional space.

Also, the magnitude of the momentum 4-vector is a Lorentz scalar, meaning it is invariant. Hence the rest mass m = |P| is invariant.

> It seems to me that before Newton, people only had a vague notion of what a force is. > When Newton stated that F=ma, he defined the concept of force precisely.

Newton wasn't the first to define the concept of force, nor the first to formulate the basic laws of motion. E.g. Galilei had formulated both the first and second laws in some sense, restricted to cases such as straight line motion on a plane. The laws came out of experiments.

Newton's brilliant work was in collecting the three laws, generalizing them and realizing their combination applied to celestial mechanics as well as the mundane brick sliding down a plane.

"I thought that F=ma is actually a definition of the concept of force."

Yes, that is part of what millstone was getting at.

The root of the problem is that for a long time, people were taught since elementary school that the equality symbol was not symmetrical, but was something more like a => symbol; you take the things on the left and can produce the things on the right, usually via some vague idea of simplification. In other words, 2 + 2 => 4; "from 2 + 2 I can produce 4", but people with this misunderstanding (which is a lot of people) will resist 4 = 2 + 2, reading it as 4 => 2 + 2. They'll often have a hard time articulating the problem, but it'll boil down to the fact that you're not "allowed" to "unsimplify" like that. It turns out you can get quite far in the standard math curriculum with that misconception, somewhere round about "factoring equations by completing the square"[1], and even farther if you just learn those handful of exceptions of one-off tricks and never go into a field of work where one way or another you need to deeply understand what is going on.

(Some people complain that you can see this misconception in the standard programming definition of =, which runs the other direction, and isn't a statement that "these two things are equal" that may be true or false, but is a command to assign this value into that slot. It's a natural outgrowth of the incorrect = understanding.)

I remember in physics that it took us collectively quite a while to get that it was not F => ma, or ma => F, that it was not that there was a "force" and a "mass acceleration", but that they were the simply the same thing. There is no free-floating "force" that can be "converted" into "mass acceleration", they are so inextricably the same thing that there was no difference. And I was in a fairly advanced and educated class at the time.

While I tend to dislike the common core in many ways, one admirable thing about it is that it is attempting to correct this mistake; as soon as = is introduced for simple arithmetic, children are given ____ = 2 + 2 as often as 2 + 2 = ____, and only slightly later than that they are getting 2 + ___ = 4.

[1]: http://www.purplemath.com/modules/solvquad3.htm It's been a while since I did this stuff, and you may not have had the exact same order as me anyhow, but this is the first time I can think of where we had to deliberately "complexify" an equation in order to solve it. And I can still recall the gears skipping in my head as I tried to deal with this, even as to me now it's nothing at all and I do an analogous thing quite often.