[semajian]

Mass is a Lorentz scalar and thus is invariant; the mass of relativistic particles does not increase as velocity increases. One can say the force necessary to keep a constant acceleration increases as velocity increase. This is a common misconception, and it appears even NASA engineers don't understand this. This is not science.

[whatshisface]

"Mass" is imprecise enough as a word to refer to either inertia, E/c^2, or rest mass. Rest mass is indeed invariant but the rest aren't.

[semajian]

I've never come across a physicist who used the term "inertia" to refer to any of the common relativistic quantities.

[whatshisface]

I used the word "inertia" to refer to the various inertia-related things that are used in mechanics. For example, the mass tensor that appears in curvilinear coordinate systems. This class of inertia-related things are usually given a name involving the word mass, and they are not usually Lorentz invariant.

[bufferoverflow]

> the mass of relativistic particles does not increase as velocity increases

What are you talking about? This stuff was taught in high school to me:

    m = m0/√(1 − v²/c²)

[gus_massa]

Your teacher lied to you :).

The problem is that you have a different "mass" when you try to accelerate the particle in the direction it is traveling and a different "mass" when you try to accelerate the mass in a perpendicular direction. (If the acceleration is not parallel or perpendicular, it's more complicated.)

Your formula is the correct one for the acceleration in the perpendicular direction, like in the magnetic field of a cyclotron, that is the typical example.

For an acceleration in the parallel direction you must add a ^3 to the correction.

  m = m0/√(1 − v²/c²)³

Most modern books of advanced electromagnetism/relativity try to avoid the change in the "mass" an use only the rest mass m0. The other "mass" is sometimes handy and sometimes misleading.

[bufferoverflow]

I doubt my teacher lied to me. This stuff is in the text books, in wikipedia.

[antonvs]

"Lie" might be a bit strong, but the concept of relativistic mass is misleading and technically incorrect in various ways.

As a result, in the last few decades or so the concept of relativistic mass has gone out of favor in physics and its teaching.

The wikipedia section for relativistic mass[1] includes the following quote from the textbook "Spacetime Physics," by Taylor and Wheeler:

> "The concept of "relativistic mass" is subject to misunderstanding. That's why we don't use it. First, it applies the name mass – belonging to the magnitude of a 4-vector – to a very different concept, the time component of a 4-vector. Second, it makes increase of energy of an object with velocity or momentum appear to be connected with some change in internal structure of the object. In reality, the increase of energy with velocity originates not in the object but in the geometric properties of spacetime itself."

In short, relativistic mass is not a good way to understand what's actually happening in these scenarios.

Essentially, the relativistic kinetic energy of the object is being counted as mass, following E=mc^2. However, that only works because E=mc^2 is a special-case simplification that is designed to apply when momentum is zero, i.e. in an object's rest frame, to its rest mass.

If you look at the full mass-energy equivalence equation[2]:

E^2 = (pc)^2 + (m0 c^2)^2

...you can see the separate momentum component p. It's possibly to "cheat" and eliminate the (pc)^2 term and bundle it into a revised value for m, but this loses information and leads to various issues of the kind described by Taylor and Wheeler.

[1] https://en.wikipedia.org/wiki/Mass_in_special_relativity#Rel...

[2] https://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalenc...

[cygx]

And many physicists (including Einstein) have argued it's a bad idea to teach things this way.

[magicalhippo]

Your m is the relativistic mass while the m0 is invariant mass, also known as rest mass, which is what GP was talking about.

AFAIK, most physicists these days will only use the invariant mass, or at least be explicit about it if they talk about the relativistic mass.

https://en.wikipedia.org/wiki/Invariant_mass

[zonidjan]

So... you're saying E=mc^2 is false? Huh, who woulda thunk it.

[gpderetta]

IANAP, but as far as I know it is true only if m is the relativistic mass which apparently is an obsolete and confusing concept which is not taught anymore (according to Wikipedia Einstein himself disliked it). Otherwise the formula only holds at rest.

[semajian]

Yes, well said.

[antonvs]

It's a simplified equation which only fully applies in the case where momentum is zero.

The full equation[1] is:

    E^2 = (pc)^2 + (m0 c^2)^2

...where p is momentum, and m0 is rest mass (mass in the rest frame of the object.)

[1] https://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalenc...

[gaze]

Yeah. Quite honestly this is embarrassing.