If we consider the special theory of relativity as a Lorentz-invariant theory, which it essentially is, then there is no place for relativistic mass in such a theory. And from the point of view of common sense, relativistic mass cannot exist, because the amount of matter that makes up a moving body cannot change in it depending on the speed at which the body is moving. This means that the amount of matter that makes up a moving body cannot change in it depending on which of the many moving inertial frames of reference the observer, who considers himself, due to the principle of relativity, to be motionless, decides to move into.

Therefore, the law E = mc^2 and the formula expressing it do not exist and cannot exist. What should I do? The theory is saved by the statement that the place of the relativistic mass m in this law should be taken by the relativistic inertia Ia, depending not only on the speed of movement, but also on the angle a between the direction of the speed and the direction of the force accelerating the body (the index a is lower here). Then we have the law E = Iac^2 of the equivalence of relativistic inertia and relativistic energy. In which the relativistic inertia Ia turns out to be almost exactly equal to the relativistic dynamic impulse inertia of the body's motion.

In addition, for the state of rest of the body, the relativistic dynamic impulse inertia turns out to be equal to zero and the proposed law of equivalence of relativistic inertia and relativistic energy turns out to be exactly reduced to the known and strictly mathematically proven law of equivalence of rest mass and rest energy.

And there are no more problems with relativistic mass in theory. In addition, the theory receives a new developed theory of inertia, in which the inertial forces cease to be fictitious quasi-Newtonian pseudo-forces (D'Alembertian or Eulerian), and become only Newtonian.

Link to my article “On the theory of inertia and the law of equivalence of relativistic inertia and energy”, in which the issue of relativistic mass is resolved once and for all in favor of relativistic inertia and the theory of inertia itself is developed:

https://www.researchgate.net/publication/374583711_On_the_theory_of_inertia_and_the_law_of_equivalence_of_relativistic_inertia_and_energy_K_teorii_inercii_i_o_zakone_ekvivalentnosti_relativistskih_inercii_i_energii_Aleksej_A_Platonov_Sankt-Peterburg

Если рассматривать специальную теорию относительности как лоренц-инвариантную теорию, каковой она, по существу, и является, то релятивистской массе в такой теории нет места. Да и с точки зрения здравого смысла релятивистская масса не может существовать, ибо количество вещества, составляющего движущееся тело, не может в нем изменяться в зависимости от того, с какой скоростью тело движется. Это означает, что количество вещества, составляющего движущееся тело, не может в нем изменяться в зависимости от того, в какую из множества движущихся инерциальных систем отсчета вздумается перейти наблюдателю, считающего себя, в силу действия принципа относительности, неподвижным.

Поэтому закон E = mc^2 и формула, его выражающая, не существуют и существовать не могут. Как быть? Теорию спасает утверждение, что место релятивистской массы m в этом законе должна занять релятивистская инерция Ia, зависящая не только от скорости движения, но и от угла a между направлением скорости и направлением действия силы, разгоняющей тело (индекс a здесь нижний). Тогда у нас появляется закон E = Iac^2эквивалентности релятивистской инерции и релятивистской энергии. В котором релятивистская инерция Iaоказывается практически точно равной релятивистской динамической импульсной инерции движения тела.

К тому же для состояния покоя тела релятивистская динамическая импульсная инерция оказывается равной нулю и предложенный закон эквивалентности релятивистской инерции и релятивистской энергии оказывается точно сведенным к известному и строго математически доказанному закону эквивалентности массы покоя и энергии покоя.

И никаких проблем с релятивистской массой в теории больше нет. К тому же теория получает новую развитую теорию инерции, в которой силы инерции перестают быть фиктивными квазиньютоновыми псевдосилами (даламберовыми или эйлеровыми), и становятся только ньютоновыми.

Ссылка на мою статью «On the theory of inertia and the law of equivalence of relativistic inertia and energy», в которой раз и навсегда решается вопрос релятивистской массы в пользу релятивисткой инерции и развивается сама теория инерции:

https://www.researchgate.net/publication/374583711_On_the_theory_of_inertia_and_the_law_of_equivalence_of_relativistic_inertia_and_energy_K_teorii_inercii_i_o_zakone_ekvivalentnosti_relativistskih_inercii_i_energii_Aleksej_A_Platonov_Sankt-Peterburg

One should learn special relativity. E=mc^2 holds in a particular reference frame. Lorentz transformations imply that E^2-|p|^2c^2 is an invariant quantity, whose sign is, also, invariant under Lorentz transformations. That leads to the definition of mass, that is invariant under such transformations.

@Stam Nicolis

Stam Nicolis, thank you for your answer! You are absolutely right. The basic equation of relativistic dynamics that you cited really shows strictly mathematically that the mass of a moving material object is invariant, that is, it does not depend on the speed of movement, and therefore does not change.

But in the equation E=mc^2 it is assumed that mass m is variable, that is, it changes with changes in speed, and therefore it cannot be an invariant. This is a contradiction with the law E^2-|p|^2c^2. From this law, the immutability of mass is proven, as you rightly noted, strictly mathematically. Therefore, the statement about the immutability of mass became part of science. The equation E=mc^2 has NOT been strictly mathematically proven (see literature (1), (2), (3), (4)). Therefore, we cannot consider it a strictly scientific statement.

You are right that the formula E=mc^2 is contained in a certain frame of reference. But it should NOT be there! There should be another formula there instead. Which? I obtained this other formula strictly mathematically.

Even A. Einstein himself in June 1948, in his letter to Lincoln Barnett, the author of the book “The Universe and Dr. Einstein,” wrote the following:

“It is not good to introduce such a concept of mass M = m/sqrt(1-v^2/c^2) of a moving body, for which a clear definition cannot be given. It is better not to introduce any other concept other than “rest mass” i.e. Instead of introducing M, it is better to give an expression for the momentum and energy of a moving body.”

Here M in A. Einstein’s letter is precisely the so-called relativistic mass, which is used in the equation E=mc^2. So, A. Einstein, in fact, called for abandoning the use of the concept of relativistic mass m in physics. And it is precisely in the equation E=mc^2. Thus, in essence, A. Einstein refused to use the equation E=mc^2 in physics.

Please note that in the above message by A. Einstein, he proposes to give not an expression for the relativistic mass m, but an expression for momentum and energy.

It is precisely the expression for momentum that I use when obtaining my new equation, which replaces the unsuccessful equation E=mc^2.

Well, in conclusion. I have studied special relativity for over 40 years. Therefore, I know her very well, I have something to say and how to say it. And I know the subject of my conversation very well.

Literature.

(1) Cahn W., “Einstein, a pictorial biography”, New York, Citadel, 1955, p. 26.

(1) Ives H.E., “Derivation of the Mass-Energy Relation,” Journal of the Optical Society of America, 1952, 42, p. 540-543.

(3) Logunov A.A., “Henri Poincaré and the theory of relativity”, Nauka, Moscow, 2004, p. 125.

(4) Jammer M., “Concepts of Mass in Classical and Modern Physics”, 1961, Harvard University Press, Cambridge-Massachusetts;

Best regards! Happy New Year!

I am adding a file with additional information on the issue under discussion.

Добавляю файл с дополнительной информацией по обсуждаемому вопросу.