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Since mass is a Lorentz invariant, it can never change to preserve the vectorial nature of the four-momentum and the other four vectors. Thus the idea that mass changes as speed increases is highly discouraged. However, Einstein in The Meaning of Relativity said that

The mass of a body is not a constant, it varies with changes in its energy.

Why is that?

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  • $\begingroup$ Einstein said it each way at different times, but his later comments generally referred to invariant mass. The "Meaning of Relativity" was 1922. $\endgroup$ Mar 19, 2023 at 2:34
  • $\begingroup$ @JerroldFranklin Einstein actually revised it quite a few times. The edition that I have is from 1954. $\endgroup$ Mar 19, 2023 at 13:29
  • $\begingroup$ I think later editions were reprints of the original edition. The original edition was a summary of lectures that Einstein had given at Princeton. $\endgroup$ Mar 19, 2023 at 17:07
  • $\begingroup$ @JerroldFranklin In the preface of 1954 Einstein states that he completely revised the fourth lecture's notes, so I think he must have read the other lectures too. $\endgroup$ Mar 19, 2023 at 19:38
  • $\begingroup$ I hadn't read the 1954 edition. So I guess he still said that in 1954. $\endgroup$ Mar 21, 2023 at 2:07

2 Answers 2

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Rest mass is Lorentz invariant.

Relativistic mass is the same thing as energy.

Einstein was referring to relativistic mass. That used to be what we called "mass". It is, however, fashionable to call rest mass "mass" nowadays.

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Since mass is a Lorentz invariant, it can never change

Yes, the rest mass is invariant. Nowadays, we usually just say "mass" instead of "rest mass." So what you wrote above is correct.

For example, when I write the symbol $m$ in the below formula, it stands for rest mass: $$ E^2 = m^2 c^4 + p^2 c^2\;. $$

Sometimes people use the symbol $m_0$ for rest mass, in which they would write: $$ E^2 = m_0^2 c^4 + p^2 c^2\;. $$

However, Einstein in The Meaning of Relativity said that

The mass of a body is not a constant, it varies with changes in its energy.

Why is that?

When Einstein used the word "mass" he didn't mean "rest mass," he presumably meant the "relativistic mass," which is the rest mass multiplied by $\gamma\equiv \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$.

For example, if I let $m$ denote the rest mass (which is invariant) we have: $$ E = mc^2 \gamma\;, $$ where the "relativistic mass" is $m\gamma$.

For example, if I let $m_0$ denote the rest mass (which is invariant) we have: $$ E = m_0c^2 \gamma\;. $$ where the "relativistic mass" is $m_0\gamma$.

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