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I was doing some practice questions regarding relativistic kinematics and I just had a question, We write total energy of an object as; $$ E^{2} = m_{0}^{2}c^{4} + p^{2}c^{2}$$

Where $m_{0}$ is the rest mass. Now if we replace $m_{0}$ with it's relativistic alternative;

$$m = \gamma m_0$$ Where $\gamma$ is the Lorentz factor.

Can I remove the kinetic energy term $(p^{2}c^{2})$, since the kinetic energy is now counted as the mass increase or am I missing something?

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I was doing some practice questions regarding relativistic kinematics and I just had a question, We write total energy of an object as; $$ E^{2} = m_{0}^{2}c^{4} + p^{2}c^{2}$$

Where $m_{0}$ is the rest mass. Now if we replace $m_{0}$ with it's relativistic alternative;

$$m = \gamma m_0$$ Where $\gamma$ is the Lorentz factor.

Can I remove the kinetic energy term $(p^{2}c^{2})$...

Yes, you can write: $$ E = \gamma m_0 c^2\;, $$ which can be squared to see that: $$ E^2 = \gamma^2 m_0^2 c^4\;. $$


As a notational aside, I would personally advise against using the notation $m$ for relativistic mass, since many many people prefer to use $m$ for rest mass. I.e., just use $m$ for rest mass and use $\gamma m$ for relativistic mass.

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  • $\begingroup$ One problem with $\gamma m$ notation is that it doesn't work if the rest mass is zero. And, of course, using it indicates no respect for the history of $E=mc^2$, its cultural significance, and its profound implications. $\endgroup$
    – John Doty
    Commented Jan 20, 2023 at 13:00
  • $\begingroup$ This is the Physics Stack Exchange, not the Hypothetical Cultural Significance Stack Exchange.... $\endgroup$
    – hft
    Commented Jan 20, 2023 at 17:13
  • $\begingroup$ How is your preference for $\gamma m$ anything beyond a cultural statement? $\endgroup$
    – John Doty
    Commented Jan 20, 2023 at 18:08
  • $\begingroup$ Why do you feel the need to die on this hill? $\endgroup$
    – hft
    Commented Jan 20, 2023 at 19:49
  • $\begingroup$ Who's dying? Not me. But physics itself is in a bit of trouble, losing sight of the fundamental phenomena. youtube.com/watch?v=aIhk9eKOLzQ $\endgroup$
    – John Doty
    Commented Jan 20, 2023 at 20:00

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