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This web page shows how to derive energy-momentum relationship, $E_{total}^2=p^{2}c^{2}+\left( mc^{2}\right) ^{2}$, given the following equations. Please note that some sources make a distinction between $m_{0}$ and $m$ where $m_{0}$ is taken to be rest mass.

$$K=\frac{mc^{2}}{\root{2}\of{1-\frac{v^{2}}{c^{2}}}}-mc^{2}$$

$$E_{total}=K+mc²=\frac{mc^{2}}{\root{2}\of{1-\frac{v^{2}}{c^{2}}}}$$

In this video, from 08:34 up to 10:23, it is shown that how energy-momentum relation could be shown to be equivalent to $E_{total}$.

Both relationships, $E=mc^{2}$ and $K=\frac{mc^{2}}{\root{2}\of{1-\frac{v^{2}}{c^{2}}}}-mc^{2}$, were found around 1905. [3]

Now coming to the question.

The Energy–momentum relation was first established by Paul Dirac in 1928 under the form $E=\root{2}\of{p^{2}c^{2}+\left( mc^{2}\right) ^{2}}+V$, where $V$ is the amount of potential energy.

Source: https://en.wikipedia.org/wiki/Energy–momentum_relation#Origins_and_derivation_of_the_equation

I have read on some other web pages as well that the energy-momentum relationship was derived by Dirac. I understand that science doesn't progress and evolve as is described in textbooks. What was the problem that it had to wait until 1928 to get energy-momentum relationship when it could have been easily derived using two fundamental relativistic equations much earlier? It might be that I'm overthinking. If my confusion is legitimate, please try to keep the answer simple.

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  • $\begingroup$ More on potential energy in SR. $\endgroup$ – Qmechanic Sep 6 at 14:55
  • $\begingroup$ @jacob1729 Okay. I would request it to be moved there. $\endgroup$ – PG1995 Sep 6 at 15:02
  • $\begingroup$ I’m voting to close this question because it's about the history of physics. $\endgroup$ – Gert Sep 6 at 15:50
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It is not that it had to wait to be "established", it is obtainable from what was known by trivial algebra, but rather that it had to wait for a reason to write it that way. In the early years of relativity the concept of the "electromagnetic mass" of electron was prominent, which suggested that said mass is velocity dependent. It was at odds with Einstein's kinematic approach in special relativity, but he reflected it nonetheless in what came to be called "relativistic mass" $m$. So it was more natural to relate total energy to that mass rather than to the rest mass $m_0$, which made for a simpler formula $E_{total}=mc^2$, see Why is Einstein's mass-energy relation usually written as $E=mc^2$, and not $\Delta E=\Delta m c^2$?

The notion of relativistic momentum $p=\frac{m_0v}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$ was introduced by Planck in 1906, but it is only natural in Minkowski's spacetime (4-vector) context, which Einstein disfavored for a long time as too fanciful, see What was the relationship between Einstein and Minkowski? Apparently, he changed his mind around 1921, as reflected in Stafford Little Lectures. Incidentally, the "relativistic mass" does not fit well with the 4-vectors (it is not Lorentz-invariant), see When and why did the concept of relativistic mass become outdated?, so it became reasonable to relate the total energy to the rest mass and momentum instead, as Dirac did. Einstein only disclaimed the "relativistic mass" explicitly in a 1948 letter to Barnett, where he also endorsed Dirac's form of the momentum-energy relation. Here is from Adler's Does mass really depend on velocity, dad?:

"The electromagnetic world view that occupied much of the first quarter of this century has been extensively and elegantly discussed elsewhere. The general idea was to construct an electromagnetic model of the extended, as opposed to the point, electron. The properties derived in that way were assumed to be extendable to bodies other than the electron. One result of this work was to predict a velocity dependent mass... Einstein's relativistic mass had its origin in the kinetics of his special theory and not in the structure of the particle. In fact he observes that "with a different definition of force and acceleration we should naturally obtain other values for the masses (meaning, longitudinal and transverse masses)".

Whatever Einstein's precise early views were on the subject, his view in later life appears clear. In a 1948 letter to Lincoln Barnett, he wrote "It is not good to introduce the concept of the mass $M=\frac{m}{\sqrt{1-\frac{v^{2}}{c^{2}}}}$ of a body for which no clear definition can be given. It is better to introduce no other mass than 'the rest mass' $m$. Instead of introducing $M$, it is better to mention the expression for the momentum and energy of a body in motion". The question naturally arises as to what motivated Einstein to this new view given his earlier use of the concept. The answer, I believe, is that by at least 1922 he had adopted Minkowski's 1908 space-time (four-vector) approach to special relativity."

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Actually, it wasn't Dirac who first found that relation. It was already used by Planck as early as in 1906 while deriving the Hamiltonian equations of motion

Planck: The Principle of Relativity and the Fundamental Equations of Mechanics (1906).

He first gave the Lagrangian function

$$ (1)\quad L={\dot {x}}{\frac {\partial H}{\partial {\dot {x}}}}+{\dot {y}}{\frac {\partial H}{\partial {\dot {y}}}}+{\dot {z}}{\frac {\partial H}{\partial {\dot {z}}}}-H={\frac {mc^{2}}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}}+const. $$

then momentum

$$ (2)\quad \xi ^{2}+\eta ^{2}+\zeta ^{2}=\varrho ^{2}$$

where

$$ \xi ={\frac {\partial H}{\partial {\dot {x}}}}={\frac {m{\dot {x}}}{\sqrt {1-{\frac {q^{2}}{c^{2}}}}}},\ etc.$$

He obtained the energy-momentum relation by combining (1) and (2):

$$ L=mc^{2}{\sqrt {1+{\frac {\varrho ^{2}}{m^{2}c^{2}}}}}+const $$

Setting the constant to zero, the previous relation can be written as:

$$ L^{2}=m^{2}c^{4}+\varrho^{2}c^{2}$$

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    $\begingroup$ +1. Also, Dirac wrote it not in 1928 but in 1926 (p. 410). $\endgroup$ – Consigliere ZARF Sep 10 at 5:44

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