Should not the energy conservation physics equation be

$$ E= - mc^2 \text{ instead of } E= m c^2 $$

because energy appears simultaneously with mass disappearance in splitting as

$$ |\Delta E|= - |\Delta m| c^2? $$

Also what in Einstein's derivation dispenses with the factor $\frac12$ from the usual kinetic energy

$$ KE= \frac12 m v^2 ?$$

Sorry about the elementary question, it remained with me for longtime.


About your first question:

$E= mc^2$ is not an equation that expresses energy conservation. It is an equation that relates the energy content of a physical object to its gravitating/inert mass. So for example, the equation says that, if you have a box filled with say some springs and a cold cup of tea and you open it, compress the springs and heat up the tea, adding some amount of energy $\Delta E$ to the system before closing the box again, then the mass of the box will increase by $\Delta E / c^2$ .

Concerning your second question:

The classical kinetic energy term arises from the relativistic energy as the leading term in the Taylor approximation that is valid for small momentum/velocity. The factor of 1/2 has its origin in the Taylor expansion of the square root. There are two ways to approach the problem. My preferred view is that the $m$ in your equation is the rest mass and the full relativistic energy momentum equation should be:

$$ E =\sqrt{m ^2 c^4 + p^2 c^2}$$

where $p$ is momentum.

There are also some heretical texts that introduce a so called "relativistic mass" $m$ which is $m = \gamma m_0$ where $m_0$ is the rest mass. In this sacrilegious notation, $E = m c^2$ is also valid outside of the rest frame of the object that you are studying and the kinetic energy term arises from a Taylor expansion of the $\gamma$ factor.


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