Einstein's $E=mc^2$ and law of conservation of energy implies that mass is a form of energy, but if it is a form of energy then why can we freely interact with mass while the other forms of energy are so abstract

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    $\begingroup$ Why do you feel you can interact more "freely" with a falling anvil than you can with a lightning bolt? $\endgroup$
    – WillO
    Nov 21 '21 at 15:28
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    $\begingroup$ How do you interact with mass? BTW, "mass" is not synonymous with "matter". $\endgroup$
    – PM 2Ring
    Nov 21 '21 at 15:50
  • $\begingroup$ Note that the $m$ in Einstein formula refers to rest mass. $\endgroup$
    – Steeven
    Nov 21 '21 at 16:23
  • $\begingroup$ @Steeven. In that case $E$ is not the total energy, just a "rest energy". Using relativistic mass $m=\gamma m_0$ the kinetic energy is included in $E$. $\endgroup$
    – md2perpe
    Nov 21 '21 at 17:03

Unfortunately $E=mc^2$ is both super-famous and usually misunderstood. It is not that mass is a form of energy, but rather that energy has mass (see below for a caveat). If you have an object that has thermal energy, electrical energy, and nuclear force energy, all of those are different forms of energy. Mass is not a separate form of energy, but rather each of those other forms of energy contribute to the object’s mass.

You can convert energy from nuclear energy to thermal energy (e.g. by nuclear decay), and that is a transformation between different forms of energy. But the mass is the same because both forms of energy have mass.

Note also that $E=mc^2$ is a special case for use only when the momentum is 0. The general form, including momentum, is $m^2 c^2=E^2/c^2-p^2$. This clearly reduces to the famous equation when $p=0$, but you can also use this to show that if $m$ is fixed and $p$ increases then $E$ also increases. This is of course linear kinetic energy, so in that sense and that sense only linear KE could be considered a different form of energy from mass. Otherwise it is only energy without momentum that has mass.

Regarding your comment, we can interact with all forms of energy pretty equally, but again mass is not a form of energy.

  • $\begingroup$ @ChiralAnomaly I thought I covered that sufficiently already in the third paragraph, but I edited to emphasize that further. $\endgroup$
    – Dale
    Nov 21 '21 at 16:19
  • $\begingroup$ "mass is not a form of energy." Why? Consider the inelastic collision of two particles. The mass of the formed particle will consist of the energies and masses of the original particles. Well, for example, the mass of a proton, which is the some sort of sum of the masses of quarks and their energies. $\endgroup$
    – Sergio
    Nov 21 '21 at 16:40
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    $\begingroup$ @Sergio I explained that already in the answer. If mass were a form of energy then electrical energy, being a different form, would not have mass. Mass is something that energy has (provided p=0), regardless of the specific form of energy. In your example, there will be an increase in some specific form of internal energy, like the EM energy in an excited atomic state. The form then is EM energy and it has mass $\endgroup$
    – Dale
    Nov 21 '21 at 16:48
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    $\begingroup$ Mass is a relativistic invariant, energy is not. So it is difficult to say that it is more or less the same thing? $\endgroup$ Nov 21 '21 at 18:11
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    $\begingroup$ @justsomeguyontheinternet the issue is that while a single photon doesn’t have mass a system of many photons does have mass. In this case, the system of two photons post-annihilation has the same mass and the same energy as the system of the original positron and electron. No mass or energy was converted, they were each individually conserved. If mass could be converted to energy then the right side of $E=mc^2$ would decrease and the left side would increase and they would no longer be equal $\endgroup$
    – Dale
    Nov 21 '21 at 20:16

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