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Meaning of Einstein's equation $E=mc^2$? How can a $1\,\mathrm g$ mass possess energy equal to $9\times10^{13}\,\mathrm J$? What does it actually mean?

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To understand the answer you should have understood well a point. Energy is conserved, mass is not. Moreover, masses possess an energy content pictured by the equation you wrote and thus this part of energy must be taken into account when writing the energy conservation law.

For instance, suppose you have a particle at rest in your (inertial) reference frame with a mass $M>0$. It may happen that the particle spontaneously breaks into a pair of different particles with masses $m_1$ and $m_2$ respectively (when measured at rest with them, I am assuming that all particles I consider are massive), but these particles are no longer at rest with you. Consequently they have kinetic energies, $K_1$, $K_2$, depending on their velocities.

Well, classically one expects the mass is conserved and thus $$M = m_1+m_2\:.$$ Instead this equation is experimentally violated.

Energy however is conserved, but part of energy is associated with the involved masses in accordance with the identity $E=mc^2$. Summing up the correct conservation rule is: $$Mc^2 = (m_1c^2 + K_1) + (m_2c^2 + K_2)$$ where $Mc^2$ is the total energetic content before the transformation of the initial particle: the energy is completely due to the initial mass. After the transformation, there are two types of energy to sum for each particle, the one due to the mass of the new particles $m_1c^2$ and $m_2c^2$ and their kinetic energy $K_1$ and $K_2$.

You see that, as $K_1+K_2 \geq 0$ you have that $m_1+m_2 \leq M$, so that the total mass decreased in this process.

There are other different kinds of transformations (for instance the inverse reaction $m_1+m_2 \to M$) also involving different types of energy (e.g. chemical or thermodynamical) but the point is that

(a) there always is part of energy due to the (rest) mass of the involved bodies, and here the celebrated equation $E=mc^2$ enters the picture;

(b) the total energy (for isolated systems in inertial reference frames) is conserved in time in any transformation process.

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It means that this mass is available for transformation in some other kind of energy. It means that there is a type of energy that is only a function of a mass of a particle. There are processes in nature in which entire mass of a particle transforms in some other type of energy...like annihilation of particle and antiparticle.How to calculate this energy? mc2 formula. It originates from relativistic form of equation for energy of a particle..if you put that speed is zero, you have mc2 term left. Like kinetic energy turns into potential or electromagnetic, so this rest energy can turn in something else. This is something that just comes out from equations. You have to know that energy is not a clear and simple topic, but regardless, it is a good device for describing processes in nature. Einsteins equation mc2 gives another proof that it is a good device indeed.

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  • $\begingroup$ "It means that this mass is available for transformation in some other kind of energy." No, conservation rules may prevent that. $\endgroup$ – pfnuesel Nov 16 '14 at 18:51
  • $\begingroup$ True, may prevent that. $\endgroup$ – Žarko Tomičić Nov 16 '14 at 20:15
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It is what it seems to be at first glance: matter intrinsically contains energy even if it is stationary and thus has no kinetic energy. This energy defined as $E=mc^2$ is called rest energy. All matter carry rest energy.

Now, due to the large numerical value of c, even the lightest objects contain such immensely huge energy. But this is not energy that is easily harnessed or transferred, because this rest energy lies within the fundamental particles that constitute matter. What the $E=mc^2$ equation really implies is the equivalence between mass and energy.

This is actually hard to explain, since I'm not yet knowledgeable on particle physics. But what I do know is that to truly extract the billions or trillions of joules in a particle would mean converting its mass into energy.

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