Theory of relativity/Special relativity/E = mc²

Physics

Special relativity

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This article presumes that the reader has read Special relativity/energy.

This article will deduce, on theoretical grounds, that the mass (rest mass, that is) of an object changes when it emits or absorbs energy, in accordance with the famous formula E=mc².

As in the previous articles, we will perform a "gedanken experiment" on an object releasing kinetic energy. An object of mass M, that consists of two parts of mass M/2 each, will burst into two separate objects. Those objects each have mass m/2, and they fly away from the original object, in opposite directions, each with speed $v\,$.

We could imagine that the two halves of the original object had a spring between them that was released, or that electric or magnetic repulsion was involved, or that an explosive charge was used, or that a nuclear disintegration took place.

Under classical mechanics, of course we expect that m = M, due to conservation of mass.

The original object, of mass M, splits into two pieces, each of mass m/2 and moving away with speed v.

The same experiment, viewed from a frame of reference moving to the left with speed v. After the event, one of the result objects is at rest and the other has speed 2v/(1+v²/c²).

In the original rest frame, there is no kinetic energy before the event. After the event, each piece has kinetic energy

${\frac {m}{2}}\ c^{2}\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)$

For a total of

$E=mc^{2}\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)$            (equation 1)

(See Special relativity/energy for the derivation of this.)

Now examine the experiment from a frame of reference moving to the left with speed $v\,$. Before the event, the object of mass M was moving to the right with speed $v\,$, so its kinetic energy was

$Mc^{2}\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)$

After the event, one of the pieces is at rest, and the other is moving to the right with speed ${\frac {2v}{1+v^{2}/c^{2}}}$ due to the addition law. (See Special relativity/space, time, and the Lorentz transform for the derivation of this.)

The kinetic energy after the event, in this frame, is

${\frac {m}{2}}\ c^{2}\left({\frac {1}{\sqrt {1-4v^{2}/((1+v^{2}/c^{2})^{2}\ c^{2})}}}-1\right)$

which is

${\frac {m}{2}}\ c^{2}\left({\frac {1+v^{2}/c^{2}}{1-v^{2}/c^{2}}}-1\right)$

or

${\frac {mv^{2}}{1-v^{2}/c^{2}}}$

The energy released, as measured in this frame, is the final energy minus the initial energy, or

$E={\frac {mv^{2}}{1-v^{2}/c^{2}}}-Mc^{2}\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)$

Now all observers, in all frames of reference, must agree on how much energy was released, so, using equation 1:

$E=mc^{2}\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)={\frac {mv^{2}}{1-v^{2}/c^{2}}}-Mc^{2}\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)$

or

$m\left(1+{\frac {v^{2}/c^{2}}{1-v^{2}/c^{2}}}-{\frac {1}{\sqrt {1-v^{2}/c^{2}}}}\right)=M\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)$

or

$m\left({\frac {1}{1-v^{2}/c^{2}}}-{\frac {1}{\sqrt {1-v^{2}/c^{2}}}}\right)=M\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)$

or

${\frac {m}{\sqrt {1-v^{2}/c^{2}}}}=M$

Now, using equation 1 again:

$E=mc^{2}\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)=c^{2}\left({\frac {m}{\sqrt {1-v^{2}/c^{2}}}}-m\right)=c^{2}\ (M-m)$

The loss in mass, times the square of the speed of light, is equal to the energy released.

Since the laws of physics work in reverse, an object's mass will increase if it absorbs energy.

This principle is often interpreted as meaning that the two classical principles of conservation of mass and conservation of energy are to be replaced with a single more general principle of conservation of "mass-energy", since mass and energy can be converted into each other. But the principle of conservation of energy was always about conservation of potential energy plus kinetic energy, and that principle still holds. Potential energy and kinetic energy can be interchanged the same way they could under classical physics. What is new is that potential energy is embodied in mass. Any object that contains potential energy in any form has extra mass because of it. In the experiment above, the initial body of mass M contained potential energy in the amount of $c^{2}\ (M-m)$, which was released when it split into two parts.

One can (and particle physicists often do) think of mass as being equivalent to potential energy, so that, for example, a Uranium atom actually "contains" 220 GeV of energy. But this potential energy is not readily convertible into kinetic energy (though 200 MeV of it can be extracted through fission.) In some cases, the notion that an object's mass is effectively all potential energy is useful. Electrons and positrons each have an intrinsic energy of 511 KeV, due to their mass of .00055 amu. When they collide and annihilate each other, that energy is totally converted into 1.022 MeV in massless photons.

The next article in this series is Special relativity/spacetime diagrams and vectors.

See also

• Special relativity/space, time, and the Lorentz transform
• Special relativity/momentum
• Special relativity/energy
• Special relativity/spacetime diagrams and vectors