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Special relativity gives us the invariance of four-vectors. Consistency with Newtonian physics implies the conservation of four-momentum.

The spatial part of four-momentum is

$P^0=m\gamma(v)$

which can be expanded to second order

$P^0 =\frac{1}{c}\left(mc^2+\frac{1}{2}mv^2\right)$

This gives good reason to suspect the equivalence of mass and energy, but is a further hypothesis required to suggest that rest energy, $mc^2$, is interchangeable with other forms of energy, as is seen in nuclear fission, etc.?

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  • $\begingroup$ Another way to put it - is it a consequence of SR that the spatial part of the momentum four vector is energy, or rather $E/c$, or is that a separate hypothesis? $\endgroup$
    – Anding
    Sep 7, 2019 at 10:25

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It is a direct consequence of Special Relativity. The equation

$$E^2-(\mathbf{p}c)^2=(mc^2)^2,$$

which expresses the invariance of the length of the energy-momentum four-vector, makes the relationship between mass and energy when $\mathbf{p}=0$ transparently obvious.

I wouldn’t call that relationship “equivalence”. Mass has energy, but energy doesn’t necessarily have mass. For example, a single photon has energy and momentum but no mass.

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  • $\begingroup$ but the distinction between mass and energy is nonexistent regarding gravity sources $\endgroup$
    – lurscher
    Sep 7, 2019 at 0:52
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    $\begingroup$ @lurscher In GR, the source of gravity is the energy-momentum tensor. Mass doesn’t even appear in the Einstein field equations. An electromagnetic field has an energy-momentum tensor. It doesn’t have mass, but it creates gravity. $\endgroup$
    – G. Smith
    Sep 7, 2019 at 1:33

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