Originally posted by @wolfgang59
But E^2=m^2c^4 is Einstein's formula.
The common E=mc^2 assumes that Energy must be positive.
Well, ignoring factors of the speed of light, the full equation is:
E^2 - p^2 = m^2
Which is just the genralization of Pythagoras' theorem to Minkowski Space applied to the energy momentum vector. It simply states that the length of the energy momentum vector is the mass. In the comoving frame the momentum is zero and we have E = m or E = -m. However, since the energy is just the length of a vector, it seems odd to assign it a negative value.
Consider a vector on the Euclidean plane, with components x and y. We have that:
L^2 = x^2 + y^2
Rotating our axes so that y=0 we get:
L = x or L = -x
Since negative length makes no sense it's normal to drop whichever one of these is negative.
Whether negative energy or mass exists is one of those empirical problems that can't really be addressed theoretically.