Popstar Tourist@octonion@tech.lgbt
@KaitlynEthylia@void.lgbt @darkwiiplayer@tech.lgbt If we're talking classical (not quantum) mechanics then the issue is that "energy" on it's own is as meaningless as "the z component" of a three dimensional vector. It's going to be completely different based on the choice of coordinates.
One should instead talk about the 4 dimensional energy-momentum vector
(E, p_x, p_y, p_z) and the thing that every observer will agree on is the dot product of this vector with itself in the appropriate metric which works out to
E² - |p|²c² = some constant :)
(where |p| is the usual 3-space vector magnitude). Considering a frame where p=0, taking the square root of both sides and revealing what this mystery constant is gives
E = mc²
Popstar Tourist@octonion@tech.lgbt
@KaitlynEthylia@void.lgbt @darkwiiplayer@tech.lgbt Another way of looking at it is that in classical mechanics the dynamics of the system are entirely determined by something called the Hamiltonian. In quantum mechanics the Hamiltonian becomes an operator and the eigenvalues of the operator are energy.
(It's time-translation symmetry of the Hamiltonian that gives rise to conservation of energy. The Hamiltonian of the universe does not have this symmetry, because the Universe is expanding, but locally or for closed systems it's a very, very good approximation.)