  Following our recent finding that the renowned formula $\vec x\times (\vec
E\times\vec B)$ is not the correct density for the electromagnetic angular
momentum, here we examine the validity of the Poynting vector $\vec E
\times\vec B$ as the electromagnetic momentum density (or energy flux). The
competitor is the gauge-invariant canonical momentum $E^i\vec \nabla
A^i_\perp$. It often gives the same result as $\vec E\times\vec B$, but we
propose that a delicate measurement (of the {\em azimuthal} energy flow in
polarized atomic radiations) can make a discrimination. By clarifying the
profound difference between two kinds of energy-momentum tensors: the canonical
(or mechanical) one and the symmetric (or gravitational) one, we predict that
it is $E^i\vec \nabla A^i_\perp$ that would pass the delicate experimental
test. Our observations have far-reaching implications for understanding the
source of gravity, and the nucleon momentum as well.
