The electromagnetic stress-energy tensor

Keep signature $(+,-,-,-)$, $x^0=ct$, and our field convention $F^{0i}=-E_i/c$, $F^{ij}=-\epsilon^{ijk}B_k$. The symmetric vacuum electromagnetic stress-energy tensor is quadratic in F. Its minus sign in the first term is required by this metric signature. Evaluating the time-time entry gives a positive sum of electric and magnetic energies. This is a useful audit: a formula copied from the opposite metric convention can otherwise make even a simple static electric field appear to have negative energy.

Every tensor component here has units of energy per volume, equivalently pascals. Let u denote field energy density, S energy flux, and g momentum density. Then $T^{00}=u$, $T^{0i}=T^{i0}=S_i/c$, and $g_i=T^{0i}/c=S_i/c^2$. Flux and density differ by factors of speed because $x^0$ is a length coordinate. Equating momentum density directly to the Poynting vector would therefore fail both dimensional analysis and the local momentum-balance equation.

Define $T^{ij}$ as the flux of ith momentum across a surface normal to direction j. With our convention it is $u\delta_{ij}-\epsilon_0E_iE_j-B_iB_j/\mu_0$. If the Maxwell stress is defined as $\sigma_{ij}=\epsilon_0E_iE_j+B_iB_j/\mu_0-u\delta_{ij}$, then $T^{ij}=-\sigma_{ij}$. Both conventions are common, but their divergence signs must travel with their definitions. A negative longitudinal momentum-flux entry in a static electric field represents field tension in this convention, rather than negative energy density.

For a plane wave propagating along x, E can point along y and B along z with $E=cB$. The electric and magnetic contributions to u are equal, the Poynting vector points along x, and $T^{xx}=u$ while the two transverse diagonal entries vanish. Its momentum flux is consequently energy flux divided by c. An absorbing surface and a reflecting surface transfer different amounts of momentum, so converting this flux into radiation pressure requires the surface interaction, not just the incident wave tensor.

Using Maxwell equations gives $\partial_\mu T^{\mu\nu}=-F^{\nu\lambda}J_\lambda$. The source on the right is minus the four-force density on matter. Its temporal component gives $\partial_tu+\nabla\cdot\mathbf S=-\mathbf j\cdot\mathbf E$. Positive electric work on matter reduces field energy unless incoming energy flux compensates. The spatial equation is $\partial_t\mathbf g-\nabla\cdot\boldsymbol\sigma=-(\rho\mathbf E+\mathbf j\times\mathbf B)$. This sign follows directly from our explicit relationship between stress and momentum flux.

In a source-free region the electromagnetic stress-energy tensor has zero divergence; with sources, field stress-energy alone generally is not conserved. Adding a compatible matter tensor restores zero total divergence for a closed system. A static field can store energy and stress even when S vanishes. Conversely, a nonzero Poynting vector can describe steady energy transport without changing the local energy density. Local storage, boundary flow, and work are separate terms of one balance, and none should be inferred from another without the actual field configuration.