Electromagnetic stress–energy tensor

ISQ convention The electromagnetic stress–energy tensor in the International System of Quantities (ISQ), which underlies the SI, is The element T^{\mu\nu} of the stress–energy tensor represents the flux of the component with index \mu of the four-momentum of the electromagnetic field, {{tmath|1= P^{\mu} }}, going through a hyperplane. It represents the contribution of electromagnetism to the source of the gravitational field (curvature of spacetime) in general relativity. == Algebraic properties ==

The electromagnetic stress–energy tensor has several algebraic properties: {{unordered list T^{\mu\nu} = T^{\nu\mu} T^{\alpha}{}_{\alpha} = 0. {{math proof T^{\mu}{}_{\mu} = \eta_{\mu\nu}T^{\mu\nu} Using the explicit form of the tensor, T^{\mu}{}_{\mu} = \frac{1}{\mu_0}\left[\eta_{\mu\nu}F^{\mu\alpha}F^{\nu}{}_{\alpha} - \eta_{\mu\nu}\eta^{\mu\nu}\frac{1}{4}F^{\alpha\beta}F_{\alpha\beta}\right] Lowering the indices and using the fact that {{tmath|1= \eta^{\mu\nu}\eta_{\mu\nu} = \delta^{\mu}_{\mu} }}, T^{\mu}{}_{\mu} = \frac{1}{\mu_0} \left[F^{\mu\alpha} F_{\mu\alpha} - \delta^{\mu}_{\mu} \frac{1}{4} F^{\alpha\beta} F_{\alpha\beta}\right] Then, using {{tmath|1= \delta^{\mu}_{\mu} = 4 }}, T^{\mu}{}_{\mu} = \frac{1}{\mu_0}\left[F^{\mu\alpha} F_{\mu\alpha} - F^{\alpha\beta} F_{\alpha\beta}\right] Note that in the first term, \mu and \alpha are dummy indices, so we relabel them as \alpha and \beta respectively. T^{\alpha}{}_{\alpha} = \frac{1}{\mu_0}\left[F^{\alpha\beta} F_{\alpha\beta} - F^{\alpha\beta} F_{\alpha\beta}\right] = 0 }} T^{00} \ge 0 }} The symmetry of the tensor is as for a general stress–energy tensor in general relativity. The trace of the energy–momentum tensor is a Lorentz scalar; the electromagnetic field (and in particular electromagnetic waves) has no Lorentz-invariant energy scale, so its energy–momentum tensor must have a vanishing trace. This tracelessness eventually relates to the masslessness of the photon. == Conservation laws ==

The electromagnetic stress–energy tensor allows a compact way of writing the conservation laws of linear momentum and energy in electromagnetism. The divergence of the stress–energy tensor is: \partial_\nu T^{\mu \nu} + \eta^{\mu \rho} \, f_\rho = 0 \, where f_\rho is the (4D) Lorentz force per unit volume on matter. This equation is equivalent to the following 3D conservation laws \begin{align} \frac{\partial u_\mathrm{em}}{\partial t} + \mathbf{\nabla} \cdot \mathbf{S} + \mathbf{J} \cdot \mathbf{E} &= 0 \\ \frac{\partial \mathbf{p}_\mathrm{em}}{\partial t} - \mathbf{\nabla}\cdot \sigma + \rho \mathbf{E} + \mathbf{J} \times \mathbf{B} &= 0 \ \Leftrightarrow\ \epsilon_0 \mu_0 \frac{\partial \mathbf{S}}{\partial t} - \nabla \cdot \mathbf{\sigma} + \mathbf{f} = 0 \end{align} respectively describing the electromagnetic energy density u_\mathrm{em} = \frac{1}{2} \left( \epsilon_0\mathbf{E}^2 + \frac{1}{\mu_0}\mathbf{B}^2 \right) and electromagnetic momentum density \mathbf{p}_\mathrm{em} = {\mathbf{S} \over {c^2}} , where \mathbf{J} is the electric current density, \rho the electric charge density, and \mathbf{f} is the Lorentz force density. == See also ==