stress-energy tensor

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Let us write Poynting's theorem and and the expression of the Lorentz force in terms of the Maxwell stress tensor together:

\left.\frac{\partial u}{\partial t}\right|_\mathrm{mech} = -\frac{\partial u}{\partial t} - \boldsymbol{\nabla}\cdot \mathbf{S}\,,
\left.\frac{\partial p_j}{\partial t}\right|_\mathrm{mech} = \sum_i \nabla_i T_{ij} + \epsilon_0 \mu_0 \frac{\partial S_j}{\partial t}\,.

Note that

\left.\frac{\partial u}{\partial t}\right|_\mathrm{mech} = \mathbf{J}\cdot\mathbf{E}\,

is the rate of change of mechanical energy of the charge, while

\left.\frac{\partial p_j}{\partial t}\right|_\mathrm{mech} = f_j\,

is the rate of change of mechanical momentum. Denote the energy momentum vector

p^\mu_\mathrm{mech} = \left(-\frac{u}{c}, p_1, p_2, p_3\right)_\mathrm{mech}\,,

and define the stress-energy tensor

T^{\mu\nu} = \begin{pmatrix} \frac{u}{c} & \frac{S_1}{c} & \frac{S_2}{c} & \frac{S_3}{c} \\ \frac{S_1}{c} & T_{12} & T_{22} & T_{13} \\ \frac{S_2}{c} & T_{22} & T_{22} & T_{23} \\ \frac{S_3}{c} & T_{32} & T_{32} & T_{33} \end{pmatrix}\,.

Then

\left.\frac{\partial p^0}{\partial x^0}\right|_\mathrm{mech} = -\frac{1}{c^2} \left.\frac{\partial u}{\partial t}\right|_\mathrm{mech} = \frac{1}{c}\left(\frac{\partial u}{\partial x^0} + \frac{1}{c} \boldsymbol{\nabla}\cdot \mathbf{S}\right) = \frac{1}{c} \partial_\mu T^{\mu 0}\,,

while

\left.\frac{\partial p^j}{\partial x^0}\right|_\mathrm{mech} = \frac{1}{c}\left.\frac{\partial p_j}{\partial t}\right|_\mathrm{mech} = \frac{1}{c}\left(\sum_i \nabla_i T_{ij} + \frac{1}{c} \frac{\partial S_j}{\partial x^0}\right) = \frac{1}{c} \partial_\mu T^{\mu j}\,

Thus

\frac{\partial p^\nu}{\partial x^0} = \frac{1}{c} \partial_\mu T^{\mu \nu}\,.
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