Electromagnetic Stress-Energy Tensor in curved space-time

Question

I found on Wikipedia that the electromagnetic stress energy tensor in curved space-time with sign convention $(-, +, +, +)$ is

$$T_{\mu\nu} = -\frac{1}{\mu_0} \left ( F_{\mu \alpha} g^{\alpha \beta} F_{\beta \nu} - \frac{1}{4} g_{\mu \nu} F_{\sigma \alpha} g^{\alpha \beta} F_{\beta \rho} g^{\rho \sigma} \right ).$$

However, I need a reputable source for this equation. Does anyone know another source for this equation? All I could find was

$$T_{\mu\nu} = \frac{1}{\mu_{0}}( F^{\beta}{}_{\mu}F_{\beta\nu} - \frac{1}{4}g_{\mu\nu}F^{\alpha\beta}F_{\alpha\beta}).$$

Answer 1

You can derive this expression by your own from the Lagrangian $$\mathscr{L} = -\frac{1}{4}\sqrt{g} F_{\mu\nu}F^{\mu\nu}$$ where $g$ is the absolute value of determinant of $g_{\mu\nu}$. Find out how the action varies when the metric is varied. Then from the general definition $$ S = \int{\text{d}^4x \sqrt{g}T^{\mu\nu}\delta g_{\mu\nu}}$$ The following relations might help you $$\delta F^{\mu\nu} = -F^{\mu\sigma}g^{\nu\lambda}\delta g_{\lambda\sigma}+F^{\nu\lambda}g^{\mu\sigma}\delta g_{\lambda\sigma}$$ $$\delta g=gg^{\lambda\sigma}\delta g_{\lambda\sigma}$$ Hope this helps.

Reminder: Because this is written in natural units you will not get the factor of $\frac{1}{\mu_0}$.

Answer 2

The metric tensor $g_{\bullet\bullet}$ is used for more than just most tensors are: it and its inverse $g^{\bullet\bullet}$ (defined so $g_{\alpha\beta}g^{\beta\gamma}=\delta_\alpha^\gamma$) are used for a canonical bijection between the vector and covector spaces.

So the vector $\vec v$ is considered to be equivalent to the covector $(\vec v\cdot)$ which takes vectors to scalars, and vice versa.

So by this bijection, tautologically, $$F^{\alpha\beta}=g^{\alpha\mu}~g^{\beta\nu}~F_{\mu\nu}.$$We could have chosen a different invertible tensor to make this correspondence, but we chose the metric because it is so ubiquitous.