# Sign problem in electromagnetic stress energy tensor

I'm having a silly problem in calculating the electromagnetic stress energy tensor: the Lagrangian is $$\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}$$

and the stress energy tensor reads

$$T^{\mu\nu}=\frac{\partial\mathcal{L}}{\partial(\partial_\nu A_\rho)}\partial^\mu A_{\rho}-g^{\mu\nu}\mathcal{L}.$$

We have that $$\frac{\partial\mathcal{L}}{\partial(\partial_\nu A_\rho)}=F^{\nu\rho}$$ and thus, the canonical stress energy tensor is

$$T^{\mu\nu}=F^{\nu\rho}\partial^{\mu}A_{\rho}+\frac{1}{4}g^{\mu\nu}F^{\alpha\beta}F_{\alpha\beta}.$$

Which we can rewrite as

$$T^{\mu\nu}=F^{\nu\rho}F^{\mu}_{\,\,\rho}+\frac{1}{4}g^{\mu\nu}F^{\alpha\beta}F_{\alpha\beta} +F^{\nu\rho}\partial_\rho A^\mu=\\=F^{\nu\rho}F^{\mu}_{\,\,\rho}+\frac{1}{4}g^{\mu\nu}F^{\alpha\beta}F_{\alpha\beta} +\partial_\rho F^{\nu\rho} A^\mu- A^\mu\partial_\rho F^{\nu\rho}=\\=F^{\nu\rho}F^{\mu}_{\,\,\rho}+\frac{1}{4}g^{\mu\nu}F^{\alpha\beta}F_{\alpha\beta} +\partial_\rho F^{\nu\rho} A^\mu+ A^\mu j^{\nu}=\tilde{T}^{\mu\nu}+\partial_\rho F^{\nu\rho} A^\mu$$

Where $$\tilde{T}^{\mu\nu}$$ is the symmetrised stress energy tensor

$$\tilde{T}^{\mu\nu}=F^{\nu\rho}F^{\mu}_{\,\,\rho}+\frac{1}{4}g^{\mu\nu}F^{\alpha\beta}F_{\alpha\beta}.$$

As you can see it differs from the version found everywhere by a sign, it should be

$$\tilde{T}^{\mu\nu}=F^{\nu\rho}F^{\mu}_{\,\,\rho} -\frac{1}{4}g^{\mu\nu}F^{\alpha\beta}F_{\alpha\beta}.$$

But I cannot see how that could be a minus sign, considering the negative sign in the Lagrangian. I cannot find my mistake, any help?

• What is your sign conventions for the metric? – Qmechanic Oct 21 '18 at 10:27
• (+,-,-,-) and characters I need to post the comment – user2723984 Oct 21 '18 at 11:13
• Comment to the post (v3): For starters, there is a sign mistake in eq. (3). – Qmechanic Oct 21 '18 at 12:31